# Dispersion and fidelity in quantum interferometry

D.S. Simon  Dept. of Electrical and Computer Engineering, Boston University, 8
St. Mary’s St., Boston, MA 02215  A.V. Sergienko  Dept. of Electrical and
Computer Engineering, Boston University, 8 St. Mary’s St., Boston, MA 02215
Dept. of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215
T.B. Bahder  Charles M. Bowden Research Facility, US Army RDECOM, Aviation and
Missile Research, Development, and Engineering Center, Redstone Arsenal, AL
35898

###### Abstract

We consider Mach-Zehnder and Hong-Ou-Mandel interferometers with nonclassical
states of light as input, and study the effect that dispersion inside the
interferometer has on the sensitivity of phase measurements. We study in
detail a number of different one- and two-photon input states, including Fock,
dual Fock, N00N states, and photon pairs from parametric downconversion.
Assuming there is a phase shift $\phi_{0}$ in one arm of the interferometer,
we compute the probabilities of measurement outcomes as a function of
$\phi_{0}$, and then compute the Shannon mutual information between $\phi_{0}$
and the measurements. This provides a means of quantitatively comparing the
utility of various input states for determining the phase in the presence of
dispersion. In addition, we consider a simplified model of parametric
downconversion for which probabilities can be explicitly computed
analytically, and which serves as a limiting case of the more realistic
downconversion model.

###### pacs:

42.50.St,42.50.Dv,07.60.Ly,03.67.-a

##  I Introduction

Interferometry is both an important tool for practical measurements and a
useful testing ground for fundamental physical principles. As a result, the
search for methods to improve the resolution of interferometers forms an
active area of study. It has been shown by a number of authors (yurke1 (1, 2,
3, 4)) that nonclassical states, in particular those with high degrees of
entanglement, when used as input to an interferometer can lead to resolutions
that approach the Heisenberg limit, the fundamental physical limit imposed by
the uncertainty principle. Most of this previous work has dealt with idealized
interferometers, with no dispersion or photon losses. Before quantum
interferometry may become a useful practical tool the question must be asked
as to how well the conclusions of these previous studies hold up in more
realistic and less idealized situations. In this paper, we will attempt to
take the next step along this road by adding dispersion to the apparatus and
examining what effect this has on the phase sensitivity of interferometry with
nonclassical input. The motivation for this work is the desire to ultimately
construct quantum sensors that can measure the values of external fields by
measuring the phases shifts they produce in an interferometer.

In particular, the nonclassical input states we will consider are (i) Fock
states $|N,0\rangle$ which have a fixed number of photons incident on one
input port, (ii) dual or twin Fock states $|N,N\rangle$ which have the same
number of photons incident on each input port, and (iii) N00N states
${1\over\sqrt{2}}\left[|N,0\rangle+|0,N\rangle\right]$. Here,
$|N_{a},N_{b}\rangle$ denotes a state with numbers $N_{a}$ and $N_{b}$ of
photons entering each of the two interferometer input ports.

There has been a great deal of recent work on the production of nonclassical
states of light with large ($N>2$) numbers of photons by means of
postselection (for example, knill (5, 6, 7, 8)), however at present the
utility of these postselection schemes for application to practical situations
is not clear. Although this work is useful for clarifying the scientific
issues involved, it is not technologically feasible at present to use these
methods to produce the desired states on demand. Rather, postselection
produces states statistically, at random times, and therefore can not be
relied upon to produce states on demand for a quantum sensor. In addition, for
large photon number, great care must be taken to distinguish between states of
$N$ photons and those of $N-1$ photons, making it difficult to prevent mixed
states from appearing, which would change the physics involved. In contrast,
two-photon entangled states with well-defined properties can be easily
produced by parametric downconversion or other methods.

Due to the current practical difficulties of producing on demand entangled
photon states with large, well-defined $N$, we save the large $N$ case for
later study and restrict ourselves in this current paper to situations which
are both simpler and of more immediate practical interest, namely the cases of
one or two photons. Furthermore, for the two-photon case, we consider two
possibilities: (i) the photons may be uncorrelated in frequency, or (ii) the
pair may be produced through spontaneous parametric downconversion (SPDC),
resulting in anticorrelation between the two frequencies.

Our goal is to compare the usefulness of each of these cases for making phase
measurements in the presence of dispersion, so we will need a means of
quantifying the sensitivity of the interferometer with respect to these
measurements. Consider a single shot consisting of a nonclassical state of
light with a fixed number of photons being injected into the input ports of
the interferometer. Suppose some phase-dependent observable $M(\phi)$ is
measured during this shot. The usual way to define the phase sensitivity of
the measurement is by computing

$\Delta\phi=\left|{{dM}\over{d\phi}}\right|^{-1}\Delta M.$ (1)

However, this is correct only if the probability distribution of the phases
has a single peak and is approximately Gaussian in shape. A more general
strategy is to take an information-theoretical approach and to define the
quantum fidelity by means of the Shannon mutual information bahdlap (9)

$H(\Phi:M)={1\over{2\pi}}\sum_{m}\int_{-\pi}^{\pi}d\phi\;P(m|\phi)\log_{2}\left[{{2\pi
P(m|\phi)}\over{\int_{-\pi}^{\pi}P(m|\phi^{\prime})d\phi^{\prime}}}\right].$
(2)

Here, $m$ and $\phi$ are the measured values of the random variables $M$ and
$\Phi$, while $P(m|\phi)$ is the conditional probability of obtaining
measurement $m$ given the phase $\phi$ on a particular shot. In this formula,
we have also assumed maximum ignorance of the phase, i.e., we have assumed a
uniform distribution for $\phi$, $p(\phi)={1\over{2\pi}}$. Suppose that the
detectors have a characteristic time-scale $T_{D}$. Then in this context, a
single shot will consist of a well-defined number of photons entering the
apparatus simultaneously (i.e. within a temporal window much smaller than
$T_{D}$) and seperated in time from any other entering photons by a time $\geq
T_{D}$. The mutual information is a measure of the information gained per shot
about the phase $\Phi$ from a measurement of the observable $M$. In our case,
the role of $M$ will be played by the number of photons detected at each of
the output ports. For $N$ input photons, output detector C will count $l$
photons, detector $D$ will detect the remaining $N-l$ photons, and the sum in
equation 2 will become a sum over $l$, where $l=0,1,\dots N$. Throughout this
paper we will use the quantum fidelity as our measure of phase sensitivity.
Besides being of very general applicability and giving a precise, calculable
measure for the utility of a measurement, the introduction of the mutual
information provides a link to the theory of quantum information processing.
Bahder and Lopata bahdlap (9) have computed the quantum fidelity as a function
of $N$ for idealized lossless and dispersionless interferometers with Fock and
N00N state inputs. In the following sections, we will see how their results
change for the cases of $N=1$ and $N=2$ when dispersion is present.

Although not the principal focus of this paper, it should be noted that the
existence of multiple peaks in the output probability distributions invalidate
the assumptions used to derive the Heisenberg bound from the Cramer-Rao lower
bound, which makes input states with multimodal distributions especially
interesting from the point of view of the study of phase sensitivity. Note
that violations of the Heisenberg limit have recently been shown to exist in
another context, distinct from the situation examined in this paper, namely in
the context of nonlinear interferometry luis (10, 11, 12).

We will assume one branch of the interferometer has a dispersive element which
gives the photon wavenumber $k$ a frequency dependence of the form

$k(\omega)=k_{0}+\alpha(\omega-\omega_{0})+\beta(\omega-\omega_{0})^{2},$ (3)

ignoring the possibility of higher order terms. The other interferometer arm
will be assumed to be of negligible dispersion. Here, $\alpha$ is the inverse
of the group velocity, and $\beta$ is the group delay dispersion per unit
length.

In addition to the Mach-Zehnder interferometer, we will examine the fidelity
of an alternate setup used in dowl (6), in which N00N states are incident on a
single beam-splitter used as a Hong-Ou-Mandel (HOM) interferometer. We will
then be in a position to compare the possible input states and interferometer
setups, with a view to gaining insight into their relative usefulness in
practical measurements. In the two-photon cases, we must distinguish between
situations in which the photon energies (or frequencies) are correlated and
those in which they are independent. Thus, after we examine the case of
energy-uncorrelated photons, we look at photon pairs anticorrelated in energy.
We further consider two subcases of the latter: (i) a simple model which can
be solved analytically and which amounts to a simplified version of
spontaneous downconversion, and (ii) a more realistic but less analytically
tractable version of downconversion.

The plan of this paper is as follows: in section II we consider the setup for
the dispersive Mach-Zehnder interferometer and define the input states we will
use in more detail. In section III, we apply the possible one-photon inputs to
the interferometer and compute the probabilities for the various possible
outcomes. In sections IV and V respectively, we do the same for the Mach-
Zehnder interferometer with several different two-photon inputs and for the
HOM interferometer with $N=2$ N00N state input. In section VI we compute and
plot the mutual information for each of the preceeding cases as functions of
bandwidth and dispersion levels; we then compare and discuss the results for
the various cases. Finally, in section VII we repeat the same calculation for
input consisting of a photon pair produced via spontaneous parametric
downconversion before arriving at final conclusions in section VIII.

For ease of reference later, table I summarizes the specific cases we will
examine over the following sections.

Table 1:  Summary of the special cases examined in the later sections of this paper. Case | # of | Interferometer | Input | Frequency
---|---|---|---|---
No. | photons | Type | State | Correlation
A | 1 | MZ | Fock | not applicable
B | 1 | MZ | N00N | not applicable
C | 2 | MZ | Fock | none
D | 2 | MZ | Dual Fock | none
E | 2 | MZ | N00N | none
F | 2 | MZ | Fock | anticorrelated
G | 2 | MZ | N00N | anticorrelated
H | 2 | MZ | Dual Fock | anticorrelated
I | 1 | HOM | N00N | none
J | 2 | HOM | N00N | none
K | 2 | HOM | N00N | anticorrelated
L | 2 | MZ | SPDC Fock | anticorrelated

##  II The Dispersive Mach-Zehnder Interferometer

Consider the Mach-Zehnder interferometer of figure 1, with 50/50
beamsplitters. Assume for the moment that there is no dispersion in the
apparatus. Let $\hat{a}_{\omega}$ and $\hat{b}_{\omega}$ be operators that
annihilate photon states in the two input ports A and B. They obey the usual
canonical commutation relations with the corresponding creation operators
$\hat{a}_{\omega}^{\dagger}$ and $\hat{b}_{\omega}^{\dagger}$:

$\left[\hat{a}_{\omega},\hat{a}_{\omega^{\prime}}^{\dagger}\right]=\left[\hat{b}_{\omega},\hat{b}_{\omega^{\prime}}^{\dagger}\right]=\delta(\omega-\omega^{\prime}),$
(4)

with all other commutators vanishing. For independent photons, the input
states to the interferometer can be described in terms of the number of
photons entering the two ports:

$\displaystyle|N_{a},N_{b};\omega_{1},\dots,\omega_{N_{a}};\omega^{\prime}_{1}\dots,\omega^{\prime}_{N_{b}}\rangle$
$\displaystyle\qquad={1\over\sqrt{N_{a}!N_{b}!}}\hat{a}^{\dagger}_{\omega_{1}}\dots\hat{a}^{\dagger}_{\omega_{N_{a}}}\hat{b}^{\dagger}_{\omega^{\prime}_{1}}\dots\hat{b}^{\dagger}_{\omega^{\prime}_{N_{b}}}|0\rangle,$
(5)

where $N_{a}$ and $N_{b}$ are the number of photons in ports A and B,
respectively, and $|0\rangle$ is the vacuum state with no photons. Similarly,
$N_{c}$, $N_{d}$, $\hat{c}_{\omega}$, and $\hat{d}_{\omega}$ will represent
the photon numbers and annihilation operators at output ports C and D.

Figure 1: Mach-Zehnder interferometer with dispersion in one arm. There is
also a phase shift $\phi_{0}$ of nondispersive origin in the same arm.

The effect of the Mach-Zehnder interferometer on a given input state may be
described in terms of the scattering matrix, $S(\phi)$. The initial and final
annihilation operators are related by a scattering matrix $S(\phi)$:

$\left(\begin{array}[]{c}\hat{c}_{\omega}(\phi)\\\
\hat{d}_{\omega}(\phi)\end{array}\right)=S(\phi)\left(\begin{array}[]{c}\hat{a}_{\omega}\\\
\hat{b}_{\omega}\end{array}\right),$ (6)

where $\phi$ is the relative phase difference experienced by photons in the
two arms. In the absence of photon losses in the system, the scattering matrix
will be unitary. Then, for an ideal Mach-Zehnder interferometer, the
scattering matrix is given by

$\displaystyle S(\phi)$ $\displaystyle=$ $\displaystyle{1\over
2}\left[e^{i\phi}e^{ikL_{1}}-e^{ikL_{2}}\right]\sigma_{z}-{i\over
2}\left[e^{i\phi}e^{ikL_{1}}+e^{ikL_{2}}\right]\sigma_{x}$ (9)
$\displaystyle=$ $\displaystyle-
ie^{ikL_{1}}e^{i\phi/2}\left(\begin{array}[]{cc}-\sin{\phi\over
2}&\cos{\phi\over 2}\\\ \cos{\phi\over 2}&\sin{\phi\over
2}\end{array}\right),$

where the Pauli matrices are

$\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)\qquad\mbox{and}\qquad\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\\\ \end{array}\right).$ (10)

In this scattering matrix we have assumed (as we will assume henceforth) that
the lengths of the two interferometer arms are equal, $L_{1}=L_{2}$. Using
this matrix in equation 6, we can invert the equation and take adjoints to
arrive at the following result:

$\displaystyle\hat{a}_{\omega}^{\dagger}$ $\displaystyle=$ $\displaystyle
i\left[\hat{c}_{\omega}^{\dagger}\sin{\phi\over
2}-\hat{d}_{\omega}^{\dagger}\cos{\phi\over 2}\right]e^{i\phi/2}$ (11)
$\displaystyle\hat{b}_{\omega}^{\dagger}$ $\displaystyle=$
$\displaystyle-i\left[\hat{c}_{\omega}^{\dagger}\cos{\phi\over
2}+\hat{d}_{\omega}^{\dagger}\sin{\phi\over 2}\right]e^{i\phi/2}$ (12)

We assume that the frequency distribution for each incoming photon is Gaussian
and that each Gaussian has the same width and central frequency, of the form
$e^{-{1\over 2}\sigma(\omega-\omega_{0})^{2}}$. Input and output states will
either be states of definite photon number in the sense that they are
eigenstates of number operators of the form $\hat{N}_{j}=\int d\omega
a^{(j)\dagger}_{\omega}a^{(j)}_{\omega}$ (where $a_{\omega}^{(j)}$ is the
annihilation operator for photons at the jth port), or else superpositions of
such states.

We introduce dispersion to the upper branch of the interferometer by giving
the wavenumber $k$ a frequency dependence of the form in equation 3. We assume
that the dispersion in the other branch of the interferometer is negligible,
i.e. that $k(\omega)=k_{0}$ in that branch. The length of the portion of the
upper arm for which $k(\omega)$ differs from $k_{0}$ will be denoted $L$,
where $0\leq L\leq L_{1}$. In addition to any phase difference resulting from
the asymmetric dispersion, we also assume that photons travelling through the
upper branch of the interferometer gain an additional phase difference
$\phi_{0}$ relative to the lower branch. $\phi_{0}$ is any phase difference of
nondispersive origin that may be present in the setup; this may be due to a
difference in path length, or an interaction of one arm of the interferometer
with an external field. Note that for our setup, the assumption of a balanced
interferometer entails no loss of generality; to account for an unbalanced
interferometer, it suffices to simply include a term of the form
$k_{0}(L_{1}-L_{2})$ inside the phase factor $\phi_{0}$.

In the presence of the dispersion, the scattering matrix will now be of the
form

$\displaystyle S(\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}e^{ik_{0}L_{1}}\left(\begin{array}[]{cc}e^{i\phi(\omega)}-1&-i\left(e^{i\phi(\omega)}+1\right)\\\
-i\left(e^{i\phi(\omega)}+1\right)&-\left(e^{i\phi(\omega)}-1\right)\end{array}\right)$
(15) $\displaystyle=$ $\displaystyle-
ie^{ik_{0}L_{1}}e^{i\phi(\omega)/2}\left(\begin{array}[]{cc}-\sin{{\phi(\omega)}\over
2}&\cos{{\phi(\omega)}\over 2}\\\ \cos{{\phi(\omega)}\over
2}&\sin{{\phi(\omega)}\over 2}\end{array}\right)$ (18)

where for future convenience we have shifted the frequency dependence into a
new phase angle by defining

$\phi(\omega)=\phi_{0}+\alpha L(\omega-\omega_{0})+\beta
L(\omega-\omega_{0})^{2}.$ (19)

Consider $N$ photons entering the interferometer and assume for now that their
frequencies are independent variables. The Fock, dual Fock, and N00N input
states are of the form:

$\displaystyle|N,0\rangle_{\sigma}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{N!}}\left({{\sigma}\over\pi}\right)^{N/4}\int
d\omega_{1}\dots d\omega_{N}e^{-{\sigma\over
2}\sum_{j=1}^{N}(\omega_{j}-\omega_{0})^{2}}$ (20)
$\displaystyle\qquad\qquad\qquad\times\hat{a}^{\dagger}_{\omega_{1}}\dots\hat{a}^{\dagger}_{\omega_{N}}|0\rangle$
$\displaystyle|N,N\rangle_{\sigma}$ $\displaystyle=$
$\displaystyle{1\over{N!}}\left({{\sigma}\over\pi}\right)^{N/2}\int
d\omega_{1}\dots d\omega_{2N}e^{-{\sigma\over
2}\sum_{j=1}^{2N}(\omega_{j}-\omega_{0})^{2}}$ (21)
$\displaystyle\qquad\qquad\times\hat{a}^{\dagger}_{\omega_{1}}\dots\hat{a}^{\dagger}_{\omega_{N}}\hat{b}^{\dagger}_{\omega_{N+1}}\dots\hat{b}^{\dagger}_{\omega_{2N}}|0\rangle$

and

$\displaystyle{1\over\sqrt{2}}\left[|N,0\rangle+|0,N\rangle\right]_{\sigma}$
$\displaystyle\quad={1\over\sqrt{N!}}{1\over\sqrt{2}}\left({{\sigma}\over\pi}\right)^{N/4}\;\int
d\omega_{1}\dots d\omega_{N}e^{-{\sigma\over
2}\sum_{j=1}^{N}(\omega_{j}-\omega_{0})^{2}}$
$\displaystyle\quad\qquad\times\left[\hat{a}^{\dagger}_{\omega_{1}}\dots\hat{a}^{\dagger}_{\omega_{N}}+\hat{b}^{\dagger}_{\omega_{1}}\dots\hat{b}^{\dagger}_{\omega_{N}}\right]|0\rangle,$
(22)

where the bandwidth of the incident beams is given by
$\Delta\omega\equiv\sigma^{-1/2}$. If the photons are produced by SPDC, then
the frequencies must occur in pairs with the photons in each pair being equal
distances above or below the pump frequency; we will consider this situation
in simplified form in section IV.2 and in a more realistic form in section
VII.

Suppose that one of the $N$-photon or $2N$-photon states described above is
input to the interferometer. Write this input state as $|\psi_{in}\rangle$.
Then, assuming that the frequencies of the final photons are not measured, we
want the joint probabilities to find $N_{c}$ photons at detector $C$ and
$N_{d}$ photons at detector $D$ (with $N_{c}+N_{d}=N_{a}+N_{b}$) for a given
nondispersive phase shift $\phi_{0}$ in the upper interferometer arm. These
probabilities can be expressed in the form

$P(N_{c},N_{d}|\phi_{0})=\langle\psi_{in}|\hat{\pi}(N_{c},N_{d};\phi_{0})|\psi_{in}\rangle,$
(23)

where the projective operator $\hat{\pi}(N_{c},N_{d};\phi_{0})$ is defined as

$\hat{\pi}(N_{c},N_{d},\phi_{0})=\int\;d\Omega|N_{c},N_{d};\Omega,\phi_{0}\rangle\langle
N_{c},N_{d};\Omega,\phi_{0}|,$ (24)

with

$|N_{c},N_{d};\Omega,\phi_{0}\rangle={1\over\sqrt{N_{c}!\;N_{d}!}}\hat{c}_{\Omega_{1}}^{\dagger}\dots\hat{c}_{\Omega_{N_{c}}}^{\dagger}\hat{d}_{\Omega_{1}^{\prime}}^{\dagger}\dots\hat{d}_{\Omega_{N_{d}}^{\prime}}^{\dagger}|0\rangle.$
(25)

Here we have suppressed the $\phi_{0}$-dependence of the $\hat{c}_{\Omega}$
and $\hat{d}_{\Omega^{\prime}}$ operators for notational simplicity, and have
represented the collection of output frequencies
$\left\\{\Omega_{1},\dots,\Omega_{N_{c}},\Omega_{1}^{\prime},\dots,\Omega_{N_{d}}^{\prime}\right\\}$
by the single symbol $\Omega$. Similarly, $d\Omega$ is being used as shorthand
for the full frequency integration measure $d\Omega_{1}\dots
d\Omega_{N_{c}}d\Omega_{1}^{\prime}\dots d\Omega_{N_{d}}^{\prime}$. These
probabilities may also be expressed in the form

$P(N_{c},N_{d}|\phi_{0})=\int d\Omega\left|\langle
N_{c},N_{d};\Omega,\phi_{0}|\psi_{in}\rangle\right|^{2}.$ (26)

From equation 2 the mutual information between the phase $\Phi$ and the output
photon numbers $M$ is then

$H(\Phi:M)={1\over{2\pi}}\sum_{N_{c},N_{d}}\int_{-\pi}^{\pi}d\phi_{0}\;P(N_{c},N_{d}|\phi_{0})\log_{2}\left[{{2\pi
P(N_{c},N_{d}|\phi_{0})}\over{\int_{-\pi}^{\pi}d\phi_{0}\;P(N_{c},N_{d}|\phi_{0})}}\right].$
(27)

We note that the probabilities $P(N_{c},N_{d},|\phi_{0})$ are also conditional
upon the values of $\alpha$, $\beta$, and $\sigma$, although we do not
explicitly include these parameters in the notation for the probabilities for
the sake of notational simplicity. We now restrict ourselves to the cases
$N=1$ and $N=2$, and proceed in the following sections to compute the mutual
information $H$ for a number of different possible input states.

##  III MZ Interferometry With One-Photon Input

In this section, we begin with the cases in which there is only one photon in
the initial state.

Case A: One-photon Fock state. We introduce the normalized input state

$|\psi_{in}\rangle_{\sigma}=|10\rangle_{\sigma}=\sqrt[4]{\sigma\over\pi}\int
d\omega\;e^{-{1\over
2}\sigma(\omega-\omega_{0})^{2}}\hat{a}_{\omega}^{\dagger}|0\rangle,$ (28)

representing a single photon incident on port A. Using relations 11 and 12,
this is equivalent to

$\displaystyle|10\rangle_{\sigma}$ $\displaystyle=$ $\displaystyle{1\over
2}\sqrt[4]{\sigma\over\pi}\int d\omega\;e^{-{1\over
2}\sigma(\omega-\omega_{0})^{2}}$ (29)
$\displaystyle\times\left[\hat{c}^{\dagger}_{\omega}\left(e^{i\phi(\omega)}-1\right)-i\hat{d}^{\dagger}_{\omega}\left(e^{i\phi(\omega)}+1\right)\right]|0\rangle.$

The photon may leave the interferometer via either port C or port D. We assume
that the detectors count the number of photons leaving the apparatus but do
not measure their frequencies. Therefore, we must integrate over the final
frequencies. The output state is then measured using the projective operators

$\displaystyle\hat{\pi}(1,0)$ $\displaystyle=$ $\displaystyle\int
d\Omega\;\hat{c}_{\Omega}^{\dagger}|0\rangle\langle 0|\hat{c}_{\Omega}$ (30)
$\displaystyle\hat{\pi}(0,1)$ $\displaystyle=$ $\displaystyle\int
d\Omega\;\hat{d}_{\Omega}^{\dagger}|0\rangle\langle 0|\hat{d}_{\Omega}.$ (31)

Expectation values of these operators give the probabilities of measurement
outcomes:

$\displaystyle P(1,0|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1-{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\cos\left(\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]$ (32)
$\displaystyle P(0,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1+{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\cos\left(\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]$ (33)

In the previous two lines, we have introduced some notation that will be
convenient for simplifying the results of this and the following sections. The
parameters $r_{1},r_{2},\theta_{1},\theta_{2}$ are defined by (see figure 2):

$\displaystyle r_{1}^{2}$ $\displaystyle=$ $\displaystyle 1+\left({{\beta
L}\over\sigma}\right)^{2}\qquad\qquad\tan\theta_{1}\;=\;{{\beta
L}\over\sigma}$ (34) $\displaystyle r_{2}^{2}$ $\displaystyle=$ $\displaystyle
1+\left({{\beta
L}\over{2\sigma}}\right)^{2}\qquad\qquad\tan\theta_{2}\;=\;{{\beta
L}\over{2\sigma}}.$ (35)

Note that these parameters depend on the second order dispersion coefficient
$\beta$, but not on $\alpha$, and that when $\beta$ vanishes we then have
$r_{1}=r_{2}=1$ and $\theta_{1}=\theta_{2}=0$.

Figure 2: Definitions of $r_{1}$, $r_{2}$, $\theta_{1}$, and $\theta_{2}$.

Case B: One-photon N00N state. The input state is

${1\over\sqrt{2}}\left[|10\rangle+|01\rangle\right]_{\sigma}={1\over\sqrt{2}}\sqrt[4]{\sigma\over\pi}\int
d\omega\;e^{-{\sigma\over
2}(\omega-\omega_{0})^{2}}\left(\hat{a}_{\omega}^{\dagger}+\hat{b}_{\omega}^{\dagger}\right)|0\rangle,$
(36)

where

$\displaystyle{1\over\sqrt{2}}\left(\hat{a}_{\omega}^{\dagger}+\hat{b}_{\omega}^{\dagger}\right)$
$\displaystyle=$
$\displaystyle{i\over{2\sqrt{2}}}\left\\{\hat{c}^{\dagger}_{\omega}\left[(i-1)-(i+1)e^{i\phi(\omega)}\right]\right.$
$\displaystyle\left.\qquad+\;\hat{d}^{\dagger}_{\omega}\left[e^{i\phi(\omega)}(i-1)-(i+1)\right]\right\\}.$

The resulting output probabilities in this case turn out to be

$\displaystyle P(1,0|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1-{{e^{-{{\alpha^{2}\beta
L^{3}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\sin\left(\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]$ (38)
$\displaystyle P(0,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1+{{e^{-{{\alpha^{2}\beta
L^{3}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\sin\left(\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right].$ (39)

##  IV MZ Interferometry With Two-Photon Input

We now consider input states with two photons distributed in assorted ways
among the input ports. However, now we must make a distinction as to whether
the two photon frequencies are independent or correlated in some manner. We
treat the uncorrelated version first. Then we will examine one particular case
of frequency-correlated photons which is of special interest for experiment:
that of photon pairs created through spontaneous parametric downconversion
(SPDC). In this section we treat only a simplified version of SPDC which will
allow us to obtain simple exact expressions for the probabilities of all of
the output states. In a later section we will compare this simplified SPDC to
a more realistic version for which only numerical results are available.

###  IV.1 Two-Photon Input with Uncorrelated Energies

Case C: Energy-uncorrelated two-photon Fock state. Sending a two-particle Fock
state into input A,

$|2,0\rangle_{\sigma}=\sqrt{\sigma\over{2\pi}}\int
d\omega_{1}d\omega_{2}\;e^{-{\sigma\over
2}\left[(\omega_{1}-\omega_{0})^{2}+(\omega_{2}-\omega_{0})^{2}\right]}\hat{a}^{\dagger}_{\omega_{1}}\hat{a}^{\dagger}_{\omega_{2}}|0\rangle,$
(40)

where we use equations 11 and 12 to write each $\hat{a}^{\dagger}$ factor in
terms of the output operators $\hat{c}^{\dagger}$ and $\hat{d}^{\dagger}$.
After a straightforward calculation, this leads to the following output
probabilities:

$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
4}\left[1-{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\cos\left(\phi_{0}+{{\theta_{1}}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]^{2}$
(41) $\displaystyle P(0,2|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
4}\left[1+{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\cos\left(\phi_{0}+{{\theta_{1}}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]^{2}$
(42) $\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1-{{e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}}\over{r_{1}}}\cos^{2}\left(\phi_{0}+{{\theta_{1}}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]$ (43)

Case D: Energy-uncorrelated two-photon dual Fock input. The normalized input
state is

$\displaystyle|1,1\rangle_{\sigma}$ $\displaystyle=$
$\displaystyle\sqrt{\sigma\over\pi}\int
d\omega_{1}\omega_{2}\;e^{-{\sigma\over
2}\left[(\omega_{1}-\omega_{0})^{2}+(\omega_{2}-\omega_{0})^{2}\right]}\hat{a}_{\omega_{1}}^{\dagger}\hat{a}_{\omega_{2}}^{\dagger}|0\rangle$
(44) $\displaystyle=$ $\displaystyle{1\over 4}\sqrt{\sigma\over\pi}\int
d\omega_{1}\omega_{2}\;e^{-{\sigma\over
2}\left[(\omega_{1}-\omega_{0})^{2}+(\omega_{2}-\omega_{0})^{2}\right]}$
$\displaystyle\times\left\\{-i\hat{c}_{\omega_{1}}^{\dagger}\hat{c}_{\omega_{2}}^{\dagger}\left(e^{i\phi(\omega_{1})}-1\right)\left(e^{i\phi(\omega_{2})}+1\right)+i\hat{d}_{\omega_{1}}^{\dagger}\hat{d}_{\omega_{2}}^{\dagger}\left(e^{i\phi(\omega_{2})}-1\right)\left(e^{i\phi(\omega_{1})}+1\right)\right.$
$\displaystyle\qquad\left.+\hat{c}_{\omega_{1}}^{\dagger}\hat{d}_{\omega_{2}}^{\dagger}\left(e^{i\phi(\omega_{1})}-1\right)\left(e^{i\phi(\omega_{2})}-1\right)-\hat{c}_{\omega_{2}}^{\dagger}\hat{d}_{\omega_{1}}^{\dagger}\left(e^{i\phi(\omega_{1})}+1\right)\left(e^{i\phi(\omega_{2})}+1\right)\right\\}|0\rangle,$

which gives the results

$\displaystyle P(2,0)$ $\displaystyle=$ $\displaystyle P(0,2)$ (46)
$\displaystyle=$ $\displaystyle{1\over
4}\left\\{1-{{e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}}\over{r_{1}}}\cos\left[2\phi_{0}+\theta_{1}-{{\alpha^{2}\beta
L^{3}}\over{2r_{1}^{2}\sigma^{2}}}\right]\right\\}$ $\displaystyle P(1,1)$
$\displaystyle=$ $\displaystyle{1\over
2}\left\\{1+{{e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}}\over{r_{1}}}\cos\left[2\phi_{0}+\theta_{1}-{{\alpha^{2}\beta
L^{3}}\over{2r_{1}^{2}\sigma^{2}}}\right]\right\\}$ (47)

Case E: Energy-uncorrelated two-photon N00N state.

For the input state

$\displaystyle|20\rangle_{\sigma}+|02\rangle_{\sigma}$ $\displaystyle=$
$\displaystyle\sqrt{\sigma\over{2\pi}}\int
d\omega_{1}\;d\omega_{2}e^{-{\sigma\over
2}[(\omega_{1}-\omega_{0})^{2}+(\omega_{2}-\omega_{0})^{2}]}$ (48)
$\displaystyle\qquad\times{1\over\sqrt{2}}\left(\hat{a}_{\omega_{1}}^{\dagger}\hat{a}_{\omega_{2}}^{\dagger}+\hat{b}_{\omega_{1}}^{\dagger}\hat{b}_{\omega_{2}}^{\dagger}\right)|0\rangle,$

the output probabilities are

$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle
P(0,2|\phi_{0})\;=\;{1\over
4}\left\\{1+{1\over{r_{1}}}e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}\right\\}$
(49) $\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left\\{1-{1\over{r_{1}}}e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}\right\\}.$
(50)

In the absence of dispersion ($\alpha=\beta=0$) or in the narrow bandwidth
limit ($\sigma\to\infty$), we see that the coincidence rate $P(1,1|\phi_{0})$
vanishes, while the other two probabilities are both equal to ${1\over 2}$.

Note that there is no dependence on $\phi_{0}$. We will see later that this
fact manifests itself in a vanishing mutual information.

###  IV.2 Two-Photon Input with Anticorrelated Energies: Simplified SPDC model

Case F: Simplified SPDC Fock states. We now examine a case with two photons
incident on the same input port and anticorrelated in energy. We do this in
the context of a simplified model of spontaneous parametric downconversion
(SPDC). Energy conservation requires that the two downconverted photons have
frequencies $\omega_{\pm}=\omega_{0}\pm\Omega$, where $2\omega_{0}$ is the
pump frequency. We again assume a Gaussian distribution of frequencies,
centered around $\omega_{0}$, of the form $e^{-{1\over
2}\sigma(\omega_{\pm}-\omega_{0})^{2}}=e^{-{\sigma\over 2}\Omega^{2}}$. We
follow essentially the same calculational procedure as before, except now we
enforce the requirement that the incoming photon frequencies satisfy
$\omega_{1}+\omega_{2}=2\omega_{0}$. In this section we impose this condition
in a manner that will allow us to obtain analytic solutions for the output
probabilities. This will serve us as a simplified version of SPDC, and we will
see in section VII that this model seems to give an upper bound to the mutual
information obtained from a more realistic model of SPDC. The input state in
this model is taken to be of the form

$|20\rangle_{\sigma}=\sqrt[4]{{2\sigma}\over\pi}\int_{-\infty}^{\infty}d\Omega\int_{-\infty}^{\infty}d\epsilon\;e^{-\sigma\Omega^{2}}f(\epsilon)a_{\omega_{+}}^{\dagger}a_{\omega_{-}}^{\dagger}|0\rangle,$
(51)

where now $\omega_{+}=\omega_{0}+\Omega$ and
$\omega_{-}=\omega_{0}-\Omega+\epsilon$. We can choose $f(\epsilon)$ to be any
function sharply peaked at zero with normalized integral (unit area under its
graph). We then compute the output probabilities according to

$P(N_{c},N_{d}|\phi_{0})=\int d\omega_{1}d\omega_{2}\left|\langle
N_{c},N_{d}|\psi_{in}\rangle\right|^{2},$ (52)

or equivalently, by applying the projection operators

$\hat{\pi}(N_{c},N_{d})=\int
d\omega_{1}\;d\omega_{2}|N_{c},N_{d}\rangle\langle N_{c},N_{d}|.$ (53)

The auxiliary function $f_{\lambda}(\epsilon)$ is necessary in this model in
order to impose the constraint $\omega_{1}+\omega_{2}=2\omega_{0}$ without
causing squares of delta functions to arise in the probability calculations. A
more correct treatment of SPDC will follow in section VII.

The measurement outcomes, integrated over final frequency, are then given by

$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
8}\left\\{2+e^{-{{\alpha^{2}L^{2}}\over{2\sigma}}}+{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right.$
$\displaystyle\qquad\left.-{4\over\sqrt{r_{2}}}e^{-{{\alpha^{2}L^{2}}\over{8r_{2}^{2}\sigma}}}\cos\left[\phi_{0}+{{\theta_{2}}\over
2}-{{\alpha^{2}\beta L^{3}}\over{16r_{2}^{2}\sigma^{2}}}\right]\right\\}$
$\displaystyle P(0,2|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
8}\left\\{2+e^{-{{\alpha^{2}L^{2}}\over{2\sigma}}}+{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right.$
$\displaystyle\qquad\left.+{4\over\sqrt{r_{2}}}e^{-{{\alpha^{2}L^{2}}\over{8r_{2}^{2}\sigma}}}\cos\left[\phi_{0}+{{\theta_{2}}\over
2}-{{\alpha^{2}\beta L^{3}}\over{16r_{2}^{2}\sigma^{2}}}\right]\right\\}$
$\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
4}\left\\{2-e^{-{{\alpha^{2}L^{2}}\over{2\sigma}}}-{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right\\}.$ (56)

As $\beta$ increases, $r_{1}$ and $r_{2}$ increase, leading to decreased
visibility of all of the oscillating terms.

Note also that in the case of zero dispersion ($\alpha=\beta=0$), the exact
expressions for energy-uncorrelated (Case C, section 4.1) and energy-
anticorrelated (downconverted) Fock states (Case F) are identical to each
other. However the probabilities begin to diverge when dispersion is turned
on. The same effect will be seen to occur for the uncorrelated and
anticorrelated N00N states in the HOM interferometer (cases J and K, below).

Case G: Simplified SPDC N00N states. Now the input state is taken to be a N00N
state,
${1\over\sqrt{2}}\left\\{|20\rangle_{\lambda,\sigma}+|02\rangle_{\lambda,\sigma}\right\\}$.
We find the measurement outcomes to be:

$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle
P(0,2)\;=\;{1\over 4}\left[1+e^{-{{\alpha^{2}L^{2}}\over{2\sigma}}}\right]$
(57) $\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1-e^{-{{\alpha^{2}L^{2}}\over{2\sigma}}}\right]$ (58)

As in the uncorrelated case, the probabilities show no dependence on
$\phi_{0}$, and so have vanishing mutual information.In this case we also see
that there is no dependence on the 2nd order dispersion coefficient $\beta$.

It is interesting to note what happens if we shift the phase of the photons in
one input port by $\pi\over 2$ before they hit the first beamsplitter. The
input to the interferometer is now proportional to
$\left|2,0\rangle\right.-\left|0,2\rangle\right..$ In this case, the
interference in $\phi_{0}$ reemerges, and the result is independent of
$\alpha$ instead of $\beta$. In fact, the counting probabilities turn out to
be very similar to those of the N00N state incident on an HOM interferometer
presented in the next section (case K). Moreover, these two cases have
identical values for the mutual information.

Case H: Simplified SPDC dual Fock state. The frequency-anticorrelated dual
Fock input state

$|1,1\rangle_{\sigma}=\sqrt[4]{\sigma\over{2\pi}}\int d\Omega\int
d\epsilon\;e^{-\sigma\Omega^{2}}f(\epsilon)\hat{a}_{\omega_{+}}^{\dagger}\hat{b}_{\omega_{-}}^{\dagger}|0\rangle$
(59)

gives the results

$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle
P(0,2|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
4}\left[1-{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right]$ $\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$
$\displaystyle{1\over
2}\left[1+{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right].$ (61)

##  V Dispersive Hong-Ou-Mandel Interferometer With N00N Input

An alternative setup has been proposed to improve phase resolution(dowl (6)).
In this section we examine this alternate version and compare it to the
previous results.

In this version, it is assumed that the N00N state is created inside the
interferometer, rather than at the input ports. Effectively, we need to remove
the first beam splitter from the interferometer and use the N00N state as
input to the remaining beamsplitter, which now acts as a Hong-Ou-Mandel (HOM)
interferometer hompaper (13). The setup is shown in figure 3. We again assume
dispersion and phase-shift $\phi_{0}$ along one of the lines entering the beam
splitter, neglecting absorption. Ignoring an overall constant phase of
$e^{ik_{0}L}$, the scattering matrix now has the form

$S(\phi_{0})={1\over\sqrt{2}}\left(\begin{array}[]{cc}e^{i\phi(\omega)}&i\\\
ie^{i\phi(\omega)}&+1\end{array}\right),$ (62)

where $\phi(\omega)$ is again given by equation 19. Thus,

$\displaystyle\hat{c}_{\omega}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\left[e^{i\phi(\omega)}\hat{a}_{\omega}+i\hat{b}_{\omega}\right]$
(63) $\displaystyle\hat{d}_{\omega}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\left[ie^{i\phi(\omega)}\hat{a}_{\omega}+\hat{b}_{\omega}\right].$
(64)

Figure 3: Hong-Ou-Mandel interferometer with dispersion and non-dispersive
phase shift $\phi_{0}$ in one arm.

Case I: Single-photon N00N state in HOM interferometer. By the same methods as
before, we can compute the counting rates for a given N00N state input. The
one-photon N00N input state is

$|\psi\rangle=C\int d\omega\;e^{-{\sigma\over
2}(\omega-\omega_{0})^{2}}(a_{\omega}^{\dagger}+b_{\omega}^{\dagger})|0\rangle,$
(65)

for which we find

$\displaystyle P(1,0|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left\\{1+{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\sin\left[\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right]\right\\}$ (66)
$\displaystyle P(0,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left\\{1-{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\sin\left[\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right]\right\\}$ (67)

Finally, assuming two-photons, with no correlation or with anticorrelation, we
arrive at two additional cases (J and K).

Case J: Energy-uncorrelated two-photon N00N state in HOM interferometer.

The input state is

$|\psi\rangle=C\int d\omega_{1}d\omega_{2}\;e^{-{\sigma\over
2}\left[(\omega_{1}-\omega_{0})^{2}+(\omega_{2}-\omega_{0})^{2}\right]}(a_{\omega_{1}}^{\dagger}a_{\omega_{2}}^{\dagger}+b_{\omega_{1}}^{\dagger}b_{\omega_{2}}^{\dagger})|0\rangle.$
(68)

From this state, we arrive at the results:

$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle
P(0,2|\phi_{0})$ (69) $\displaystyle=$ $\displaystyle{1\over
4}\left[1-{{e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}}\over{r_{1}}}\cos\left(2\phi_{0}+\theta_{1}-{{\alpha^{2}\beta
L^{3}}\over{2r_{1}^{2}\sigma^{2}}}\right)\right]$ $\displaystyle
P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1+{{e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}}\over{r_{1}}}\cos\left(2\phi_{0}+\theta_{1}-{{\alpha^{2}\beta
L^{3}}\over{2r_{1}^{2}\sigma^{2}}}\right)\right]$ (70)

Case K: Simplified SPDC two-photon N00N state in HOM interferometer.

For the input

$|\psi\rangle=C\int
d\Omega\;e^{-\sigma\Omega^{2}}(a_{\omega_{+}}^{\dagger}a_{\omega_{-}}^{\dagger}+b_{\omega_{+}}^{\dagger}b_{\omega_{-}}^{\dagger})|0\rangle,$
(71)

we compute:

$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle
P(0,2|\phi_{0})$ (72) $\displaystyle=$ $\displaystyle{1\over
4}\left[1-{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right]$ $\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$
$\displaystyle{1\over
2}\left[1+{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right]$ (73)

In this last case, the results turn out to be independent of the first order
dispersion coefficient, $\alpha$.

##  VI Comparison and Discussion of Cases A to K

The detection probabilities of the previous sections can now be combined with
equation 2 to compute the mutual information for each of the experimental
setups and inputs states. Plotting the results as functions of $\alpha$,
$\beta$, and $\sigma$, we find the results in figures 4-6 for single photon
input and figures 7-9 for two photons. $\alpha L$ is given in units of
$\omega_{0}^{-1}$, while $\beta L$ and $\sigma$ are in units of
$\omega_{0}^{-2}$. In the dispersionless limit, $\alpha,\beta\to 0$, we find
Shannon mutual information values that agree with those previously calculated
in bahdlap (9).

Figure 4: (color online). Mutual information versus alpha for single photon
cases (cases A, B, and I), plotted for the values $\beta=0$, $\sigma=1$. The
mutual information is the same for all three cases. ($\alpha$, $\beta$, and
$\sigma$ are in units of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and
$\omega_{0}^{-2}$, respectively.) Figure 5: (color online). Mutual information
versus $\beta$ for single photon cases (cases A, B, and I), for the values
$\alpha=.5$, $\sigma=1$. ($\alpha$, $\beta$, and $\sigma$ are in units of
$L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$,
respectively.) Figure 6: (color online). Mutual information versus squared
inverse bandwidth $\sigma$ for single photon cases (cases A, B, and I), for
the values $\alpha=1$, $\beta=.1$. ($\alpha$, $\beta$, and $\sigma$ are in
units of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and
$\omega_{0}^{-2}$, respectively.) Figure 7: (color online). Mutual information
versus alpha for two-photon cases, for the values $\sigma=1$, $\beta=0$. Cases
E and G vanish identically. Cases J and D are identical, as are cases H and K.
($\alpha$, $\beta$, and $\sigma$ are in units of $L^{-1}\omega_{0}^{-1}$,
$L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$, respectively.) Figure 8:
(color online). Mutual information versus beta for two-photon cases, for the
values $\sigma=1$, $\alpha=.5$. ($\alpha$, $\beta$, and $\sigma$ are in units
of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$,
respectively.) Figure 9: (color online). Mutual information versus squared
inverse bandwidth $\sigma$ for two-photon cases, for the values $\alpha=1$,
$\beta=.1$. ($\alpha$, $\beta$, and $\sigma$ are in units of
$L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$,
respectively.)

Only positive values of $\beta$ were graphed. However, the formulas of the
previous sections work equally in the anomalous dispersion (negative $\beta$)
region.

Note also that the four parameters $\alpha,\beta,\sigma,L$ appear in all
equations only through the dimensionless quantities

$\Lambda_{1}={\sigma\over{\beta
L}}\qquad\qquad\mbox{and}\qquad\qquad\Lambda_{2}={\sigma\over{\alpha^{2}L^{2}}}.$
(74)

Thus, other parameter ranges can easily be obtained from those graphed here
via appropriate rescaling of variables with the dimensionless ratios held
fixed.

A few conclusions are immediately clear from these graphs and from the
equations of the previous sections. (i) First, the dual Fock states entering
the Mach-Zehnder interferometer give identical results as the N00N states
entering the Hong-Ou Mandel interferometer (compare equations IV.2 and 61 to
72 and 73, or compare 46 and 47 to 69 and 70). This is to be expected, since
for $N=2$ the two cases are equivalent: the first beam splitter in the Mach-
Zehnder interferometer turns a dual Fock input state into a N00N output state,
which then strikes the second beamsplitter. The second beamsplitter can then
be viewed as an HOM interferometer. Thus cases J and D are equivalent, as are
cases H and K. (This equivalence will not for $N>2$.) (ii) Second, the single-
photon cases (cases A, B, and I) all give identical curves for the mutual
information as functions of $\alpha$, $\beta$, and $\sigma$. The explanation
for this is clear if the action of the first beam splitter on the input is
examined. Cases B and I are equivalent for the same reason mentioned in the
previous point: they both lead to a one-particle N00N state in the portion of
the interferometer before the dispersive element is reached, and so give the
same output. Meanwhile, in case A, the output of the first beam splitter is
the state ${1\over\sqrt{2}}\left(|01\rangle+i|10\rangle\right);$ this is
similar to a N00N state, except one term is shifted in phase by $\pi\over 2$
relative to the other. This converts the sines in the probabilities of cases B
and I into the cosines of case A (equations 32 and 33), but has no other
effect. Since the mutual information involves integrals from $-\pi$ to $\pi$,
interchanging sines and cosines inside the integrals has no effect on the
mutual information. Unsurprisingly, the single photon cases generally result
in lower mutual information than the two-photon cases. (iii) We see from the
graphs for the two-photon states that the energy-uncorrelated and energy-
anticorrelated version of each input give identical results for zero
dispersion or zero bandwidth ($\sigma=\infty$); however, the uncorrelated
versions all drop off rapidly to zero fidelity as the dispersion increases,
whereas the anticorrelated (downconverted) input leads to a much slower drop.
(iv) Two-photon N00N states incident on the MZ interferometer (cases E and G)
have zero mutual information as anticipated earlier. (v) For fixed bandwidth
and fixed quadratic dispersion coefficient $\beta$, the two-photon
downconverted N00N state in the HOM interferometer (case K) is independent of
the linear coefficient $\alpha$. However, it decays rapidly with increasing
$\beta$. (vi) Overall, the simplified SPDC-generated Fock states (case F) seem
to hold up best in the presence of dispersion. This case starts off with a
higher value of $H$ at zero dispersion and decays more slowly as $\alpha$ and
$\beta$ increase. The only exception to this statement is when $\beta$ is
small, in which case the anticorrelated HOM N00N state (case K) works better
at large $\alpha$.

A bit of insight into some of the properties of the 2-photon results may be
obtained by considering the exponential decay factor

$\zeta\equiv
e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}=e^{-{{\Lambda_{2}}\over{4r_{1}^{2}}}}=e^{-{{\Lambda_{2}}\over{4(1+\Lambda_{1}^{-2})}}}.$
(75)

$\Lambda_{1}$ and $\Lambda_{2}$ are the dimensionless quantities defined in
equation 74. In frequency-uncorrelated cases such as cases C and D, all of the
$\phi_{0}$-dependent terms are multiplied by a factor of $\zeta$ which arises
from interference between $e^{i\phi(\omega_{1})}$ and $e^{i\phi(\omega_{2})}$
terms, where $\omega_{1}$ and $\omega_{2}$ are the frequencies of the photons
entering the input ports. The relevant term is of the form
$e^{i[\phi(\omega_{1})+\phi(\omega_{2})]}$. As $\alpha\to\infty$ or $\sigma\to
0$, we find that $\Lambda_{2}\to 0$ and $\zeta\to 0$, so that only constant
($\phi_{0}$-independent) terms survive in the limit. Thus, for large $\alpha$
or small $\sigma$, the dependence of the probability distributions on
$\phi_{0}$ decays exponentially, causing the mutual information to also decay
rapidly.

In contrast, for the frequency-anticorrelated cases, such as F and H, the term
$e^{i(\phi(\omega_{1})+\phi(\omega_{2}))}$ becomes

$e^{i[\phi(\omega_{+})+\phi(\omega_{-})]}=e^{i[2\phi_{0}+2\beta\omega^{2}]},$
(76)

with the $\alpha$-dependence cancelling. As a result, $\phi_{0}$-dependent
terms occur without the exponentially decaying $\zeta$ factor, allowing much
slower decay of $H$ at large $\alpha$ (or even no decay at all, as in case H).
The slower decay at large dispersion is therefore a direct consequence of the
quantum-mechanical correlations present in the initial state.

As for the $\beta$ dependence, we see that as $\beta$ becomes large, both
$\zeta$ and $r_{1}$ become $\beta$ independent, with $\zeta\to
e^{-{{\Lambda_{2}}\over 4}}$ and $r_{1}\to 1$; thus all the curves approach
constant values at large $\beta$, with slopes ${{dH}\over{d\beta}}$ of
comparable order of magnitude.

We turn now to one additional case, that of more realistic SPDC photon pairs,
which we then proceed to compare with the simplified SPDC model already
examined.

##  VII Case L: SPDC

Now we present results for the mutual information using a more realistic model
for the parametric downconversion process. Numerically, the results turn out
qualitatively (and for some parameter ranges quantitatively as well) to be
very similar to those of the simplified SPDC model in the previous section;
however we no longer will be able to present explicit analytic expressions for
the measurement outcomes.

There are many possible cases that could be considered, but we restrict
ourselves here to the single case of collinear type-II SPDC in a nonlinear
crystal, with both of the outgoing photons entering port A of the dispersive
Mach-Zehnder interferometer. We now have to consider the parameters of both
the interferometer and the crystal. We allow the pump frequency to vary around
central frequency $2\omega_{0}$, with the deviation from the center of the
distribution represented by $2\Omega_{p}$; in other words, the pump frequency
is represented as

$\omega_{p}=2(\omega_{0}+\Omega_{p}).$

We once again assume a Gaussian distribution of frequencies, in this case
represented by a weighting factor $e^{-{{\sigma}\over
2}(\omega_{p}-2\omega_{0})^{2}}=e^{-2\sigma\Omega_{p}^{2}}$. The signal and
idler frequencies are then

$\displaystyle\omega_{s}$ $\displaystyle=$ $\displaystyle{{\omega_{p}}\over
2}+\Omega=\omega_{0}+\Omega_{p}+\Omega$ (77) $\displaystyle\omega_{i}$
$\displaystyle=$ $\displaystyle{{\omega_{p}}\over
2}-\Omega=\omega_{0}+\Omega_{p}-\Omega,$ (78)

with $\omega_{s}+\omega_{i}=\omega_{p}$. Suppose that the crystal is cut so
that exact phase matching occurs at the central frequency

$k_{p}(2\omega_{0})=k_{s}(\omega_{0})+k_{i}(\omega_{0}).$ (79)

Then, assuming that terms quadratic and higher in the frequencies are small,
the phase matching condition for the crystal gives us a condition on the wave-
vectors of the form ou (14)

$\Delta k\equiv
k_{p}(\omega_{p})-k_{s}(\omega_{s})-k_{i}(\omega_{i})=\Lambda_{p}\Omega_{p}+\Lambda\Omega,$
(80)

where
$\Lambda_{p}=2k_{p}^{\prime}(2\omega_{0})-k_{s}^{\prime}(\omega_{0})-k_{i}^{\prime}(\omega_{0})$
and $\Lambda=k_{i}^{\prime}(\omega_{0})-k_{s}^{\prime}(\omega_{0})$.

The wavefunction for the biphoton state entering the interferometer is now

$|\psi_{in}\rangle=\int
d\Omega\;d\Omega_{p}\Phi(\Omega_{p},\Omega)\hat{a}_{\omega_{0}+\Omega_{p}+\Omega}^{\dagger}\hat{a}_{\omega_{0}+\Omega_{p}-\Omega}^{\dagger}|0\rangle,$
(81)

where

$\Phi(\Omega_{p},\Omega)={\cal
N}e^{-2\sigma\Omega_{p}^{2}}\left({{\sin{{\Delta kL_{c}}\over 2}}\over{{\Delta
kL_{c}}\over 2}}\right)e^{-i\Delta k\;L_{c}/2},$ (82)

with normalization constant ${\cal N}$. Here, $L_{c}$ is the length of the
nonlinear crystal. Using this wavefunction, we can compute output
probabilities as before. Denoting the frequencies at the detectors by $\omega$
and $\omega^{\prime}$, we have

$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle\int
d\omega\;d\omega^{\prime}\left|\Phi({{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over
2})+\Phi(-{{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over 2})\right|^{2}$ (83)
$\displaystyle\times\left|1+e^{i[\phi(\omega)+\phi(\omega^{\prime})]}-e^{i\phi(\omega)}-e^{i\phi(\omega^{\prime})}\right|^{2}$
$\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle\int
d\omega\;d\omega^{\prime}\left|\Phi({{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over
2})\left[1-e^{i[\phi(\omega)+\phi(\omega^{\prime})]}-e^{i\phi(\omega)}+e^{i\phi(\omega^{\prime})}\right.\right.$
(84) $\displaystyle+\left.\Phi({{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over
2})\left[1-e^{i[\phi(\omega)+\phi(\omega^{\prime})]}+e^{i\phi(\omega)}-e^{i\phi(\omega^{\prime})}\right]\right|^{2}$
$\displaystyle P(0,2|\phi_{0})$ $\displaystyle=$ $\displaystyle\int
d\omega\;d\omega^{\prime}\left|\Phi({{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over
2})+\Phi(-{{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over 2})\right|^{2}$ (85)
$\displaystyle\times\left|1+e^{i[\phi(\omega)+\phi(\omega^{\prime})]}+e^{i\phi(\omega)}+e^{i\phi(\omega^{\prime})}\right|^{2},$

where $\phi(\omega)$ is as defined in equation 19. Note that
$\Phi\left({{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over 2}\right)$ depends only on the
crystal properties, while $\phi(\omega)$ depends only on the properties of the
interferometer. The integrands of $P(0,2|\phi_{0})$ and $P(2,0|\phi_{0})$
factor in their dependence on these two sets of parameters; that of
$P(1,1|\phi_{0})$ does not, indicating the entangled nature of the
$|11\rangle$ state.

Given these output probabilities, the mutual information can once again be
computed. In contrast to the previous sections, the analytic forms of the
probabilities are too complicated to be enlightening, so we proceed to
numerical calculations. Some examples are graphed in figures 10 to 14. The
plots are expressed in terms of the new parameters $b=\Lambda_{p}L_{c}$ and
$\lambda={\Lambda\over{\Lambda_{p}}}$.

Examples of the dependence of $H$ on the parameters of the pump beam
($\sigma$), interferometer ($\alpha$, $\beta$), and nonlinear crystal
($\lambda$, $b$) are given in figures 10 through 14. We see that, although $H$
decays overall with increasing values of the dispersion paramaters in the
interferometer, $\alpha$ and $\beta$, there are oscillations superimposed on
the decay, which are especially noticeable at low values of $\alpha$ and
$\beta$. This effect was in fact also present in the simplified SPDC model of
the previous sections, but in the latter case the oscillations were too weak
to be visible on the graphs. We see also that as either $b$ or $\lambda$
increases (or equivalently, as $\Lambda$ or $\Lambda_{p}$ increases), the
plots approach those of the simplified SPDC model. Since $\Lambda$ and $b$ are
proportional to the crystal length, this means that the simplified SPDC model
is an increasingly better approximation to real SPDC for longer crystals. It
also appears from the numerical simulations that for a given set of parameter
values $\alpha$, $\beta$, $L$, and $\sigma$, the simplified SPDC model
provides an upper bound to $H$ for the real SPDC cases with the same parameter
values. The maximum information content clearly occurs for low dispersion in
the interferometer, long nonlinear crystals, and large mismatch at
$\omega_{0}$ between signal and idler inverse group velocities in the crystal
(large $\Lambda$).

##  VIII Conclusions

In this paper, we have examined the effect of dispersion on the mutual
information that interferometric photon-detection measurements carry about
phase shifts. We have looked at a number of different situations involving two
interferometer set-ups and several different types of non-classical input
states. Comparing the results, we now have a precise and quantitative means to
measure the relative merits of different input states for various input-
parameter ranges. As a by-product, we have shown that in some circumstances,
parametric downconversion can be approximated by a much simpler model that is
amenable to exact analytical analysis.

Returning to the original question of which input state yields the most
information about the phase shift, the graphs of the previous sections yield
fairly clear results. Restricting discussion to MZ interferometers for
simplicity, we can see that for quantum interferometry in the presence of
dispersion the entangled photon pair produced by downconversion has a clear
advantage over other cases when input to a single port (Fock state input).
This advantage does not exist in the case of an dispersionless interferometer,
in which case the presence or absence of frequency correlations becomes
irrelevant for the information content. The only situation we have found in
which another input is superior to the frequency-anticorrelated Fock input is
when $\alpha$ is large but $\beta$ small, in which case the anticorrelated
dual Fock input is superior. These conclusions all hold when the simplified
downconversion model of section IV.2 is a good approximation; the results of
section VII imply that such conclusions weaken as the crystal becomes shorter.

Figure 10: (color online). Mutual information versus squared inverse bandwidth
$\sigma$ for SPDC. ($\alpha$, $\beta$, and $\sigma$ are in units of
$L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$,
respectively. $b$ is in units of $\omega_{0}^{-2}$, while $\lambda$ is
dimensionless.) Figure 11: (color online). Mutual information versus alpha for
SPDC with $\sigma=1$, $\beta=.1$. ($\alpha$, $\beta$, and $\sigma$ are in
units of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and
$\omega_{0}^{-2}$, respectively. $b$ is in units of $\omega_{0}^{-2}$, while
$\lambda$ is dimensionless.) Figure 12: (color online). Mutual information
versus beta for SPDC with $\sigma=1$, $\alpha=.3$. ($\alpha$, $\beta$, and
$\sigma$ are in units of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and
$\omega_{0}^{-2}$, respectively. $b$ is in units of $\omega_{0}^{-2}$, while
$\lambda$ is dimensionless.) Figure 13: (color online). Mutual information
versus lambda ($\lambda=\Lambda/\Lambda_{p}$) for SPDC. ($\alpha$, $\beta$,
and $\sigma$ are in units of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$,
and $\omega_{0}^{-2}$, respectively. $b$ is in units of $\omega_{0}^{-2}$,
while $\lambda$ is dimensionless.) Figure 14: (color online). Mutual
information versus b ($b={{L_{c}}\over 2}\Lambda_{p}$) for SPDC. ($\alpha$,
$\beta$, and $\sigma$ are in units of $L^{-1}\omega_{0}^{-1}$,
$L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$, respectively. $b$ is in units
of $\omega_{0}^{-2}$, while $\lambda$ is dimensionless.)

###### Acknowledgements.

This work was supported by a U. S. Army Research Office (ARO)
Multidisciplinary University Research Initiative (MURI) Grant; by the Bernard
M. Gordon Center for Subsurface Sensing and Imaging Systems (CenSSIS), an NSF
Engineering Research Center; by the Intelligence Advanced Research Projects
Activity (IARPA) and ARO through Grant No. W911NF-07-1-0629.

## References

  * (1) B. Yurke, Phys. Rev. Lett. 56, 1515 (1986).
  * (2) B. Yurke, S.L. McCall, J.R. Klauder, Phys. Rev. A 33, 4033 (1986).
  * (3) M.J. Holland, K. Burnett, Phys. Rev. Lett. 71, 1355 (1993).
  * (4) J.J. Bollinger, W.M. Itano, D.J. Wineland, D.J. Heinzen, Phys. Rev. A 54, R4649 (1996).
  * (5) E. Knill, R. Laflamme, G.J. Milburn, Nature 409, 46 (2001).
  * (6) H. Lee, P. Kok, N.J. Cerf, J.P. Dowling, Phys. Rev. A 65, 030101(R) (2002).
  * (7) P. Walther, J.W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, A. Zeilinger, Nature 429, 158 (2004).
  * (8) M.W. Mitchell, J.S. Lundeen, A.M. Steinberg, Nature 429, 161 (2004).
  * (9) T.B. Bahder, P.A. Lopata, Phys. Rev. A 74, 051801(R) (2006), arXiv:quantu-ph/0602123.
  * (10) A. Luis, Phys. Lett. A 329, 8 (2004).
  * (11) J. Beltrán, A. Luis, Phys. Rev. A 72, 045801 (2005).
  * (12) S. Boixo, A. Datta, M.J. Davis, S.T. Flammia, A. Saji, C.M. Caves, Phys. Rev. Lett. 101, 040403 (2008).
  * (13) C.K. Hong, Z.Y. Ou, L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
  * (14) Z.Y. Ou, Multi-Photon Quantum Interference, Springer, N.Y. (2007).