# Interferometry and higher-dimensional phase measurements with directionally-
unbiased linear optics

David S. Simon<EMAIL_ADDRESS>Dept. of Physics and Astronomy, Stonehill
College, 320 Washington Street, Easton, MA 02357  Dept. of Electrical and
Computer Engineering & Photonics Center, Boston University, 8 Saint Mary’s
St., Boston, MA 02215, USA  Christopher R. Schwarze<EMAIL_ADDRESS>Dept. of
Electrical and Computer Engineering & Photonics Center, Boston University, 8
Saint Mary’s St., Boston, MA 02215, USA  Alexander V. Sergienko
<EMAIL_ADDRESS>Dept. of Electrical and Computer Engineering & Photonics
Center, Boston University, 8 Saint Mary’s St., Boston, MA 02215, USA  Dept. of
Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, USA

###### Abstract

Grover multiports are higher-dimensional generalizations of beam-splitters, in
which input to any one of the four ports has equal probability of exiting at
any of the same four ports, including the input port. In this paper, we
demonstrate that interferometers built from such multiports have novel
features. For example, when combined with two-photon input and coincidence
measurements, it is shown that such interferometers have capabilities beyond
those of standard beam splitter-based interferometers, such as easily-
controlled interpolation between Hong-Ou-Mandel (HOM) and anti-HOM behavior.
Further, it is shown that the Grover-based analog of the Mach-Zehnder
interferometer can make three separate phase measurements simultaneously. By
arranging the transmission lines between the two multiports to lie in
different planes, the same interferometer acts as a higher-dimensional Sagnac
interferometer, allowing rotation rates about three different axes to be
measured with a single device.

##  I Introduction

One of the chief characteristics of the Maxwell wave equation is that it is
_linear_ , and that its solutions therefore obey the superposition principle:
given two solutions to the wave equation, any linear combination of those
solutions is also a valid solution. The result is that electromagnetic waves
can interfere. More explicitly, since the intensity is proportional to the
magnitude squared of the electric field, $I\sim|E|^{2}$, the intensity
produced when two fields $E_{1}$ and $E_{2}$ are combined on a screen or
viewing device is

$I\sim|E_{1}+E_{2}|^{2}=I_{1}+I_{2}+2\;\mbox{Re}(E_{1}^{\ast}E_{2}).$ (1)

If the two fields differ in phase by $\phi$, the last term is the interference
term, and is of the form

$2|E_{1}E_{2}|\cos\phi\sim 2\sqrt{I_{1}I_{2}}\cos\phi.$ (2)

This classical interference has long formed the basis of a diverse array of
precision measurement methods, with applications ranging from the
determination of the structure of DNA via Bragg diffraction to the measurement
of stellar radii with Michelson stellar interferometers.

Quantum mechanics is also based on a linear wave equation, the Schrödinger
equation, so all quantum systems obey the superposition principle and exhibit
interference between quantum amplitudes. As a result, optical interference
effects can occur at the single photon level. So, for example, in a Young two-
slit experiment a single photon impinging on a pair of slits has an amplitude
for passing through each slit. As long as there is no measurement of which
slit the photon passes through, the two amplitudes interfere with each other;
after many trials, a visible interference pattern can be built up one photon
at a time. Any measurement of which slit the photon traversed will destroy the
interference pattern by effectively erasing one of the two interfering
amplitudes. But as long as this “which path” information is absent, the
resulting quantum interference pattern will look exactly like the classical
interference pattern. This is _single_ -photon interference in the sense that
what is interfering are multiple different amplitudes for the _same_ photon,
not amplitudes for multiple different photons.

It is well-known that single-photon interference patterns can always be
mimicked by interference of classical electromagnetic waves. To produce
interference effects that are distinctly quantum in nature, it is necessary to
go beyond single-photon interference. For two photon interference to occur,
the state must be a superposition of two different two-photon states. Consider
a _pair_ of photons (labeled by subscripts 1 and 2), that can be in two
possible states (for example two polarization states or two spatial modes,
labeled by the letters a and b). If there is no knowledge available about
which photon is in which state, then the system is in a superposition of two
possibilities:

$|\psi\rangle={1\over\sqrt{2}}\left(|\psi_{a}\rangle_{1}|\psi_{b}\rangle_{2}+e^{i\phi}|\psi_{b}\rangle_{1}|\psi_{a}\rangle_{2}\right).$
(3)

Such as state can be created, for example, using type II spontaneous
parametric down conversion (SPDC) if the two oppositely-polarized photons are
fed into separate arms of a Mach-Zehnder interferometer. If a polarization-
dependent phase shift is included in the upper arm of the interferometer and
we don’t know which polarization traveled which path through the system, then
the output state will be of the form above. The phase $\phi$ in equation 3
would then be the extra phase shift one polarization gains relative to the
other when passing through the upper arm. This two-particle superposition,
which cannot be factored into a pair of single-particle states, is an example
of an entangled state. Interference between the pair of superposed two-photon
amplitudes leads to effects that cannot be mimicked classically.

The archetypal example of a two-photon interference effect is the Hong-Ou-
Mandel effect HOM , shown schematically in Fig. 1. The two identical photons
in this case are indistinguishable and are initially in a product state (non-
entangled). They are sent into different ports of a beam splitter, and a
coincidence measurement is made at the two output ports. Each photon can
either be transmitted or reflected at the beam splitter, leading to four
possible outcomes, as shown in the Figure. However, the two outcomes in the
middle (both reflecting or both transmitting) are indistinguishable and appear
with amplitudes of opposite signs, so they exhibit destructive interference.
When the photons enter at widely separated times, they are distinguishable
simply by their exit times, and so there is no interference. But as the delay
$\tau$ decreases, they become less distinguishable and begin to destructively
interfere. The indistinguishability is complete when $\tau=0$, and so at this
point there are no coincidence counts: the two photons can still leave at
either exit port, but they always exit at the _same_ randomly-chosen port,
never at different ports. The result is the famous HOM dip HOM , in which the
coincidence rate dips sharply to zero as the time delay between the photons
vanishes, and which has been verified experimentally in both photonic and
atomic systems beugnon ; wal ; lopes . For recent reviews of two-photon
interference, see jach ; bouchard ; hoch .

Note that here, even though the input is in a product state, what matters is
that the _output_ state of the beam splitter is entangled: it is a
superposition of two two-photon amplitudes
$|\psi_{a}\rangle_{3}|\psi_{b}\rangle_{3}+|\psi_{a}\rangle_{4}|\psi_{b}\rangle_{4}$,
where $3$ and $4$ label the beam splitter output ports. The two coincidence or
cross terms that could potentially arise in this superposition,
$|\psi_{a}\rangle_{3}|\psi_{b}\rangle_{4}$ and
$|\psi_{a}\rangle_{4}|\psi_{b}\rangle_{3}$, are indistinguishable by the
detectors, and so are capable of canceling each other. The key point is
indistinguishability of the interfering amplitudes _at the detection stage_ ,
regardless of the presence or absence of indistinguishability or entanglement
at the input. In the setups described in subsequent sections, the
distinguishability of multiple outcomes will effectively be erased by making
it impossible for the detector to determine the path taken through the system
by the photons, which then allows two-photon interference to take place.

Figure 1: The HOM effect: when two correlated photons are sent into different
ports of a beam splitter, the middle two amplitudes destructively interfere,
so that there are no coincidences detected at the two output ports. Even
though the photons don’t interact with each other, both of them always exit at
the same port, but _which_ port is completely random.

Beam splitters are often an integral part of interferometers. But beam
splitters are essentially one-way devices: the photons can never reverse
direction to exit at the same port from which they entered. Recently, optical
multiports have been studied simon1 ; osawa2 ; kim ; osawa1  that do not share
this directional bias: light entering a given port can exit at _any_ port,
including the one through which it entered. The three-port versions of these
unbiased multiports have been implemented in bulk devices in two different
manners in osawa2  and kim , while plans are currently underway to implement
four-ports on integrated chips. The directionally-unbiased nature of these
devices has implications for reducing the resources required in optical
networks, as well as introducing new capabilities such as novel multiphoton
interference effects, as has been demonstrated, for example, in simon2 ;
osawa3 ; osawa4 . Further examples of the new capabilities introduced by these
devices are discussed in this paper: in the following it will be shown that
measurements with interferometers containing directionally-unbiased multiports
can extract more information than the analogous standard beam splitter-based
interferometers.

Of special interest is the directionally-unbiased four-port, which acts as a
physical implementation of the four-dimensional Grover coin operation moore
that appears in studies of quantum walks carneiro . When two correlated
photons are input to such a four-port device, it has been shown simon2 ;
osawa3  that higher-dimensional analogs of the HOM effect occur, in which both
photons always exit at the same _pair_ of ports: either both reverse direction
and exit at the two input ports, or they both continue in the forward
direction and exit at the other two ports; there is never one photon
reflecting back and one exiting in the forward direction. By introducing
additional phase shifters this effect can be used to control the flow of two-
photon entanglement through multiple branches of an optical network osawa4 .

In this paper, we demonstrate additional two-photon interference effects in
interferometers that contain Grover four-ports. In the next section, the
effect of these four-ports on two-photon amplitudes is reviewed, then in
Section III it is shown that a simple Grover multiport arrangement allows
continuous interpolation between HOM dips (clustered, boson-like behavior) and
anti-HOM peaks (anti-clustered, fermion-like behavior). Unlike previous
approaches to demonstrating an anti-HOM effect, neither prior preparation of
polarization-entangled states mechler  nor non-Hermiticity li ; vetlugin  in
the system is required. Switching between the different types of behavior can
be done in real time simply by varying a phase shift. In Section IV it is
shown that by joining two Grover multiports together it is possible to measure
up to three separate phases simultaneously with a single interferometer that
has the same topology as the Mach-Zehnder interferometer. This effect is then
applied in Section V, showing that a slight variation of the same setup can
simultaneously measure rotations of the device about three orthogonal axes,
providing a three-dimensional generalization of the standard Sagnac
interferometer with a single device. Conclusions are briefly summarized in
Section VI.

Although we are not assuming in this paper that the input photons are
entangled, we do assume that the input consists of temporally-coincident pairs
of indistinguishable photons. Currently, the easiest way to produce such time-
correlated photon pairs is using SPDC, so that the entanglement is already
present for free, so to speak, without any additional effort. As is well-
known, the use of entangled input states can lead to greatly enhanced
resolution in imaging, interferometry, and lithography boto ; israel ; yurke ;
huver ; untern , though at the expense of increased fragility against
environmental disturbances. Particularly relevant to the proposed setup in
this paper, entanglement can improve the resolution of Sagnac interferometry
fink . So the setups described below are naturally well-suited to entangled-
photon generalizations. But we leave the study of entanglement-based
enhancements for study elsewhere, and simply assume the use here of
classically time-correlated pairs of indistinguishable photons as input.
Throughout this paper, “correlated photons” will always mean two
indistinguishable photons arriving in a time-correlated pair, and will not
necessarily imply entanglement.

##  II Grover four-ports

The Grover coin operator, originally introduced for quantum walk applications,
has a four-dimensional version that takes the matrix form:

$U={1\over 2}\left(\begin{array}[]{cccc}-1&1&1&1\\\ 1&-1&1&1\\\ 1&1&-1&1\\\
1&1&1&-1\end{array}\right).$ (4)

A recent recipe for physically implementing directionally-unbiased $n$-ports
simon1  includes the four-dimensional Grover coin matrix as a special case for
$n=4$. In this case, the rows and columns represent the amplitudes entering
and exiting from the four optical input/output ports of the device, so that
$U$ represents the evolution matrix from the input optical state before
encountering the multiport to the output state after the multiport,
$|\psi\rangle_{out}=U|\psi\rangle_{in}$. Labeling the four ports as shown in
Fig. 2, the optical states are of the form

$|\psi\rangle=\left(\begin{array}[]{c}\phi_{1}\\\ \phi_{2}\\\ \phi_{3}\\\
\phi_{4}\end{array}\right),$ (5)

where $\phi_{i}$ is the amplitude at port $i$. Here, we are not explicitly
including directional labels on the states, indicating whether the photon
amplitude is moving inward or outward motion the multiport. In what follows,
we will continue to supress these directional labels, but it should be
understood that the direction flips from inward to outward at each multiport
encounter.

Figure 2: The Grover optical multiport converts input at the four ports into
output amplitudes at all four ports via unitary four-by-four matrix $U$.

Suppose that a pair of correlated photons are input, one on each of two lines
of the Grover multiport (at lines 1 and 2 of Fig. 2, say). Then equal
amplitudes appear for a transmitting state $|\psi_{t}\rangle$ which exits out
the opposite pair of ports (3 and 4) and a state $|\psi_{r}\rangle$ which
reflects back out the original two ports (1 and 2). Here, $|\psi_{t}\rangle$
and $|\psi_{r}\rangle$, defined below, are states that propagate unchanged
during encounters with the multiports, aside from a reflection (in the case of
$|\psi_{r}\rangle$) or a translation from one side of the multiport to the
other (in the case of $|\psi_{t}\rangle$). Switching to notation that will be
more convenient here, let $|n_{1}n_{2}n_{3}n_{4}\rangle$ denote the state with
$n_{j}$ photons on the $j$th line. Let $\hat{a}_{j}^{\dagger}$ be the creation
operator for a photon on the $j$th line, so that

$\displaystyle|n_{1},n_{2},n_{2},n_{3}\rangle$ (6)
$\displaystyle\qquad={1\over\sqrt{n_{1}!n_{2}!n_{3}!n_{4}!}}(a_{1}^{\dagger})^{n_{1}}(a_{2}^{\dagger})^{n_{2}}(a_{3}^{\dagger})^{n_{3}}(a_{4}^{\dagger})^{n_{4}}|0\rangle.$

Then the input state with one photon on each of two lines,

$|\psi\rangle_{in}=|1100\rangle$ (7)

splits into a superposition of two states,

$|\psi\rangle_{in}\to|\psi\rangle_{out}={1\over\sqrt{2}}\left(|\psi_{t}\rangle+|\psi_{r}\rangle\right),$
(8)

where the arrow indicates one time step of evolution, taking the state before
entering the multiport into the state exiting it. Aside from the reversal of
direction (ingoing to outgoing), and of a translation one step to the right in
the case of $|\psi_{t}\rangle$, these states are unchanged by the multiport:

$\displaystyle|\psi_{t}\rangle$ $\displaystyle=$ $\displaystyle{1\over
2}\left(|0020\rangle+|0002\rangle\right)+{1\over\sqrt{2}}|0011\rangle$ (9)
$\displaystyle=$
$\displaystyle{1\over{2\sqrt{2}}}\left(\hat{a}_{3}^{\dagger}+\hat{a}_{4}^{\dagger}\right)^{2}|0\rangle$
(10)

continues to be transmitted unchanged each time it encounters a multiport,
while the state

$\displaystyle|\psi_{r}\rangle$ $\displaystyle=$ $\displaystyle-{1\over
2}\left(|2000\rangle+|0200\rangle\right)+{1\over\sqrt{2}}|1100\rangle$ (11)
$\displaystyle=$
$\displaystyle-{1\over{2\sqrt{2}}}\left(\hat{a}_{1}^{\dagger}-\hat{a}_{2}^{\dagger}\right)^{2}|0\rangle$
(12)

simply reflects backward at each multiport encounter simon2 .

So, sending two photons into the multiport via optical circulators, the output
has equal amplitudes for states $|\psi_{t}\rangle$ and $|\psi_{r}\rangle$.
Until a measurement is made, each of the two two-photon amplitudes exists
simultaneously, moving in opposite directions, as shown in Fig. 3. In other
words, the two-photon state exiting the multiport is now entangled in the
spatial degree of freedom.

Figure 3: (a) Optical circulators are used to input two photons into the
multiport, via lines 1 and 2. The multiport produces an output that is a 50-50
superposition of two two-photon amplitudes: $|\psi_{t}\rangle$ exits on lines
3 and 4, while $|\psi_{r}\rangle$ exits on lines 1 and 2. (b) To produce new
interference effects, the lines 2 and 4 are merged, as are the lines 1 and 3.
A common phase shift has been added to lines 1 and 2, relative to 3 and 4.

##  III Tuning between HOM and anti-HOM interference effects

We now continue to consider a multiport with two input photons inserted via
optical circulators. As in the previous section, this results in a
superposition of a state $|\psi_{r}\rangle$ exiting at the top two ports and
$|\psi_{t}\rangle$ at the bottom two. But, suppose now that pairs of
amplitudes are brought together and allowed to interfere. Start by imposing a
phase shift $\phi$ on each of the photons in the top line
($|\psi_{r}\rangle\to e^{2i\phi}|\psi_{r}\rangle$), and then merge the blue
$\hat{a}_{1}$ and $\hat{a}_{3}$ lines. Merging here means that they both are
directed to a common detector that can’t distinguish whether the detected
photon came from port $1$ or port $3$. Both modes trigger the same detection
event at the end. As a practical matter, we can then effectively replace both
$\hat{a}_{1}^{\dagger}$ and $\hat{a}_{3}^{\dagger}$ by a single operator
representing the mode that the detector sees (call it $\hat{a}^{\dagger}$) .
This is represented schematically by the merger of the blue lines in Fig. 3.
Similarly, combine the two red $\hat{a}_{2}$ and $\hat{a}_{4}$ lines at a
single detector, effectively replacing both $\hat{a}_{2}^{\dagger}$ and
$\hat{a}_{4}^{\dagger}$ by a single detected mode $\hat{b}^{\dagger}$. The
desired merger of the beams can be accomplished by means of a beam splitter,
although this would entail the loss of some light out the unused ports. A
better approach is to merge them via evanescent coupling within a fiber
Y-coupler, which can be designed to transfer nearly all of the light from the
two incoming paths into a single output path with minimal loss. Or, if working
in free space, mirrors can simply be used to direct the two beams to the same
detector, since all that is needed is indistinguishability between the two
paths the photons could take to each detector.

Then at the $a$ and $b$ ports on the right, the output state will be:

$|\psi_{out}\rangle=e^{i\phi}\left[-{i\over\sqrt{2}}\left(|02\rangle+|20\rangle\right)\sin\phi+|11\rangle\cos\phi\right],$
(13)

where $|n,m\rangle$ now means the state with $n$ photons in the $a$ line and
$m$ photons in the $b$ line.

For the special case of $\phi={\pi\over 2}$, we find

$|\psi_{out}\rangle={1\over\sqrt{2}}\left(|02\rangle+|20\rangle\right).$ (14)

In other words, we find the usual HOM effect, with an entangled state of two
photons either in one line or in the other, and _no_ coincidences between the
$a$ and $b$ lines.

On the other hand, if we set $\phi=0$, then we get an anti-HOM effect, in
which we have a product state containing _only_ coincidences:

$|\psi_{out}\rangle=|11\rangle.$ (15)

There is now a coincidence _peak_ at zero delay, instead of a dip.

So, by turning a dial to change the value of $\phi$ we can continuously, and
in real time, interpolate between _no_ coincidences and _all_ coincidences, or
between maximal spatial-mode entanglement and a product state. Unlike previous
methods for interpolating between HOM and anti-HOM effects, there is no need
for loss or non-unitary evolution to be imposed li ; vetlugin , and there is
no need for different polarization-entangled states to be prepared for each
type of behavior mechler . The same interferometer and light source can be
toggled easily between the two types of behavior without reconfiguring the
setup.

The source of the effect is easy to see. After merging the pairs of lines, the
outputs of the top pair and of the bottom pair are, respectively,

$-e^{2i\phi}(\hat{a}^{\dagger}-\hat{b}^{\dagger})^{2}|0\rangle\quad\mbox{and}\quad\hat{(}a^{\dagger}+\hat{b}^{\dagger})^{2}|0\rangle.$
(16)

Adding the two terms with $\phi={\pi\over 2}$, the cross terms cancel and we
get zero coincidence:

$\left[(\hat{a}^{\dagger}-\hat{b}^{\dagger})^{2}+(\hat{a}^{\dagger}+\hat{b}^{\dagger})^{2}\right]|0\rangle=2(\hat{a}^{\dagger
2}+\hat{b}^{\dagger 2})|0\rangle.$ (17)

But for $\phi=0$, the squared terms cancel, leaving _only_ coincidences:

$\left[-(\hat{a}^{\dagger}-\hat{b}^{\dagger})^{2}+(\hat{a}^{\dagger}+\hat{b}^{\dagger})^{2}\right]|0\rangle=4\hat{a}^{\dagger}\hat{b}^{\dagger}|0\rangle.$
(18)

The continuous tuning between HOM and anti-HOM can be made clear
experimentally by introducing several time delays into the system. Imagine,
for example, $\tau_{1}$ and $\tau_{2}$ are delays between the two lines within
a single two-photon amplitude; $\tau_{0}$ is a delay between one two-photon
amplitude and the other. The phase shifts are now due to the time delays,
$\phi_{j}=\omega\tau_{j}$, where $\omega$ it the common frequency of the
indistinguishable photons.

Assuming that all of the photons have the same spectral amplitude,
$\Phi(\omega)$, a little work leads to the coincidence probability between the
two outputs:

$\displaystyle p$ $\displaystyle=$ $\displaystyle{1\over 2}+{1\over 2}\cos
2\phi\int d\omega_{a}\;d\omega_{b}|\Phi(\omega_{a})\Phi(\omega_{b})|^{2}$ (19)
$\displaystyle\qquad\qquad\qquad\qquad\times\cos\left[\omega_{b}\Delta\tau+(\omega_{a}+\omega_{b})\tau_{0}\right]$
$\displaystyle\qquad+{1\over 2}\sin 2\phi\int
d\omega_{a}\;d\omega_{b}|\Phi(\omega_{a})\Phi(\omega_{b})|^{2}$
$\displaystyle\qquad\qquad\qquad\times\sin\left[-\omega_{b}\Delta\tau+(\omega_{a}-\omega_{b})\tau_{0}\right],$

where $\Delta\tau=\tau_{2}-\tau_{1}$. Imagine holding either $\tau_{0}$
constant while varying $\Delta\tau$, or else holding $\Delta\tau$ fixed while
varying $\tau_{0}$. In either case, the result is an HOM dip or anti-HOM peak
with a width similar to the standard HOM dip for the same assumed
$\Phi(\omega)$ (Fig. 4). But notice that in the first case (varying
$\Delta\tau$), the dip doesn’t measure the delays between photons within the
same input pair; rather, it measures the _difference_ in delays between two
different pairs of paths. Effectively it measures the _gradient_ of the inter-
photon delay between the pairs in the upper and lower two-photon paths. In
other words, if $\Delta\phi_{r}$ is the difference between photons in the
upper two lines (in the reflected amplitude) and $\Delta\phi_{t}$ is the
difference between the two photons in the lower path (in the transmitted
amplitude), then the dip is measuring the difference of these two differences,
$\Delta^{2}\phi\equiv\Delta\phi_{r}-\Delta\phi_{t}$.

Figure 4: By tuning the value of $\phi$, the coincidence count can be made to
exhibit boson-like behavior (an HOM dip for $\phi={\pi\over 2}$, red, dashed
curve), classical-like behavior (no two-photon interference, for
$\phi={\pi\over 4}$, black, dotted line), or anti-clustered fermion-like
behavior (a coincidence peak, for $\phi=0$, blue, solid curve). The plots here
are for scans over $\tau_{0}=\phi/\omega$, but scans over
$\Delta\tau=\tau_{2}-\tau_{1}$ show similar behavior. The simulations shown
here use a spectral amplitude $\Phi(\omega)$ proportional to a $sinc$
function.

##  IV Multiple simultaneous phase measurements: the Grover-Mach-Zehnder
Interferometer

The effect above results from placing a Grover four-port into an
interferometer. This naturally leads to the question of what other effects
might be possible by building other interferometers from four-ports. In Fig.
5, a Grover four-port analog of the Mach-Zehnder interferometer is shown. A
two-photon state of the form of Eq. 7 is input to the left-hand four-port
($G_{1}$), which splits it into two two-photon amplitudes travelling along the
upper and lower branches to the second four-port, $G_{2}$. The photons are
coupled into $G_{1}$ by a pair of circulators (the red circles in Fig. 5) at
the left, and at the right output is extracted by additional circulators
coupled to four detectors (A, B, C, D). Tagging photon arrival at each
detector with a time stamp, we can measure the coincidence rates
$R_{AB},R_{AC}$, etc. between any pair of detectors. There will also be events
where two photons arrive at the same detector, but in what follows we are only
interested in the coincidence events.

Figure 5: (a) Mach-Zehnder interferometers can be generalized, replacing the
beam splitters by Grover four-ports. The two four-ports are connected by two
lines in the upper branch and two in the lower branch. Phase shifts
(represented by the grey-shaded circles) are added as shown. (b) Photons are
coupled in and out of the system with circulators (the red circles). The
output lines are fed to four detectors, labeled A, B, C, D.

Suppose three phase shifts are added as in Fig. 5. $\phi_{0}$ is the shift
between upper and lower branches (specifically, the shift of $b_{2}$ relative
to $b_{1}$), while $\phi_{1}$ and $\phi_{2}$ are the relative phase shifts
between the two lines within each of those branches (between $b_{1}$ and
$a_{1}$ in the first case, and between $b_{2}$ and $a_{2}$ in the second). So,
for example, the phase of $a_{1}$ is shifted higher than that of $b_{2}$ by
$\phi_{0}+\phi_{1}$.

Some of the coincidence rates are equal ($R_{AC}=R_{BD}$ and $R_{AD}=R_{BC}$),
so $R_{BD}$ and $R_{BC}$ no longer need to be considered. That leaves four
independent coincidence rates, which are readily calculated:

$\displaystyle R_{AC}$ $\displaystyle=$ $\displaystyle R_{BD}$ (20)
$\displaystyle=$ $\displaystyle
R_{0}\left|e^{i(2\phi_{0}+\phi_{1})}\sin\phi_{1}-e^{i\phi_{2}}\sin\phi_{2}\right|^{2}$
$\displaystyle=$ $\displaystyle
R_{0}\left[\sin^{2}\phi_{1}+\sin^{2}\phi_{2}\right.$
$\displaystyle\quad-\left.2\sin\phi_{1}\sin\phi_{2}\cos(2\phi_{0}+\phi_{1}-\phi_{2})\right]$
$\displaystyle R_{AD}$ $\displaystyle=$ $\displaystyle R_{BC}$ (21)
$\displaystyle=$ $\displaystyle
R_{0}\left|e^{i(2\phi_{0}+\phi_{1})}\sin\phi_{1}+e^{i\phi_{2}}\sin\phi_{2}\right|^{2}$
$\displaystyle=$ $\displaystyle
R_{0}\left[\sin^{2}\phi_{1}+\sin^{2}\phi_{2}\right.$
$\displaystyle\left.+2\sin\phi_{1}\sin\phi_{2}\cos(2\phi_{0}+\phi_{1}-\phi_{2})\right]$
$\displaystyle R_{AB}$ $\displaystyle=$ $\displaystyle
R_{0}\left|e^{i(2\phi_{0}+\phi_{1})}(\cos\phi_{1}+1)+e^{i\phi_{2}}(\cos\phi_{2}+1)\right|^{2}$
$\displaystyle=$ $\displaystyle
R_{0}\left[(\cos\phi_{1}+1)^{2}+(\cos\phi_{2}+1)^{2}\right.$
$\displaystyle\left.+2(\cos\phi_{1}+1)(\cos\phi_{2}+1)\cos(2\phi_{0}+\phi_{1}-\phi_{2})\right]$
$\displaystyle R_{CD}$ $\displaystyle=$ $\displaystyle
R_{0}\left|e^{i(2\phi_{0}+\phi_{1})}(\cos\phi_{1}-1)+e^{i\phi_{2}}(\cos\phi_{2}-1)\right|^{2}$
$\displaystyle=$ $\displaystyle
R_{0}\left[(\cos\phi_{1}-1)^{2}+(\cos\phi_{2}-1)^{2}\right.$
$\displaystyle\left.+2(\cos\phi_{1}-1)(\cos\phi_{2}-1)\cos(2\phi_{0}+\phi_{1}-\phi_{2})\right],$

where $R_{0}$ is a fixed constant determined by the rate at which photons are
entering the initial multiport. Once these coincidence rates are measured, the
formulas can be inverted to solve for the four unknowns, $R_{0}$, $\phi_{0}$,
$\phi_{1}$, $\phi_{2}$.

Simplifications can be obtained by taking sums and differences of rates:

$\displaystyle{{R_{AC}+R_{AD}}\over{R_{0}}}$ $\displaystyle=$ $\displaystyle
2(\sin^{2}\phi_{1}+\sin^{2}\phi_{2})$ (24)
$\displaystyle{{R_{AD}-R_{AC}}\over{R_{0}}}$ $\displaystyle=$ $\displaystyle
4\sin\phi_{1}\sin\phi_{2}\cos(2\phi_{0}+\phi_{1}-\phi_{2})$ (25)
$\displaystyle{{R_{AB}+R_{CD}}\over{R_{0}}}$ $\displaystyle=$ $\displaystyle
2(1+\cos^{2}\phi_{1})+2(1+\cos^{2}\phi_{2})$ (26)
$\displaystyle\qquad+4\cos(2\phi_{0}+\phi_{1}-\phi_{2})$
$\displaystyle\qquad\qquad\times(1+\cos\phi_{1}\cos\phi_{2})$
$\displaystyle{{R_{AB}-R_{CD}}\over{R_{0}}}$ $\displaystyle=$ $\displaystyle
4(\cos\phi_{1}+\cos\phi_{2})$ (27)
$\displaystyle\qquad\times[1+\cos(2\phi_{0}+\phi_{1}-\phi_{2})],$

where $R_{0}$ is some constant that will depend on the rate at which photons
enter the system.

This set of four independent equations allows the three phase shifts
$\phi_{0},\phi_{1},\phi_{2}$ and the constant $R_{0}$ to be determined from
the experimentally-measured coincidence rates. They can be readily solved
numerically for a given set of measured coincidence rates.

In practice, $R_{0}$ can be measured independently for the given source (it is
essentially the total down conversion rate). So, if it is assumed that $R_{0}$
is known, then in fact the equations simplify to the point where it is
possible to give exact solutions; Mathematica, for example, can solve to
provide a set of exact analytic solutions. However, these solutions are more
than a page long when written out, and so we do not choose to reproduce them
here. But, despite their complexity, the exact solutions could be used instead
of numerical solutions, meaning that the only errors would be due to
imprecision in the measured data.

Figure 6: Solving for the phases as functions of the measured coincidence
rates gives multiple possible solution surfaces, depicted schematically in
this figure as curves. Calibration of the system before use by measuring the
coincidence rates at zero phase shifts allows determination of which solution
surface the system lies upon. However, in most cases, this will still leave a
sign ambiguity (between the upper and lower branches of the surface), allowing
only the absolute value of the phase to be determined in these cases.

A more serious issue is that, because of the squares in the equations, there
are multiple solutions for the phases at a given set of coincidence rates, and
so the correct solution for the given physical problem must be singled out
from among the mathematical possibilities. In practice, this can be done by
carrying out an initial calibration before the system is used: measure all the
coincidence rates for the given source when the system has no phase shifts
introduced. Only one of the multiple solution surfaces will pass through the
correct coincidence values. See Fig. 6 for a simplified picture of this. This
uniquely determines the solution curve, which in turn determines the phase up
to an overall sign.

In general, there could still be overall sign ambiguities left over in the
phases (the choice between upper and lower branch of each curve in Fig. 6)
after determining the correct solution curve. Note that these sign ambiguities
often don’t appear in simplified cases where only two phase shifts are
independent; this can be seen in the example given below.

The end result is that with a Grover four-port generalization of the Mach-
Zehnder interferometer, _measurements of the coincidence rates allow for the
simultaneous determination of the magnitudes (although possibly not the signs)
of three phase shifts with a single interferometer_. It should also be pointed
out that because of the factors of 2 that arise in the sines and cosines of
Eqs. 24-27, the resolution with which $\phi_{0}$ is determined will be double
the resolution of the usual Mach-Zehnder.

In special cases where there are only two independent phases, a simple
analytic solution is readily obtained. For example, suppose that
$\phi_{1}=\phi_{2}$, leaving $\phi_{0}$ and $\phi_{1}$ to be determined. It is
straightforward to show that:

$\displaystyle\cos 2\phi_{0}$ $\displaystyle=$ $\displaystyle\Lambda_{1}$ (28)
$\displaystyle\cos\phi_{1}$ $\displaystyle=$
$\displaystyle{{1\pm\sqrt{1-\Lambda_{1}^{2}}}\over{\Lambda_{2}}},$ (29)

where

$\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle{{R_{AD}-R_{AC}}\over{R_{AD}+R_{AC}}}$ (30)
$\displaystyle\Lambda_{2}$ $\displaystyle=$
$\displaystyle{R_{AB}-R_{CD}}\over{R_{AB}+R_{CD}}.$ (31)

Since these equations are linear, the aforementioned sign ambiguities do not
occur.

Figure 7: Schematic of the standard Sagnac interferometer. As the apparatus
rotates, the detector sees an interference pattern due to the fact that the
two counter-propagating beams gain a relative phase shift $\delta\phi$ that is
proportional to the rotational speed $\omega$.  Figure 8: The Grover-Mach-
Zehnder interferometer of Fig. 5 can be put into a non-planar configuration,
allowing rotations about two axes to be measured via the coincidence rates.
Although not drawn in, detectors are coupled in via circulators after $G_{2}$
as in Fig. 5.

##  V The Grover-Sagnac interferometer

The well-known Sagnac effect sag ; post ; log ; hari  allows rotation rates to
be measured via interference, and has a number of applications to fiber-
optical gyroscopes herve , measurement of gravitational effects sun , and
quantum sensing paiva . Sagnac-based system have even been devised rest ;
matos  to test quantum effects such as HOM interference and the Aharonov-Bohm
effect in non-inertial frames. A schematic depiction of the standard Sagnac
interferometer is shown in Fig. 7. The incident beam is split into two parts
by $BS_{1}$, and these parts propagate around the loop in opposite directions.
If the loop is rotating at angular speed $\omega$ then the beams in the two
directions take different times to make a complete loop and reach the
detector. The time difference is readily shown to be

$\delta
t={{4A\omega}\over{c^{2}-r^{2}\omega^{2}}}\approx{{4A\omega}\over{c^{2}}},$
(32)

where $A$ is the area enclosed by the loop. This leads to a phase difference
of

$\delta\phi=\omega\;\delta t\approx{{8\pi A\omega}\over{\lambda c}}$ (33)

between the two counter-propagating beams. Since $\delta\phi$ is proportional
to $\omega$, measurement of the phase shift amounts to a measurement of the
rotation rate about the axis perpendicular to the plane of the interferometer.

A variation on the ideas of the previous section allows simultaneous
measurement of rotations about three orthogonal axes with a single
interferometer. Consider measurements about just two axes first. The Grover-
Mach-Zehnder interferometer above can be configured so that $a_{1}$ and
$b_{1}$ lines form a loop in the $xy$-plane, while the $a_{2}$ and $b_{2}$
lines form a loop in the $yz$-plane. Schematically (leaving out the
circulators needed to couple in the input), this is shown in Fig. 8.

Then the phase shifts $\phi_{1}$ and $\phi_{2}$ of the previous section are
caused by rotations about the $z$\- and $x$-axes, respectively, as in a pair
of standard Sagnac interferometers:

$\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle\phi_{z}\;=\;{{8\pi}\over{\lambda c}}\omega_{z}A_{z}$ (34)
$\displaystyle\phi_{2}$ $\displaystyle=$
$\displaystyle\phi_{x}\;=\;{{8\pi}\over{\lambda c}}\omega_{x}A_{x},$ (35)

where $\omega_{j}$ and $A_{j}$ are the frequencies of rotations about the
$j$th axes and the areas enclosed by the corresponding loops. If a loop is not
confined to a single plane, then the relevant area is the area of the loop’s
projection onto the plane perpendicular to the corresponding rotation axis.

To allow rotations about the $y$-axis to be measured as well, suppose now that
an optical circulator is inserted into the $a_{2}$ line (Fig. 9), allowing a
loop in the $xz$-plane to be introduced into that line. Now, each passage from
$G_{1}$ to $G_{2}$ exiting $a_{1}$ or $b_{1}$ comprises half a loop about the
$z$-axis, and each passage from $G_{1}$ to $G_{2}$ leaving from $b_{2}$
comprises half a loop about $x$, as before, but each passage leaving $a_{2}$
introduces a half loop about $x$ _and_ a _full_ loop about $y$.

$\phi_{0}$, $\phi_{1}$, $\phi_{2}$ correspond, respectively, to closed loops
exiting and reentering $G_{1}$ at ports $b_{1}$-$b_{2}$, $a_{1}$-$b_{1}$, and
$a_{2}$-$b_{2}$, respectively, so the phase shifts measured by the coincidence
rates in the previous section now correspond to phase shifts induced by
rotations about the $x$, $y$, and $z$ axes through the relations:

$\displaystyle\phi_{0}$ $\displaystyle=$ $\displaystyle{1\over
2}(\phi_{x}-\phi_{z})$ (36) $\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle\phi_{z}$ (37) $\displaystyle\phi_{2}$ $\displaystyle=$
$\displaystyle\phi_{x}+\phi_{y}.$ (38)

In matrix form,

$\left(\begin{array}[]{c}\phi_{0}\\\ \phi_{1}\\\
\phi_{2}\end{array}\right)=\left(\begin{array}[]{ccc}{1\over 2}&0&-{1\over
2}\\\ 0&0&1\\\ 1&1&0\end{array}\right)\left(\begin{array}[]{c}\phi_{x}\\\
\phi_{y}\\\ \phi_{z}\end{array}\right).$ (39)

Since the square matrix in the previous equation has nonzero determinant,
measurement of $\phi_{0}$, $\phi_{1}$, $\phi_{2}$ via coincidence rates allows
a unique solution for the rotation-induced shifts; namely:

$\displaystyle\phi_{x}$ $\displaystyle=$ $\displaystyle 2\phi_{0}+\phi_{1}$
(40) $\displaystyle\phi_{y}$ $\displaystyle=$
$\displaystyle-2\phi_{0}-(\phi_{1}+\phi_{2})$ (41) $\displaystyle\phi_{z}$
$\displaystyle=$ $\displaystyle\phi_{1}.$ (42)

Thus, by measuring $\phi_{0},\;\phi_{1},$ and $\phi_{2}$ via coincidence
rates, as in the previous section, it is straightforward to reconstruct the
rotation rates about all three axes simultaneously. The main drawback of this
approach is that, due to the sign ambiguities mentioned in the previous
section, in most cases only the rotation _rate_ (not the rotation _direction_)
can be determined.

It is useful to consider the resource requirements of the single proposed
higher-dimensional interferometer, compared to three separate standard Sagnac
interferometers. The Grover-based interferometer proposed here requires four
detectors, whereas three Sagnacs would require three, meaning that only one
additional detector would be needed. But consider that in applications where
high resolution is essential, interferometers using quantum state input are
useful, as mentioned in the introduction. A single Grover interferometer with
one entangled-state input source would then certainly be less hardware-
intensive than three standard interferometers, each with its own separate
entangled-state input. In principle, one could try to use a single entangled
source with a pair of beam splitters to direct the source output into the
three interferometers. But with the three interferometers rotating
independently along different axes, this would be very difficult to arrange in
practice. In the Grover-based proposal, a single source shares the rotations
of the single interferometer. So the resource savings due to having a single
entangled source instead of three would more than compensate for the need for
an extra detector. Thus, the proposal here may actually be feasible for ultra-
high precision measurements, especially if the interferometers were fabricated
on integrated chips. Such on-chip interferometers would remove potential
stability and alignment problems.

Figure 9: The three-dimensional Grover-Sagnac interferometer: Measurement of
rotations about a third axis can be added by altering the setup of the
previous Figure using an optical circulator in one line to introduce a loop in
the $y$-$z$ plane. Rotations of the system about the $y$-axis produce phase
shifts proportional to the area of this loop. The circulator is shown expanded
in the inset. Detectors are again understood to be coupled in via circulators
after $G_{2}$ as in Fig. 5.

##  VI Conclusion

In this paper we have shown that the use of linear-optical interferometers
containing unbiased four-ports in place of beam splitters allows new effects
to arise, such as continuous real-time tuning between HOM and anti-HOM
effects, and the simultaneous measurements of multiple phase shifts, possibly
induced by rotation rates, with reduced resources. These new possibilities
arise from the increased dimensionality of the multiports and from their
directionally-unbiased nature, allowing reflection backward toward the input
direction. These interference effects are inherently quantum mechanical
phenomena, requiring the use of two-photon states and coincidence counting.
Whether additional interesting effects can be obtained in other interferometer
topologies by similarly generalizing their vertices from beam splitters to
directionally-unbiased multiports is an open avenue for future investigation.

As mentioned in the introduction, allowing the input optical states to be
entangled allows improved phase resolution in many interferometers. The setups
proposed here are naturally suited for such entanglement-based approaches;
however, the effects of entanglement on the measurement resolution of these
setups remain to be explored elsewhere.

## Acknowledgements

This research was supported by the Air Force Office of Scientific Research
MURI award number FA9550-22-1-0312 and DOD/ARL grant number W911NF2020127.

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