Compiled

# Quantum imaging and metrology with undetected photons: a tutorial

Gabriela Barreto Lemos  Instituto de Física, Universidade Federal do Rio de
Janeiro, Av. Athos da Silveira Ramos 149, Rio de Janeiro, CP: 68528, Brazil.
Corresponding author<EMAIL_ADDRESS>Mayukh Lahiri  Department of
Physics, Oklahoma State University, 145 Physical Sciences Bldg, Stillwater,
Oklahoma 74078, USA  Sven Ramelow  Humboldt-Universität zu Berlin, Institut
für Physik, Newtonstraße 15, 12489 Berlin, Germany  IRIS Adlershof, Humboldt-
Universität zu Berlin, Zum Großen Windkanal 6, 12489 Berlin  Radek Lapkiewicz
Institute of Experimental Physics, Faculty of Physics, University of Warsaw,
Pasteura 5, Warsaw 02-093, Poland  William N. Plick  University of Dayton,
Department of Physics, Department of Electro-Optics, Dayton, OH, 45469, United
States

###### Abstract

We present a tutorial on the phenomenon of induced coherence without induced
emission, and specifically its application to imaging and metrology. It is
based on a striking effect where two nonlinear crystals, by sharing a coherent
pump and one of two output beams each, can induce correlations between the
other two individual, non-interacting beams. This can be thought of as a type
of quantum-erasure effect, where the “welcher-weg” (which-way), or in this
case “which-source” information is erased when the shared beams are aligned.
With the correct geometry this effect can allow an object to be imaged using
only photons which have never interacted with the object – in other words the
image is formed using undetected photons. Interest in this and related setups
has been accelerating in recent years due to a number of desirable properties,
mostly centered around the fact that the fields for detection and imaging
(since separate) may have different optical properties, entailing significant
advantages to various applications. The purpose of this tutorial is to
introduce researchers to this area of research, to provide practical tools for
setting up experiments as well as understanding the underlying theory, and to
also provide a comprehensive overview of the sub-field as a whole.

††journal: josab

##  1 Introduction

One thousand years ago, in 1021, Ḥasan Ibn al-Haytham (a.k.a. Alhazen)
completed the “Book of Optics” in which he laid the foundations of modern
optics and detailed the apparatus that later Kepler dubbed camera obscura
(pinhole camera). In the intervening ten centuries imaging technology has
progressed immeasurably – and up until the present day, developing ever-better
imaging and sensing devices remains an extremely active field of research.
Evolving approaches to imaging technology have enabled new abilities like
extreme sensitivity and resolutions, imaging at non-visible wavelengths, and
many others too diverse to fully enumerate.

One of the very-latest areas of research is a suite of new technologies
enabled by the development of quantum theory, which contains the most accurate
description of optical fields. Devices that take advantage of the co-called
“quantum nature of light” include new imaging schemes which may beat the
classical limits of sensitivity [1, 2, 3, 4], spatial resolution [5, 6, 7, 8,
9, 10], and phase metrology [11, 12, 13]. Quantum Optics has also enabled the
emergence of imaging schemes where the light that interacts with the sample is
_not_ captured by the pixelated detector/camera, e.g. Interaction Free Imaging
[14], Ghost Imaging [15, 16, 17, 18, 19, 20, 21], and Quantum Imaging with
Undetected Photons (QIUP) [22, 23, 24].

With ever-more-sophisticated camera and photon-source technologies emerging in
recent years, quantum imaging and quantum-inspired imaging techniques have
become even more promising avenues for research and development – and will
likely be a key component of the twenty-first century quantum revolution,
alongside quantum computing, quantum communication and quantum metrology.

Here we focus on Quantum Imaging with Undetected Photons (QIUP) and its spin-
offs. In these experiments, coherence is induced between light produced in two
twin-photon sources placed within a nonlinear interferometer. Quantum imaging
with nonlinear interferometers was introduced in [22] and applications of this
method to bio imaging, spectroscopy, optical coherence tomography and moving
images were first proposed in [25, 26]. The most important advantage of this
kind of technique is that one can obtain information about an object probed by
a light beam of one wavelength by only detecting a _separate_ light field at a
different wavelength. The light field which illuminates the sample is not
detected at all. This is especially useful when the illumination wavelength is
one for which detectors are not available or unsatisfactory, and in the case
of delicate samples that require low intensity illumination.

How to read this tutorial: In this tutorial we will prepare you both
theoretically and experimentally to investigate a few methods of quantum
imaging and metrology with undetected photons, i.e., quantum imaging based
technologies using induced coherence without induced emission within a
nonlinear interferometer in the low gain regime. We will assume basic
knowledge of the quantum optics formalism and familiarity with spontaneous
parametric down-conversion (SPDC). This tutorial does not have to be read in a
linear fashion. Readers with different interests might jump to different
sections. In particular, those interested in theory may skip sections 7, 8.
Those who are not planning on doing detailed calculations, but are interested
in building an experiment for imaging or metrology with undetected photons may
skip sections 3–5.

If this is your first encounter with imaging and metrology using induced
coherence without induced emission (ICWIE), we suggest you start by reading
Sec. 2. In Sec. 3 we give a basic overview of the main quantum optics states
and operators, for those who have not encountered quantum optics formalism
before or would like a recap. Readers interested in the application of ICWIE
to phase metrology can learn about it in Sec. 4. In Sec. 5 we will provide the
theoretical description for a multi-mode nonlinear interferometer and its
application to phase and absorption imaging. In Sec. 6 we describe optical
coherence tomography, holography and spectral imaging with undetected photons.
The following sections, 7 and 8, are aimed at giving experimental guidelines
to researchers who are building a Zou-Wang-Mandel (ZWM) interferometer or a
SU(1,1) interferometer in the low gain regime. In Sec. 9, we give an outlook
of some interesting research directions one could explore.

Before diving into the tutorial, we would like to point out that in our
exposition we use the language of squeezed correlated fields, such as states
produced in Parametric Down Conversion. However, the interferometers we
analyse and the mathematical formalism we introduce can also be applied to a
variety of other systems, such as atomic spin waves [27], superconducting
microwave cavities [28, 29], and four wave mixing [30].

##  2 Induced Coherence without Induced Emission

Richard Feynman considered quantum interference the biggest mystery in quantum
mechanics [31]. Quantum interference is a phenomenon observed at a detector if
and only if it is impossible to associate each detected quantum (e.g. photon,
electron, atom, molecule) to a particular path, among the two or more paths
connecting that detection apparatus to the quantum source [31, 32]. Moreover,
the fringe visibility in a two-way interferometer gives an upper bound on the
available which-way (welcher weg) information [33]. The quintessential example
is the double slit experiment, where no interference is observed if the
alternative paths between the source and the detector are distinguishable and
interference is observed if those paths are indistinguishable [31, 34]. Other
important examples in optics are the Michelson, Mach-Zehnder and Sagnac
interferometers [35, 36, 37].

Figure 1: (a) Mach-Zehnder interferometer. A light field is split at the
$50$:$50$ beam splitter, BS1, and is recombined at BS2. A phase shifter is
placed in path $A$ and an object with complex field transmittance $T$ placed
in path $B$. Interference is analysed at the detector. (b) Two-Particle
Interferometer. A pump is split into paths $P_{1}$ and $P_{2}$ at a $50$:$50$
beam splitter (BS1) and illuminates two nonlinear sources, $Q_{1}$ and
$Q_{2}$, producing correlated particle pairs. When a pair is produced in
source $Q_{1}$ ($Q_{2}$), the so-called signal particle is emitted into path
$S_{1}$ ($S_{2}$) and so-called idler particle is emitted into path $I_{1}$
($I_{2}$). Signal paths $S_{1}$ and $S_{2}$ are combined at BS2 and idler
paths $I_{1}$ and $I_{2}$ are combined at BS3. No interference is observed in
the signal intensity at the detector because which way information is in
principle obtainable. One can only observe interference using post-selection,
i.e. by detecting idlers at one output of BS3 in coincidence with signals at
one output of BS2.

When setting up any interferometer in the laboratory, in order to observe good
interference visibility, one aligns the beams incoming to the detector and
adjusts the path lengths of the interferometer so that they differ by not more
than the coherence length of the quanta. In the language of quantum
information, the alignment and the length adjustment of the interferometer
paths amount to ensuring indistinguishability between quanta arriving at the
detector, thus enabling interference [32].

The Mach Zehnder Interferometer. Let us consider what happens to light in a
Mach-Zehnder interferometer (MZI), illustrated in Fig.1. The incoming light
field is split at the first beam splitter BS1 and recombined at a second beam
splitter, BS2. For interference to be seen the optical path lengths of paths
$A$ and $B$ between the two beam splitters must be equal to within the
coherence length of the light field, such that such that each photon is
described as being in a superposition,
$\left(|1\rangle_{A}+e^{-i{\phi_{A}}}|1\rangle_{B}\right)/\sqrt{2}$, where
$|1\rangle_{x}$ denotes a photon in path $x$. The phase $\phi_{A}$ is
associated with relative optical delays between the two modes of the
interferometer and can be adjusted by slightly shifting a mirror or beam
splitter or by inserting a slab of a transparent material, such as silica. The
count rate at a detector placed after BS2 in path $A$ (or $B$) is given by

$\mathcal{R}_{A/B}=\frac{\left(1+|T|^{2}\right)\pm 2|T|\cos(\phi-\gamma)}{4},$
(1)

where $T=|T|e^{i\gamma}$, with $0\leq|T|\leq 1$ and $0\leq\gamma<2\pi$, is the
complex field field transmittance of an object placed in path $B$. The
interference visibility obtained by scanning $\phi_{A}$, is defined as

$\mathcal{V}=\frac{\mathcal{R}_{\max}-\mathcal{R}_{\min}}{\mathcal{R}_{\max}+\mathcal{R}_{\min}}.$
(2)

Assuming equal optical path lengths, the interference visibility of the MZI is
thus

$\mathcal{V}_{MZ}=\frac{2|T|}{1+|T|^{2}}.$ (3)

In the language of quantum information, the reduction of visibility for
$|T|<1$ is due to the path distinguishability introduced by the object. One
application of the MZI is interaction free measurements [38, 39], which can be
used for imaging [14]. An object is placed in one arm of a MZI where photons
are sent one at a time. The object affects the interference pattern at the
output and, in a fraction of the experimental runs, the presence of the object
can be deduced without the photon having interacted with it.

The Zou-Wang-Mandel (ZWM) Interferometer. In the MZI, as well as the Michelson
Interferometer and the Sagnac, classical wave models, including classical
electromagnetism, can describe the interference visibility due to
(mis)alignment, path length difference and/or an object placed in a path of
the interferometer. Seeking to unravel the connection between quantum
indistinguishability and interference, in $1991$ Zou, Wang and Mandel, with an
essential insight from Jeff Ou, created an interferometer which cannot be
described by classical wave optics models [40, 41].

A ZWM setup uses two identical sources of photon pairs, e.g. nonlinear
crystals that can generate photon pairs through spontaneous parametric down-
conversion (SPDC). The sources are prepared such that the biphoton fields
emerging from them are mutually coherent. These crystals are weakly pumped by
mutually coherent laser beams, for example, generated by splitting a laser
beam into two as shown in Fig. 2. Let us denote the two sources $Q_{1}$ and
$Q_{2}$ and the emitted beams will be referred to as signal beams ($S_{1}$ and
$S_{2}$) and idler beams ($I_{1}$ and $I_{2}$). The beams $S_{1}$ and $S_{2}$
are combined at a beam splitter, the outputs of which are sent to detectors.
Both sources emit the idler beams into the same spatial mode, $I$. Considering
that the the idler photons are not detected at all, do you expect that
interference fringes can be observed in the detected signal outputs? Why or
why not?

Figure 2: Three architectures of the Zou-Wang-Mandel Interferometer. In (a)
and (b) the idler paths produced in sources $Q_{1}$ and $Q_{2}$ are aligned
and a $50$:$50$ beam splitter, BS2, combines signal paths $S_{1}$ and $S_{2}$,
which path information is erased and single-particle interference can be
observed in the detector, even though idler particles are not detected at all.
The field transmittance function, $T$, of an object illuminated by the idler
field $I$ can be observed in the interference pattern of the signal beams at
the detector, even though signal particles have not interacted with that
object. The phase $\phi$ can be tuned by adjusting signal, idler or pump
optical path lengths. In (b) and (c) signal and idler are emitted in the same
direction as the pump (collinear emission), and if they are all at distinct
frequencies, dichroic mirrors can be used to separate them. In (b) the
dichroic mirror DM1 reflects idler photons and transmits signal wavelength,
whereas DM2 transmits the pump and reflects idler photons. In (c), single
crystal is pumped from both sides. Dichroic mirror, DM1, reflects the pump and
transmits both signal and idler, whereas DM2 reflects idler and transmits the
signal. Finally, DM4 reflects signal and idler while transmitting the pump.
Notice that in this architecture, undetected light goes twice through the same
sample ($T^{\prime}=T^{2}$).

A few simple calculations can help us understand what is going on. In a first
approach to this problem, it is instructive to write down the particle states
(here photons) in the device as relatively-simple state vectors. In this
picture the action of a nonlinear source is simply to add one photon each to
the appropriate modes. For example, the action of first source, $Q_{1}$ in
Fig. 2 is modelled as taking the vacuum input in modes $S_{1}$ and $I_{1}$ and
transforming it as
$|0\rangle_{S_{1}}|0\rangle_{I}\rightarrow|1\rangle_{S_{1}}|1\rangle_{I}$,
where $|1\rangle_{S_{1}},\;|1\rangle_{I}$ represent a photon occupying mode
$S_{1},\;I$, respectively. We keep the kets as single-photon number states so
as to maintain the same notation as the rest of the paper, but since first
quantization does _not_ use the Fock basis (all modes are assumed to have only
one photon) we could equally-well omit the occupation number – that is
$|1\rangle_{I}=|I\rangle$, for example. Given this assumption it is also very
unlikely that _both_ nonlinear sources, $Q_{1}$ and $Q_{2}$, fire in sync. We
will see in Sec. 5 that the object/sample in the idler path is modelled as a
beam splitter with complex field field transmittance function of that object
by $T=|T|e^{i\gamma}$, with $0\leq|T|\leq 1$ and $0\leq\gamma<\pi/2$. In this
picture, the action of the object in Fig. 2 is to transform the state produced
in source $Q_{1}$ as
$|1\rangle_{S_{1}}|1\rangle_{I}\rightarrow|1\rangle_{S_{1}}\left(T|1\rangle_{I}+i\sqrt{1-|T|^{2}}|1\rangle_{0}\right)$,
where $|1\rangle_{0}$ represents a photon absorbed or scattered by the object.

Assuming both sources are identical and their emissions are coherent, the
state of a photon pair just before BS2 in Fig. 2 can be written as

$\frac{\left(T|1\rangle_{I}+i\sqrt{1-|T|^{2}}|1\rangle_{0}\right)|1\rangle_{S_{1}}+e^{-i\phi}|1\rangle_{I}|1\rangle_{S_{2}}}{\sqrt{2}},$
(4)

where $0\leq\phi<2\pi$. Note that mode $I$ does not acquire a second photon as
the single photon in that mode is assumed to have come from _either_ the
second or first crystal. At this point the origin of this photon could be
determined by seeing which of modes $S_{1}$ and $S_{2}$ contain a photon.

The final beam splitter BS2 combines the signal fields, after which the state
of the twin particles can be written as $|\psi_{f}\rangle=$

$\displaystyle|\psi_{f}\rangle=$
$\displaystyle\frac{1}{2}\left(\left(T+ie^{-i\phi}\right)|1\rangle_{I}+i\sqrt{1-|T|^{2}}|1\rangle_{0}\right)|1\rangle_{S_{1}}$
$\displaystyle+\frac{1}{2}\left(\left(e^{-i\phi}+iT\right)|1\rangle_{I}+\sqrt{1-|T|^{2}}|1\rangle_{0}\right)|1\rangle_{S_{2}}$
(5)

By tracing out the idler modes $I$ and $O$ in the state above, we can obtain
the count-rate at a detector placed at either output of BS2:

$\displaystyle\mathcal{R}_{S_{1}(S_{2})}=\frac{1\pm|T|\cos(\phi+\gamma)}{2}.$
(6)

Where – strikingly – an interference pattern modulated by the object $T$ can
now be observed, despite the fact that _neither_ $S_{1}$ or $S_{2}$ interacted
with that object. Coherence is thus “induced” between the two modes as the
result of aligning as precisely as possible the shared idler mode. The idler
mode $I$, which contains no phase information, is typically discarded.

In the ZWMI, which we have just described, the interference visibility is
directly proportional to the absolute value of the field transmission
coefficient:

$\mathcal{V}_{ZWM}\propto|T|.$ (7)

This relationship holds even if the intensities of the signal beams are not
equal. In fact, any photon loss in the idler arm results in a reduction of the
total visibility, as it introduces partial path distinguishability (welcher-
weg information).

Let us compare this with the nonlinear effect of loss ($|T|<1$) in an arm of a
MZI (Eq.3). The linear relation between interference visibility and loss in
the undetected idler photon path between the two sources is a distinguishing
feature of the ZWMI interferometer. It is shown in [40, 42] that this
characterizes the non-classicality of induced coherence without induced
(stimulated) emission. This is a very important point: stimulated emission at
the second source $Q_{2}$ due to the input idler field $I$ is not necessary
for induced coherence (interference), a fact which highlights the non-
classicality of the phenomenon [43]. In the cases where stimulated emission is
not negligible in $Q_{2}$, the interference visibility is a nonlinear function
of the field field transmittance $|T|$ [42, 44]. This regime can be achieved
using very high gain sources (e.g using very high pump power), or by seeding
$Q_{1}$ and $Q_{2}$ via mode $I$ with a coherent state (a laser beam) with the
idler beam wavelength [45, 46].

The ZWM interferometer was generalized to many spatial modes using spatially
correlated photon pairs to produced Quantum imaging with undetected photons
(QIUP) [22, 23]. Quantum interference and spatial correlations between signal
and idler photons [47] together produce in the detected signal photons images
of an object placed in the idler beam (path $d$) [23, 48, 49, 24]. Both the
real part and the imaginary part of the object’s spatially varying field
transmittance function can be observed. The real part $|T|$ is encoded in the
interference visibility. The imaginary part $\gamma$ appears as an
interferometric phase.

Two-Particle Interferometer. A natural question to ask is what happens if,
instead of both sources emitting into the same spatial idler beam, $I$ (Fig.
2), the idler fields from $Q_{1}$ and $Q_{2}$ were emitted into separate
spatial modes, $I_{1}$ and $I_{2}$, and only later combined at a beamsplitter
($BS_{3}$), as illustrated in Fig. 1(b). A straightforward calculation shows
that in this case, interference is not directly observed in intensity
measurements at the signal detector without any post-selection (coincidence
detection). The two photon state after BS2 and BS3 in Fig. 1(b) is

$\displaystyle\left(\frac{f_{-}|1\rangle_{I_{1}}+if_{+}|1\rangle_{I_{2}}+i\sqrt{2(1-|T|^{2})}|1\rangle_{0}}{2\sqrt{2}}\right)|1\rangle_{S_{1}}$
$\displaystyle+\left(\frac{if_{+}|1\rangle_{I_{1}}-f_{-}|1\rangle_{I_{2}}-\sqrt{2(1-|T|^{2})}|1\rangle_{0}}{2\sqrt{2}}\right)|1\rangle_{S_{2}},$
(8)

where $f_{\pm}\equiv\left(Te^{i\phi}\pm 1\right)$.

If the idler photons remain undetected, mathematically we perform a partial
trace over the idler modes and obtain a constant (=$1/2$) signal photon
counting rate at either output of BS2. In other words, no interference is
observed in signal intensities. An intuitive explanation for this is that
which-source information is retrievable in principle, for example, one could
(hypothetically) add a fourth Beam Splitter combining the two idler outputs,
which would reveal which idler came from each source. In other words, by not
detecting idlers we are leaving "lose ends" of information which inhibit
signal intensity modulation due to interference.

Note that if one detects one output of BS2 in coincidence with an output of
BS3, i.e. by using post-selection, it is possible to observe interference [45,
50], and the visibility is $\mathcal{V}=2|T|/(|T|^{2}+1)$.

Figure 3: Three architectures for the SU(1,1) Interferometer. In (a) both
signal and idler paths $S$ and $I$ from the first nonlinear source ($Q_{1}$)
are aligned with signal and idler paths originating in the second nonlinear
source ($Q_{2}$). In (b) and (c) the laser pumps a crystal and is reflected
back through the same crystal The undetected light traverses twice through the
same sample ($T^{\prime}=T^{2}$). In (c) the pump, signal and idler leave the
crystal collinear with each other and if they are all at different frequencies
they can be separated using three dichroic mirrors: DM1 transmits the pump but
reflects signal and idler photons; DM1 transmits the pump and idler photons,
but reflects signal photons; DM3 transmits the pump and signal photons, but
reflects idler photons. In all three architectures, single-particle
interference is seen in signal and idler outputs without post-selection
(coincidence detection). The field transmittance function of an object placed
in either paths $S$ or $I$ can be seen in the interference pattern appearing
in a camera placed in either output path.

An SU(1,1) Interferometer. Let’s now consider that both signal and idler
photons from source $Q_{1}$ are fed into source $Q_{2}$ [51, 52], as shown in
Fig.3. Here we will refer to this interferometer as an “SU(1,1)
interferometer,” also known as a "non-linear Mach-Zehnder[51, 53, 54]. This
experiment can be thought of as a nonlinear adaptation of the MZI, which has
two beam splitters, whereas the SU(1,1) has two nonlinear media, $Q_{1}$ and
$Q_{2}$. The object with transmission function $T=|T|e^{i\gamma}$ is placed in
the idler mode between the crystals. At the output modes the two photon state
can be written as

$|\psi\rangle=\frac{(|T|e^{\gamma+\phi}+1)|1\rangle_{S}|1\rangle_{I}+\sqrt{1-|T|^{2}}|1\rangle_{S}|1\rangle_{0}}{2}.$
(9)

The (singles) counting rate at detectors placed on either output path, is
therefore

$\mathcal{R}_{S/I}=\frac{1+|T|\cos(\phi+\gamma)}{2},$ (10)

giving the intererence visibility $\mathcal{V}_{Y}=|T|$, just as in the case
of the ZWMI (Eq.7). As the interferometer path of the signal, idler or pump is
adjusted, both signal and idler count rates oscillate, a clear manifestation
of interference. The very unique feature of this particular interferometer is
that the single photon output ports are in phase with each other, though out
of phase with the laser output port. That means that if maximum(minimum)
counts are observed in output mode $S$, maximum(minimum) counts are
simultaneously observed in output mode $I$. This leads to the curious
phenomenon of Frustrated Down-Conversion, analysed in Ref.[52]. If one
introduces which path information in the signal or idler paths between the
crystal, for example by unaligning the modes, interference is reduced or even
lost in both signal and idler modes, showing complementarity between welcher
weg (which-path) information and interference visibility.

The experiments described above and illustrated in Figs. 2 and 3 can be viewed
as “quantum eraser” experiments [55, 56]. In the ZWMI interferometer (Fig.2)
after the crystals but before the final beam splitter no interference would
appear in either of the detection modes since the path itself marks which
crystal experienced the the photon-generating down-conversion, however _after_
the final beam splitter this information is erased (a photon in either mode
could have come from either crystal) and thus interference appears. In the
SU(1,1) interferometer (Fig.3) after the first crystal but before the second
crystal no interference would appear in either of the detection modes since
the path itself marks which crystal experienced the the photon-generating
down-conversion, however _after_ the second crystal this information is erased
and interference is observed. Also note that, unlike in interferometers such
as Mach–Zehnder, Michelson and Sagnac, where only a single phase shift is
possible, in the nonlinear interferometers we discussed here, one can
independently change the phases of the pump, of the signal field, and of the
idler fields, and usually these have different frequencies.

Notice that in the setup in Fig.4a signal and idler go though the object, and
in Fig.4a all three fields, signal, idler and pump, go through the imaged
object. In that case the equations in the theory sections must be adapted
accordingly. In addition, we have shown in this tutorial the imaging due to
light transmitted through an object, but it is trivial to adapt the theory to
the case of a reflective object, which is the case of the Optical Coherence
Tomography setup (Fig.14b), described in subsec.6.2.

Figure 4: These are two alternative versions of the SU(1,1) interferomter. In
(a) the same crystal acts as sources $Q_{1}$ and $Q_{2}$, as pump, signal and
idler are reflected back into that crystal. the Dichroic Mirror (DM) only
reflects the signal field and the Long Pass (LP) only reflects the pump. In
(b) we show a setup for collinear non-degenerate downconversion, where both
signal and idler pass through the sample. Dichroic mirror $DM_{1}$ separates
the signal $S$ from the other fields, and dichroic mirror $DM_{2}$ seperates
the idler $I$ from the pump.

It turns out that, in addition to the ZWMI and the SU(1,1) configurations,
there is a whole range of "nonlinear interferometers", i.e interferometric
architectures that involve two or more nonlinear sources[51, 53]. nonlinear
interferometers have proved to be interesting and useful for imaging,
spectroscopy, metrology and other applications[57, 58, 59, 60, 61, 62, 63,
64]. In this tutorial we will explain how to build and model some of these.

##  3 Brief introduction to the main quantum optics states and operators

The basic description of interferometric devices is given in terms of the so-
called creation and annihilation operators of the quantum-optical field, which
are the ladder operators that work on the photon number degree of freedom as

$\displaystyle|n\rangle=\frac{(\hat{a}^{\dagger})^{n}}{\sqrt{n!}}|0\rangle,$
(11)

where $|0\rangle$ has the standard definition of the vacuum state. A mode in a
vacuum state represents a physical electromagnetic mode with only the zero-
point energy.

A monochromatic light field in a pure state (light is most commonly found in
the form of pure states, as mixed states are difficult to produce with objects
that do not typically interact with each other) may exist in a quantum
mechanical superposition of containing different numbers of photons. Thus when
a measurement is performed on the state it is found to have a specific number
of photons. Mathematically this may be expressed, for some state
$|\psi\rangle$, as

$\displaystyle|\psi\rangle=\sum_{n=0}^{\infty}p_{n}|n\rangle,$ (12)

where the $p_{n}$’s are the probability amplitudes of the various definite
photon number states. Since any quantum state may be decomposed in the number
basis we can call it complete. Also, since orthonormality includes linear
independence we can state that the the basis of definite number states (also
called Fock states after Vladimir Fock, the Soviet Physicist) constitute a
true and complete mathematical basis.

One can write the operator corresponding to the electric field in the
following simplified form [65]

$\displaystyle\hat{E}(t,z)=E_{0}\sin(kz)[\hat{a}e^{-i\omega
t}+\hat{a}^{\dagger}e^{i\omega t}].$ (13)

We can rewrite this in terms of two dimensionless quantities called quadrature
operators defined as

$\displaystyle\hat{q}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\hat{a}+\hat{a}^{\dagger}),$ (14)
$\displaystyle\hat{p}$ $\displaystyle=$
$\displaystyle\frac{1}{2i}(\hat{a}-\hat{a}^{\dagger}).$ (15)

Inverting these equations and substituting into Eq. (13) yields

$\displaystyle\hat{E}(t,z)=2E_{0}\sin(kz)\left[\hat{q}\cos(\omega
t)+\hat{p}\sin(\omega t)\right].$ (16)

From this it is clear that the two quadrature operators are always $\pi/2$ out
of phase, and thus always in different quadratures (hence the name).

It is standard to define an uncertainty relation using the generalized
Heisenberg uncertainty principle for non-commuting operators $\Delta A\Delta
B\geq\frac{1}{2}|\langle[\hat{A},\hat{B}]\rangle|$. This yields

$\displaystyle\Delta\hat{q}\Delta\hat{p}\geq\frac{1}{4}.$ (17)

If the equality in this expression is obtained we have a so-called “minimum
uncertainty state”.

It’s helpful to represent quantum-optical states as shapes in the quadrature
space defined by Eqs.(14,15). Specific single-mode states are represented in
this diagram by their variances (the physical extent and shape of the states)
and by their average values (coordinate position of the states). The diagram
can also be interpreted as an intensity-phase plot using the circular-polar
coordinate system (with radial distance as intensity, and angle as phase).
States rotate about the origin as they evolve with time. So for example the
coherent state will trace out sinusoids in its quadrature values. A number of
such visualisations can be seen in Fig.5. The shapes represent the quadrature
variances as defined in Eq.(17), technically they can be thought of as a slice
through the Wigner distribution at half-maximum. The states themselves with be
briefly described, and referenced, in the rest of this section and the next.

Firstly, there is the vacuum state, representing an optical mode unoccupied by
photons. Since in the quantum regime the zero point energy is always present,
this state still has finite quadrature variances. Secondly, and vital for the
description of interferometers, are the _coherent_ and _squeezed_ states.
Coherent states can be defined in three different ways: as the state that
obtains the equality in the uncertainty relation with $\Delta p=\Delta q$, as
a displaced vacuum state with the displacement operator defined as

$\displaystyle\hat{D}(\alpha)\equiv
e^{(\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a})}$ (18)

which displaces the state it operates on in quadrature space by an amount
$\alpha$ and generates a coherent state as
$\hat{D}(\alpha)|0\rangle=|\alpha\rangle$ (see Fig.5), and as eigenstates of
the annihilation operator $\hat{a}|\alpha\rangle=\alpha|\alpha\rangle$. For
optical fields the definitions are equivalent. Coherent states are also the
most “classical” in the sense that their electric fields have a coherent
waveform resembling a classical harmonic oscillator and that they are
unaffected by the removal of a quanta of light (i.e. that they are eignstates
of annihilation).

The squeezed states take their name from the uncertainty relation shown above.
If $p$ and $q$ are associated with coordinates in quadrature space then
$\Delta p$ and $\Delta q$ can be thought of as distances and the product
$\Delta p\Delta q$ as an area. If the equality is obtained in the uncertainty
relation this sets a specific minimum area of uncertainty in quadrature space.
However though this area may be not reduced beyond this minimum its shape may
be altered, allowing a reduction of the uncertainty along one quadrature at
the cost of increasing it along another. This “squeezing” of the area of
uncertainty gives squeezed states their name. Quantum-optical states may also
be squeezed along other bases (for example photon number and phase). In Fig.5
two squeezed states are displayed, one is a vacuum state that has been
quadrature-squeezed, and the other is a coherent state that has been phase-
squeezed.

Figure 5: Quadrature Diagram: A configuration space defined by the two
quantum-optical quadratures $\hat{p}$ and $\hat{q}$. Five states are shown:
the fundamental vacuum state (gray), a displaced vacuum state / coherent state
(red) – both symmetric minimum uncertainty states, a number state (green) with
completely defined intensity and completely undefined phase, a phase state
with opposite uncertainties from the number state (orange-brown, note – in
principle this state should infinitesimally thin, but then you couldn’t see
it); and two squeezed states, a vacuum state squeezed in the quadrature
direction (blue), and a coherent state squeezed in the phase direction
(purple).

Mathematically the squeezing operation on a single quantum-optical mode may be
described as

$\displaystyle\hat{S}(\xi)=\mathrm{exp}\left[\frac{1}{2}\left(\xi^{*}\hat{a}^{2}-\xi\hat{a}^{\dagger
2}\right)\right],$ (19)

where $\xi$ is known as the squeezing parameter and quantifies the amount of
squeezing.

Acting on a mode operator this produces

$\displaystyle\hat{S}^{\dagger}(\xi)\hat{a}\hat{S}(\xi)$ $\displaystyle=$
$\displaystyle\hat{a}\cosh(r)-\hat{a}^{\dagger}e^{i\theta}\sinh(r),$ (20)
$\displaystyle\hat{S}^{\dagger}(\xi)\hat{a}^{\dagger}\hat{S}(\xi)$
$\displaystyle=$
$\displaystyle\hat{a}^{\dagger}\cosh(r)-\hat{a}e^{-i\theta}\sinh(r),$ (21)

where we have used the polar decomposition $\xi=re^{i\theta}$. Equivalently we
can have the squeezing operator acting on a state. Here the vacuum,

$\displaystyle\hat{S}(\xi)|0\rangle\equiv|\xi\rangle=\frac{1}{\sqrt{\cosh(r)}}\sum_{n=0}^{\infty}(-1)^{n}\frac{\sqrt{(2n)!}}{2^{n}n!}e^{in\theta}[\tanh(r)]^{n}|2n\rangle.$
(22)

So the squeezing operation adds a superposition of even numbers of photons to
the vacuum.

Since squeezed light is a fundamental resource in nonlinear interferometry
it’s important to see how it is produced physically. Inside of certain
materials, where light fields spatially coincide, their component electric
(and magnetic) fields may no longer simply add as vectors like they do in free
space. Consider the expansion of the polarization of a general dielectric
material

$\displaystyle\frac{P_{i}}{\epsilon_{o}}=\sum_{j}\chi_{ij}^{(1)}E_{j}+\sum_{jk}\chi_{ijk}^{(2)}E_{j}E_{k}+\sum_{jkl}\chi_{ijkl}^{(3)}E_{j}E_{k}E_{l}+...$
(23)

The polarization $P$ represents how a dielectric material reacts to the
presence of electric fields. The index $i$ runs over the three-dimensional
vector components. The constant $\chi_{ij}^{(1)}$ is called the first order
(or linear) susceptibility and it is a complex vector constant. Likewise
$\chi_{ijk}^{(2)}$ is the second order susceptibility (a tensor constant), and
on and on in that manner. Most materials have only a non-negligible
$\chi^{(1)}$. In this case as an electric field interacts with the material it
induces an oscillating dipole moment, which in turn creates an oscillating
electric field, and so on. Therefore the light propagates through the material
with a dispersion and absorption determined by the real and imaginary parts of
$\chi^{(1)}$, respectively.

However, now consider the case where the other terms in this series are non-
negligible. Take the case of a material with a large $\chi^{(2)}$, for
example. Take an electric field that has two separate frequency components

$\displaystyle
E(t)=E_{1}\left(e^{i\omega_{1}t}+e^{-i\omega_{1}t}\right)+E_{2}\left(e^{i\omega_{2}t}+e^{-i\omega_{2}t}\right),$
(24)

and input it into the second term of the polarization (ignoring the tensor
nature of $\chi^{(2)}$, i.e. assuming co-linearity).

$\displaystyle P^{(2)}(t)$ $\displaystyle=$
$\displaystyle\epsilon_{o}\chi^{(2)}\left[E_{1}^{2}\left(e^{2i\omega_{1}t}+e^{-2i\omega_{1}t}\right)\right.$
$\displaystyle+E_{2}^{2}\left(e^{2i\omega_{2}t}+e^{-2i\omega_{2}t}\right)$
$\displaystyle+2E_{1}E_{2}\left(e^{i(\omega_{1}+\omega_{2})t}+e^{-i(\omega_{1}+\omega_{2}t)}\right)$
$\displaystyle\left.+2E_{1}E_{2}\left(e^{i(\omega_{1}-\omega_{2})t}+e^{-i(\omega_{1}-\omega_{2}t)}\right)+2E_{1}^{2}+2E_{2}^{2}\right].$

Here we have assumed that the electric amplitudes are real. Eq. (3) gives rise
to several interesting phenomena, but we will be interested in the effect
caused by the terms that oscillate as $\omega_{1}-\omega_{2}$. This tells us
that, if two light beams of different frequencies, $\omega_{1}$ and
$\omega_{2}$, pump a material with a large second order susceptibility (a
tensor constant, $\chi^{(2)}$) a new light beam at a frequency of
$\omega_{3}=\omega_{1}-\omega_{2}$ is generated. This is called “difference
frequency generation”. Quantum mechanically the $\omega_{2}$ mode need not be
populated by photons (that is, it may be in the vacuum state) for the process
to occur. In this case the interpretation is that a photon from a strong beam
(called the pump) splits into two daughter photons inside the optical
nonlinearity. If the two daughter photons are in the same spatial and spectral
modes then we can write the interaction Hamiltonian for this process as

$\displaystyle\hat{H}_{I}=i\hbar\chi^{(2)}\left(\hat{a}^{2}\hat{b}^{\dagger}-\hat{a}^{\dagger
2}\hat{b}\right),$ (26)

where $\hat{b}$ represents the pump mode and $\hat{a}$ represents the mode of
the daughter photons. The second term expresses one photon being transformed
into two, the first term is present because the Hamiltonian must be Hermitian.
Suppose the pump beam is in a coherent state (this is called the parametric
approximation, meaning that the pump is undepleted: no photons are lost), then
we can write for the daughter fields alone

$\displaystyle\langle\beta|\hat{H}_{I}|\beta\rangle$ $\displaystyle=$
$\displaystyle
i\hbar\chi^{(2)}|\beta|\left(\hat{a}^{2}e^{i\omega_{1}t}-\hat{a}^{\dagger
2}e^{-i\omega_{1}t}\right),$ $\displaystyle=$ $\displaystyle
i\hbar\chi^{(2)}|\beta|\left(\hat{a}^{2}e^{i(\omega_{1}-2\omega_{2})t}-\hat{a}^{\dagger
2}e^{-i(\omega_{1}-2\omega_{2})t}\right).$

Where in the second line the time dependence of $\hat{a}$ and
$\hat{a}^{\dagger}$ has been made explicit. Here we choose
$\omega_{1}=2\omega_{2}$, so the Hamiltonian is in fact time-independent and
we can write the time evolution operator for the system simply as

$\displaystyle\hat{U}(t)=e^{-i\hat{H}_{I}t/\hbar}=e^{\hbar\chi^{(2)}t|\beta|\left(\hat{a}^{2}-\hat{a}^{\dagger
2}\right)}.$ (28)

Now compare this to Eq. (19) and we see that we have the single mode squeezing
operator where $\xi=2\hbar\chi^{(2)}t|\beta|$. Furthermore we could get two
mode squeezing for the case where the daughter photons are not in the same
modes. So we have a way to physically squeeze vacuum (and other) states.

This is the process that occurs in both Fig.2 and Fig.3 in the nonlinear
crystals creating squeezed light between the two non-pump output modes,
typically called “signal” and “idler” for historical reasons not relevant
here.

It should be remarked that in realistic settings the Hamiltonian at different
times will not commute since frequency values are not completely sharp and
have some distribution. Also higher-order terms do indeed contribute. So the
above is an approximation, though one that is almost-universally standard and
in good agreement with experiments involving two-photon coincidence
measurements.

Now, if we take $\hat{U}(t)$ acting on the vacuum and expand it as a power
series

$\displaystyle\hat{U}(t)|0\rangle=|0\rangle+\hbar\chi^{(2)}t|\beta||2\rangle+(\hbar\chi^{(2)}t|\beta|)^{2}|4\rangle+...$
(29)

For a coherent pump $|\beta|$ which is strong we can get several of these
terms. In this case one pump photon can split into two (second term), two pump
photons can combine and then split into four (third term) and so on. However
if the pump is not very strong only the first two terms will be non-
negligible. In this case we get the state vector for spontaneous parametric
down-conversion.

Now we move away from the description of the states created inside the
nonlinear interferometers and look at the other components. The transformation
that represents a phase shift is

$\displaystyle\hat{a}_{2}=e^{i\phi}\hat{a}_{1}.$ (30)

The theoretical description also needs the operator transformations for beam
splitters, which are given as

$\displaystyle\hat{a}_{2}=\frac{1}{\sqrt{2}}\left[\hat{a}_{1}+i\hat{b}_{1}\right],\quad\hat{b}_{2}=\frac{1}{\sqrt{2}}\left[\hat{b}_{1}+i\hat{a}_{1}\right],$
(31)

where the first two transformations are the fields after a 50:50 beam
splitter, with two input fields. The field operators $\hat{a}$ and $\hat{b}$
represent the two modes (both input and output), and the subscripts 1 and 2
represent before and after the beam-splitter (input and output), respectively.
The $i$ factor on the opposite mode is the result of the phase shift of the
mode under reflection.

Given these transformations the propagation of the quantum state of the device
from input to output is straightforward, if sometimes somewhat burdensome.
There are two basic methods for analyzing the optical modes. We can take the
second term in Eq.(29) – that is assume spontaneous parametric downconversion.
Then the mode operators acting on the vacuum are transformed according to the
relations above and nonlinearities act on the overall state by adding creation
operators to the relevant modes.

This approach is only an approximation since the full description of the
nonlinearities is given by all terms in Eq.(29) and the action of the
operators representing all optical elements in the device. However, performing
these operations on the full state is highly non-trivial. It is much more
efficient to start with a target detection operator and propagate them
_backwards_ through the device to the coherent or vacuum input states. Then
all of the elements of the optical device become linear transformations on the
set of all creation and annihilation operators.

With these basic building blocks we can understand how the fields, represented
by operators or states or diagrams propagate through the device and we can use
these as tools to understand the various effects present. In the next section
we will examine one of the most common applications of interferometer: phase
detection. And in the sections after that we will see how the basic single-
mode description generalizes to the multi-mode case and allows the study of
imaging.

##  4 Theory of Phase Metrology with Undetected Photons

A typical task in interferometry is to determine the relative phase delay
between two (or more) modes of the device. For example in an MZI (Fig.1) we
can measure the delay between the two modes which yields information about the
difference of path lengths between modes A and B (or, perhaps, the index of
refraction of some intervening transparent material, or some other similar
thing). In the simplest case this information can be abstracted out as a phase
shift, and in fundamental studies of interferometers the actual mechanism of
the shift is typically ignored. Devices are then characterized by the minimum
phase shift which can be observed – corresponding to the most sensitive
configuration.

The fundamental limit for _classical_ interferometers (typically classified as
those that use coherent light only) is called the “standard quantum limit"
(SQL), which itself is the combination of two effects.

The first is the radiation-pressure noise, which is a result of the light beam
imparting a fluctuating amount of momentum into the mirror. Obviously, the
imparted momentum causes “jitter”, fowling up the very sensitive phase
measurements an interferometer might otherwise perform. This noise increases
as the power of the light increases. Conversely it decreases as the mass of
the mirror increases. Much of the research into reducing radiation pressure
noise is concerned with mirror stabilization.

The second is shot noise, which is a result of the photon number fluctuations
from “shot to shot”. Shot noise _decreases_ as the intensity increases, in
contrast to radiation-pressure noise. This is the noise source that is
typically the limiting factor. In principle the mass of the mirrors in an
interferometer may be made very large, such that radiation pressure becomes
small when compared to the shot noise. Though in practice this may be very
difficult, there are a number of systems where the dominant source of noise is
indeed the shot noise and it is often the case that the terms SQL and shot
noise limit (SNL) are used interchangeably.

By using simple arguments about the statistics of coherent states the limiting
case for classical nterferometry may be found to be

$\displaystyle\Delta\phi_{\mathrm{min}}=\frac{1}{\sqrt{\bar{n}}},$ (32)

which is the shot noise limit on the minimum detectable phase shift. This is
the best that can be done classically. The factors $\Delta\phi_{\mathrm{min}}$
and $\bar{n}$ are the minimum detectable phase shift, and average photon
number, respectively.

Now the obvious questions are “Can this be improved upon using quantum
resources? And, if so, what new fundamental limit constrains quantum devices?”
The answers to these questions are commonly agreed to be “yes” and “the
Heisenberg limit,” respectively.

The Heisenberg limit presumes to be the absolute limit on the phase
sensitivity of an interferometer. Unlike the shot noise limit in Eq. (32), it
makes no assumptions about the specific kind of light being used. Instead the
Heisenberg limit draws upon the fundamental laws of quantum mechanics to place
a bound on how accurately we may measure a light field’s (or matter wave’s)
phase. However, it must be noted that it is not straightforward to provide a
rigorous derivation of the Heisenberg limit for the phase measurement, which
is accepted by all. The root of the problem lies in the definition of the
phase operator. Below we provide a brief and lucid description of the issue.

A phase operator was originally introduced by Dirac in his celebrated paper on
quantum theory of radiation [66]. To understand Dirac’s approach, let us take
the annihilation operator acting on a coherent state

$\displaystyle\hat{a}|\alpha\rangle=\alpha|\alpha\rangle=e^{i\phi}|\alpha||\alpha\rangle=e^{i\phi}\sqrt{\bar{n}}|\alpha\rangle.$
(33)

The annihilation operator is a purely quantum mechanical object with no
classical analogue. However, it can be decomposed into quantities that are
familiar in classical optics: the average intensity, $\bar{n}$, and the phase,
$\phi$, of an optical field. Dirac took this to mean that the creation and
annihilation operators could be factored into Hermitian observables as

$\displaystyle\hat{a}=e^{i\hat{\phi}}\sqrt{\hat{n}}$ (34a)
$\displaystyle\hat{a}^{\dagger}=\sqrt{\hat{n}}e^{-i\hat{\phi}},$ (34b)

where the second equation is obtained by simply taking the conjugate transpose
of the first. He thus defined the phase operator $\hat{\phi}$. This combined
with the commutation relation $[\hat{a},\hat{a}^{\dagger}]=1$ yields
$[\hat{n},\hat{\phi}]=i$.

Dirac thus concluded that photon number and phase are conjugate (canonical)
observables. Therefore, in order to become more certain about one, we must
become less certain about the other. This relationship can be made
quantitative by employing the generalized Heisenberg uncertainty principle for
non-commuting operators: $\Delta A\Delta
B\geq\frac{1}{2}|\langle[\hat{A},\hat{B}]\rangle|$. Using this we have $\Delta
n\Delta\phi\geq 1/2$. So in order to know as much as we can about the phase,
we must reduce by as much as possible our knowledge of the number of photons
in the field. Since we are concerned with fundamental limits we will take the
case where photon number is as uncertain as possible: when $\Delta
n=\bar{n}/2$. The uncertainty can not be made any larger than this because
then there would be a non-zero probability of detecting a negative number of
photons in the field (which is physically meaningless). Therefore, we get the
following expression for the Heisenberg limit:

$\displaystyle\Delta\phi_{\mathrm{min}}=\frac{1}{\bar{n}}.$ (35)

However, there is a flaw in this argument. The problem exists in the
definition of the phase operator that was later shown to be non-Hermitian [67,
68]. In fact, if we attempt to write down an eigenstate for this phase
operator we can visualize the issue.

Eigenstates of the number operator (representing a light field with an
exactly-known number of photons but a completely undefined phase) are well-
defined as a ring in quadrature space with a diameter equal to the intensity
of the field and a finite area (and thus energy), see Fig. 5. However, in
quadrature space, a phase eigenstate is a wedge radiating out from the origin
to infinity (all of the space that exists at a particular angle). Such a state
has infinite area and thus infinite energy, and thus is not normalizable (see
Fig. 5 again).

Since the problem with Dirac’s phase operator was pointed out, there have been
numerous discussions and proposals on this issue (see, for example, [68, 69,
70, 71]). Furthermore, there exists another approach to understand the
achievable precision of phase measurement from the perspective of quantum
estimation theory [72, 73]. Nevertheless, the Heisenberg limit
($\Delta\phi_{\mathrm{min}}=1/\bar{n}$) is widely used and commonly regarded
as an approximate bound, in the limit of high photon number [67, 71]. The
Heisenberg limit remains a useful and common goalpost for studies in
interferometry.

Now suppose we wish to consider a _specific_ device that probes the abstract
phase shift by imprinting it on some measurable quantity. Mathematically this
means we have some quantity that is a function of the phase $M(\phi)$. So we
ask the question “Given that we are measuring $M$ what is the smallest change
we can detect in $\phi$?” To answer this question start with the Taylor series
expansion of a function $M$ with the variable as $\phi$ about a point
$\phi_{o}$

$\displaystyle M(\phi)=M(\phi_{0})+(\phi-\phi_{0})\left.\frac{\partial
M}{\partial\phi}\right|_{\phi\rightarrow\phi_{0}}+\quad...$ (36)

Where $\partial(\phi-\phi_{0})=\partial\phi$ since $\phi_{0}$ is a constant.
The smallest detectable phase shift would be equal to the smallest we could
make $\phi-\phi_{0}$. Since we are considering this quantity to be very small,
we can truncate the series after the second term and recast the above as

$\displaystyle\frac{M(\phi)-M(\phi_{0})}{\left.\frac{\partial
M}{\partial\phi}\right|_{\phi\rightarrow\phi_{0}}}=\phi-\phi_{0}=\Delta\phi_{\mathrm{min}}.$
(37)

If we take $\phi_{0}$ to be the average value of the phase then
$M(\phi)-M(\phi_{0})$ gains the interpretation of being the variance of $M$
for a _single_ measurement, as $M(\langle\phi\rangle)=\langle M(\phi)\rangle$
– and the derivative becomes the derivative with respect to phase of $\langle
M\rangle$. However, we want the statistically averaged variance for a _series_
of measurements (the standard deviation) of $M$, so we take $(M-\langle
M\rangle)^{2}$ and average it (which is the standard deviation squared), which
yields $\langle M^{2}\rangle-\langle M\rangle^{2}$.

Then, squaring both sides of Eq.(37), making the substitutions above, and
identifying $\langle M^{2}\rangle-\langle M\rangle^{2}$ as the square of the
variance we arrive at

$\displaystyle\Delta\phi_{\mathrm{min}}=\frac{\Delta\hat{M}}{\left|\frac{\partial\langle\hat{M}\rangle}{\partial\phi}\right|}.$
(38)

Where we have promoted $M$ to a quantum mechanical observable, and taken the
root of both sides. This is the minimum detectable phase shift. To calculate
this we need both a choice of measurement operator, and the quantum-mechanical
state that operator works on (in order to take the expectation values).

Now, recall the limit of a standard Mach-Zehnder Interferometer (MZI) with
coherent light input – Eq.(32). We wish to use this new formula to calculate
the sensitivity of this device to changes in the abstract phase $\phi$. We
need the relationship between the input modes and the output modes given by
Eq.(31). These will allow us to write the operators at the detection end of
the MZI in terms of the operators at the input end. We then take the
expectation values of these operators at the input end.

The most important question, with regards to our sensitivity formula and the
MZI, is the choice of the detection scheme $\hat{M}$. Which for a MZI is the
difference of the intensities at the bright-port and the dark-port

$\displaystyle\hat{M}=\hat{b}_{f}^{\dagger}\hat{b}_{f}-\hat{a}_{f}^{\dagger}\hat{a}_{f}.$
(39)

Where the subscript indicates that these operators act on the final state.
This corresponds to an intensity difference measurement between modes A and B
in Fig1. Using this information and Eq. (38) we find for this setup
$\Delta\phi_{\mathrm{min}}=1/|\alpha|=1/\sqrt{\bar{n}}$. Which is,
unsurprisingly, the shot noise limit in Eq.(32).

An analysis of the phase sensitivity has also been recently performed for the
ZWMI by some of us and others [74]. Several interesting effects are found.
Firstly, as might be expected for a “highly quantum” device, the minimum
detectable phase shift reaches below the classical bound of the shot noise
limit, meaning that the device is “super-sensitive”. This effect is maintained
regardless of gain regime (at least in principle). Furthermore, when the
initial crystal $Q_{1}$ is seeded with a strong coherent (laser) beam the
sensitivity is further increased (“boosted”) into the bright-light regime
while still maintaining some aspects of the super-sensitive scaling. Though
the general equations produced by this calculation are very large, a simple
case of the minimum detectable phase-shift (squared) can be presented for
coherent light injection into one of the modes (in this case the one that does
not pass through the sample, corresponding to mode $S_{1}$ in Fig.2) and
intensity difference subtraction measurement between the two detected modes
(modes $S_{1}$ and $S_{2}$ in the same), when the gains are very large (and
equal to each other), and with the probe phase (and all other phases) set to
zero:

$\displaystyle\Delta\phi_{\mathrm{min}}^{2}=\frac{e^{-2r}}{4(1+\beta^{2})}.$
(40)

Note that this equation is not optimal – rather it is presented because of its
tractability. (for detailed discussion see Ref.[74]). From this equation it is
clear that both the squeezing due to the nonlinearities and the coherent light
injection improve the sensitivity. The improvement is exponential for the gain
and inverse-squared for the coherent light injection.

The SU(1,1) interferometer, Fig.3, has become a commonly-studied device [51,
52, 75, 76, 77] due to the fact that it allows super-sensitive detection with
bright light. Recently these interferometers have also been modified so that a
MZI is nested inside [78], which has some conceptual similarity to the ZWMI
configuration in the sense that both beam-splitter and squeezing operations
are performed.

An argument against the use of all non-linear interferometers could be
paraphrased as “If we need a bright laser to pump the non-linearity, wouldn’t
it just be better to use that bright light in an MZI?”. To confront this
criticism, any light that is used to pump a non-linear source is added to the
MZI “light budget”, making the comparison “fair”. In figure Fig.6 the phase
sensitivity of the ZWM, SU(1,1) and MZ interferometers are compared. This
graph uses the concept of a “fair comparison state”. We see that, though the
SU(1,1) configuration performs best, the ZWMI type also displays super-
sensitivity, beating the MZI after even a modest non-linear gain.

Figure 6: Metrology with undetected photons. Figure taken from [74]. The
minimum detectable phase-shift squared of several fair comparison
interferometric setups and detection schemes as a function of gain (of the
first crystal for ZWMI and of _both_ crystals for SU(1,1)). Here we display
the boosted ZWMI setup with intensity detection at mode $B$ (green), intensity
difference detection between modes $B$ and $C$ (brown), the boosted SU(1,1)
setup (black), and a standard coherent-light-seeded MZI with the extra light
needed to create the aforementioned squeezings added to the initial input
(red). The latter is equivalent to the shot-noise limit. All other parameters
are numerically optimized at each point. The circular points (upper set)
represent injected coherent light of about the same intensity as would be
needed for a high-gain non-linearity, and the square points (lower set)
represent a much-brighter coherent input.

Surprisingly, this super-sensitive phase detection is available regardless of
which of the two initial input modes is “boosted” into the bright-light regime
(corresponding to injecting coherent light either into mode $S_{1}$ or into
mode $I$ in Fig.2). Therefore one can choose not to shine the extra laser
through the sample and achieve the same sensitivity increase as if it had
been. Likewise, one can shine the light through the sample, into the mode that
is discarded and thus avoiding hitting the detectors. This technique could
prove very useful in cases where either the detectors or sample is sensitive
to bright coherent light. Furthermore, when contrasted with the SU(1,1)
configuration, which requires adaptive intensity measurements (where the
signal is produced by summing the intensities of the two output modes) or
homodyning for detection, the ZWMI geometry uses intensity subtraction, making
it a more stable and thus at least in some cases more experimentally
desirable.

The ZWMI has also been studied from the perspective of the signal-to-noise
ratio. In Ref.[79] the authors theoretically study a ZWMI with a variable-
field transmittance beam splitter inserted in the idler path between the two
non-linear sources, with different pump intensities for the non-linear sources
as well. They look at the signal-to-noise and visibility of the output as a
function of the gain in the non-linearities and the field transmittance of the
aforementioned beam splitter, paying special attention to the qualitative and
quantitative difference between various regimes of gain. They also find that
the visibility of the system may be optimized by proper choice of the field
transmittance of the beam splitter.

So far we have examined only “single mode” descriptions of the ZWMI and the
SU(1,1). That is, we do not take into account the real momentum and frequency
distributions. We have assumed that fields can differ only in path. In the
following section we build on the previous to create a more complete picture,
especially as it relates to imaging.

##  5 Theory of Imaging with Undetected Photons

Amplitude and phase imaging using a Zou-Wang-Mandel interferometer was
introduced in [22] using collinear SPDC, as shown in Fig.2(b). Three objects
were placed in the undetected idler arm with wavelength $1550$ nm: A cardboard
cut-out, an etched fused silica plate and an etched silicon plate. The images
of these objects were retrieved in the interference pattern of the combined
$810$ nm signal field using an Electron Multiplying Charged Coupled Device
(EMCCD), although a scientific Complementary metal-oxide-semiconductor (sCMOS)
camera could have been used instead. Indeed, there is no requirement of single
photon detection, although it is important to have a camera or spatial
scanning detector that has low-light sensitivity at the signal wavelength.

In this section, we present a rigorous theoretical description of the image
formation in the Zou-Wang-Mandel interferometer. In order to have a thorough
idea of the imaging we must consider the multi-mode structure of optical
fields.

Quantum Imaging with Undetected Photons (QIUP) relies on transverse spatial
correlations between signal and idler photons. Although SPDC photon pairs in
general exhibit transverse spatial entanglement [47], we will see that spatial
entanglement is in general not necessary for imaging via induced coherence. In
the first experiment, lenses in the idler path between the crystals (not shown
in the figures) were used such that the object was in the far field relative
to $Q_{1}$ and $Q_{2}$ [22, 23]. The object was imaged onto the camera also
using lenses in the signal fields. In that case, the imaging was enabled by
momentum correlation between the twin photons. Alternatively, the object and
the camera can be placed in the near field (source plane) relative to the twin
photon sources and, in this case, the imaging is enabled by the position
correlation between twin photons [24]. (Evanescent fields play no role in this
configuration and, therefore, must not be confused with the conventional near-
field imaging.) Images generated in these two cases have distinct features. We
discuss the two configurations separately in Secs. 55.2 and 55.3. We stress
here that the theory is not restricted to twin photons generated by SPDC; it
applies to spatially correlation twin photons generated by any source.

###  5.1 Multi-mode twin photon states

Throughout the analysis we assume that photons propagate as paraxial beams and
are always incident normally on both the object and the detector. Under these
assumptions, the two-photon quantum state can be written as (see, for example,
[47])

$|\widetilde{\psi}\rangle=\int d\textbf{q}_{s}\ d\textbf{q}_{I}\
C(\textbf{q}_{s},\textbf{q}_{I})|\textbf{q}_{s}\rangle_{s}|\textbf{q}_{I}\rangle_{I},$
(41)

where
$|\textbf{q}_{s}\rangle_{s}\equiv\hat{a}_{s}^{\dagger}(\textbf{q}_{s})|vac\rangle$
denotes a signal photon Fock state labeled by the transverse component
$\textbf{q}_{s}$ of the wavevector $\textbf{k}_{s}$. Similarly,
$|\textbf{q}_{I}\rangle_{I}\equiv\hat{a}_{I}^{\dagger}(\textbf{q}_{I})|vac\rangle$
denotes an idler photon Fock state labeled by the transverse component
$\textbf{q}_{I}$ of the wavevector $\textbf{k}_{I}$. The complex quantity
$C(\textbf{q}_{S},\textbf{q}_{I})$ ensures that $|\psi\rangle$ is normalized,
i.e.,

$\int d\textbf{q}_{s}d\textbf{q}_{I}|C(\textbf{q}_{s},\textbf{q}_{I})|^{2}=1.$
(42)

Imaging enabled by both momentum correlation (Sec. 5.2) and position
correlation (Sec. 5.3) can be described by the quantum state given by Eq.
(41). Such a quantum state is usually generated by SPDC at a nonlinear
crystal. However, the theoretical analysis applies to any source that can
generate such a state.

The state 41 can be entangled, i.e. $C(\textbf{q}_{s},\textbf{q}_{I})\neq
C_{s}(\textbf{q}_{s})C_{i}(\textbf{q}_{I})$. However, transverse spatial
entanglement is not a requirement for imaging with undetected photons.
Consider, for example, two separate SU(1,1)s (or two ZWMIs), like in Fig.7a.
One interferometer is placed close to the other such that one camera can
capture at once the signal outputs of both interferometers. In the top SU(1,1)
once places a sample with field transmittance $T_{1}$ and in the other one
places a sample with field transmittance $T_{2}$. An image of $T_{1}$ and
$T_{2}$, and their spatial separation, can be observed on the camera. This
shows that IUP requires transverse spatial correlation between signal and
idler fields at the plane where the object is placed, but spatial entanglement
is not necessary. An experimental result that proves that imaging in a
nonlinear interferometer is possible with only classical spatial correlations
is described in ref.[80]. In that experiment, a seeding laser was injected
through the first pass of a SU(1,1) (Fig.7b), where it stimulated emission
into the signal mode. This laser then went through the object, was reflected
on the mirror and passed again through the crystal, stimulating into the same
signal mode. A picture of the object was seen on the camera. The signal and
idler fields in this case are not entangled, but present classical transverse
spatial correlations [81].

Figure 7: Classical transverse spatial correlations are sufficient for
imaging.(a) Two spatially separated interferometers can together produce an
image on the camera. Transverse spatial entanglement is not necessary for
QIUP. (b)In classical imaging with undetected light [80], two spatially
separated interferometers can together produce an image on the camera.
Transverse spatial entanglement is not necessary for IUP.

###  5.2 Imaging Enabled by Momentum Correlation

####  5.2.1 General theory

It is evident from Eq. (41) that the joint probability density of detecting a
signal photon with transverse momentum $\hbar\textbf{q}_{s}$ and and idler
photon with transverse momentum $\hbar\textbf{q}_{I}$ is given by

$P(\textbf{q}_{s},\textbf{q}_{I})\propto|C(\textbf{q}_{s},\textbf{q}_{I})|^{2}.$
(43)

This probability density characterizes the momentum correlation between the
twin photons. We now show that in the far field configuration, this momentum
correlation enables image formation.

Figure 8: Quantum Imaging with Undetected Photons. (a) In a collinear non-
degenerate ZWMI[82], non-degenerate photon pairs are emitted along the
propagation axis of a Laser at each source. Waveplates and a polarizing beam
splitter (PBS) in the pump are used to control the relative phases and
amplitudes of the two-photon states generated in sources $Q_{1}$ and $Q_{2}$.
Dichroic mirrors or long pass filters can be used to separate the pump from
the daughter fields after each crystal. A lens $L_{0}$ is used to control the
pump waist at the crystals, which affects the twin photon transverse momentum
correlations, which in turn affects the image resolution, as shown in section
5.2.3. Imaging system $A$ ensures a good overlap of the combined signal fields
and optical system $C$ is used to image the plane of the object with spatial
features $T(\rho)$ onto the plane of the camera or scanning detector. (b) In
the case of imaging enabled by momentum correlation, the optical systems $B$
and $B^{\prime}$ guarantee that the object $T(\rho)$ is at the Fourier plane
of sources $Q_{1}$ and $Q_{2}$. An effective positive lens with focal length
$f_{c}$ associates the plane on the camera with the Fourier Plane of the
sources $Q_{1}$ and $Q_{2}$. A plane wave vector $\boldsymbol{q}_{s}$ makes an
angle $\theta$ with the optical axis and is focused along a circle of radius
$|\boldsymbol{\rho}_{c}|$. (c) In the case of imaging enabled by position
correlation, optical systems $B$ and $B^{\prime}$ are imaging systems; a point
$\boldsymbol{\rho}_{s}$ on $Q_{j}(j=1,2)$ is imaged onto
$\boldsymbol{\rho}_{c}=M_{s}\boldsymbol{\rho}_{s}$ on the camera by $A$.

The experimental setup is illustrated in Fig. 8. There are two sources,
$Q_{1}$ and $Q_{2}$, each of which can emit a photon pair. $Q_{1}$ emits the
signal and idler photons into beams $S_{1}$ and $I_{1}$, respectively.
Likewise, $S_{2}$ and $I_{2}$ represent the beams into which the signal and
idler photons are emitted by $Q_{2}$. The two sources ($Q_{1}$ and $Q_{2}$)
almost never emit simultaneously and almost never produce more than two
photons individually. Furthermore, the two sources emit coherently. Under
these circumstances, the quantum state of light generated by the two sources
is given by the superposition of the states generated by them individually,
i.e., by

$\displaystyle|\psi\rangle=$ $\displaystyle\int
d\textbf{q}_{s}d\textbf{q}_{I}\leavevmode\nobreak\
C(\textbf{q}_{s},\textbf{q}_{I})$
$\displaystyle\quad\times\left[\alpha_{1}\hat{a}^{\dagger}_{s_{1}}(\textbf{q}_{s})\hat{a}^{\dagger}_{I_{1}}(\textbf{q}_{I})+\alpha_{2}\hat{a}^{\dagger}_{s_{2}}(\textbf{q}_{s})\hat{a}^{\dagger}_{I_{2}}(\textbf{q}_{I})\right]|vac\rangle,$
(44)

where $\alpha_{1}$ and $\alpha_{2}$ are complex numbers satisfying the
condition $|\alpha_{1}|^{2}+|\alpha_{2}|^{2}=1$, and $|vac\rangle$ represents
the vacuum state.

As shown in Fig. 8, lens systems are used to place both the object and the
camera in the far field (Fourier plane) of the sources. These lens systems
must also ensure that source $Q_{1}$ is imaged onto $Q_{2}$ and the camera is
on the image plane of the object. For the best possible alignment of idler
beams, it is important that $Q_{1}$ is imaged onto $Q_{2}$ with unit
magnification. Here, we present a generic treatment which does not consider
any specific lens systems and provides a full understanding of the imaging
mechanism. The theory can certainly be tailored to specific systems and this
may only result into incremental differences such as a different sign of the
image magnification.

A lens system $B$ that places the object in the Fourier plane of $Q_{1}$ can
be effectively modeled by a single positive lens, where $Q_{1}$ and the object
are located on the back and front focal planes of this positive lens,
respectively. Similarly, the lens system $B^{\prime}$ that places $Q_{2}$ on
the Fourier plane of the object can be modeled by another positive lens. In
order for $Q_{1}$ to be imaged onto $Q_{2}$ with unit magnification, the focal
length these two lenses must be equal, and shall here be denoted _effective
focal length_ , $f_{I}$. Since a transverse wave vector $\textbf{q}_{I}$ is
focused on a point $\boldsymbol{\rho}_{o}$ on the object, within the paraxial
approximation we obtain from lens rules

$\displaystyle\boldsymbol{\rho}_{o}=\frac{\lambda_{I}f_{I}}{2\pi}\textbf{q}_{I},$
(45)

where $\lambda_{I}$ is the mean wavelength of the idler photon.

The interaction of the idler field with the object can be represented by the
transformation made by a beam splitter and we, therefore, have the following
expression [23]:

$\displaystyle\hat{a}_{I_{2}}(\textbf{q}_{I})=e^{i\phi_{I}}\left[T(\boldsymbol{\rho}_{o})\hat{a}_{I_{1}}(\textbf{q}_{I})+R(\boldsymbol{\rho}_{o})\hat{a}_{0}(\textbf{q}_{I})\right],$
(46)

where $\phi_{I}$ is the phase due to propagation of the idler beam from
$Q_{1}$ to $Q_{2}$, the operator $\hat{a}_{0}$ represents vacuum field at the
unused port of the beam splitter (object), $T(\boldsymbol{\rho}_{o})$ is the
amplitude transmission coefficient of the object at a point
$\boldsymbol{\rho}_{o}$ that is related to $\textbf{q}_{I}$ by Eq. (45), and
$|T(\boldsymbol{\rho}_{o})|^{2}+|R(\boldsymbol{\rho}_{o})|^{2}=1$. The
quantity, $R(\boldsymbol{\rho}_{o})$, can be interpreted the amplitude
reflection coefficient at the same point while illuminated from the other
side.

The quantum state of light generated by the system is obtained by combining
Eqs. (5.2.1) and (46). It is given by

$\displaystyle|\Psi\rangle=$ $\displaystyle\int
d\textbf{q}_{s}d\textbf{q}_{I}\leavevmode\nobreak\
C(\textbf{q}_{s},\textbf{q}_{I})[\alpha_{1}|\textbf{q}_{s}\rangle_{s_{1}}+e^{-i\phi_{I}}\alpha_{2}T^{\ast}(\boldsymbol{\rho}_{o})|\textbf{q}_{s}\rangle_{s_{2}}]|\textbf{q}_{I}\rangle_{I_{1}}$
$\displaystyle+\int d\textbf{q}_{s}d\textbf{q}_{I}\leavevmode\nobreak\
C(\textbf{q}_{s},\textbf{q}_{I})e^{-i\phi_{I}}\alpha_{2}R^{\ast}(\boldsymbol{\rho}_{o})|\textbf{q}_{s}\rangle_{s_{2}}|\textbf{q}_{I}\rangle_{0},$
(47)

where
$\hat{a}^{{\dagger}}_{0}(\textbf{q}_{I})|vac\rangle=|\textbf{q}_{I}\rangle_{0}$.

Since the camera is placed in the combined signal field, at the far field
relative to the sources, we can once again use the concept of an effective
positive lens, as shown in Fig. 8(b). Suppose that the focal length of this
lens is denoted by $f_{c}$. Following an argument, which is similar to the one
used to obtain Eq. (45), we find that a point $\boldsymbol{\rho}_{c}$ on the
camera is related to the transverse signal wave vector $\textbf{q}_{s}$ by the
following formula:

$\displaystyle\boldsymbol{\rho}_{c}=\frac{\lambda_{s}f_{c}}{2\pi}\textbf{q}_{s},$
(48)

where $\lambda_{s}$ is the mean wavelength of the signal photon. The quantized
field at a point, $\boldsymbol{\rho}_{c}$, on the camera plane can now be
represented by

$\displaystyle\hat{E}^{(+)}_{s}(\boldsymbol{\rho}_{c})\propto
e^{i\phi_{s_{1}}}\hat{a}_{s_{1}}(\textbf{q}_{s})+e^{i\phi_{s_{2}}}\hat{a}_{s_{2}}(\textbf{q}_{s}),$
(49)

where $\phi_{s_{1}}$ and $\phi_{s_{2}}$ are phases due to propagation of the
signal beams from $Q_{1}$ and $Q_{2}$, respectively, to the camera. The
single-photon counting rate (intensity) at a point $\boldsymbol{\rho}_{c}$ the
camera can be determined by the standard formula
$\mathcal{R}(\boldsymbol{\rho}_{c})\propto\langle\Psi|\hat{E}^{(-)}_{s}(\boldsymbol{\rho}_{c})\hat{E}^{(+)}_{s}(\boldsymbol{\rho}_{c})|\psi\rangle$.
It now follows from Eqs. (5.2.1) and (49) that

$\displaystyle\mathcal{R}(\boldsymbol{\rho}_{c})\propto\int
d\textbf{q}_{I}\leavevmode\nobreak\
P(\textbf{q}_{s},\textbf{q}_{I})\left[1+\left|T(\boldsymbol{\rho}_{o})\right|\cos(\phi_{in}-\text{arg}\\{T(\boldsymbol{\rho}_{o})\\})\right],$
(50)

where
$\phi_{in}=\phi_{s_{2}}-\phi_{s_{1}}-\phi_{I}+\text{arg}\\{\alpha_{2}\\}-\text{arg}\\{\alpha_{1}\\}$,
arg represents argument of a complex number, and $\boldsymbol{\rho}_{o}$ &
$\textbf{q}_{I}$ and $\boldsymbol{\rho}_{c}$ & $\textbf{q}_{s}$ are related by
Eqs. (45) and (48) respectively, and we have assumed
$|\alpha_{1}|=|\alpha_{2}|$ for simplicity.

It is evident from Eq. (50) that the information about the object (both
magnitude and phase of the amplitude transmission coefficient) appears in the
interference pattern observed on the camera, even though the photons probing
with the object are not detected by the camera. The presence of
$P(\textbf{q}_{s},\textbf{q}_{I})$ in Eq. (50) shows that the momentum
correlation between the twin photons enables the image acquisition. For
example, when there is no correlation between the momenta,
$P(\textbf{q}_{s},\textbf{q}_{I})$ can be expressed in the product form
$P(\textbf{q}_{s},\textbf{q}_{I})=P_{s}(\textbf{q}_{s})P_{I}(\textbf{q}_{I})$.
It can be checked from Eq. (50) that in this case no interference pattern will
be observed and the information of the object will be absent in the photon
counting rate measured by the camera. Therefore, in order the imaging scheme
to work there must be some correlation between the momenta of the twin
photons. Furthermore, the momentum correlation also determines the image
quality. In fact, we will see in Sec. 5.2.3 below that this momentum
correlation limits the image resolution.

In the ideal scenario, when the momenta of the twin photons are perfectly
correlated, the probability density $P(\textbf{q}_{s},\textbf{q}_{I})$ can be
effectively replaced by a Dirac delta function. Consequently, it follows from
Eq. (50) that

$\displaystyle\mathcal{R}(\boldsymbol{\rho}_{c})\propto
1+\left|T\left(\boldsymbol{\rho}_{o}\right)\right|\cos\big{[}\phi_{in}-\text{arg}\\{T(\boldsymbol{\rho}_{o})\\}\big{]},$
(51)

The phase $\phi_{in}$ is varied experimentally and consequently an
interference pattern is observed at each point $\boldsymbol{\rho}_{c}$ on the
camera. It is evident from Eq. (51) that the information of a point
($\boldsymbol{\rho}_{o}$) on the object appears in the interference pattern
observed at a point ($\boldsymbol{\rho}_{c}$) on the camera. Extraction of
this information results in imaging.

We illustrate the imaging by first considering an absorptive object for which
we can set $\text{arg}\\{T(\boldsymbol{\rho}_{o})\\}=0$. It from Eqs. (2) and
(51) that the visibility of the single-photon interference pattern at a point
$(\boldsymbol{\rho}_{c})$ on the camera is given by [24]

$\displaystyle\mathcal{V}(\boldsymbol{\rho}_{c})=\left|T(\boldsymbol{\rho}_{o})\right|.$
(52)

Clearly, the spatially dependent visibility provides an image of the object.

Alternatively, one can acquire the image of an absorptive object by
subtracting the minimum intensity from the maximum intensity, i.e., by
determining the quantity

$\displaystyle
G(\boldsymbol{\rho}_{c})=\mathcal{R}_{\text{max}}(\boldsymbol{\rho}_{c})-\mathcal{R}_{\text{min}}(\boldsymbol{\rho}_{c}).$
(53)

In Ref. [22], images were acquired using this method. In Fig.9 IA and IB,
interference is seen in the body of the cat, corresponding to regions of the
idler field that are transmitted through a cardboard cutout. No interference
is seen outside the cat, because the corresponding idler modes are blocked by
the cardboard. If one sums the two outputs the cat disappears (Fig.9ID), and
the Gaussian profile of the signal field is seen. This shows that the total
signal field intensity is not affected by the absorptive object. Subtracting
the two outputs results in a high contrast absorption image of the sample
(Fig.9IC and IIC).

We call $G(\boldsymbol{\rho}_{c})$ the _image function_. It can be readily
checked from Eq. (51) that when the momenta of the twin photons are maximally
correlated,
$G(\boldsymbol{\rho}_{c})\propto\left|T(\boldsymbol{\rho}_{o})\right|$.

Figure 9: Absorption and phase imaging enabled by momentum correlations.
Figures I(A-D) and II(A-B) have been adapted from Ref.[82], which used the
setup in Fig.8(a) and realized imaging enabled by momentum correlations. In IA
and IB are shown two intensity signal outputs of a collinear non-degenerate
ZWMI. The detection wavelength was $810\pm 1.5nm$, the sample was a cardboard
cutout place at the Fourier plane of the sources and illuminated by an idler
beam with wavelength centered at $1550nm$. The difference (sum) of those two
outputs is shown in IC (ID). Phase imaging of an etched silica plate using the
same setup is shown in IIA and IIB. Momentum correlation enabled absorption
(IIC) and phase (IID) images (adapted from Ref.[60]) from a sample of a mouse
heart. The setup was that shown in Fig.3(c) with the addition of lenses and an
off-axis parabolic mirror. The detection and illumination central wavelengths
were $0.8\mu m$ and $3.8\mu m$, respectively.

Since $\text{arg}\\{T(\boldsymbol{\rho}_{o})\\}$ appears in Eq. (51), phase
imaging is also possible using this scheme (Fig.9 IIA, IIB and IID). For
objects with relatively simple phase distribution as the ones considered in
Ref. [22], the image can be obtained by intensity subtraction.

####  5.2.2 Image magnification

It follows from Eqs. (51) and (52) that a point $\boldsymbol{\rho}_{o}$ on the
object is imaged at a point $\boldsymbol{\rho}_{c}$ on the camera. Therefore,
the image magnification ($M$) is equal to the ratio
$|\boldsymbol{\rho}_{c}|/|\boldsymbol{\rho}_{o}|$. It now follows from Eqs.
(45) and (48) that

$M\equiv\frac{|\boldsymbol{\rho}_{c}|}{|\boldsymbol{\rho}_{o}|}=\frac{f_{c}\lambda_{s}|\textbf{q}_{s}|}{f_{I}\lambda_{I}|\textbf{q}_{I}|}.$
(54)

We now note that image blurring must be neglected for defining the
magnification. Therefore, we must only consider the ideal case in which the
momenta of the twin photons are perfectly correlated. As mentioned above, in
this case the probability density governing the momentum correlation can be
effectively replaced by a Dirac delta function. In fact, Eqs. (51) and (52)
are obtained with this condition. In particular, we consider twin photons
generated by SPDC, for which
$P(\textbf{q}_{s},\textbf{q}_{I})\propto\delta(\textbf{q}_{s}+\textbf{q}_{I})$.
Consequently, for determining the image magnification, we need to use the
condition $\textbf{q}_{s}=\textbf{q}_{I}$. Applying this condition to Eq.
(54), we find that the image magnification is given by

$M=\frac{f_{c}\lambda_{s}}{f_{I}\lambda_{I}}.$ (55)

An interesting feature of QIUP in the far field configuration (i.e., enabled
by momentum correlation) is that the image magnification depends on the
wavelengths of twin photons. This fact is illustrated by Fig. 10, which shows
experimental observations presented in Ref. [83].

Figure 10: Image magnification in momentum correlation enabled QIUP. The same
object is imaged for two sets of values of $\lambda_{s}$ and $\lambda_{I}$,
while other parameters such as focal lengths and distances are unchanged.
Higher value of the ratio $\lambda_{s}/\lambda_{I}$ resulted in larger image
magnification (right). (Adapted from Fig. 2 of Ref. [83].)

The wavelength dependence of magnification is also observed in the experiments
reported in Refs. [22, 83, 60, 62].

Thus far, we have not considered the sign of the magnification. This is
because the sign will depend on the details of the lens systems using in the
setup. For example, if the lens system is exactly as chosen in Refs. [22], the
image magnification will have positive sign, i.e., the image will be erect. A
detailed analysis of the image magnification for this case is presented in
Ref. [23].

####  5.2.3 Spatial resolution

The resolution limit of momentum correlation enabled QIUP can be studied by
applying theory discussed in Sec. 5.2.1. A detailed description of this topic
can be found in Ref. [83]. Here, we discuss one resolution measure, namely the
edge-spread function (ESF). We will consider another resolution measures in
Sec. 5.3.3 when we will discuss the resolution limit of the position
correlation enabled QIUP.

We pointed out in Sec. 5.2.1 that the image of an absorptive object can be
obtained by determining the image function, $G(\boldsymbol{\rho}_{c})$. It
follows from Eqs. (50) and (53) that

$\displaystyle G(\boldsymbol{\rho}_{c})\propto\int
d\textbf{q}_{I}\leavevmode\nobreak\
P(\textbf{q}_{s},\textbf{q}_{I})\left|T(\boldsymbol{\rho}_{o})\right|,$ (56)

where $\boldsymbol{\rho}_{o}$ & $\textbf{q}_{I}$ and $\boldsymbol{\rho}_{c}$ &
$\textbf{q}_{s}$ are related by Eqs. (45) and (48), respectively. We will use
the image function for determining the resolution because the mathematical
analysis becomes simpler. We stress that the results remain the same if one
obtains the image from the visibility.

It is evident from Eq. (56) that when the momenta of the twin photons are not
perfectly correlated, information about a range of points on the object plane
appears at a single point on the camera. The broader the probability
distribution $P(\textbf{q}_{s},\textbf{q}_{I})$, the larger is the range of
the points on the object plane. Therefore, it can be readily guessed that a
weaker momentum correlation results in reduced resolution.

To study the resolution quantitatively, we need to know the form of the
probability density function, $P(\textbf{q}_{s},\textbf{q}_{I})$. To this end,
we consider twin photons generated through SPDC and assume that the pump beam
has a Gaussian profile. In this case, the probability density function can be
approximated in the following form (see, for example, [81, 47])

$\displaystyle
P(\textbf{q}_{s},\textbf{q}_{I})\propto\exp\left(-\frac{1}{2}|\textbf{q}_{s}+\textbf{q}_{I}|^{2}w_{p}^{2}\right),$
(57)

where $w_{p}$ represents the waist of the Gaussian pump beam. Clearly, the
standard deviation of the probability distribution is inversely proportional
to the pump waist. Consequently, a larger pump-waist ($w_{p}$) results in a
narrower probability distribution, i.e., enhanced momentum correlation.

To determine the ESF a knife-edge can be used as an object. The image, which
turns out to be a blurred edge, effectively represents the ESF. Without any
loss of generality, we assume that the knife-edge is placed parallel to the
$y_{o}$ axis and along the line $x_{o}=x_{0}^{\prime}$, such that the idler
field is blocked form $x_{o}\leq x_{0}^{\prime}$. Therefore, we can write

$\displaystyle T(\boldsymbol{\rho}_{o})\equiv
T(x_{o},y_{o})=\left\\{\begin{array}[]{ll}0&x_{o}\leq x_{0}^{\prime},\\\
1&x_{o}>x_{0}^{\prime},\\\ \end{array}\right.\quad\forall\leavevmode\nobreak\
y_{o}.$ (60)

It now follows from Eqs. (56), (57), and (60) that the ESF is given by

$\displaystyle\text{ESF}(x_{c})\propto
G(\boldsymbol{\rho}_{c})\propto\text{Erfc}\left(\frac{\sqrt{2}\pi
w_{p}}{f_{c}\lambda_{s}}\left(x_{c}-Mx_{0}^{\prime}\right)\right),$ (61)

where Erfc is the complementary error function and $M$ is the wavelength
dependent image magnification given by Eq. (55). We stress that in order to
determine the ESF one can also measure the position dependent visibility
instead of the image function.

Figure 11: Edge-spread function (ESF) and resolution. a, The image of a knife-
edge is obtained by measuring the position dependent visibility on the camera
(left). The visibility measured along an axis ($x_{c}$) is fitted with error
function to experimentally determine the edge-spread function (right). The
blurring ($\sigma$) is determined from the ESF. b, Experimentally measured
values (data points) of $\sigma$ are compared with theoretical prediction
(solid lines) for two sets of wavelengths, $\lambda_{I}=1550$ nm,
$\lambda_{s}=810$ nm (red), and $\lambda_{I}=780$ nm, $\lambda_{s}=842$ nm
(blue). Since the detected wavelengths are close to each other, the blurring
appears to be almost equal despite wide difference between the illuminating
(undetected) wavelengths. c, The resolution ($\sigma/M$) is measured
experimentally (data points) and compared with theoretical results (solid
curves) for the same sets of wavelengths. Shorter illumination wavelength
results in higher resolution. The resolution enhances with increasing pump-
waist ($w_{p}$), i.e., with stronger momentum correlation between twin-
photons. (Adapted from Fig. 4 of Ref. [83])

Figure 11a shows an experimentally observer image of a knife-edge [83]. The
experimental results are in full accordance with the theoretical predictions
made by Eq. (61). A measure of image blurring is how steeply the complementary
error function representing the ESF rises. A sharper rise means less blurring.
Mathematically, the blurring can be quantified by the inverse of the
coefficient of $x_{c}$ inside the Erfc in Eq. (61), i.e., by

$\displaystyle\sigma=\frac{f_{c}\lambda_{s}}{\sqrt{2}\pi w_{p}}.$ (62)

This quantity can be determined from the experimentally obtained ESF (Fig.
11a): one can check from the properties of the complementary error function
that $\sigma$ is the distance for which the value of ESF rises from $24\%$ to
$76\%$ of the maximum attainable value.

Equation (62) shows how the resolution depends on the momentum correlation
between the twin photons. As mentioned below Eq. (57), a larger value of the
pump waist ($w_{p}$) implies a stronger momentum correlation between the twin
photons. It follows from Eq. (62) that a larger value of $w_{p}$ results in a
smaller value $\sigma$, i.e., less blurring implying higher resolution. Figure
11b shows the experimentally measured values of $\sigma$ for two experimental
setups [83]. The solid lines represent theoretical predictions made from Eq.
(62). Figure 12a demonstrates how the image of a collection of three slits
gets blurred when the momentum correlation between twin photons is reduced.

We now discuss the wavelength dependence of the resolution. Equation (62)
shows that $\sigma$ does not depend on the wavelength ($\lambda_{I}$) of the
undetected photon that interacts with the object; it instead depends on the
wavelength ($\lambda_{s}$) of the detected photon that never interacts with
the object. However, it must _not_ be concluded from this observation that the
resolution depends on the detected wavelength.

Figure 12: Resolution of momentum correlation enabled QIUP. a, Resolution
enhances as momentum correlation becomes higher. A set of slits are imaged for
five values of pump waist ($w_{p}$) in the decreasing order (left to right). A
bigger value of $w_{p}$ means a stronger momentum correlation between the twin
photons, which results in higher resolution. (Wavelengths are kept the same
for each measurement.) b, A smaller value of the undetected wavelength
($\lambda_{I}$) results in better resolution. The same set of slits are imaged
for $\lambda_{I}=1550$ nm (left) and $\lambda_{I}=780$ nm (right), while the
pump waist is kept the same. (Adapted from Figs. 3b and 5b of Ref. [83].)

This is because $\sigma$ is the blurring measured in the camera coordinates
and the resolution is basically the minimum resolvable distance on the object
place. Therefore, a more accurate measure of resolution is obtained if one
divides $\sigma$ by the magnification, $M$, which essentially results in
expressing $\sigma$ in object coordinates. The division by the magnification
is essential for understanding the wavelength dependence of the resolution
because in this imaging configuration the magnification depends on wavelength.
It now follows from Eqs. (55) and (62) that

$\displaystyle{\rm res}=\frac{\sigma}{M}=\frac{f_{I}\lambda_{I}}{\sqrt{2}\pi
w_{p}},$ (63)

which is a measure of resolution of this imaging scheme. It follows from Eq.
(63) that the resolution depends only on the undetected wavelength, i.e., the
wavelength that probes the object. Here, we note that the resolution depends
on the momentum correlation in the same way $\sigma$ does. Therefore, our
conclusions regarding the dependence of resolution on the momentum correlation
remains unchanged.

The wavelength dependence of resolution has been verified experimentally by
building two experimental setups for which the values of detected wavelength
are very close (810 nm and 842 nm), whereas the undetected wavelengths are
widely separated (1550 nm and 780 nm) [83]. The experimental results are
displayed in Fig. 11b, which confirms the prediction made by Eq. (62): the
values of $\sigma$ for the two setups are very close to each other because the
values of detected wavelength are also very close. However, the experimentally
obtained value of $\sigma/M$ shows that the resolution of the two setups are
significantly different (Fig. 11c). This is because the undetected wavelength
for the two setups are widely separated. The wavelength dependence of the
resolution has also been experimentally tested by using a 1951 USAF resolution
test chart and it has been found that the results matches accurately with
theoretical predictions [83]. In Fig. 12b, we show images of the same
collection of slits for two values of undetected wavelength while the pump
waists (i.e., the momentum correlation) is kept fixed. It is evident from this
figure that the resolution is higher for a shorter undetected wavelength.

In refs. [60] and [62], the same imaging resolution Eq.63 was verified for the
$SU(1,1)$ interferometer with illumination photons at the mid-infrared ( $3\mu
m$) and detection wavelength suitable for silicon based cameras.

####  5.2.4 Resolution and Field-of-View

A useful parameter to calculate is the number of spatial modes per direction,
also sometimes referred to as "number of spatial modes" is typically estimated
by the Field of View (FoV) divided by the spatial resolution. The FoV is
straight-forwardly given by the emission angle of the down-converted idler
light that defines the size of the illuminating area:

${\rm FOV_{MC}}=2f_{I}\tan(\theta_{I})\approx 2f_{I}\theta_{I},$ (64)

where $f_{I}$ denotes the focal length of the collimating optical element
adjacent to the crystal, $\theta_{I}$ is the idler divergence angle. Defined
as half-width at half-maximum (HWHM) it is given by

$\theta_{I}=\lambda_{I}\sqrt{\frac{2.78n_{s}n_{I}}{\pi
L(n_{s}\lambda_{I}+n_{I}\lambda_{s})}},$ (65)

where $n_{s}$ $(n_{I})$ is the index of refraction of the signal (idler) field
in the crystal (see supplementary material of ref.[60]).

The number of spatial modes per direction can therefore be estimated as:

$m_{MC}=\frac{\rm FOV_{MC}}{{\rm res}^{\rm FWHM}}\propto
w_{p}\sqrt{\frac{n}{L(\lambda_{I}+\lambda_{s})}},$ (66)

where ${\rm res}^{\rm FWHM}=2\sqrt{ln2}\;{\rm res}$ and $n=n_{s}\approx
n_{I}$. Unsurprisingly, the number of spatial modes per direction does not
depend on $f_{I}$ or the magnification ($M$) - provided the intermediate
optics features a sufficiently large numerical aperture.

We conclude this section by summarizing the key features of the resolution of
momentum correlation enabled QIUP. The resolution enhances if the momentum
correlation between twin photon becomes stronger. For photon generated by
SPDC, the resolution is linearly proportional to the standard deviation of the
probability distribution that governs the momentum correlation. The resolution
is linearly proportional to the wavelength of the undetected photon (i.e., the
photon that probes the object). Therefore, a shorted undetected wavelength
results in a higher correlation.

Finally, the method described in this section applies to any twin-photon
state. Here, we considered a Gaussian probability distribution. For other
forms of probability distribution, an exact mathematical expression may not be
obtained and resorting to numerical simulation may be necessary.

###  5.3 Imaging Enabled by Position Correlation

####  5.3.1 General theory

We now analyze the complementary scenario in which both the object and the
camera are placed in the near field relative to the source. In this case, the
imaging is enabled by position correlation between the twin photons. We stress
that we do not use evanescent fields for the image acquisition and therefore
the our method is not to be confused with conventional near-field imaging. We
simply place the object and the camera at separate image planes of the
sources.

The joint probability density of detecting the signal and the idler photons at
positions (transverse coordinates) $\boldsymbol{\rho}_{s}$ and
$\boldsymbol{\rho}_{I}$, respectively, on the source plane is given by [47]

$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I})\propto\left|\int
d\textbf{q}_{s}\ d\textbf{q}_{I}\ C(\textbf{q}_{s},\textbf{q}_{I})\
e^{i(\textbf{q}_{s}.\boldsymbol{\rho}_{s}+\textbf{q}_{I}.\boldsymbol{\rho}_{I})}\right|^{2},$
(67)

where $C(\textbf{q}_{s},\textbf{q}_{I})$ is introduced in Eq. (41). The
position correlation between the two photons is governed by this joint
probability density. If $P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I})$ can
be expressed as a product of a function of $\boldsymbol{\rho}_{s}$ and a
function of $\boldsymbol{\rho}_{I}$, there is no position correlation. In the
other extreme case, when the positions of the two photons are maximally
correlated, the joint probability density is proportional to a Dirac delta
function.

The schematic of the imaging setup is given in Fig. 8. In contrast to the case
of momentum correlation enabled QIUP, in this case, the propagating part of
the source field is recreated on the object by an imaging system.

As usual, beam $I_{1}$ from source $Q_{1}$ illuminates the object, passes
through source $Q_{2}$, and gets perfectly aligned with beam $I_{2}$. An
imaging system, $B$, is placed between the source $Q_{1}$ and the object ($O$)
such that the idler field at $Q_{1}$ is imaged onto the object with
magnification $M_{I}$. Another imaging system, $B^{\prime}$, images the idler
field at the object onto source $Q_{2}$ with magnification $1/M_{I}$ (i.e.,
demagnified by the equal amount). These two imaging systems also ensure that
$Q_{2}$ lies on the image plane of $Q_{1}$. For simplicity, we have assumed
that magnifications of $B$ and $B^{\prime}$ have the same sign. In order to
obtain the best possible alignment of beams $I_{1}$ and $I_{2}$, it is
essential that the magnitude of the total magnification due to the combined
effect of $B$ and $B^{\prime}$ is 1.

The object is once again characterized by its complex amplitude transmission
coefficient $T(\boldsymbol{\rho}_{o})$, where $\boldsymbol{\rho}_{o}$
represents a point on the object plane. Since the object is treated like a
beam splitter, the quantum field associated with the idler photon at $Q_{2}$
is related to that at $Q_{1}$ by the following formula [24]

$\displaystyle\hat{E}_{I_{2}}^{(+)}(\boldsymbol{\rho}_{I})=e^{i\phi_{I}^{\prime}(\boldsymbol{\rho}_{I})}\big{[}e^{i\phi_{I}(\boldsymbol{\rho}_{o})}T(\boldsymbol{\rho}_{o})\hat{E}_{I_{1}}^{(+)}(\boldsymbol{\rho}_{I})+R(\boldsymbol{\rho}_{o})\hat{E}_{0}^{(+)}(\boldsymbol{\rho}_{I})\big{]},$
(68)

where $\hat{E}_{0}^{(+)}(\boldsymbol{\rho}_{I})$ is the corresponding vacuum
field, $|T(\boldsymbol{\rho}_{o})|^{2}+|R(\boldsymbol{\rho}_{o})|^{2}=1$,
phases $\phi_{I}(\boldsymbol{\rho}_{o})$ and
$\phi_{I}^{\prime}(\boldsymbol{\rho}_{I})$ are introduced by the imaging
systems $B$ and $B^{\prime}$, respectively, and due to the presence of these
imaging systems, $\boldsymbol{\rho}_{o}=M_{I}$ and $\boldsymbol{\rho}_{I}$ are
related by the formula

$\displaystyle\boldsymbol{\rho}_{o}=M_{I}\boldsymbol{\rho}_{I}.$ (69)

It follows from Eq. (68) that (see [24] for a detailed proof)

$\displaystyle\hat{a}_{I_{2}}(\textbf{q}_{I})=\int$ $\displaystyle
d\textbf{q}_{I}^{{}^{\prime}}\frac{1}{M_{I}^{2}}\big{[}\widetilde{T}^{\prime}\left(\frac{\textbf{q}_{I}-\textbf{q}_{I}^{{}^{\prime}}}{M_{I}}\right)\
\hat{a}_{I_{1}}(\textbf{q}_{I}^{{}^{\prime}})$
$\displaystyle+\widetilde{R}^{\prime}\left(\frac{\textbf{q}_{I}-\textbf{q}_{I}^{{}^{\prime}}}{M_{I}}\right)\
\hat{a}_{0}(\textbf{q}_{I}^{{}^{\prime}})\big{]},$ (70)

where $\widetilde{T}^{\prime}(\textbf{q}_{I}/M_{I})$ and
$\widetilde{R}^{\prime}(\textbf{q}_{I}/M_{I})$ are the Fourier transforms of
$\text{exp}[i\\{\phi_{I}(M_{I}\boldsymbol{\rho}_{I})+\phi_{I}^{\prime}(\boldsymbol{\rho}_{I})\\}]T(M_{I}\boldsymbol{\rho}_{I})$
and
$\text{exp}[i\phi_{I}^{\prime}(\boldsymbol{\rho}_{I})]R(M_{I}\boldsymbol{\rho}_{I})$,
respectively. We encourage the readers to convince themselves that
$\hat{a}_{0}$ is related to $\hat{E}_{0}^{(+)}$ in the same way
$\hat{a}_{I_{j}}$ is related to $\hat{E}_{I_{j}}^{(+)}$.

It becomes evident by comparing Eq. (5.3.1) with Eq. (46) that the conditions
due to the alignment of the idler beams are significantly different in near
and far field QIUP. Although the initial quantum state generated by the two
sources is once again given by Eq. (5.2.1), the difference between the
alignment conditions ensures that the final quantum state of light generated
by the imaging system is distinct in the two configurations. In the present
scenario, the quantum state generated by the system is obtained by combining
Eqs. (5.2.1) and (5.3.1) and is given by [24]

$\displaystyle|\psi\rangle=$ $\displaystyle\alpha_{1}\int d\textbf{q}_{I_{1}}\
d\textbf{q}_{s_{1}}\ C(\textbf{q}_{I_{1}},\textbf{q}_{s_{1}})\
|\textbf{q}_{I_{1}}\rangle_{I_{1}}|\textbf{q}_{s_{1}}\rangle_{s_{1}}$
$\displaystyle+\alpha_{2}\int d\textbf{q}_{I_{2}}\ d\textbf{q}_{s_{2}}\
d\textbf{q}_{I}^{{}^{\prime}}\ C(\textbf{q}_{I_{2}},\textbf{q}_{s_{2}})$
$\displaystyle\qquad\times\frac{1}{M_{I}^{2}}\Big{[}\widetilde{T}^{\prime*}\left(\frac{\textbf{q}_{I_{2}}-\textbf{q}_{I}^{{}^{\prime}}}{M_{I}}\right)\
|\textbf{q}^{{}^{\prime}}_{I}\rangle_{I_{1}}$
$\displaystyle\qquad\qquad+\widetilde{R}^{\prime*}\left(\frac{\textbf{q}_{I_{2}}-\textbf{q}_{I}^{{}^{\prime}}}{M_{I}}\right)\
|\textbf{q}^{{}^{\prime}}_{I}\rangle_{0}\Big{]}|\textbf{q}_{s_{2}}\rangle_{s_{2}},$
(71)

where $|\textbf{q}\rangle_{0}=\hat{a}_{0}^{{\dagger}}(\textbf{q})|vac\rangle$.

The two signal beams ($S_{1}$ and $S_{2}$) are superposed by a $50:50$ beam
splitter (BS) and one of the outputs of BS is detected by a camera. An imaging
system ($A$) with magnification $M_{S}$ ensures that the signal field at the
sources is imaged onto the camera (Fig. 8 a & b). If we represent the signal
field at each source ($z=0$) by its angular spectrum ([84], Sec. 3.2; see also
[81]), the positive frequency part of the total signal field at a point,
$\boldsymbol{\rho}_{c}$, on the camera is given by [24]

$\hat{E}_{s}^{(+)}(\boldsymbol{\rho}_{c})\propto\int d\textbf{q}_{s}\left[\
\hat{a}_{s_{1}}(\textbf{q}_{s})+i\
e^{[i\phi_{s0}+\phi_{s}(\boldsymbol{\rho}_{c})]}\
\hat{a}_{s_{2}}(\textbf{q}_{s})\
\right]e^{i\textbf{q}_{s}\cdot\boldsymbol{\rho}_{s}},$ (72)

where the phase difference between the two signal fields is expressed as a sum
of $\phi_{s0}$ and $\phi_{s}(\boldsymbol{\rho}_{c})$; the former is a
spatially independent phase that can be varied to obtain interference patterns
and the latter is a spatially dependent phase that can arise due to the
presence of imaging system $A$. The presence of this imaging system results
into the following relationship between the coordinates

$\displaystyle\boldsymbol{\rho}_{c}=M_{s}\boldsymbol{\rho}_{s}.$ (73)

The photon counting rate at a point $\boldsymbol{\rho}_{c}$ on the camera is
determined by the standard formula
$\mathcal{R}(\boldsymbol{\rho}_{c})\propto\langle\psi|\hat{E}_{s}^{(-)}(\boldsymbol{\rho}_{c})\hat{E}_{s}^{(+)}(\boldsymbol{\rho}_{c})|\psi\rangle,$
where
$\hat{E}_{s}^{(-)}(\boldsymbol{\rho}_{c})=[\hat{E}_{s}^{(+)}(\boldsymbol{\rho}_{c})]^{\dagger}$.
Using Eqs. (67), (69), and (5.3.1) – (73), we find that [24]

$\displaystyle\mathcal{R}(\boldsymbol{\rho}_{c})\propto$ $\displaystyle\int
d\boldsymbol{\rho}_{I}P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I})\big{(}1+|T(\boldsymbol{\rho}_{o})|$
$\displaystyle\quad\times\text{cos}[\phi_{in}+\phi_{s}(\boldsymbol{\rho}_{c})-\phi_{I}(\boldsymbol{\rho}_{o})-\phi_{I}^{\prime}(\frac{\boldsymbol{\rho}_{o}}{M_{I}})-\phi_{T}(\boldsymbol{\rho}_{o})]\big{)},$
(74)

where
$\phi_{in}=\phi_{s0}+\text{arg}\\{\alpha_{2}\\}-\text{arg}\\{\alpha_{1}\\}$
and we have assumed $|\alpha_{1}|=|\alpha_{2}|=1/\sqrt{2}$ for simplicity.

It follows from Eq. (5.3.1) that the information about the object (both
magnitude and phase of the amplitude transmission coefficient) appears in the
interference pattern observed on the camera, even though the photons probing
with the object are not detected by the camera. The presence of
$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I})$ in Eq. (5.3.1) shows that the
position correlation between the twin photons enables the image acquisition.
For example, when there is no correlation between the momenta,
$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I})$ can be expressed in the
product form
$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I})=P_{s}(\boldsymbol{\rho}_{s})P_{I}(\boldsymbol{\rho}_{I})$.
It can be checked from Eq. (5.3.1) that, in this case, no interference pattern
will be observed and the information of the object will be absent in the
photon counting rate measured by the camera. Therefore, in order the imaging
scheme to work there must be some correlation between the positions of the
twin photons. Furthermore, the position correlation also determines the image
resolution; we will elaborate on this in Sec. 5.3.3.

In order to illustrate the image formation, we consider the special case in
which the positions of the photon pair are maximally correlated, i.e.,
$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I})\propto\delta(\boldsymbol{\rho}_{s}-\boldsymbol{\rho}_{I})$.
We now have from Eq. (5.3.1) that [24]

$\displaystyle\mathcal{R}(\boldsymbol{\rho}_{c})\propto
1+\left|T\left(\boldsymbol{\rho}_{o}\right)\right|\cos\big{[}\phi_{in}-\phi_{T}\left(\boldsymbol{\rho}_{o}\big{)}\right],$
(75)

where, for simplicity, we have assumed that the phases introduced by the
imaging systems are spatially independent and have included them in
$\phi_{in}$. It is evident that if $\phi_{in}$ is varied, the photon counting
rate (intensity) at each point on the camera varies sinusoidally, i.e., a
single-photon interference pattern is observed at each point on the camera.

The image is acquired from these interference patterns in the same way as
discussed in Sec. 5.2.1. For example, when the object is purely absorptive,
i.e., when $T(\boldsymbol{\rho}_{o})=|T(\boldsymbol{\rho}_{o})|$, the image is
given by the spatially dependent visibility measured on the camera [24]:

$\displaystyle\mathcal{V}(\boldsymbol{\rho}_{c})=\left|T(\boldsymbol{\rho}_{o})\right|.$
(76)

Alternatively, the image can also be obtained by measuring the image function
that is given by Eq. (53). It can be verified from Eq. (5.3.1) that both the
visibility and the image function are independent of the phases introduced by
the imaging systems. Therefore, the image of an absorptive object can be
acquired using this method even if the phases introduced by the imaging
systems are spatially dependent. However, image acquisition of a phase object
may require information about these phases.

####  5.3.2 Image magnification

In order to determine the magnification, we once again need to neglect image
blurring. Therefore, we must consider the case in which the twin photons are
maximally position-correlated, i.e.,
$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I})\propto\delta(\boldsymbol{\rho}_{s}-\boldsymbol{\rho}_{I})$.
It now follows from Eqs. (69) and (73) that the image magnification is given
by [24]

$M=\frac{M_{s}}{M_{I}}.$ (77)

Clearly, the magnification does not have any explicit dependence on the
wavelengths of the photons. This fact marks one important distinction between
the momentum correlation enabled and position correlation enabled imaging.

####  5.3.3 Spatial resolution

We note from Eq. (5.3.1) that information about a range of points on the
object plane, averaged by the joint probability density
$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I})$, appears at a single point on
the camera. Since this probability distribution characterized the position
correlation between the twin photons (see Eq. (67)), it becomes evident that
the position correlation plays the key role in the image formation in this
case.

The resolution of position correlation enabled QIUP will be studied following
the method described in Sec. 5.2.3. However, we choose a different resolution
measure, namely the minimum resolvable distance. In this case, we consider two
radially opposite points, separated by a distance $d$ and located on an axis
(say, $X_{o}$) on the object plane. These two points can be represented by the
amplitude transmission coefficient

$\displaystyle T(\boldsymbol{\rho}_{o})\equiv
T(x_{o},y_{o})\propto\delta(y_{o})[\delta(x_{o}-d/2)+\delta(x_{o}+d/2)],$ (78)

where $x_{o}$ and $y_{o}$ represent the position along two mutually orthogonal
Cartesian coordinate axes $X_{o}$ and $Y_{o}$, respectively.

The image of the pair of points can be obtained by determining the image
function. It follows from Eqs. (53) and (5.3.1) that for the position
correlation enabled QIUP, the image function takes the form

$\displaystyle G(\boldsymbol{\rho}_{c})\propto\int
d\boldsymbol{\rho}_{o}P\left(\frac{\boldsymbol{\rho}_{c}}{M_{s}},\frac{\boldsymbol{\rho}_{o}}{M_{I}}\right)|T(\boldsymbol{\rho}_{o})|.$
(79)

To assume a form of the joint probability density, we once again assume that
the twin photons are generated through SPDC. In this case, one can represent
the probability density function in the following form [85]:

$\displaystyle
P\left(\frac{\boldsymbol{\rho}_{c}}{M_{s}},\frac{\boldsymbol{\rho}_{o}}{M_{I}}\right)\propto\exp\left[-\frac{4\pi}{L(\lambda_{I}+\lambda_{s})}\left|\frac{\boldsymbol{\rho}_{c}}{M_{s}}-\frac{\boldsymbol{\rho}_{o}}{M_{I}}\right|^{2}\right],$
(80)

where $L$ represents the length of the nonlinear crystal. The standard
deviation of this probability distribution is linearly proportional to
$\sqrt{L}$. Therefore, a shorter crystal generates stronger position
correlation between twin photons.

Using Eqs. (78) – (80), we find that the image of the two points is given by
the image function

$\displaystyle G(\boldsymbol{\rho}_{c})\propto$
$\displaystyle\exp\left[-\frac{4\pi
y_{c}^{2}}{M_{s}^{2}L(\lambda_{I}+\lambda_{s})}\right]$
$\displaystyle\times\Bigg{\\{}\exp\left[-\frac{4\pi}{L(\lambda_{I}+\lambda_{s})}\left(\frac{x_{c}}{M_{s}}-\frac{d}{2M_{I}}\right)^{2}\right]$
$\displaystyle\qquad+\exp\left[-\frac{4\pi}{L(\lambda_{I}+\lambda_{s})}\left(\frac{x_{c}}{M_{s}}+\frac{d}{2M_{I}}\right)^{2}\right]\Bigg{\\}}.$
(81)

Figure 13a displays a simulated image of pair of points separated by $70$
$\mu$m for $L=2$ mm. The other parameters are given in the figure caption.

Figure 13: Resolution and position correlation between twin photons. a,
Simulated camera image of two points separated by a distance of $d=70$ $\mu$m
for the following choice of parameters: $L=2$ mm, $\lambda_{s}=810$ nm,
$\lambda_{I}=1550$ nm, and $M_{s}=M_{I}=1$. b, The image function,
$G(x_{c},0)$, plotted against $x_{c}$ the same set of parameters. The ratio
($\beta$) of its value at the dip to that at one of the peaks is $\beta\approx
0.08$. c, The minimum resolvable distance ($d_{\text{min}}$) plotted against
crystal length ($L$) for $M_{I}=1$ and $M_{I}=2$ using Eq. (83) (solid lines).
The filled circles represent simulated data points for a pair of square
pinholes with side length $1$ $\mu$m. The minimum resolvable distance
increases (i.e., resolution reduces) as the position correlation becomes
weaker. The resolution also decreases as the imaging magnification, $M_{I}$,
from the source to the object increases. (Remaining parameters are same as in
a and b.) (Adapted from Figs. 3c, 3d, and 4c of Ref. [85].)

In order to quantify how resolved the two points are we consider the function
$G(x_{c},0)$, which is obtained by setting $y_{c}=0$ in the image function. If
we plot $G(x_{c},0)$ against $x_{c}$, we get a double-humped curve which is
illustrated by Fig. 13b. A measure of how well two points are resolved can be
given by the ratio ($\beta$) of the value of $G$ at the dip ($G_{\text{dip}}$)
to that at one of the peaks ($G_{\text{peak}}$), i.e.,

$\displaystyle\beta\equiv\frac{G_{\text{dip}}}{G_{\text{peak}}}.$ (82)

The lower the value of $\beta$, the better resolved the two points are.

The two points cannot be resolved after $\beta$ exceeds a certain value, say,
$\beta_{\text{max}}$. We say that the two points are just resolved when
$\beta=\beta_{\text{max}}$; in this case, the separation between the two
points becomes the minimum resolvable distance (i.e., $d=d_{\text{min}}$). It
follows from Eqs. (5.3.3) and (82) that $d_{\text{min}}\propto
M_{I}\sqrt{L(\lambda_{I}+\lambda_{s})}$. (A detailed explanation is provided
in Ref. [85].) In order to obtain a specific value of $d_{\text{min}}$, one
needs to specify the value of $\beta$. There is no strict rule to choose the
value of $\beta_{\text{max}}$. For the purpose of illustration, we choose
$\beta_{\text{max}}=0.81$ which appears in the study of fine structure of the
spectral lines with a Fabry-Perot interferometer ([86], Sec. 7.6.3). In this
case, we numerically obtain the value of the proportionality constant and find
it to be approximately 0.53. That is, the minimum resolvable distance defined
by setting $\beta_{\text{max}}=0.81$ is given by the formula

$\displaystyle d_{\text{min}}\approx
0.53M_{I}\sqrt{L(\lambda_{I}+\lambda_{s})},$ (83)

To test the formula for minimum resolvable distance, we consider a pair of
identical square apertures, each with side length $1$ $\mu$m, placed radially
opposite on the $X_{o}$ axis (object plane). We choose nine values of the
crystal length ($L$) and for each crystal length (i.e., fixed amount of
position correlation), we choose two values of $M_{I}$. In each case, we
numerically simulate the distance between the centers of the apertures by
setting $\beta_{\text{max}}=0.81$. In Fig. 13c, we compare these numerically
simulated distances (data points represented by filled circles) with
theoretically predicted minimum resolvable distances (solid curves) that are
predicted by Eq. (83). Clearly, the simulated data are in excellent agreement
with the theoretical prediction.

In Ref. [87] the same resolution was found by employing an SU(1,1)
interferometer. Instead of the minimum resolvable distance [see Eq. (83)] they
define the resolution through the corresponding full width half maximum of the
edge spread function on the camera ($\sigma^{\rm FWHM}_{PC}$):

${\rm res}_{\rm PC}^{\rm FWHM}=\frac{\sigma_{\rm PC}^{\rm
FWHM}}{M}=0.44M_{I}\sqrt{\frac{L(\lambda_{I}+\lambda_{s})}{n}},$ (84)

where we assumed the same index of refraction, $n$, for signal and idler
fields in the crystal. Equations (83) and (84) reveal important features of
the resolution limit of the position correlation enabled QIUP. We find that
the resolution linearly proportional to the square root of the crystal length.
Since a shorter crystal length implies a stronger position correlation between
the twin photons, it becomes evident that a stronger position correlation
between the twin photons results in a higher spatial resolution.

Furthermore, the resolution is also linearly proportional to the magnification
($M_{I}$) of the imaging system, $B$, placed on the path of the undetected
photon. Therefore, if the cross-section of the undetected beam (at source)
demagnified while illuminating the object, the spatial resolution enhances,
i.e., the resolution can be enhanced at the cost of the field of view (FoV).
It is to be noted that a smaller value of $M_{I}$ results in a higher
magnification of the imaging system (see Eq. (77)). Therefore, if the image
magnification is enhanced by using the optical components placed on the path
of the undetected photon, the resolution also enhances. However, if one
enhances the magnification using the optical elements placed in the path of
the detected photon, resolution does not change.

####  5.3.4 Resolution and Field of View

For a Gaussian pump beam, the Field of View (FoV) is a Gaussian distribution
with a full width at half maximum (FWHM) given by [87]

$FOV_{PC}=\sqrt{2ln2}M_{I}w_{p}.$ (85)

The ratio of the FoV and ${\rm res}_{\rm PC}^{\rm FWHM}$ approximates the
number of spatial modes per direction,

$m_{PC}=\frac{FOV_{PC}}{{\rm res}_{\rm PC}^{\rm FWHM}}\propto
w_{p}\sqrt{\frac{n}{L(\lambda_{I}+\lambda_{s})}}.$ (86)

We conclude this section by summarizing the main results relating to the
resolution of position correlation enabled QIUP. The resolution enhances if
the position correlation between twin photon becomes stronger. For photon
generated by SPDC, the resolution is linearly proportional to the square root
of the crystal-length, which is linearly proportional to standard deviation of
the probability distribution corresponding to the position correlation. Both
the detected and undetected wavelengths play symmetric role in determining the
resolution and the resolution can be enhanced at the cost of the field of
view.

The method described in this section applies to any twin-photon state with in
idealized angular distribution such as the here considered Gaussian
probability distribution. For other forms of probability distribution (see,
for example, [88]), an exact mathematical expression may not be obtained and
resorting to numerical simulation may be necessary.

###  5.4 Comparison between momentum and position correlations enabled QIUP

Equations (66) and (86) show that the number of spatial modes per direction
has the same dependence on experimental parameters in position correlation and
in momentum correlation enabled QUIP. However, these results were obtained
considering the Gaussian approximation for the $sinc^{2}$-shaped angular
emission probability, which for the case of standard collinear SPDC leads to
significant deviations in the exact number of spatial modes per direction[87].
This is particularly noteworthy in the case of position correlation enabled
imaging, where the number of spatial modes per direction is significantly
reduced, compared to imaging via momentum correlation[87].

In Table 1, we summarize the comparison between QIUP schemes that utilize
momentum correlation and position correlation between twin photons.

| Imaging with Momentum Correlations | Imaging with Position Correlations
---|---|---
|  Formula (cf. [83]) |  Experiment (Theory) (cf. [83]) |  Experiment (Theory) (cf. [60]) |  Formula (cf. [85]) |  Experiment (Theory) (cf. [87])
resFWHM | $\frac{\sqrt{2ln2}f_{I}\lambda_{I}}{\pi w_{p}}$ | $366\leavevmode\nobreak\ \mu m\leavevmode\nobreak\ (366\leavevmode\nobreak\ \mu\text{m})$ | $320\leavevmode\nobreak\ \mu\text{m}\leavevmode\nobreak\ (330\leavevmode\nobreak\ \mu\text{m})$ | $0.44M_{I}\sqrt{\frac{L(\lambda_{I}+\lambda_{s})}{n}}$ | $9\leavevmode\nobreak\ \mu\text{m}\leavevmode\nobreak\ (8\leavevmode\nobreak\ \mu\text{m})$
FOV | $2f_{I}\theta_{i}$ | — (3.7 mm) | $9\leavevmode\nobreak\ \text{mm}\leavevmode\nobreak\ (10\leavevmode\nobreak\ \text{mm})$ | $\sqrt{2ln2}w_{p}M_{I}$ | $160\leavevmode\nobreak\ \mu\text{m}\leavevmode\nobreak\ (130\leavevmode\nobreak\ \mu\text{m})$
$m$ | $5w_{p}\sqrt{\frac{n}{L(\lambda_{I}+\lambda_{s})}}$ | — (10) | $28\leavevmode\nobreak\ (30)$ | $2.7w_{p}\sqrt{\frac{n}{L(\lambda_{I}+\lambda_{s})}}$ | $18\leavevmode\nobreak\ (16)$
Table 1: Experimental and theoretical comparison between QUIP enabled by
momentum and position correlations. Experimental parameters for QIUP enabled
by momentum correlations in ref.[83]:
$\lambda_{I}=1550nm,\lambda_{s}=810nm,n_{s}\approx
n_{I}=1.4,f_{I}=75,w_{p}=119\mu m$, experimentally measured res=
res${}^{FWHM}/(2\sqrt{ln2})=220\mu m.$ Experimental parameters for QIUP
enabled by momentum correlations in ref.[60]: L=2mm, $n_{s}\approx n_{I}=1.8$,
$w_{p}$= 430$\mu m$, $M_{I}=1$, $f_{I}$=100mm, $\lambda_{I}$=3.8$\mu m$,
$\lambda_{s}$=0.8, $\theta_{i}=50mrad$. Experimental parameters for QUIP
enabled by position correlations in ref.[87]: L=2mm, $n_{s}\approx n_{I}=1.8$,
$w_{p}$= 430$\mu m$,$M_{I}=0.25$, $\lambda_{I}$=3.8$\mu m$,
$\lambda_{s}$=0.8,$\theta_{i}=50m$rad.

##  6 Further Applications

A main theme of applications for nonlinear interferometers is to enable
measurements with high-performance Silicon-based cameras or linecameras
(arrays) for wavelength regions, where these and also corresponding sources
are not easily available. These are mainly regions in the infra-red. The
applications can be divided into using spatial or spectral correlations or
both.

Using spatial correlations enables imaging techniques such as phase-imaging,
amplitude-imaging, as we have shown in the previous section, which can also
can be extended to microscopy and holography, which is a combination of phase
and amplitude imaging. Spectral correlations enable spectroscopy and optical
coherence tomography (OCT). Applying nonlinear interferometers to
hyperspectral imaging represents the combination of using spatial and spectral
correlations. Note that for spectral measurements using nonlinear
interferomenters based on SPDC one can provide very large spectral bandwidths
with specially designed crystals[89]. Such large bandwidths are typically
otherwise complex and costly to achieve.

Also applications accessing the THz-region are possible, addressing mainly
OCT-type measurements[90] with a single spatial mode, since multi-spatial mode
correlation is extremely challenging to achieve (see "Limitations").

Another special class of applications is refractometry for not easily
accessible wavelength regions[91]. Here the reflectivity and/or field
transmittance of a dielectric medium can be measured (wavelength dependently),
allowing to infer its (wavelength-dependent) refractive index.

###  6.1 Holography and 3D imaging

The first quantitative phase imaging with undetected photons,i.e. extraction
of the value of phase at each pixel of the image, was realized in ref.[83] and
was later realized in ref.[92]. Simple subtraction of the the outputs is not
enough to realize quantitative phase imaging. Fortunately, the interferogram
described by Eqs 51 and 75 has a form very similar to standard interferograms,
and so the well-known digital holography methods can be used to extract the
phase of the object $\text{arg}\\{T(\boldsymbol{\rho}_{o})\\}$.

The phase stepping method requires recording at least three interference
patterns with different values of $\phi_{in}$ given by
$\phi_{in}^{j}=\frac{2\pi j}{K}$, where K is the number of phase steps. The
interferograms obtained for different phases need to be added with
appropriately chosen complex prefactors. The resultant complex-valued function
of position contains the phase information as its argument.

Disadvantages of phase stepping include the necessity of precisely controlling
$\phi_{in}$ and recording multiple images from which a single phase image is
reconstructed. The sample and the setup must not change during the acquisition
of the images.

The Fourier off-axis holography method dispenses with the need of acquiring
multiple frames at a cost of slightly more complex image processing and a
small modification to the experimental setup. Here an extra tilt, or
equivalently a linear phase $\phi_{in}$, is introduced between the interfering
beams at the object plane. This linear phase $\phi_{in}(\rho_{0})=a\rho_{0}$
enables the isolation of the second term in Eqs 51 and 75 , from which the
phase can be extracted by taking its argument. The isolation of the phase is
performed by filtering the interferogram in the Fourier domain. The linear
phase has to be subtracted from the reconstructed phase and therefore it has
to be pre-calibrated.

Interestingly, digital holography can enable three-dimensional ($3$D) imaging.
Multiple phase images, taken at different illumination angles can be combined
algorithmically to obtain a single $3$D image [93].

###  6.2 Optical Coherence Tomography

Optical coherence tomography (OCT) is an interferometric technique for
3D-imaging. It has a host of applications in non-destructive testing as well
as medical imaging, e.g. in ophthalmology. In its most common modality
(Fourier-domain OCT), spectral phases on broadband light proportional to the
depth of reflections from inside a sample can be read out after interference
with a reference arm using fast grating spectrometer. Fourier-transforming the
spectra then yields an axial depth-profile of the sample, which after x-y-
scanning of the sample and/or of the probing beam can be used to reconstruct a
full 3D-image. Since nonlinear interferometers by definition measure
interferometrically encoded information, they are naturally suited to
implement OCT - both in its time-domain[94, 95] and frequency-domain
flavour[61]. The working principle of the latter is shown in Fig.14.

Figure 14: Working principle of Fourier-Domain OCT and its adaptation using a
nonlinear interferometer.

As with imaging applications, the main advantage of OCT with undetected light
is that one can illuminate the sample with light of a wavelength for which the
detectors are not suitable. For example, infrared illumination can be
desirable in the case of highly scattering (but water-free) samples, because
scattering is strongly suppressed for longer wavelengths enabling high
penetration depths. In the practicle implementation of OCT with nonlinear
interferometers, one of the crucial goals often is to reach a high axial
resolution $\Delta z$, which is directly related to the idler spectral
bandwidth $\Delta\lambda$. For a Gaussian spectrum it is given by:

$\Delta z=\frac{0.44\lambda_{I}^{2}}{\Delta\lambda}$ (87)

which will be slightly modified for differently shaped spectra. Formulated in
the frequency domain, as a rule of thumb, one can estimate that a $10THz$
spectral bandwidth corresponds to about $20\mu m$ axial resolution. Typical
non-degenerate SPDC sources, however, feature bandwidths on the order of
$1THz$, resulting in an axial resolution $\Delta z>100\mu m$, which is not
sufficient for most real-world applications. Notwithstanding, there are
several strategies to engineer sources with much broader spectra:

  * •

Choose crystals with a short length, L, because the bandwidth normally scales
as $1/L$. Note, though, that this severely reduces the total brightness, which
scales as $L^{2}$. Thus a 10-fold improvement in axial resolution would result
in a 100-fold reduction in brightness making this a sub-optimal design
strategy.

  * •

Choose chirped poling periods, which can lead to drastically increased
bandwidth. Here, again, the penalty to pay is in spectral brightness compared
to a crystal of the same length. Nevertheless, this reduction in brightness is
is not as severe as for short crystals, making this a possible option,
especially for reaching ultra-broad bandwidths.

  * •

Use signal-idler group velocity matched phase-matching[89], which yields
ultra-broadband spectra without sacrificing spectral brightness. Also, the
bandwidth and total brightness is still dependent on the crystal length but
the trade-off is different here: as a special trait of this type of phase-
matching, the bandwidth scales only with $1/\sqrt{L}$ , whereas the spectral
brightness still scales with $L^{2}$ and therefor the total brightness with
$L^{3/2}$. Thus, trading off a factor of 2 in axial resolution (with a 4x
longer crystal) results in 8x more photons per pump power.

Other relevant parameters for implementing FD-OCT with nonlinear
interferometers are the sensitivity (SNR), imaging depth, SNR-roll-off and
speed. They are highly analogous to conventional FD-OCT, for which there is
extensive introductory and overview literature, e.g. [96].

A special feature of FD-OCT with a undetected photons is: SPDC has no spectral
fluctuations beyond the shot noise, as it is induced by the vacuum with a
temporally constant spectrum. Moreover, any intensity noise of the pump laser
does not reduce the sensitivity, because it only affects the absolute
intensity of the whole spectrum. On the other hand, frequency noise of the
pump laser would need to be as large as the spectrometer resolution to have an
effect on the sensitivity. Thus, the shot-noise-limit for the sensitivity of
FD-OCT can be reached quite straight-forwardly[89]. Note that by harnessing
the high-gain regime of SPDC it is even possible to surpass the shot-noise
limit in OCT [97].

For the practical implementation of FD-OCT with or without a nonlinear
interferometer, the large bandwidths used mandate careful dispersion
management. This can be done numerically, but in practice it is better to
physically compensate as much dispersion as possible directly in the setup.
Interestingly and sometimes usefully, the dispersion in nonlinear
interferometers can be compensated both in the idler arm, and/or in the signal
arm. Note, that due to phase-mismatch away from the center of the SPDC
bandwidth there is an additional dispersion term from the SPDC crystal to
consider, in addition to the dispersion of the optical components (filters,
lenses, dichroic mirrors) signal and idler photons pass through.

Another practical requirement for implementation is the use of a suitably
narrowband pump laser: in order not to affect the imaging depth and
sensitivity roll-off, it should be at least a factor of 2 narrower than the
resolution of the spectrometer used to acquire the signal spectra.

###  6.3 Spectral imaging

One of the first applications of sensing with nonlinear interferometers has
been the pioneering work of Kalashnikov et al.[58]. They used a highly multi-
mode geometry, where a large, collimated laser pumps two relatively thin ppLN
crystal in a gas-cell. In between the crystals the generated mid-IR light is
partially absorbed by a pressure-controlled CO2-gas. Leveraging the spectral-
correlations between the visible and mid-IR light and the correlations between
angle and wavelength in the SPDC process, the absorption spectrum of CO2 at
4.3$\mu m$ as well as wavelength dependent refractive index could be deduced
from the non-trivial circular interference fringes imaged on a CCD camera. A
sensitivity of around $10^{-5}$ for measuring (n-1) and $0.1cm^{-1}$for the
absorption coefficient could be reached with this method.

Extending imaging to spectral imaging faces the "challenge of dimensionality"