# Hybrid approximation approach to generation of atomic squeezing with quantum
nondemolition measurements

Ebubechukwu O. Ilo-Okeke  New York University Shanghai, 1555 Century Ave,
Pudong, Shanghai 200122, China  Department of Physics, School of Physical
Sciences, Federal University of Technology, P. M. B. 1526, Owerri 460001,
Nigeria  NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road
North, Shanghai 200062, China  Manikandan Kondappan  State Key Laboratory of
Precision Spectroscopy, School of Physical and Material Sciences, East China
Normal University, Shanghai 200062, China  New York University Shanghai, 1555
Century Ave, Pudong, Shanghai 200122, China  Ping Chen  State Key Laboratory
of Precision Spectroscopy, School of Physical and Material Sciences, East
China Normal University, Shanghai 200062, China  New York University Shanghai,
1555 Century Ave, Pudong, Shanghai 200122, China  Yuping Mao  State Key
Laboratory of Precision Spectroscopy, School of Physical and Material
Sciences, East China Normal University, Shanghai 200062, China  New York
University Shanghai, 1555 Century Ave, Pudong, Shanghai 200122, China
Valentin Ivannikov  New York University Shanghai, 1555 Century Ave, Pudong,
Shanghai 200122, China  NYU-ECNU Institute of Physics at NYU Shanghai, 3663
Zhongshan Road North, Shanghai 200062, China  Tim Byrnes  State Key Laboratory
of Precision Spectroscopy, School of Physical and Material Sciences, East
China Normal University, Shanghai 200062, China  New York University Shanghai,
1555 Century Ave, Pudong, Shanghai 200122, China  NYU-ECNU Institute of
Physics at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute,
New York University Abu Dhabi, UAE  Department of Physics, New York
University, New York, NY 10003, USA

###### Abstract

We analyze a scheme that uses quantum nondemolition measurements to induce
squeezing of a two-mode Bose-Einstein condensate in a double well trap. In a
previous paper [Ilo-Okeke et al. Phys. Rev. A 104, 053324 (2021)], we
introduced a model to solve exactly the wavefunction for all atom-light
interaction times. Here, we perform approximations for the short interaction
time regime, which is relevant for producing squeezing. Our approach uses a
Holstein-Primakoff approximation for the atoms while we treat the light
variables exactly. It allows us to show that the measurement induces
correlations within the condensate, which manifest in the state of the
condensate as a superposition of even parity states. In the long interaction
time regime, our methods allow us to identify the mechanism for loss of
squeezing correlation. We derive simple expressions for the variances of
atomic spin variables conditioned on the measurement outcome. We find that the
results agree with the exact solution in the short interaction time regime.
Additionally, we show that the expressions are the sum of the variances of the
atoms and the measurement. Beyond the short interaction time regime, our
scheme agrees qualitatively with the exact solution for the spin variable that
couples to light.

##  I Introduction

Squeezed states improve precision measurements beyond the standard quantum
limit attainable by classical means. As squeezed states possess entanglement
Einstein et al. (1935), they surpass the standard quantum limit set by the
quantum noise of individual uncorrelated quantum particles. Consequently,
squeezed states are increasingly used for enhanced measurement and detection
in many areas of physics such as image reconstruction Brida et al. (2010),
magnetometry and electric-field sensing Brask et al. (2015); Fan et al.
(2015), optical interferometry Dowling and Seshadreesan (2015); Schnabel
(2017), gravitational wave detection Eberle et al. (2010); Pitkin et al.
(2011), and used as a resource Julsgaard et al. (2001); Kuzmich and Kennedy
(2004); Estève et al. (2008) for heralding entanglement in many-body spin
systems.

Squeezed states Walls (1983); Kitagawa and Ueda (1993); Wineland et al.
(1994); Ma et al. (2011); Schnabel (2017); Pezzè et al. (2018) are realized in
many different physical systems via interactions that induce correlations
between quantum particles. For instance, correlations and squeezing arise
naturally from interactions found in the quantum system, such as the two-body
interactions in atomic Bose-Einstein condensates Kitagawa and Ueda (1993);
Estève et al. (2008); Böhi et al. (2009) and ions Leibfried et al. (2004); Ge
et al. (2019), or the long-range interactions of the Rydberg atoms Saffman et
al. (2010); Gil et al. (2014). Additionally, squeezing and correlations are
induced by interactions between two different quantum systems. A typical
example is the generation of optical squeezed states Slusher et al. (1985); Wu
et al. (1986); Josse et al. (2003) using the interactions between light and
barium borate crystal.

Quantum nondemolition (QND) measurements have been shown to be an effective
way of producing squeezing in a variety of systems Unruh (1979); Braginsky et
al. (1980); Roch et al. (1992); Grangier et al. (1998); Takahashi et al.
(1999); Kuzmich et al. (2000); Higbie et al. (2005); Lecocq et al. (2015). In
atomic systems, for instance, the quantum states of light and atoms become
entangled as a result of interactions between atoms and light. Upon the
detection of the light, the entanglement between quantum states of atom and
light is transferred to that among the individual atoms Higbie et al. (2005);
Meppelink et al. (2010); Ilo-Okeke and Byrnes (2014, 2016), manifesting itself
as the squeezed states of the atoms Kitagawa and Ueda (1993). Such states
possessing squeezing are viewed as a resource in quantum metrology Pezzè et
al. (2018), and quantum information Cerf et al. (2007); Abdelrahman et al.
(2014); Pyrkov and Byrnes (2014); Byrnes et al. (2015); Feng et al. (2021).
Stroboscopic monitoring has been used to squeeze the atomic spins Thorne et
al. (1978); Braginsky et al. (1980); Vasilakis et al. (2015); Møller et al.
(2017). In this method, a measurement is timed to coincide with the period of
the oscillator. These ensure that the spin component of the atom coupled to
light experiences a minimum disturbance due to measurement—an effect termed
measurement back-action Thorne et al. (1978); Braginsky et al. (1980).

In a previous work (Ref. Ilo-Okeke et al. (2021)), we proposed a scheme that
does not depend on timed measurements to achieve minimal disturbances due to
measurement while achieving squeezing of the atomic states. In this scheme,
the continuous monitoring of the atoms induces random motion on the mean spin
whose projection is confined to a plane perpendicular mean field direction. In
addition, the photon detection constricts the motion of the mean spin along
the spin variable that couples to light thereby resulting in spin squeezing.
This squeezing manifests as the shearing of the distribution in phase-space
Ilo-Okeke et al. (2021).

Squeezing correlations emerge in interacting many-particle systems where the
interaction in the system is weak. It is a precursor to other forms of
correlations found in the system. As interactions within the system become
strong, the squeezing correlations give way to other forms of correlation
Pezzè et al. (2018); Ilo-Okeke et al. (2021), such as the Schrödinger-cat
state Ilo-Okeke et al. (2021). The Holstein-Primakoff (HP) approximation
Holstein and Primakoff (1940) is used to study the dynamics of many-particle
systems in the large particle number limit. These include correlations, like
squeezing, generated by weak interactions. In atom-light systems, usually
described within QND measurements, there are many studies Hald et al. (1999);
Duan et al. (2000); Julsgaard et al. (2001); Kuzmich et al. (2000); Wang and
Sanders (2003); Madsen and Mølmer (2004); Hammerer et al. (2010); Vasilakis et
al. (2015) on the squeezing of atoms employing the HP approximation Holstein
and Primakoff (1940). However, all these studies to date focus on using the
system’s variables, such as spins or atom number, in showing the squeezing of
atoms. To date, within the HP approximation, there are no works that give
evidence of the squeezing correlation in atom systems using their quantum
state as has been done for light Barnett and Radmore (1997); Scully and
Zubairy (1997); Walls and Milburn (2008). In this paper, for the first time,
we derive the quantum squeezed state of an atom oscillator interacting weakly
with light within the HP approximation. The state we derive is an even parity
state akin to what is obtained using light Barnett and Radmore (1997); Scully
and Zubairy (1997); Walls and Milburn (2008).

The rest of the paper is organized as follows. We begin by presenting the
technique and a summary of the main result of this work in Sec. II. Next, we
present the double-well oscillator model in Sec. III. The hybrid approximation
is presented in Sec. IV. The hybrid approximation is used to analyze an
initial state that is perfectly polarized in Sec. V. In Sec. VI, we analyze an
initial state with a small but finite population in the excited state using
the hybrid approximation, and compare the results with the exact solution. We
present the general feature of the conditional variance in the short time
regime in Sec. VII, and finally present our summary and conclusions in Sec.
IX.

Figure 1: Atoms in a double-well trap are imaged using identical laser light
in a Mach-Zehnder interferometer geometry. The available states in the trap
are the symmetric ground state $\lvert g\rangle$ and the anti-symmetric first
excited state $\lvert e\rangle$.

##  II Main result of this paper

We first give a basic introduction of the aim, main technique, and main result
of this paper. As illustrated in Fig. 1, the number of atoms in the system is
macroscopically large $N\gg 1$, with virtually all of them found in the ground
state of the trap, and an atomic Bose-Einstein condensate placed in the double
well realizes this. Since the number of atoms in the ground state is large, a
classical number $\sqrt{N}$, it can be used to replace the mode operators of
the ground state. Doing this gives rise to operators mathematically equivalent
to photon operators. This step is the Holstein-Primakoff approximation, where
the mode operators of the excited state map to the $\hat{X}$ and $\hat{P}$ of
a harmonic oscillator. However, the light operators are not approximated. The
resulting Hamiltonian is that of a displaced oscillator for the atomic
operators because of the presence of light,

$H=-\frac{\hbar\Omega}{2}(\hat{X}^{2}+\hat{P}^{2})+\sqrt{2N}\hbar
g\hat{X}S_{z},$

where $\Omega$ is the tunneling frequency between the wells, $g$ is coupling
frequency between atom and light, and $S_{z}$ is the relative photon number
operator. The approach where we treat the mode operators of the double-well
trap within the Holstein-Primakoff approximation while not approximating the
photon operators is the key to deriving the squeezed state of an atom
oscillator. We call this the “hybrid approximation”, which allows us to
calculate the quantum state of the atom oscillator. It also marks the point of
departure of this work from previous studies Hald et al. (1999); Duan et al.
(2000); Kuzmich et al. (2000); Julsgaard et al. (2001); Wang and Sanders
(2003); Madsen and Mølmer (2004); Hammerer et al. (2010); Vasilakis et al.
(2015) using the Holstein-Primakoff approximation Holstein and Primakoff
(1940).

Our calculation then proceeds as follows. Let $\lvert\psi_{0}\rangle$ be the
initial state of the condensate, while
$\lvert\alpha_{l}\rangle,\,\lvert\alpha_{r}\rangle$ are the initial coherent
state of light in the left and right arms of the interferometer, respectively.
The combined state of the atom and light,
$\lvert\psi_{0},\alpha_{l},\alpha_{r}\rangle=\lvert\psi_{0}\rangle\otimes\lvert\alpha_{l}\rangle\otimes\lvert\alpha_{r}\rangle$,
evolves under the Hamiltonian as prescribed by Schrödinger equation,
$e^{-iHt/\hbar}\lvert\psi_{0},\alpha_{l},\alpha_{r}\rangle$. At the
beamsplitter, the light beams in the two arms of the interferometer interfere.
The unitary operator that describes the action of the beam splitter is
$e^{-i\pi S_{x}/2}$. Counting the photons in the light beams by the detectors,
$c$ and $d$, collapses the quantum state of light. The collapse of the light
state described by a projection operator $\lvert n_{c},n_{d}\rangle\langle
n_{c},n_{d}\rvert$ gives the state of the condensate conditioned on counting
$n_{c}$ and $n_{d}$ photons at the detectors $c$ and $d$, respectively, as

$\lvert\psi_{n_{c},n_{d}}\rangle=\langle n_{c},n_{d}\rvert e^{-i\pi
S_{x}/2}e^{-iHt/\hbar}\lvert\psi_{0},\alpha_{l},\alpha_{r}\rangle.$

At this point we assume that the relative photon number difference is smaller
than the total number of photons collected at the detectors $c$ and $d$,
$\lvert n_{d}-n_{c}\rvert\ll(n_{c}+n_{d})$. It is the condition for which
correlation is maximized. Additionally, to minimize the effect of measurement
on the quantum state of the condensate containing $N$ atoms, we consider
balanced detection: the average photon number in the arms of the
interferometer is the same, $\alpha_{r}=\alpha_{l}$. At this condition, the
effects due to the total photon number $n_{c}+n_{d}$ in a measurement outcome
vanish, leaving off only those from the relative photon number difference,
$n_{d}-n_{c}$.

We state the main result that we find from the above calculation. For a
perfectly polarized initial state $\lvert\psi_{0}\rangle$ of the condensate,
where all the $N$ atoms are in the ground state of the double-well, the
normalized state of the condensate conditioned on the counting $n_{c}$ and
$n_{d}$ photons at the detectors $c$ and $d$, respectively, is

$\lvert\psi_{n_{c},n_{d}}\rangle\propto\sum_{n=0}^{N_{\mathrm{max}}}C_{n}\left(\lvert\alpha_{r}\rvert,gt,N\right)\lvert
2n\rangle,$

where $N_{\mathrm{max}}=N/2$ for $N$ even, and $N_{\mathrm{max}}=(N-1)/2$ for
$N$ odd. The coefficients $C_{n}\left(\lvert\alpha\rvert,gt,N\right)$ depend
on the average photon number at the input $\lvert\alpha_{r}\rvert^{2}$, the
atom-light interaction strength $gt$ and the total number of atoms $N$,
expressed in polynomial powers of $n$. Its explicit form is given in (20). The
above preceding equation is the main results of this paper.

In the remainder of the paper, we give a detailed derivation of this state and
use it to examine properties of the oscillator by calculating the averages
such variance and expectation value of the oscillator quadratures $\hat{X}$
and $\hat{P}$, and compare the results to the exact solution. Additionally, we
use the Q-function to highlight the various contributions from the initial
state of the light and the detection of photons to the quantum state of the
condensate. Given an imperfectly polarized state of the condensates, where a
small but finite number of atoms are found in the excited state as shown in
Fig. 1, we showed that the normalized quantum state of the condensate
conditioned on the detection of $n_{c}$ and $n_{d}$ is also an even parity
state.

##  III The double-well model

The model we consider has been previously described in Ref. Ilo-Okeke et al.
(2021) which we summarize briefly here. We consider an atomic Bose-Einstein
condensate (BEC) confined in a double-well potential with the lowest energy
states being the symmetric and antisymmetric states as shown in Fig. 1. We
further assume that fluctuation about the mean number density is small so that
the resulting mean-field Hamiltonian can be written in second quantization as

$\hat{H}=\frac{1}{2}(\omega_{e}+\omega_{g})\hat{N}+\hbar\Omega J_{z},$ (1)

where
$\hat{N}=\hat{b}^{\dagger}_{g}\hat{b}_{g}+\hat{b}^{\dagger}_{e}\hat{b}_{e}$ is
the total atom number operator, $\hat{b}_{e,g}$ are the annihilation operators
for the excited and ground states, respectively, and
$\Omega=\omega_{e}-\omega_{g}$ is the particle tunneling frequency. The spin
operators are defined as

$\begin{split}J_{x}&=\frac{\hat{b}^{\dagger}_{e}\hat{b}_{g}+\hat{b}^{\dagger}_{g}\hat{b}_{e}}{2},\\\
J_{y}&=-i\frac{\hat{b}^{\dagger}_{e}\hat{b}_{g}-\hat{b}^{\dagger}_{g}\hat{b}_{e}}{2},\\\
J_{z}&=\frac{\hat{b}^{\dagger}_{e}\hat{b}_{e}-\hat{b}^{\dagger}_{g}\hat{b}_{g}}{2}.\end{split}$
(2)

The atoms in the double-well potential interact with laser light that is
detuned from atomic resonance transition. Such interactions are described by
the QND Hamiltonian Takahashi et al. (1999); Julsgaard et al. (2001); Kuzmich
and Kennedy (2004); Ilo-Okeke and Byrnes (2014); Behbood et al. (2014). Hence
the total Hamiltonian of the atom-light system is

$\hat{H}=\hbar\Omega J_{z}+2\hbar gJ_{x}S_{z},$ (3)

where $g$ is the atom-light coupling strength, and we have neglected the
constant energy term that contributes a global phase shift in the states of
atom and light in (3). The dependence of the atom-light interactions on the
spins arises from writing the sum on atom-light interactions in both wells in
terms of the relative photon number
$S_{z}=(\hat{a}^{\dagger}_{l}\hat{a}_{l}-\hat{a}^{\dagger}_{r}\hat{a}_{r})/2$.

The initial state of $N$ atoms in the BEC is taken to be a spin coherent state
formed from the superposition of the excited and ground state as Byrnes and
Ilo-Okeke (2021)

$\lvert\alpha,\beta\rangle\rangle=\frac{(\alpha b^{\dagger}_{e}+\beta
b^{\dagger}_{g})^{N}}{\sqrt{N!}}\lvert\mathrm{vac}\rangle,$ (4)

where $\lvert\mathrm{vac}\rangle$ is the vacuum state, a state with no atoms
or photons. The amplitudes $\alpha$ and $\beta$ give the fractional population
of atoms in the excited and ground state, respectively. Similarly, the quantum
state of the laser light in the two arms of the interferometer is a coherent
state of the form $\lvert\alpha_{l}\rangle\otimes\lvert\alpha_{r}\rangle$,
where

$\lvert\alpha_{s}\rangle=e^{-\frac{|\alpha_{s}|^{2}}{2}}e^{\alpha_{s}\hat{a}^{\dagger}_{s}}\lvert\mathrm{vac}\rangle,$
(5)

and $s=r,l$.

Given the initial states of atoms (4) and light (5), the combined quantum
state of atom and light at any other time is governed by (3) as

$\lvert\Psi\rangle=e^{-i\frac{\hat{H}t}{\hbar}}\lvert\alpha,\beta\rangle\rangle\otimes\lvert\alpha_{l}\rangle\otimes\lvert\alpha_{r}\rangle,$
(6)

and is a solution of the Schrödinger equation

$i\hbar\frac{d\lvert\Psi\rangle}{dt}=\hat{H}\lvert\Psi\rangle.$ (7)

This entangles the atoms and the light. After this, the light in the Mach-
Zehnder interferometer are interfered and measured. We previously derived an
exact solution of this valid for all interaction times Ilo-Okeke et al.
(2021). Although exact, the resulting expressions are typically complex due to
the many-body entangled wavefunction that results from (6). Our aim in this
paper will be to obtain some simple formula for the relevant quantities, so
that one can obtain a greater understanding of the important features and
parameters governing the behavior of a quantum system. For instance, in the
case at hand, it will be interesting to understand how short the evolution
time could be to sustain squeezing. One could also be interested in describing
the squeezing effect in terms of parameters of the atom and light in the short
time regime. Such understanding offers insight into the control and
manipulation of squeezed states of atoms.

##  IV The large _N_ limit: The hybrid approximation

In this section, we describe the quantum state of an atom within the Holstein-
Primakoff approximation. This is motivated by the fact that at short atom-
light interaction times, the ground state population is nearly undepleted
$\langle J_{z}\rangle\sim N/2$, even in the presence of a small dephasing
parameter, $\gamma<g$ (see Figs. 5 and 8 of Ref. Ilo-Okeke et al. (2021)). The
results of this section will be used to analyze various physical quantities
for two different initial states of the atoms, a perfect and an imperfectly
spin polarized atoms, in Secs. V and VI.

Because of the large number $N$ of atoms $N\gg 1$, and given that at short
times, the atoms are nearly in the ground state of the double-well trap, the
Hamiltonian (3) within the Holstein-Primakoff approximation, reduces to

$H\approx-\frac{\hbar\Omega}{2}\left(\hat{X}^{2}+\hat{P}^{2}\right)+\sqrt{2N}\hbar
g\hat{X}S_{z},$ (8)

where $\hat{X}=J_{x}/\sqrt{N/2}$, $\hat{P}=J_{y}/\sqrt{N/2}$ and a term
proportional to the total atom number $N$ is neglected in writing (8). The
operators $\hat{X}$ and $\hat{P}$ in terms of the harmonic oscillator
operators are respectively

$\hat{X}=\frac{\hat{b}^{\dagger}_{e}+\hat{b}_{e}}{\sqrt{2}},\qquad\hat{P}=i\frac{\hat{b}^{\dagger}_{e}-\hat{b}_{e}}{\sqrt{2}},$
(9)

and satisfies commutation relation $[\hat{X},\hat{P}]=i$.

Given an initial state of the atoms, $\lvert\psi_{0}\rangle$, and in the
presence of atom-light interactions $g\neq 0$, the initial state of atom-light
system,
$\lvert\psi_{0},\alpha_{l},\alpha_{r}\rangle=\lvert\psi_{0}\rangle\otimes\lvert\alpha_{l}\rangle\otimes\alpha_{r}\rangle$,
evolves under the Hamiltonian (8) as

$\lvert\psi(t)\rangle=e^{i\Omega
t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}-\sqrt{2}\kappa\hat{X}(\hat{a}^{\dagger}_{l}\hat{a}_{l}-\hat{a}^{\dagger}_{r}\hat{a}_{r})\right]}\lvert\psi_{0},\alpha_{l},\alpha_{r}\rangle,$
(10)

where $\kappa=g\sqrt{N}/(2\Omega)$. This may also be written as

$\displaystyle\lvert\psi(t)\rangle$
$\displaystyle=e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}\sum_{n_{l}=0}^{\infty}\sum_{n_{r}=0}^{\infty}e^{i\Omega
t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}-\sqrt{2}\kappa(n_{l}-n_{r})\hat{X}\right]}$
$\displaystyle\times\frac{\left(\alpha_{l}\right)^{n_{l}}}{n_{l}!}(\hat{a}^{\dagger}_{l})^{n_{l}}\frac{\left(\alpha_{r}\right)^{n_{r}}}{n_{r}!}(\hat{a}^{\dagger}_{r})^{n_{r}}\lvert\psi_{0}\rangle\lvert\mathrm{vac}\rangle,$
(11)

where $n_{l}$ and $n_{r}$ are the number of photons in the left and right arms
of the interferometer, respectively. It is readily seen that the phase of the
photons contains information about the atomic BEC. This information carried by
the photons is accessed by interfering the photons at the beamsplitter as
shown in Fig. 1. The interference of photons performed by beamsplitter
corresponds to the following transformation

$\hat{a}^{\dagger}_{l}=\frac{\hat{a}^{\dagger}_{c}+i\hat{a}^{\dagger}_{d}}{\sqrt{2}},\quad\hat{a}^{\dagger}_{r}=\frac{i\hat{a}^{\dagger}_{c}+\hat{a}^{\dagger}_{d}}{\sqrt{2}}.$
(12)

Hence, the wave function immediately after recombination becomes

$\displaystyle\lvert\psi_{\mathrm{BS}}(t)\rangle$
$\displaystyle=e^{-\frac{\lvert\alpha_{l}\rvert^{2}}{2}-\frac{\lvert\alpha_{r}\rvert^{2}}{2}}\sum_{n_{l}=0}^{\infty}\sum_{n_{r}=0}^{\infty}e^{i\Omega
t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}-\sqrt{2}\kappa(n_{l}-n_{r})\hat{X}\right]}$
$\displaystyle\times\frac{\left(\alpha_{l}\right)^{n_{l}}}{n_{l}!}\left(\frac{\hat{a}^{\dagger}_{c}+i\hat{a}^{\dagger}_{d}}{\sqrt{2}}\right)^{n_{l}}\frac{\left(\alpha_{r}\right)^{n_{r}}}{n_{r}!}\left(\frac{i\hat{a}^{\dagger}_{c}+\hat{a}^{\dagger}_{d}}{\sqrt{2}}\right)^{n_{r}}$
$\displaystyle\times\lvert\psi_{0}\rangle\lvert\mathrm{vac}\rangle.$ (13)

The state (IV) that emerges after the interference of photons on the
beamsplitter is a superposition of the states of atoms and light. Counting
$n_{c}$ and $n_{d}$ photons at the detector collapses the states of light.
Such a measurement is described by the projection operator $\mathcal{P}=\rvert
n_{c},n_{d}\rangle\langle n_{c},n_{d}\lvert$. The state of atoms (not
normalized)
$\lvert\psi_{n_{c},n_{d}}\rangle=\mathcal{P}\rvert\psi_{\mathrm{BS}}(t)\rangle$
conditioned on photon detection is

$\displaystyle\lvert\tilde{\psi}_{n_{c},n_{d}}\rangle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\frac{\alpha_{r}}{\sqrt{2}}\right)^{n_{c}+n_{d}}\sum_{n_{l}=0}^{\infty}e^{in_{l}\phi}$
$\displaystyle\times I(n_{c},n_{d},n_{l})e^{i\Omega
t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}+\sqrt{2}\kappa(n_{c}+n_{d}-2n_{l})\hat{X}\right]}\lvert\psi_{0}\rangle,$
(14)

where $\phi=\arg(\alpha_{l})-\arg(\alpha_{r})$, and the Fourier coefficient
$I(n_{c},n_{d},n_{l})$ is defined as

$\displaystyle I(n_{l},n_{c},n_{d})$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}d\phi\,e^{-i\,n_{l}\phi}\left(1+i\frac{|\alpha_{l}|}{|\alpha_{r}|}e^{i\phi}\right)^{n_{c}}\times$
$\displaystyle\left(i+\frac{|\alpha_{l}|}{|\alpha_{r}|}e^{i\phi}\right)^{n_{d}},$
(15)

see Appendix C for details. For the analysis of the Fourier coefficient, the
interval $\left[0,2\pi\right]$ is convenient when the largest contribution of
the integrand comes from points around $\phi\approx\pi$, whereas the interval
will be shifted to $\left[-\pi,\pi\right]$ when the largest contribution of
the integrand comes from points around $\phi\approx 0$.

##  V Perfect initial spin polarization

In this section, we derive the quantum state of the atoms after detection
using (IV). This allows us to establish the squeezing correlations induced by
the measurement. Secondly, it allows us to estimate how short the measurement
should be, that is the region of validity of Holstein-Primakoff approximation
performed on the atoms. We then solve exactly the state of atom and light
using the Schrödinger equation (7), followed by the interference of the
photons on a beamsplitter as shown in Fig. 1. The exact wave function
conditioned on the detection of photons is used to calculate the relevant
averages of the atomic operators. We derive expressions for the mean and
variance of the atomic variables under the hybrid approximation and compare
them with the exact solution. Additionally, we visualize the evolution of the
conditional squeezing in phase-space using the Husimi _Q_ -distribution.

###  V.1 State Of Condensate Conditioned On Photon Detection

We consider the case where all the atoms are initially in the ground state of
the double-well trap. The fractional probability amplitudes become $\alpha=0$
and $\beta=1$. In this case, the initial state (4) becomes

$\lvert\psi_{0}\rangle=\lvert 0,1\rangle\rangle=\lvert 0,N\rangle\equiv\lvert
0\rangle,$ (16)

where the state $\lvert 0,N\rangle$ is a Fock state defined as

$\lvert
0,N\rangle=\frac{\left(b^{\dagger}_{g}\right)^{N}}{\sqrt{N!}}\lvert\mathrm{vac}\rangle.$
(17)

Equation (16) is the vacuum state of the excited state of the double-well
trap. This state describes a perfectly polarized state along _z_. It is easy
to see from (9) that although the atoms in the excited state execute harmonic
motion, on average, the number of atoms in the excited state at any time for
$g=0$ is zero, $\langle 0\lvert\hat{X}\rvert 0\rangle=0$. These point to the
type of motion executed by the atoms is mostly due to fluctuation. Hence, the
variance is $(\Delta X)^{2}=1/2$. In the presence of light $g\neq 0$, the
initial state $\lvert\psi_{0}\rangle$ evolves as prescribed by (10), and is
modified further by the detection of photons.

The unnormalized state of the atoms after detection of photons is given by
(IV). The operator $e^{i\Omega
t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}+\sqrt{2}\kappa(n_{c}+n_{d}-2n_{l})\hat{X}\right]}$
in (IV) can be written as a product of operators
$e^{ip(\theta)\hat{b}^{\dagger}_{e}}e^{iq(\theta)\hat{b}^{\dagger}_{e}\hat{b}_{e}}e^{ir(\theta)\hat{b}_{e}}e^{is(\theta)}$,
see (51). Of these only the terms having $s(\theta)$ and
$\hat{b}^{\dagger}_{e}$ in the exponent have an effect on the initial state
(16) — a state with zero spin variance along the _z_ -axis. The measurement
has two effects on the state $\lvert 0\rangle$. It creates a state with finite
width via the term containing $s(\theta)$ from the initial state. Secondly,
the term having $\hat{b}^{\dagger}_{e}$ then introduces correlations when the
sum over all the input photon states $n_{l}$ is performed (IV).

To see this, we evaluate the probability amplitude explicitly for the initial
state (16). Detailed calculations are provided in Appendices C and D, we show
the final results here. To minimize the effect of measurement, we examine the
case where the measurement is balanced $\alpha_{l}=\alpha_{r}$. At this
condition, the effects due to the total photon number $n_{c}+n_{d}$ is
suppressed, leaving off those only from the relative photon number difference,
$n_{d}-n_{c}$.

Using the completeness relation for the atomic state $\sum_{n}\lvert
n\rangle\langle n\rvert=\hat{\mathbbm{1}}$, the state of the atoms conditioned
on detection of $n_{c}$, $n_{d}$ photons is

$\displaystyle\lvert\tilde{\psi}_{n_{c},n_{d}}\rangle$
$\displaystyle\propto\sum_{n=0}^{N}\sum_{k=0}^{\frac{n}{2}}\frac{n!}{k!(n-2k)!}\left(\frac{2\lvert\alpha_{r}\rvert^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\right)^{k}$
$\displaystyle\left(\frac{2ix_{0}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\right)^{n-2k}\lvert
n\rangle,$

where $x_{0}=\arcsin\left[(n_{d}-n_{c})/(n_{d}+n_{c})\right]$. The behavior of
the sum over $k$ dictates how strong the correlations would be. For instance,
where the relative photon number difference $\rvert n_{d}-n_{c}\lvert$ is
large and comparable to total photon number $n_{c}+n_{d}$, the magnitude of
$x_{0}$ is large with a maximum value $\pi/2$. However,
$\lvert\alpha_{r}\rvert^{2}$ is large compared to $\pi/2$. Thus, the term with
the power $n-2k$ is small compared to the term with power $k$, and is treated
as a perturbation. Hence, maximum correlation is obtained in the limit $\lvert
n_{d}-n_{c}\rvert\ll n_{c}+n_{d}$ for which $\lvert x_{0}\rvert\approx 0$, and
$k$ takes on a value of $n/2$. The unnormalized state of the atoms conditioned
on the detection of photons (IV) then becomes

$\displaystyle\lvert\tilde{\psi}_{n_{c},n_{d}}\rangle=$
$\displaystyle\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\,\sqrt{n_{d}!}}\left(\frac{n_{c}}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}\left(\frac{n_{d}}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}$
$\displaystyle\times\frac{(\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2})^{\frac{n_{c}+n_{d}}{2}}}{\sqrt{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}}\sum_{n=0,2,4,\cdots}\frac{\sqrt{n!}}{\left(\dfrac{n}{2}\right)!}$
$\displaystyle\times
i^{n}\left(\frac{2\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\right)^{\frac{n}{2}}\lvert
n\rangle.$ (18)

A global phase factor has been neglected in writing (V.1). The final state
obtained is not a spin coherent state. More so, the state
$\lvert\psi_{n_{c},n_{d}}\rangle$ accepts only even integer numbers — the
final state conditioned on photon detection is a superposition of even number
states only. Such a superposition of the same parity states is characteristic
of a squeezed state. The term
$\dfrac{2\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}$
determines the depth of the squeezing. For short atom-light interaction times,
$\dfrac{2\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}$
is small (less than unity), and the probability amplitude of the $\lvert
0\rangle$ is largest and goes to zero for larger $n$. Similarly, for long
interaction times,
$\dfrac{2\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}$
is appreciable (greater than unity), and the state at $n=N$ has the largest
probability amplitude, while the amplitude of $n=0$ is smallest. In this case,
it is obvious that the Holstein-Primakoff approximation of the atomic state is
invalid. Hence, the validity of the Holstein-Primakoff approximation performed
on the atomic state implies that
$\dfrac{2\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}<0.5$
and is the regime where the atomic state is squeezed. The weak form of the
condition is $gt<2(\lvert\alpha_{r}\rvert\sqrt{N})^{-1}$, which we put in the
form that is amenable to the results obtained later below as $\Omega
t<(\lvert\alpha_{r}\rvert\kappa)^{-1}$ where $\kappa$ is defined in (10).

The probability
$P(n_{c},n_{d})=\langle\psi_{n_{c},n_{d}}\lvert\psi_{n_{c},n_{d}}\rangle$ of
getting $n_{c}$, $n_{d}$ photons in the measurement is obtained by summing
over the _n_ th atom state, which converges for
$gt<2(\lvert\alpha_{r}\rvert\sqrt{N})^{-1}$. Hence, the probability
$P(n_{c},n_{d})$ becomes

$\displaystyle P(n_{c},n_{d})$
$\displaystyle=\frac{e^{-\lvert\alpha_{l}\rvert^{2}-\lvert\alpha_{r}\rvert^{2}}}{n_{c}!\,n_{d}!}\left(\frac{n_{c}}{n_{c}+n_{d}}\right)^{n_{c}}\left(\frac{n_{d}}{n_{c}+n_{d}}\right)^{n_{d}}$
$\displaystyle\times(\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2})^{n_{c}+n_{d}}\frac{1}{\sqrt{1+8\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}}.$
(19)

Consequently the normalized state (V.1) of the atoms conditioned on photon
detection is

$\begin{split}\lvert\psi_{n_{c},n_{d}}\rangle&=\frac{\left(1+8\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}\right)^{1/4}}{\sqrt{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}}\sum_{n=0}^{N_{\mathrm{max}}}\frac{\sqrt{(2n)!}}{n!}\\\
&\times\left(\frac{-2\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\right)^{n}\lvert
2n\rangle,\end{split}$ (20)

where $N_{\mathrm{max}}=N/2$ for $N$ even, and $N_{\mathrm{max}}=(N-1)/2$ for
$N$ odd.

Previously, Refs. (Ilo-Okeke and Byrnes, 2014, 2016) reported that Faraday
imaging is minimally-destructive for atom-light interaction strengths that are
sufficiently small compared to the square-root of the number of atoms in the
condensate, $gt\ll 1/\sqrt{N}$ which provides an upper bound. Here, the
strength of the measurement and the point of observation (the photon number
$n_{c}$ and $n_{d}$) would determine the required coupling strength $gt$. For
a particular measurement to be minimally destructive, it is the total number
of atoms $N$ and the average photon number $\lvert\alpha_{r}\rvert^{2}$ that
determine the strength of the atom-light coupling,
$gt<2/(|\alpha_{r}|\sqrt{N})$. Although the inequality is obtained in the
undepleted pump approximation and for optimal performance, it provides the
condition of applicability of an ac Stark shift to be minimally destructive in
a conditional measurement.

###  V.2 Hybrid Variance

The normalised state immediately after the detection of $n_{c}$ and $n_{d}$
photons is thus

$\displaystyle\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\frac{\alpha_{r}}{\sqrt{2}}\right)^{n_{c}+n_{d}}\frac{1}{\sqrt{P(n_{c},n_{d})}}$
$\displaystyle\times\sum_{n_{l}=0}^{\infty}e^{in_{l}\phi}I(n_{c},n_{d},n_{l})$
$\displaystyle\times e^{i\Omega
t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}+\sqrt{2}\kappa(n_{c}+n_{d}-2n_{l})\hat{X}\right]}\lvert\psi_{0}\rangle,$
(21)

where $P(n_{c},n_{d})$ is the probability (V.1), and $I(n_{c},n_{d},n_{l})$ is
the Fourier coefficient (IV). The normalized conditional expectation value
$\bar{B}$ of any of the atom operator $\hat{B}=\hat{X},\,\hat{P}$ subject to
having counted $n_{c}$ and $n_{d}$ photons at the detector is calculated using
$\lvert\psi_{n_{c},n_{d}}\rangle$ as

$\displaystyle\bar{B}$
$\displaystyle=\sqrt{\frac{2}{N}}\langle\psi_{n_{c},n_{d}}\lvert\hat{B}\rvert\psi_{n_{c},n_{d}}\rangle.$
(22)

Similarly, the normalized conditional variance of the atom operators is found
using (22) as

$\Delta\bar{B}=2\left[\langle\psi_{n_{c},n_{d}}\lvert\hat{B}^{2}\lvert\psi_{n_{c},n_{d}}\rangle-\left(\langle\psi_{n_{c},n_{d}}\lvert\hat{B}\lvert\psi_{n_{c},n_{d}}\rangle\right)^{2}\right].$
(23)

In the text, equations (22) and (23) will be referred to as the hybrid
expectation values and hybrid variance, respectively. The variances are
normalized such that for $g=0$, $\Delta\bar{B}$ gives unity,
$\Delta\bar{B}=1$.

###  V.3 Approximate Variance

The details of calculating the mean values and the variances are given in
Appendix E which we summarize here. Using (16) in (22), we first calculate the
approximate expression for the hybrid mean values of $\hat{X}$, and $\hat{P}$
(9). The expectation values are evaluated at
$n_{c},\,n_{d}=(\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2})/2$, and
$\alpha_{l}=\alpha_{r}$. Because $\lvert\psi_{n_{c},n_{d}}\rangle$ selects
only state with even parity, we find for an _n_ th state $\lvert n\rangle$,
$\langle n\rvert\hat{\mathcal{O}}\lvert\psi_{n_{c},n_{d}}\rangle=0$ for _n_
even where $\hat{\mathcal{O}}=b^{\dagger}_{e},\,b_{e}$, while $\langle
n\rvert\psi_{n_{c},n_{d}}\rangle\neq 0$ for _n_ even, and vice versa. Hence,
the expectation value of operator $\hat{\mathcal{O}}$ is easily calculated by
summing over $n$,
$\langle\psi_{n_{c},n_{d}}\rvert\hat{\mathcal{O}}\lvert\psi_{n_{c},n_{d}}\rangle=\sum_{n}\langle\psi_{n_{c},n_{d}}\lvert
n\rangle\langle n\lvert\hat{\mathcal{O}}\rvert\psi_{n_{c},n_{d}}\rangle$. It
is immediately deduced from (22) that the expectation values of $\hat{X}$ and
$\hat{P}$ are

$\langle\psi_{n_{c},n_{d}}\lvert\hat{X}\rvert\psi_{n_{c},n_{d}}\rangle=\langle\psi_{n_{c},n_{d}}\lvert\hat{P}\rvert\psi_{n_{c},n_{d}}\rangle=0,$
(24)

for all time. For the operators
$\hat{\mathcal{O}}=b^{\dagger}_{e}b_{e},\,(b^{\dagger}_{e})^{2},\,\mathrm{and}\,b^{2}_{e}$
needed in the calculation of variances, $\langle
n\lvert\hat{\mathcal{O}}\rvert\psi_{n_{c},n_{d}}\rangle\neq 0$ for _n_ even,
and zero otherwise. Hence, their expectation values are evaluated as
$\langle\psi_{n_{c},n_{d}}\lvert\hat{\mathcal{O}}\rvert\psi_{n_{c},n_{d}}\rangle=\sum_{n}\langle\psi_{n_{c},n_{d}}\lvert
n\rangle\langle n\lvert\hat{\mathcal{O}}\rvert\psi_{n_{c},n_{d}}\rangle$.
Using these, the normalized approximate variances of $\hat{X}$ and $\hat{P}$
calculated using (23) evaluate to

$\displaystyle\Delta\bar{X}$
$\displaystyle=\frac{1}{1+8\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}},$
(25) $\displaystyle\Delta\bar{P}$
$\displaystyle=1+8\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}.$ (26)

Because of the normalization implemented in (23), the approximate variances
(25) and (26) are such that for $g=0$, they give $1$. Hence a division by $2$
recovers the expected values of $1/2$. We note that the variances of the
quadrature $X$ and $P$ [(25) and (26), respectively] do not depend on the
tunneling frequency $\Omega$. This is because the parameter $\kappa$ defined
in (10) is proportional to the inverse of the tunneling frequency $\Omega$,
$\kappa=g\sqrt{N}/(2\Omega)$. However, to be able to make comparisons with the
results of Ref. Ilo-Okeke et al. (2021), we will keep their present form.

###  V.4 Numerical Evaluation

We now numerically solve the Schrödinger equation (7) to compare the results
that we have obtained in the previous section to exact results. At the end of
the evolution, the photons are interfered by the beamsplitter according to the
transformation (12). The detection of $n_{c}$ and $n_{d}$ photons following
immediately their interference collapses the wave function of light and is
described by the projection operator $\mathcal{P}=\lvert
n_{c},n_{d}\rangle\langle n_{c},n_{d}\rvert$. The details of the numerical
method for solving the Schrödinger equation is given in Appendix A. The state
$\lvert\Psi_{n_{c},n_{d}}\rangle$ of the atoms, as given in Eq. (A), is
conditioned upon the detection of light and is used to calculate the
normalized averages of atomic spin variables $J_{x,y}$:

$\displaystyle\bar{J}_{i}$
$\displaystyle=\frac{2}{N}\langle\Psi_{n_{c},n_{d}}\lvert
J_{i}\rvert\Psi_{n_{c},n_{d}}\rangle,$ (27) $\displaystyle\Delta\bar{J}_{i}$
$\displaystyle=\frac{4}{N}\left(\langle\Psi_{n_{c},n_{d}}\lvert
J_{i}^{2}\rvert\Psi_{n_{c},n_{d}}\rangle-\frac{N^{2}}{4}\bar{J}_{i}^{2}\right),$
(28)

for $i=x,\,y$. These are used later in Secs. V.5, VI.2 and VI.3, and are the
exact solution for the mean values and variances, respectively.

###  V.5 Exact Solution Versus The Hybrid Approximation Results

Figure 2: The normalized conditional variance of $J_{x}$, $J_{y}$. In (a), the
solid line is exact solution (7) calculated as described in the text. The
dashed line is hybrid variance obtained using (16) in (23), while the dashed-
dotted line is the approximate variance (25). In (b), the solid line is the
exact variance (7) calculated as described in the main text. The dashed line
is the hybrid variance obtained using (16) in (23), while the dashed-dotted
line is the approximate variance (26). The parameters for plots are
$\alpha_{l}=\alpha_{r}=\sqrt{20}$, $N=200$, $\Omega=\pi/4$, and
$g=0.1\Omega/N$.

The results for the variance calculations are presented in Fig. 2. The
variance of $\hat{X}$ in Eq. (23) is evaluated using the hybrid approximation,
decreases below unity with an increase in time before returning to unity after
longer time scales as shown in Fig. 2(a). These are in perfect agreement with
the exact variance $J_{x}$ in Eq. (28). On the other hand, the hybrid variance
$\Delta\bar{P}$ is never below unity and agrees with the exact variance
$\Delta\bar{J}_{y}$ for the short time interval $\Omega
t<(\lvert\alpha\rvert\kappa)^{-1}$ as shown in Fig. 2(b). To better understand
this behavior we turn to approximate expression for the variance of $\hat{X}$
and $\hat{P}$ given in (25) and (26), respectively. In Fig. 2(b), it is seen
that in the short time regime, the approximate variance $\Delta\bar{P}$ (Eq.
(26)) and hybrid variance (23) agree with the exact solution
$\Delta\bar{J}_{y}$ (Eq. (28)). The variance of $J_{y}$ increases
quadratically with time at a rate given by
$\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}$ as given by the approximate
variance $\Delta\bar{P}$ (Eq. (26)).

Similarly, for short times, the approximate variance $\Delta\bar{X}$ (Eq.
(25)) starts out decreasing quadratically with time below unity roughly at a
rate $\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}$ in good agreement with
the hybrid variance (23), and the exact solution $\Delta\bar{J}_{x}$ (Eq.
(28)), as shown in Fig. 2(a). The decrease of the variance below unity shows
that the state is squeezed, as the error in the mean value is below the
classically expected, which in normalized units is unity. This confirms the
existence of correlations in the condensate as discussed in Sec. V.1, see
(20). However, at longer times $\Omega t\geq(|\alpha_{r}|\kappa)^{-1}$, the
approximate variance (Eq. (25)) decreases inversely with the square of time
roughly at a rate $(\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2})^{-1}$, and
goes to zero as $1/t^{2}$. In the same time regime, the hybrid variance
$\Delta\bar{X}$ and exact solution $\Delta\bar{J}_{x}$ reach their minimum
value and then increase towards unity. It is not surprising that the
approximate variance $\Delta\bar{X}$ breaks down. As already discussed in Sec.
V.1, the validity of approximate results hold for $\Omega
t<(\lvert\alpha_{r}\rvert\kappa)^{-1}$. It is in this neighborhood that the
exact variance $\Delta\bar{J}_{x}$ reaches its minimum (best) squeezing value
and the approximate variances $\Delta\bar{X}$ (Eq. (25)) and $\Delta\bar{P}$
(Eq. (26)) break down.

Note that in the region of importance where there is a squeezing effect, where
the exact solution $\Delta\bar{J}_{x}$ is decreasing with time, the
approximate variance $\Delta\bar{X}$ (Eq. (25)) does well predicting its
behavior. It starts failing in the neighborhood of the minimum squeezing as
predicted by the exact solution $\Delta\bar{J}_{x}$ (Eq. (28)), where the
gains made in error reduction are beginning to be lost. Hence one can always
predict the onset of losing squeezing gained from the measurement for times
not fulfilling the condition $gt<2(\sqrt{N}\lvert\alpha_{r}\rvert)^{-1}$.
Similarly, the hybrid variance (23) and approximate expression (26) of
$\Delta\bar{P}$ start to deviate from the exact solution $\Delta\bar{J}_{y}$
(Eq. (28)) in the same time neighbourhood, $gt\sim
2(\sqrt{N}\lvert\alpha_{r}\rvert)^{-1}$ as shown in Fig. 2(b). Surprisingly,
the hybrid variance (23) does well in predicting the exact solution (28) even
at long times, $gt>2(\sqrt{N}\lvert\alpha_{r}\rvert)^{-1}$.

Ordinarily one would expect the system to exhibit oscillations for atoms
tunneling in a double-well trap. The lack of oscillations in the solutions of
$J_{x}$ and $J_{y}$ is due to the initial state (16) used in their
calculation. For this state, the contribution of
$\hat{X}^{2}+\hat{P}^{2}\propto\hat{b}^{\dagger}_{e}\hat{b}_{e}$, which is
responsible for oscillation, is zero, see Appendix. B. Thus its exponential
operator does not affect the vacuum state and does not contribute to the
averages of spin operators.

###  V.6 Husimi _Q_ -distribution

Figure 3: The conditional Husimi _Q_ -function. The parameters for the plots
are $\alpha_{l}=\alpha_{r}=\sqrt{12}$, $N=1000$, $\Omega=\pi/4$, and
$g=0.1\Omega/N$. The time $\Omega t$ for each plot is as shown in the figure.

As shown in Sec.V.1, measurements induce correlations in the condensates. They
manifest as a reduction in the variance below the standard quantum limit for
the atomic spin variable coupled to light, as shown in Sec.V.5. Additionally,
correlations affect the width of the distribution representing the quantum
state. To visualize the effect of correlation among the atoms on the state
$\lvert\psi_{n_{c},n_{d}}\rangle$ (V.2) we plot the Husimi _Q_ -function
defined as

$Q_{n_{c},n_{d}}(\nu,t)=\frac{\lvert\langle\nu\lvert\psi_{n_{c},n_{d}}\rangle\rvert^{2}}{\pi},$
(29)

where $\lvert\nu\rangle=e^{-\tfrac{|\nu|^{2}}{2}}e^{\nu b^{\dagger}_{e}}\lvert
0\rangle$, is the coherent state of atoms within the Holstein-Primakoff
approximation Byrnes and Ilo-Okeke (2021) [see also (33)], and
$\nu=\lvert\nu\rvert e^{i\varphi}$. The _Q_ -function is numerically
calculated and is shown in Fig. 3. The distribution starts out at $\Omega t=0$
as a circle on _x_ – _y_ plane. This is the typical behavior of a coherent
state with a normalized width of unity. As time progress, the circle becomes
squeezed along the _x_ –axis. However, at time $\Omega t=200$, the width of
the distribution starts showing a subtle increase in the width along _x_
–axis. The squeezing along _x_ –axis is lost at $\Omega t=300$. To better
understand these effects, we explicitly calculate the conditional Husimi _Q_
-function

$\displaystyle Q_{n_{c},n_{d}}(\nu,t)$
$\displaystyle=\frac{e^{-|\alpha_{l}|^{2}-|\alpha_{r}|^{2}-|\nu|^{2}}}{\pi\,n_{c}!\,n_{d}!}\frac{\left(|\alpha_{r}|^{2}+|\alpha_{l}|^{2}\right)^{n_{c}+n_{d}}}{P(n_{c},n_{d})}$
$\displaystyle\times\left(\frac{n_{c}}{n_{c}+n_{d}}\right)^{n_{c}}\left(\frac{n_{d}}{n_{c}+n_{d}}\right)^{n_{d}}\frac{1}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}$
$\displaystyle\times
e^{-\frac{\Omega^{2}t^{2}\kappa^{2}(n_{c}+n_{d}-2a)^{2}}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}e^{-\frac{\sigma(x_{0}-\phi)^{2}}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times e^{-\frac{4\sigma\Omega
t\kappa\lvert\nu\rvert(x_{0}-\phi)\cos\varphi}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}e^{-\frac{1+8\sigma\Omega^{2}t^{2}\kappa^{2}\cos^{2}\varphi}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}\lvert\nu\rvert^{2}}$
$\displaystyle\times e^{\frac{2\Omega
t\kappa(n_{c}+n_{d}-2a)|\nu\rvert\sin\varphi}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}.$
(30)

Equation (V.6) shows that there are contributions to the _Q_ function due to
the meter (laser light) and the detection in addition to the effect that we
intend to observe that is proportional to $\nu^{2}$ in the exponent. The
exponents with term $(x_{0}-\phi)$ contain contributions from the relative
phase of the measurement arising from the relative phase
$\phi=\phi_{l}-\phi_{r}$ of the light beams in the arms of the interferometer,
and the measured (detected) phase $x_{0}$ (see (63)) and $\sigma$ is defined
in (68). The exponents with the term
$n_{c}+n_{d}-2a=(n_{c}+n_{d})(\rvert\alpha_{r}\rvert^{2}-\rvert\alpha_{l}\rvert^{2})/(\rvert\alpha_{r}\rvert^{2}+\rvert\alpha_{l}\rvert^{2})$
represent the contributions from the observed total photon number
$n_{c}+n_{d}$. These contributions can be made to vanish hence limiting the
effect of measurement on the state of the atoms Ilo-Okeke et al. (2021). For
instance, at balanced detection, the amplitude and phase of light beams in the
two arms of the interferometer are same, $\phi_{l}=\phi_{r}$ and
$\lvert\alpha_{r}\rvert=\lvert\alpha_{l}\rvert$. Consequently, the relative
phase $\phi$ and the fractional total population $n_{c}+n_{d}-2a$ identically
vanish, $\phi=0=n_{c}+n_{d}-2a$. Additionally, where the measured relative
photon number difference is small compared to the total photon number,
($\rvert n_{d}-n_{c}\lvert\ll n_{c}+n_{d}$), $x_{0}\approx 0$. With these
conditions and upon substituting (V.1) in (V.6), it simplifies to

$\displaystyle
Q_{n_{c},n_{d}}(\nu,t)=\frac{1}{\pi}\frac{\sqrt{1+8\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\,e^{-\frac{1+8\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}\cos^{2}\varphi}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\nu^{2}},$
(31)

where the effect of measurement on _Q_ function is contained in its width
$\Delta\nu$.

It is then evident that the _Q_ function has a peak at $\nu=0$ with width
$\Delta\nu$ given by

$\Delta\nu^{2}=\frac{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+8\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}\cos^{2}\varphi}.$
(32)

As expected, the measurement has no effect on the width of the _Q_ function
for $t=0$. In this case, the width is simply unity, the same to that of the
coherent state $\lvert\psi_{0}\rangle$ (Eq. (16)), which is equal in all
directions. This is shown in Fig. 3 at $\Omega t=0$. The limit $\Delta\nu=1$
is the so-called standard quantum limit. For $t>0$, the phase-angle $\varphi$
controls the direction on _x_ – _y_ plane. For instance
$\varphi=\pi/2,\,3\pi/2$ corresponds to the positive and negative _y_ –axis
respectively. Similarly, $\varphi=0,\,\pi$ corresponds to the positive and
negative _x_ –axis. It then becomes clear that the width along the _y_ –axis
(both negative and positive) increases quadratically with time, roughly at the
rate $\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}$.

Similarly, the width along the _x_ –axis (both positive and negative)
decreases quadratically with time from unity for small times $0<\Omega t\ll 1$
at a rate $\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}$. These cause the
width shrink along the _x_ –axis while extending along the _y_ –axis. Hence
the quantum state $\lvert\psi_{n_{c},n_{d}}\rangle$ is said to be squeezed
along _x_ –axis. This is shown in Fig. 3 at $\Omega=50,100,150$. Additionally,
(32) predicts finite squeezing as the width along _x_ –axis tends to
$\Delta\nu^{2}=1/2$ as $\Omega t$ becomes very large, which is the case as
seen from Fig. 3 at $\Omega t=150$. A careful study of the plot at $\Omega
t=200$ shows a subtle increase in the width of the distribution along the _x_
–axis. These imply that the gains made are gradually being eroded with an
increase in time. This is seen clearly in Fig. 3 at $\Omega t=300$. That (32)
breaks down—predicting finite squeezing with infinite time $\Omega
t\gg(|\alpha_{r}|\kappa)^{-1}$—is not surprising as the expression is only
valid for $gt<2(\sqrt{N}\lvert\alpha_{r}\rvert)^{-1}$, which is the region
where the width of the distribution is squeezed as discussed in Secs. V.1 and
V.5.

##  VI Imperfect initial spin polarization

In many experimental situations, there may be a finite but small atomic
population in the excited state of the double-well trap. In these situations,
the initial state is not perfectly polarized. Such a situation could lead to
results qualitatively different from perfectly polarized states, as discussed
in Sec. V. In this section, using a coherent state as the initial state for
imperfect polarization, we establish the mechanism for the existence of
quantum squeezing correlations at short interaction time and their loss at
long interaction time. We then calculate the averages of atomic spin
variables.

###  VI.1 Emergence And Disappearance Of Squeezing Correlations

Consider the case where there is a finite but small fractional population of
atoms in the excited state of the double-well trap, $0<|\alpha|^{2}\ll 1$. In
this limit, the initial state (4) reduces to Byrnes and Ilo-Okeke (2021)

$\lvert\psi_{0}\rangle=e^{-N\frac{\lvert\alpha\rvert^{2}}{2}}\sum_{m=0}^{N}\frac{(\sqrt{N}\alpha)^{m}}{\sqrt{m!}}\lvert
m\rangle,$ (33)

for the undepleted pump approximation.

The passage of light through the condensate and subsequent detection of
photons as shown in Fig. 1 gives the unnormalized state of the atoms (IV)
where the initial state $\lvert\psi_{0}\rangle$ is (33). The action of the
exponential operator $e^{i\Omega
t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}+\sqrt{2}\kappa(n_{c}+n_{d}-2n_{1})\hat{X}\right]}$
on the state $\lvert\psi_{0}\rangle$ is to first impart a phase proportional
to the tunneling frequency $\Omega$, then displace it by an amount $-i\Omega
t\kappa(n_{c}+n_{d}-2n_{1})$, see (51),

$\displaystyle\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle\propto\sum_{n_{l}=0}^{\infty}e^{in_{l}\phi}I(n_{c},n_{d},n_{l})e^{-i\Omega
t\kappa(n_{c}+n_{d}-2n_{l})\sqrt{N}\alpha}$ $\displaystyle\times
e^{-\frac{N\lvert\alpha\rvert^{2}+\Omega^{2}t^{2}\kappa^{2}(n_{c}+n_{d}-2n_{l})^{2}}{2}}$
$\displaystyle\times\sum_{m=0}^{N}\frac{\left(-i\Omega
t\kappa(n_{c}+n_{d}-2n_{l})+\sqrt{N}\alpha e^{i\Omega
t}\right)^{m}}{\sqrt{m!}}\lvert m\rangle.$ (34)

Additionally, the exponential term in the second line proportional to
$N\lvert\alpha\rvert^{2}$ broadens the width of the initial state. A spin-
coherent-like state emerges if $N\rvert\alpha\lvert^{2}$ is dominant while a
correlated state emerges where
$\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}t^{2}$ is the dominant term, as
we show below.

The unnormalized state of the atoms obtained by summing over $n_{l}$ under the
conditions that the measurement be balanced $\alpha_{l}=\alpha_{r}$ and the
relative photon number difference in a measurement be small in comparison with
the total number of photons, $\rvert n_{d}-n_{c}\lvert\ll n_{d}+n_{d}$ is

$\displaystyle\lvert\tilde{\psi}_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\,\sqrt{n_{d}!}}\left(\frac{n_{c}}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}\left(\frac{n_{d}}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}$
$\displaystyle\times\frac{(\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2})^{\frac{n_{c}+n_{d}}{2}}}{\sqrt{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}}e^{-\frac{N\lvert\alpha\rvert^{2}}{2}}e^{-\frac{2\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}N\alpha^{2}}{(1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2})}}$
(35)
$\displaystyle\times\sum_{n=0}^{N}\sum_{k=0}^{n/2}\frac{\sqrt{n!}}{k!(n-2k)!}\left(\frac{-2\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\right)^{k}$
$\displaystyle\times\left(\sqrt{N}\alpha\left[e^{i\Omega
t}-\frac{4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\right]\right)^{n-2k}\lvert
n\rangle,$

where we have used the completeness relation $\sum_{n}\lvert n\rangle\langle
n\rvert=\hat{\mathbbm{1}}$ in evaluating the sum. Equation (VI.1) presents two
interesting limits for the state $\lvert\psi_{n_{c},n_{d}}\rangle$. In one of
the limits where $\kappa=0$ (or $gt=0$), there is no interaction between the
atoms and light, the unnormalized state (VI.1) is simply the product of the
collapsed state $\lvert n_{c},n_{d}\rangle$ and the initial state (33).
Similarly, for $\sqrt{N}\alpha$ being small but large in comparison with atom-
light coupling strength $\lvert\sigma\rvert\Omega t\kappa$, then the terms
with exponent $n-2k$ is the dominant term ($|e^{i\Omega t}|\sim 1$). Hence,
the terms with exponent $k$ is treated as a perturbation, and $k$ takes on
values $0,1,\cdots,n/2$ for $n$ even and $0,1,\cdots,(n-1)/2$ for $n$ odd.
Nevertheless, whether $n$ is even or odd only a few terms $k=0$ and possibly
$k=1$ are sufficient to describe the amplitude of the state $\lvert n\rangle$
in this limit. Hence no correlations exist among the atoms, and the physics of
the system can be described classically.

In the other limit where $\sqrt{N}\alpha$ is small compared to the atom light
coupling strength $\lvert\alpha_{r}\rvert\Omega t\kappa$, then the term with
the exponent $n-2k$ is treated as a perturbation. In this limit, there exist
correlations among the atoms. As a special case where $\alpha=0$, $k$ must be
$n/2$ and one recovers (V.1), see also (20). For
$\sqrt{N}\lvert\alpha\rvert\ll 1$ and
$gt<2(\lvert\alpha_{r}\rvert\sqrt{N})^{-1}$, and considering leading and the
next term in $k$ ($k=n/2,n/2-1$), the normalized state conditioned on the
detection of photons becomes

$\begin{split}\lvert\psi_{n_{c},n_{d}}\rangle&\approx\frac{\left(1+8\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}\right)^{1/4}}{\sqrt{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}}\frac{1}{\sqrt{P_{\epsilon}}}\sum_{n=0}^{N_{\mathrm{max}}}\Bigg{[}\frac{\sqrt{(2n)!}}{n!}\\\
&\times\left(\frac{-2\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\right)^{n}+N\alpha^{2}\\\
&\times\frac{\sqrt{(2n)!}}{2(n-1)!}\left(\frac{-2\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\right)^{n-1}\\\
&\times\left(e^{i\Omega
t}-\frac{4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\right)^{2}\Bigg{]}\lvert
2n\rangle,\end{split}$ (36)

where $N_{\mathrm{max}}=N/2$ for $N$ even, $N_{\mathrm{max}}=(N-1)/2$ for $N$
odd and the corrections to the probability is

$P_{\epsilon}\approx
1-N\rvert\alpha\lvert^{2}\frac{4\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}{1+8\lvert\alpha_{r}\rvert^{2}\Omega^{2}t^{2}\kappa^{2}}\cos\left(2\Omega
t+2\varepsilon\right),$ (37)

and $\varepsilon$ is the argument of $\alpha$. Again, we see that the case
$\alpha=0$ (20) is a special case. It is then evident that the state (36) is a
superposition of even number states thereby affirming the existence of
correlations in the condensate. The state (36) is a squeezed state of an
atomic oscillator.

At small atom-light interaction strengths, the effect that is observed depends
on which term has the greater value—the initial population
$N\lvert\alpha\rvert^{2}$ or the atom-light interaction strength $gt$. For
instance, starting with the conditions that the atom-light interaction
strength is smaller than the initial population $N\lvert\alpha\rvert^{2}$,
there are no correlations. However, because the parameter $gt$ depends on
time, these conditions are dynamic. Eventually, correlations emerge in the
regime where atom-light interaction strengths become greater than the initial
population of atoms. Hence, having very small population of atoms in the
excited state $N\lvert\alpha\rvert^{2}\ll 1$ ensures that correlations always
exist at small interaction times.

There is a loss of squeezing correlations at long interaction times, as
already noted in Sec. V.1. This is because at long atom interaction times, the
Fourier term (IV) presents other peaks besides $x_{0}$, see for instance
$x_{0}^{\prime}$ in (C). These other peaks that were exponentially small at
short interaction times and thus negligible in the evaluation of atomic wave
function are now comparable to $x_{0}$. Physically, this implies that $x_{0}$
is no longer the only dominant detectable amplitude—other amplitudes are now
possible. The interference (superposition) of these amplitudes scrambles the
squeezing correlations which leads to their loss at a long interaction times.
This effect was also observed in Fig. 2(b) of Ref. Ilo-Okeke et al. (2021).
However, the superposition of amplitudes gives rise to other forms of non-
Gaussian correlations, such as the Schrödinger cat state Ilo-Okeke et al.
(2021) that emerge at long interaction times. Hence, the loss of squeezing is
a precursor to the emergence of other non-Gaussian correlations.

We thus see that for systems which are not perfectly polarized, when the
population executing the oscillation $N\lvert\alpha\rvert^{2}$ exceeds the
atom-light coupling strength $gt$, no correlations will be observed. The
photon statistics dominates its physics and the system behaves classically.
Correlations begin to emerge where the population $\sqrt{N}\alpha$ is
comparable to interaction strength $gt$. Consequently, correlations dominate
the physics of the system only in the regime where
$\sqrt{N}\lvert\alpha\rvert\ll 1$. Since the coherent state is a state of an
oscillator, it then implies that quantum effects such as correlations may be
observed in oscillators using QND measurement, if the number of oscillators
are small, $N\lvert\alpha\rvert^{2}\ll 1$.

###  VI.2 The Conditional Expectation Values

Figure 4: The normalized conditional mean values of $J_{x}$, and $J_{y}$. In
(a), the exact mean (solid line) is calculated using (27) while the hybrid
mean (dashed line) is calculated using (33) in (22). (b) shows the behavior of
the exact mean value $\bar{J}_{y}$ for the entire evolution. In (c), the exact
mean (solid line) is calculated using (27) while the hybrid mean (dashed line)
is calculated using (33) in (22). The parameters for plots are
$\alpha_{l}=\alpha_{r}=\sqrt{20}$, $N=200$, $\Omega=\pi/4$ and
$g=0.1\Omega/N$, $\beta=\sqrt{9999}\times 10^{-2}$ and
$\lvert\alpha\rvert=0.01$.

Using the initial state (33), we numerically calculate the averages for
$\hat{X}$ and $\hat{P}$ and compare the results with the exact solution. It is
seen from Fig. 4 that the exact mean values of $J_{x}$ and $J_{y}$ oscillate
about the hybrid mean, the solid line at $\bar{J}_{x}=0$ and $\bar{J}_{y}=0$,
respectively, when $\alpha=0$. The measurement provides a modulation envelope
while the tunneling is responsible for the oscillations. As shown in Fig.
4(c), the exact solution tends to zero in the region where modulation by
measurement is strong. In addition, the exact solution oscillates about zero
and only approaches it linearly, as shown in Figs. 4(a) and (b).

Some of this behavior can be deduced from the hybrid mean values. For
instance, the measurement modulates the oscillations of the exact solution
$\bar{J}_{x}$ and $\bar{J}_{y}$. At short interaction times, $\Omega
t<(\lvert\alpha_{r}\rvert\kappa)^{-1}$, we see from Fig. 4(c) that the
oscillations in the hybrid calculation are in phase with the exact solution.
Additionally, the amplitude of oscillations of the hybrid means is not
symmetric about zero. This is because the modulation by the measurement causes
the hybrid mean value to approach zero for the amplitude of oscillation
greater than zero, while the amplitude below zero is unaffected. The
approximate mean obtained in the short time limit, $\Omega
t<(\lvert\alpha_{r}\rvert\kappa)^{-1}$, while treating the oscillations as
perturbation $\sqrt{N}\lvert\alpha\rvert\ll 1$, equally shows this behavior

$\displaystyle\bar{X}_{\alpha}\approx$
$\displaystyle\frac{\lvert\alpha\rvert}{(1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}t^{2})^{2}}\Big{[}-8\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}t^{2}\cos\phi_{\alpha}$
$\displaystyle+2(1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}t^{2})\cos(\Omega
t+\phi_{\alpha})\Big{]},$ (38)

where $\phi_{\alpha}$ is the phase of the fractional population in the excited
state. It is zero in all cases considered in this work. For $\cos(\Omega
t+\phi_{\alpha})$ positive, notice that there are cancellations between the
first and second term that leads to the reduction in the amplitude of the
hybrid mean. However, when $\cos(\Omega t+\phi_{\alpha})$ is negative, the
terms add. This effect brings about the one-sided modulation effect exhibited
by the expectation value of the hybrid mean in Fig. 4(c).

On the other hand, we see that in Fig. 4(a) the amplitude of the hybrid mean
value increases with every oscillation while being modulated by the
measurement, with modulation strength proportional to
$\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}t^{2}$. This effect is opposite
to that exhibited by the amplitudes of the exact mean value. The oscillations
decrease very slowly, as shown in Fig. 4(b). The poor prediction of the
behavior of the exact solution by hybrid mean value is on display in the
approximate mean value, which under the same conditions as (VI.2), gives

$\displaystyle\bar{P}_{\alpha}$
$\displaystyle\approx\frac{\lvert\alpha\rvert\left(1+8\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}t^{2}\right)}{\left(1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}t^{2}\right)^{2}}\Big{[}-8\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}t^{2}\sin\phi_{\alpha}$
$\displaystyle+2(1+4\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}t^{2})\sin(\Omega
t+\phi_{\alpha})\Big{]}.$ (39)

Since $\phi_{\alpha}=0$, it is immediately evident that
$\langle\hat{P}\rangle$ is modulated by the measurement and increases
quadratically with time in the short time regime. Additionally, the
oscillations in the hybrid mean value is $\pi$ out of phase with the exact
solution as shown in Fig. 4(a).

The hybrid approximation does not capture all the features of the exact mean
values of atom operators. It is notably worse for predictions of the hybrid
mean value about $\bar{J}_{y}$. The failure of the hybrid approximation to
describe the exact mean values in the presence of oscillations stems from
approximating the operators of the ground state by a c-number. Such an
approximation is good when the ground state of the oscillator contributes no
more than random fluctuations. In this case, the excited state alone
sufficiently describes the physics of the system. However, for a small but
finite population of atoms in the excited state
$0<\sqrt{N}\lvert\alpha\rvert\ll 1$, the ground state contribution can no
longer be approximated by a c-number. Consequently, the excited state alone is
insufficient in describing the mean values of the system. Including
corrections to hybrid approximation would not improve the results since the
next order correction is proportional to order three in terms of the
operators, $\propto b^{\dagger 2}_{e}b_{e},\propto b^{\dagger}_{e}b_{e}^{2}$
and vice versa, hence will give rise to terms of order
$N^{3/2}\lvert\alpha\rvert^{3}$ which is less than
$\sqrt{N}\lvert\alpha\rvert$ for $\sqrt{N}\lvert\alpha\rvert<1$.

Figure 5: The normalized conditional variance of $J_{x}$ and $J_{y}$. (a) The
exact solution $J_{x}$ (solid line) is calculated from (28) as described in
the main text. The dashed line is the hybrid variance calculated using (33) in
(23), while the dashed-dotted line is the approximate variance (25). (b) The
exact variance (solid line)is calculated in same way as in (a) and similarly
for the hybrid variance (dashed line). The approximate variance (dashed-dotted
line) is from (25) while the hybrid variance (dotted line) calculated using
(16) in (23). The straight line at $\Delta\bar{J}_{x}=1$ is a guide for the
eye. In (c), the exact variance (solid line) is calculated from (28) as
described in the main text. The hybrid variance (dashed line) is calculated
using (33) in (23), while the dashed-dotted is the approximate variance (26).
In (d), the exact variance of $J_{y}$ (solid line) is calculated in same way
as in (c) and similarly for the hybrid variance (dashed line). The dashed-
dotted line is approximate variance (26) while the hybrid variance (dotted
line) is calculated using (16) in (23). The parameters for plots are
$\alpha_{l}=\alpha_{r}=\sqrt{20}$, $N=200$, $\Omega=\pi/4$ and
$g=0.1\Omega/N$. For the first column, $\beta=\sqrt{9999}\times 10^{-2}$ and
$\lvert\alpha\rvert=0.01$, while for the second column $\beta=\sqrt{0.999}$
and $\alpha=\sqrt{0.001}$.

###  VI.3 Conditional Variances Of Atomic Operators

We now compare the normalized variance $\Delta\bar{J}_{y}$ and its hybrid
approximation in Fig. 5(c). In the short time regime, $\Omega
t<(\lvert\alpha_{r}\rvert\kappa)^{-1}$, there is a very good agreement between
the exact solution (solid line) and the hybrid variance (dashed line). For a
better understanding of the results, we turn to the approximate expression
(26) valid for short times and obtained by assuming that the oscillator
amplitude is small in comparison to unity, $\sqrt{N}\lvert\alpha\rvert\ll 1$.
At short times, there is no dependence of the variances on the oscillations,
and the approximate variance (26) predicts that the exact solution (28) and
hybrid variance (23) increase quadratically with time. This is in remarkable
agreement with the results of the hybrid approximation and the exact solution,
as shown in Fig 5(c). For long times, $\Omega
t>(\lvert\alpha_{r}\rvert\kappa)^{-1}$, however, the hybrid variance and the
approximate variance are qualitatively close but do not agree precisely with
the exact variance. In particular, there are oscillations present in the
hybrid variance that are not present in either approximate variance or the
exact solution (28). In the regime where the amplitude of oscillation is
large, $\sqrt{N}\lvert\alpha\rvert<1$, as shown in Fig. 5(d), there is still
agreement between the hybrid variance and the exact solution in the short time
regime, $\Omega t<(\lvert\alpha_{r}\rvert\kappa)^{-1}$. But still, there are
no oscillations in the exact solution for a long time, which are very
prominent in the hybrid variance.

The exact variance $\Delta\bar{J}_{x}$ (solid line), and hybrid approximation
are compared in Fig. 5(a). There is very good agreement between the exact
solution (28) and hybrid variance (23) in the squeezing regime, $\Omega
t<(\lvert\alpha_{r}\rvert\kappa)^{-1}$. These results are understood by
turning to the approximate variance (25) obtained for short times,
$gt<2(\sqrt{N}\lvert\alpha_{r}\rvert)^{-1}$, while treating the oscillation
amplitude as a perturbation, $\sqrt{N}\lvert\alpha\rvert\ll 1$. There are no
oscillations in this regime. The approximate variance predicts that the
variance of $J_{x}$ decreases quadratically with time which is in remarkable
agreement with the exact solution and hybrid variance in the squeezing regime
$gt<2(\sqrt{N}\lvert\alpha_{r}\rvert)^{-1}$. It starts failing at the onset of
losing squeezing where $gt<2(\sqrt{N}\lvert\alpha_{r}\rvert)^{-1}$ is no
longer satisfied. Beyond this regime, squeezing is being lost, and there are
oscillations in the exact solution and the hybrid variance. However, the
oscillations in the hybrid variance occur at every other cycle of the exact
solution. Hence, the exact solution oscillates at approximately twice the
frequency of the hybrid variance. Notice that the amplitude of these
oscillations is small and hence is well below unity.

However, for an appreciable number of atoms in the excited state, the
oscillator amplitude will be appreciable though small,
$\sqrt{N}\lvert\alpha\rvert<1$. This leads to squeezing and anti-squeezing
effects occurring in the regime $\Omega t>(\lvert\alpha_{r}\rvert\kappa)^{-1}$
as shown in Fig. 5(b). The minima of the variances shown in Fig. 5(a) or Fig.
5(b) are points where there are no oscillations. For instance, as shown in
Fig. 5(b) the minima of oscillations in exact solution and hybrid variance
correspond to the values of the hybrid variance calculated using (16) in (23).
It suggests treating oscillations as a perturbation to the non-oscillatory
state $\alpha=0$. Hence, systems with large oscillations are strongly
perturbed. In general, we posit that the hybrid approximation does better in
predicting the variances of operators than it did in predicting their mean
values.

The success of the hybrid approximation in describing variance within the
short time regime is understood in the following way. The conditional variance
of the system is one-half, irrespective of the value of $\alpha$. In the
presence of atom-light interactions, the correction term happens to be the
contribution from the measurement. Higher order corrections consisting of the
product of
$\lvert\alpha_{r}\rvert^{2}\Omega^{2}\kappa^{2}t^{2}N\lvert\alpha\rvert^{2}$
are negligible. Hence the zeroth order terms from the oscillator and first-
order correction provided by the measurement are sufficient to describe the
variances of system operators, especially in the regime where there is
squeezing. In the long interaction time regime, terms in Fourier coefficient
$I(n_{c},n_{d},n_{l})$ (IV) which were exponentially small or not present in
the short term regime are no longer small and become comparable to the term
used in the short interaction time regime, see for instance (C). These terms
contributed to corrections that improved the overall behavior of the hybrid
approximation, especially for the spin variable coupled to light.

##  VII Composition of Variance in Squeezing Regime

From the foregoing analysis of the variances with or without oscillations, it
is apparent that the variance of any of the system’s operator $\hat{C}$ in the
squeezing regime, $gt<(\sqrt{N}\lvert\alpha_{r}\rvert)^{-1}$, can, in general,
be written as the sum of the variance of the operators $(\Delta C)_{0}^{2}$ in
the absence of measurement and the variance of the meter

$(\Delta C)^{2}=(\Delta C)_{0}^{2}+(\Delta\mathrm{meter})^{2}.$ (40)

This is exemplified in the approximate expressions given in (25) and (26). For
a collection of a large number $N$ of particles, the variance of system
operators will be near their classical limit, $(\Delta C)_{0}^{2}\sim N^{-1}$.
Squeezing is realized for the system operator that couples to the meter by a
suitable choice of the meter parameters such that there are cancellations
between the variance of that system operator $(\Delta C)_{0}^{2}$ and the
variance of the meter $(\Delta\mathrm{meter})^{2}$. Squeezing is lost in the
regime where the measurement strength is comparable to the inverse product of
the average photon number and the square root of the number of particles,
$gt\sim 2(\sqrt{N}\lvert\alpha_{r}\rvert)^{-1}$. Beyond the squeezing regime,
one can still observe squeezing even though squeezing is being lost as
discussed in Sec. V.2. For an initial state of the system prepared with a
finite but small population in the excited state, there will be oscillations
beyond the squeezing regime as discussed in Sec. VI.2. If the population of
atoms in the excited state is finite but small $N\alpha\ll 1$, it is still
possible to observe squeezing beyond the squeezing regime as the amplitude of
the oscillations will be small. However, for an excited state that is
appreciably populated $N\alpha<1$, there will be oscillations that may be
present in the squeezing regime. Because of the large amplitude of
oscillations in the regime where squeezing is being lost, there will be times
when the variance $(\Delta C)^{2}$ is greater than the initial variance
$(\Delta C)_{0}^{2}$, thus being anti-squeezed as discussed in Sec VI.1. To
observe squeezing in this regime may require filters or timing the measurement
to do away with the anti-squeezing.

##  VIII Experimental Implementation

We previously detailed possible implementations of our model in experiments in
Ref. Ilo-Okeke et al. (2021). Here, we briefly describe an experimental
realization of our model. A BEC in a radio-frequency double-well potential
Harte et al. (2018); Barker et al. (2020) realizes the atom oscillator
described in Fig. 1. Squeezing the quantum state of the atom oscillator can
then be done by working in the frequency domain. Because atoms in the minima
of the double well experience different trapping frequencies, they modulate an
off-resonant light field Jammi et al. (2018) passing through them at different
frequencies, thus realizing single-site addressing. A balanced polarimeter
detects the signal at each frequency Jammi et al. (2018). We did not consider
effects due to photodetection efficiency that limits the efficacy of resolving
the number of photons collected by the detectors Barnett et al. (1998); Lee et
al. (2004). It affects the quantum state of photons by decreasing the linear
superposition of the average photon numbers
$(\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2})$ by an amount
proportional to detection efficiency $\eta_{E}$,
$(\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2})\rightarrow\eta_{E}(\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2})$.
By doing so, it masks the number of photons at the readout so that the actual
values $n_{c},\,n_{d}$ are never known, thereby introducing randomness in the
readout. As a result, each QND measurement readout is random and thus is
affected by the single-particle resolution of the detectors. Here, we do not
need to know the actual values of the number of photons at the readout, and
our results retain their form in the presence of inefficient photodetection.
As such, squeezing would still be observed in the presence of imperfect
photodetection. However, decoherence from the spontaneous emission of the
atoms would modify the results of this work Ilo-Okeke et al. (2021).

##  IX Summary and conclusions

We analyzed the squeezing effect in a double-well potential using the hybrid
approximation, where the spin variables of the atoms are treated within the
Holstein-Primakoff approximation, but the photonic degrees of freedom are
calculated exactly. We considered an initial state where all the atoms are in
the ground state of the double-well trap. The main result of this paper is
(20), which shows that correlations develop because of the measurement. The
correlations manifest in the state of the atoms as a superposition of even
number states. We calculated simple expressions for the variances of atom
operators, (25) and (26). We showed that in the squeezing regime, the variance
is the sum of the variances of the system and the meter (40). The hybrid
approximation agrees very well with the exact solution obtained by solving the
Schrödinger equation in the short interaction time regime. We found the short
interaction time regime to be where the atom-light interaction strength $gt$
is comparable to the inverse of the product of the average photon number
$\lvert\alpha_{r}\rvert$ and the square-root of the total atom number $N$ in
the trap, $gt\sim 2(\lvert\alpha_{r}\rvert\sqrt{N})^{-1}$. It marks the onset
of losing squeezing, and the approximate results failed beyond the short
interaction time regime.

Additionally, we calculated the Husimi _Q_ -distribution and highlighted
various contributions from the light and its detection to the _Q_
-distribution. We showed that these contributions vanished for a balanced
detection and a readout performed at zero photon number difference. The Husimi
_Q_ -distribution showed that the quantum state of the atom condensates is
squeezed. We obtained a simple expression (32) that explains the variations in
the width of the distribution in phase space as the BEC state goes from a
coherent state to being squeezed.

We further checked the validity of the hybrid approximation in the presence of
a small but finite population of atoms in the excited state of the double-well
trap. We showed that correlations emerge depending on the atom-light
interaction strength being greater than the population of atoms in the excited
state (VI.1). The implication is that quantum features such as correlations
are not observed in oscillators using QND measurement unless the number of
oscillators is small. We calculated the averages of the spin variables. The
approximate variances agreed very well with the exact solution in the
squeezing regime, $gt<2(\lvert\alpha_{r}\rvert\sqrt{N})^{-1}$. However, the
hybrid mean values performed poorly in predicting the exact solution. The
hybrid mean value for the spin variable coupled to light agreed well with the
modulation of the exact solution. For the spin variable not coupled to light,
the hybrid mean was $\pi/2$ out of phase with the exact solution.
Additionally, the hybrid mean predicted increases in the modulation with time,
while the exact solution decreased linearly over time.

The emergence of superposition of amplitudes in the quantum state of the
condensate, beyond the short time regime
$gt>2(\lvert\alpha_{r}\rvert\sqrt{N})^{-1}$, led to the loss of squeezing
correlations, and heralds the emergence of non-Gaussian correlations Pezzè et
al. (2018); Ilo-Okeke et al. (2021). Additionally, it provides corrections to
the variance of the spin variable coupled to light. Hence, the hybrid variance
well reproduces the results of the exact solution. However, the superposition
of amplitudes did not improve the variance of the spin variables not coupled
to light. Hence, the hybrid approximation failed to replicate the exact
variance for the spin variable not coupled to light. Nonetheless, the onset of
losing squeezing predicted as $gt\sim 2/(\sqrt{N}|\alpha_{r}|)$ in the hybrid
approximation worked very well for the exact solution.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(62071301); NYU-ECNU Institute of Physics at NYU Shanghai; the Joint Physics
Research Institute Challenge Grant; the Science and Technology Commission of
Shanghai Municipality (19XD1423000,22ZR1444600); the NYU Shanghai Boost Fund;
the China Foreign Experts Program (G2021013002L); the NYU Shanghai Major-
Grants Seed Fund; Tamkeen under the NYU Abu Dhabi Research Institute grant
CG008.

##  Appendix A Solving The Schrödinger Equation

To find the evolution of the the state vector $\Psi(t)$, we use the ansatz

$\lvert\Psi(t)\rangle=\sum_{k}\psi_{k}(t)\lvert
k;\alpha_{k,l}(t),\alpha_{k,r}(t)\rangle,$ (41)

where for each atom number state $\lvert k\rangle=\lvert k,N-k\rangle$ defined
in the $J_{x}$ basis, there is a coherent state
$\lvert\alpha_{k,s}\rangle=e^{\tfrac{-|\alpha_{k,s}|^{2}}{2}}e^{\alpha_{k,s}\hat{a}^{\dagger}_{s}}\lvert
0\rangle$, $s=l,r$, and $\lvert k;\alpha_{k,l},\alpha_{k,r}\rangle=\lvert
k\rangle\otimes\lvert\alpha_{k,r}\rangle\otimes\lvert\alpha_{k,l}\rangle$.
Substituting the ansatz, (41) into (7) produces two states of light
$\lvert\alpha_{k,s}\rangle$ and $\hat{a}^{\dagger}\lvert\alpha_{k;l,s}\rangle$
that are not necessarily orthogonal. Decomposing the state
$\hat{a}^{\dagger}\lvert\alpha_{k,s}\rangle$ into its orthogonal form
$\hat{a}^{\dagger}\lvert\alpha_{k,s}\rangle=\lvert\alpha_{\perp,k,s}\rangle+\alpha^{*}_{k,s}\lvert\alpha_{k,s}\rangle$
where $\langle\alpha_{\perp,k,s}\lvert\alpha_{k,s}\rangle=0$ gives the
equations for the evolution of $\alpha_{k,s}$ as

$\displaystyle\frac{d\alpha_{k,l}}{dt}$
$\displaystyle=-\frac{ig(2k-N)}{2}\alpha_{k,l},$
$\displaystyle\frac{d\alpha_{k,r}}{dt}$
$\displaystyle=\frac{ig(2k-N)}{2}\alpha_{k,r},$ (42)

whose solutions are

$\displaystyle\alpha_{k,l}(t)$
$\displaystyle=\alpha_{l}e^{-i(2k-N)\tfrac{gt}{2}},$
$\displaystyle\alpha_{k,r}(t)$
$\displaystyle=\alpha_{r}e^{i(2k-N)\tfrac{gt}{2}}.$ (43)

Clearly we see that the effect of the atom-light interaction is to impact a
phase on the photons. This phase carrying information can be extracted in an
interference measurement.

Eliminating the $\alpha_{k,s}$ in the evolution of $\rho_{kk^{\prime}}$ using
(A) and their solutions (A) García-Ripoll et al. (2005); Hussain et al. (2014)
give the following equation for the matrix elements $\rho_{kk^{\prime}}$

$\displaystyle\frac{d\psi_{k}}{dt}$
$\displaystyle=-i\Omega\frac{\sqrt{k(N-k+1)}}{2}\langle\alpha_{k}\lvert\alpha_{k-1}\rangle\psi_{k-1}$
$\displaystyle-i\Omega\frac{\sqrt{(k+1)(N-k)}}{2}\langle\alpha_{k}\lvert\alpha_{k+1}\rangle\psi_{k+1},$

where $\langle\alpha_{k}\lvert\alpha_{k\pm 1}\rangle$ is the overlap of light
coherent state that interacted with $k$ atoms and light coherent state that
interacted with $k\pm 1$, respectively,

$\displaystyle\langle\alpha_{k}\lvert\alpha_{k+1}\rangle$
$\displaystyle=e^{-(|\alpha_{l}|^{2}+|\alpha_{r}|^{2})}e^{|\alpha_{l}|^{2}e^{-igt}}e^{|\alpha_{r}|^{2}e^{igt}},$
$\displaystyle\langle\alpha_{k}\lvert\alpha_{k-1}\rangle$
$\displaystyle=e^{-(|\alpha_{l}|^{2}+|\alpha_{r}|^{2})}e^{|\alpha_{l}|^{2}e^{igt}}e^{|\alpha_{r}|^{2}e^{-igt}}.$
(45)

Equation (A) is a time dependent differential equation as determined by the
phases (A).

At the beamsplitter, photons that left each well are recombined and sorted
into bins $c$ and $d$ according to (12). The state after recombination at the
beamsplitter becomes

$\displaystyle\lvert\Psi\rangle_{\mathrm{BS}}(t)$
$\displaystyle=\sum_{k}\psi_{k}(t)e^{-\left(\frac{\lvert\alpha_{k,l}\rvert^{2}}{2}+\frac{\lvert\alpha_{k,r}\rvert^{2}}{2}\right)}e^{\frac{\alpha_{k,l}+i\alpha_{k,r}}{\sqrt{2}}\hat{a}^{\dagger}_{c}}$
$\displaystyle\times
e^{\frac{i\alpha_{k,l}+\alpha_{k,r}}{\sqrt{2}}\hat{a}^{\dagger}_{d}}\lvert
k,0,0\rangle$ (46)

The probability of counting $n_{c}$ and $n_{d}$ photons in the bins $c$ and
$d$, respectively, after recombination at the beamsplitter is obtained by the
expectation of the projection operator $\mathcal{P}=\lvert
n_{c},n_{d}\rangle\langle n_{c},n_{d}\rvert$ in the atoms subspace,

$\displaystyle P(n_{c},n_{d})$
$\displaystyle=\mathrm{Tr}[\lvert\Psi\rangle_{\mathrm{BS}}(t)\langle\Psi\rvert_{\mathrm{BS}}(t)\mathcal{P}],$
$\displaystyle=\frac{1}{n_{c}!}\frac{1}{n_{d}!}\sum_{k}\psi_{k}^{*}\psi_{k}(t)e^{-\left(\lvert\alpha_{k,l}\rvert^{2}+\lvert\alpha_{k,r}\rvert^{2}\right)}$
$\displaystyle\times\left(\frac{\lvert\alpha_{k,l}\rvert^{2}+i\alpha_{k,r}\alpha^{*}_{k,l}-i\alpha_{k,l}\alpha^{*}_{k,r}+\lvert\alpha_{k,r}\rvert^{2}}{2}\right)^{n_{c}}$
$\displaystyle\times\left(\frac{\lvert\alpha_{k,l}\rvert^{2}-i\alpha_{k,r}\alpha^{*}_{k,l}+i\alpha_{k,l}\alpha^{*}_{k,r}+\lvert\alpha_{k,r}\rvert^{2}}{2}\right)^{n_{d}}.$
(47)

Thus the state of the atoms given that $n_{c}$ and $n_{d}$ photons have been
detected becomes

$\displaystyle\lvert\Psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{1}{\sqrt{n_{c}!}}\frac{1}{\sqrt{n_{d}!}}\frac{1}{\sqrt{P(n_{c},n_{d})}}\sum_{k}\psi_{k}(t)\times$
$\displaystyle
e^{-\left(\frac{\lvert\alpha_{k,l}\rvert^{2}}{2}+\frac{\lvert\alpha_{k,r}\rvert^{2}}{2}\right)}\left(\frac{\alpha_{k,l}+i\alpha_{k,r}}{\sqrt{2}}\right)^{n_{c}}$
$\displaystyle\times\left(\frac{i\alpha_{k,l}+\alpha_{k,r}}{\sqrt{2}}\right)^{n_{d}}\lvert
k,N-k\rangle.$ (48)

##  Appendix B Disentangling Of Exponential Operators

Given the Hamiltonian (8), we would like to write the exponential of the
Hamiltonian as product of exponential of each term appearing in it. Such
simplification allows for easy analysis of the exponential operator on a
quantum state. We consider an operator of the form $U=e^{i\Omega
t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}-\sqrt{2}\kappa n\hat{X}\right]}$
appearing in Sec. IV and Sec. V. The operator may also be written in terms of
$\hat{b}_{e}$ and $\hat{b}^{\dagger}_{e}$

$U=e^{i\Omega t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}-\sqrt{2}\kappa
n\hat{X}\right]}=e^{i\Omega t\left[b^{\dagger}_{e}b_{e}-\kappa
n(\hat{b}^{\dagger}_{e}+\hat{b}_{e})\right]},$ (49)

while we have ignored a constant factor of one-half that introduced an overall
global phase in writing the second equality. The operators
$\hat{b}^{\dagger}_{e}\hat{b}_{e}$, $\hat{b}^{\dagger}_{e}$ and $\hat{b}_{e}$
satisfy the following commutation relation

$\displaystyle[\hat{b}^{\dagger}_{e}\hat{b}_{e},\hat{b}^{\dagger}_{e}]$
$\displaystyle=\hat{b}^{\dagger}_{e},$
$\displaystyle[\hat{b}^{\dagger}_{e}\hat{b}_{e},\hat{b}_{e}]$
$\displaystyle=-\hat{b}_{e},$ (50)
$\displaystyle[\hat{b}_{e},\hat{b}^{\dagger}_{e}]$ $\displaystyle=1.$

Hence, any equivalent representation of $U$ would consist exponential
operators made up of these operators and a constant. Thus, we seek to write
$U$ in normally ordered form as

$U=e^{ip(\theta)\hat{b}^{\dagger}_{e}}e^{iq(\theta)\hat{b}^{\dagger}_{e}\hat{b}_{e}}e^{ir(\theta)\hat{b}_{e}}e^{is(\theta)},$
(51)

where $\theta=\Omega t$. Differentiating (49) with respect to $\theta$ gives

$\frac{dU}{d\theta}=i\left(b^{\dagger}_{e}b_{e}-\kappa
n(\hat{b}^{\dagger}_{e}+\hat{b}_{e}\right)U.$ (52)

Similarly, differentiating (51) with respect to $\theta$ gives

$\displaystyle\frac{dU}{d\theta}$
$\displaystyle=i\Bigg{(}s^{\prime}+p^{\prime}\hat{b}^{\dagger}_{e}+q^{\prime}e^{ip\hat{b}^{\dagger}_{e}}b^{\dagger}_{e}\hat{b}_{e}e^{-ip\hat{b}^{\dagger}_{e}}$
$\displaystyle+r^{\prime}e^{ip\hat{b}^{\dagger}_{e}}e^{iq\hat{b}^{\dagger}_{e}\hat{b}_{e}}\hat{b}_{e}e^{-iq\hat{b}^{\dagger}_{e}\hat{b}_{e}}e^{-ip\hat{b}^{\dagger}_{e}}\Bigg{)}U$
(53)

where the primes on $p$, $q$, $r$ and $s$ means differentiation with respect
to $\theta$.

Applying Baker-Campbell-Hausdorff relation to (B), and equating coefficients
of operators in (52) and (B), one obtains the following coupled differential
equation for the functions,

$\displaystyle p^{\prime}-iq^{\prime}p$ $\displaystyle=-\kappa n,$
$\displaystyle r^{\prime}e^{-iq}$ $\displaystyle=-\kappa n,$ (54)
$\displaystyle q^{\prime}$ $\displaystyle=1,$ $\displaystyle
s^{\prime}-ir^{\prime}pe^{-iq}$ $\displaystyle=0.$

Solving the differential equations (B) with the initial condition
$p(\theta=0)=r(\theta=0)=q(\theta=0)=s(\theta=0)=0$ gives

$\displaystyle p$ $\displaystyle=i\kappa n\left(e^{i\theta}-1\right),$
$\displaystyle q$ $\displaystyle=\theta,$ (55) $\displaystyle r$
$\displaystyle=i\kappa n\left(e^{i\theta}-1\right),$ $\displaystyle s$
$\displaystyle=\kappa^{2}n^{2}\left(\frac{e^{i\theta}}{i}-\theta-\frac{1}{i}\right).$

Expanding the exponential $e^{i\theta}$ and keeping the leading order term
gives

$\displaystyle p$ $\displaystyle=-\Omega t\kappa n,$ $\displaystyle q$
$\displaystyle=\Omega t,$ (56) $\displaystyle r$ $\displaystyle=-\Omega
t\kappa n,$ $\displaystyle s$
$\displaystyle=-\frac{\Omega^{2}t^{2}\kappa^{2}n^{2}}{2i}.$

Using (B) in (51), we obtain equivalent form of $e^{i\Omega
t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}-\sqrt{2}\kappa n\hat{X}\right]}$ that
was used in Secs. V and VI.

##  Appendix C Evaluation Of Fourier Coefficient

The detection of photons is described by the projection operator
$\mathcal{P}=\lvert n_{c},n_{d}\rangle\langle n_{c},n_{d}\rvert$. The
collapsed state of light at the measurement for an nth state of the atom is
$\langle n\lvert\mathcal{P}\rvert\psi_{BS}(t)\rangle=\langle
n_{c},n_{d},n\lvert\psi_{BS}(t)\rangle\rvert n_{c},n_{d}\rangle$, where
$\lvert\psi_{BS}(t)\rangle$ is given in (IV). The probability amplitude
$\langle n_{c},n_{d},n\lvert\psi_{BS}(t)\rangle$ then becomes

$\displaystyle\langle
n_{c},n_{d},n\lvert\psi_{BS}\rangle=e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}\left(\frac{1}{\sqrt{2}}\right)^{n_{c}+n_{d}}$
$\displaystyle\times\sum_{n_{l},n_{r}}\alpha_{l}^{n_{l}}\alpha_{r}^{n_{r}}\delta_{n_{l}+n_{r}-n_{c},n_{d}}$
$\displaystyle\times\langle n\rvert e^{i\Omega
t\left[\frac{\hat{X}^{2}+\hat{P}^{2}}{2}-\sqrt{2}\kappa(n_{l}-n_{r})\hat{X}\right]}\lvert
0\rangle$
$\displaystyle\times\sum_{k=\mathrm{max}(0,n_{l}-n_{d})}^{\mathrm{min}(n_{c},n_{l})}\frac{\sqrt{n_{c}!}\sqrt{n_{d}!}\,i^{n_{l}+n_{c}-2k}}{k!(n_{l}-k)!(n_{c}-k)!(n_{d}-n_{l}+k)!}$
(57)

In the absence of atom-light interaction, the photon states after
recombination at the beamsplitter has a probability amplitude given as Ilo-
Okeke and Byrnes (2016)

$\displaystyle\langle n_{c},n_{d}\lvert\psi\rangle$
$\displaystyle=e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}\frac{1}{\sqrt{n_{c}!}}\frac{1}{\sqrt{n_{d}!}}\left(\frac{i\alpha_{l}+\alpha_{r}}{\sqrt{2}}\right)^{n_{c}}\times$
$\displaystyle\left(\frac{\alpha_{l}+i\alpha_{r}}{\sqrt{2}}\right)^{n_{d}}$
(58)

Similarly, (C) in the same limit ($g=0$) becomes

$\displaystyle\langle n_{c},n_{d}\lvert\psi_{\mathrm{BS}}\rangle$
$\displaystyle=e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}\left(\frac{\alpha_{r}}{\sqrt{2}}\right)^{n_{c}+n_{d}}\sqrt{n_{c}!}\sqrt{n_{d}!}$
$\displaystyle\sum_{n_{l}}\left(\frac{\alpha_{l}}{\alpha_{r}}\right)^{n_{l}}S(n_{c},n_{d},n_{l})$
(59)

where

$S(n_{l},n_{c},n_{d})=\sum_{\mathrm{k_{min}}}^{\mathrm{k_{max}}}\frac{i^{n_{l}+n_{c}-2k}}{k!(n_{l}-k)!(n_{c}-k)!(n_{d}-n_{l}+k)!},$

$\mathrm{k_{min}}=\mathrm{max}(0,n_{l}-n_{d})$, and
$\mathrm{k_{max}}=\mathrm{min}(n_{c},n_{l})$. Equations (C) and (C) though may
be functionally different, they represent the same probability amplitude and
hence are the same. Equating (C) and (C), the Fourier transform immediately
gives

$\displaystyle
S(n_{l},n_{c},n_{d})=\left(\frac{\lvert\alpha_{l}\rvert}{\lvert\alpha_{r}\rvert}\right)^{-n_{l}}\frac{1}{n_{c}!}\frac{1}{n_{d}!}\frac{1}{2\pi}\times$
$\displaystyle\int_{0}^{2\pi}e^{-in_{l}\phi}d\phi\left(1+i\frac{|\alpha_{l}|}{|\alpha_{r}|}e^{i\phi}\right)^{n_{c}}\left(i+\frac{|\alpha_{l}|}{|\alpha_{r}|}e^{i\phi}\right)^{n_{d}},$
(60)

where $\phi=\phi_{l}-\phi_{r}$ is the relative phase between $\alpha_{l}$ and
$\alpha_{r}$.

From (C) and (C) the sum over $k$ in Eq. (C) can be replaced and the
probability amplitude describing the detection of photons at detectors $c$ and
$d$ becomes

$\displaystyle\langle n_{c},n_{d},n\lvert\psi_{\mathrm{BS}}(t)\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\frac{\alpha_{r}}{\sqrt{2}}\right)^{n_{c}+n_{d}}\sum_{n_{l}}e^{in_{l}\phi}$
$\displaystyle I(n_{c},n_{d},n_{l})$ (61) $\displaystyle\times\langle n\lvert
e^{i\Omega
t\left(\frac{\hat{X}^{2}+\hat{P}^{2}}{2}+\sqrt{2}\kappa(n_{c}+n_{d}-2n_{l})\hat{X}\right)}\lvert
0\rangle,$

where the Fourier coefficient $I(n_{c},n_{d},n_{1})$ is defined in (IV). The
evaluation of Fourier coefficient (IV) gives Ilo-Okeke and Zozulya (2010);
Ilo-Okeke and Byrnes (2016)

$\displaystyle
I(n_{c},n_{d},n_{1})=\frac{1}{\sqrt{2\pi\sigma}}\left[\frac{2n_{c}}{n_{c}+n_{d}}(1+\eta^{2})\right]^{\frac{n_{c}}{2}}\times$
$\displaystyle\left[\frac{2n_{d}}{n_{c}+n_{d}}(1+\eta^{2})\right]^{\frac{n_{d}}{2}}\exp\left[-\frac{(n_{c}\phi^{\prime}_{c,0}+n_{d}\phi^{\prime}_{d,0}-n_{l})^{2}}{2\sigma}\right]\times$
$\displaystyle\left\\{e^{i(n_{c}\phi_{c,0}+n_{d}\phi_{d,0}-n_{l}x_{0})}+(-1)^{n_{d}}e^{-i(n_{c}\phi_{c,0}+n_{d}\phi_{d,0}+n_{l}x^{\prime}_{0})}\right\\},$
(62)

where $x^{\prime}_{0}=\pi-x_{0}$,
$\eta=\tfrac{\lvert\alpha_{l}\rvert}{\lvert\alpha_{r}\rvert}$

$\displaystyle x_{0}$
$\displaystyle=\arcsin\left(\frac{n_{d}-n_{c}}{n_{c}+n_{d}}\frac{1+\eta^{2}}{2\eta}\right),$
(63) $\displaystyle\phi_{d,0}$
$\displaystyle=\arctan\left(\frac{2(n_{c}+n_{d})+(n_{d}-n_{c})(1+\eta^{2})}{\sqrt{4\eta^{2}(n_{c}+n_{d})^{2}-(n_{d}-n_{c})^{2}(1+\eta^{2})^{2}}}\right),$
(64) $\displaystyle\phi_{c,0}$
$\displaystyle=\arctan\left(\frac{\sqrt{4\eta^{2}(n_{c}+n_{d})^{2}-(n_{d}-n_{c})^{2}(1+\eta^{2})^{2}}}{2(n_{c}+n_{d})-(n_{d}-n_{c})(1+\eta^{2})}\right),$
(65) $\displaystyle\phi^{\prime}_{d,0}$
$\displaystyle=\left(\frac{2\eta^{2}(n_{c}+n_{d})+(n_{d}-n_{c})(1+\eta^{2})}{4n_{d}(1+\eta^{2})}\right),$
(66) $\displaystyle\phi^{\prime}_{c,0}$
$\displaystyle=\left(\frac{2\eta^{2}(n_{c}+n_{d})-(n_{d}-n_{c})(1+\eta^{2})}{4n_{c}(1+\eta^{2})}\right).$
(67)

and

$\sigma=\frac{4\eta^{2}(n_{c}+n_{d})^{2}-(n_{d}-n_{c})^{2}(1+\eta^{2})^{2}}{8n_{c}n_{d}(1+\eta^{2})^{2}}(n_{c}+n_{d}).$
(68)

##  Appendix D Evaluating Photon Probability Amplitude

To understand the behavior of photon detection probability distribution, we
begin our analysis by evaluation of the probability amplitude (IV). Upon
substitution of $I(n_{c},n_{d},n_{l})$, and (51) in (IV) and summing over the
photon input states $n_{1}$ gives

$\displaystyle\langle n\rvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\frac{\alpha_{r}}{\sqrt{2}}\right)^{n_{c}+n_{d}}\frac{\left(i\Omega
t\kappa\right)^{n}}{\sqrt{n!}}$
$\displaystyle\times\left(\frac{2n_{c}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}\left(\frac{2n_{d}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}$
$\displaystyle\times\sqrt{\frac{1}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times\Bigg{\\{}e^{i[b-a(x_{0}-\phi)]}e^{-\frac{2i\sigma\Omega^{2}t^{2}\kappa^{2}(x_{0}-\phi)(n_{c}+n_{d}-2a)}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times
e^{-\frac{\Omega^{2}t^{2}\kappa^{2}(n_{c}+n_{d}-2a)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}e^{-\frac{\sigma(x_{0}-\phi)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}$
$\displaystyle\times\sum_{k=0}^{\frac{n}{2}}\frac{n!}{k!(n-2k)!}\left(\frac{2\sigma}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\right)^{k}$
$\displaystyle\times\left(n_{c}+n_{d}-2(a+q)\right)^{n-2k}+(-1)^{n_{c}}$
$\displaystyle\times
e^{-i[b+a(x^{\prime}_{0}-\phi)]}e^{-\frac{2i\sigma\Omega^{2}t^{2}\kappa^{2}(x^{\prime}_{0}-\phi)(n_{c}+n_{d}-2a)}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times
e^{-\frac{\Omega^{2}t^{2}\kappa^{2}(n_{c}+n_{d}-2a)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}e^{-\frac{\sigma(x^{\prime}_{0}-\phi)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}$
$\displaystyle\times\sum_{k=0}^{\frac{n}{2}}\frac{n!}{k!(n-2k)!}\left(\frac{2\sigma}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\right)^{k}$
$\displaystyle\times\left(n_{c}+n_{d}-2(a+q^{\prime})\right)^{n-2k}\Bigg{\\}}$
(69)

where $a=n_{c}\phi^{\prime}_{c,0}+n_{d}\phi^{\prime}_{d,0}$, and
$b=n_{c}\phi_{c,0}+n_{d}\phi_{d,0}$,
$q=\sigma(2\Omega^{2}t^{2}\kappa^{2}(n_{c}+n_{d}-2a)-i(x_{0}-\phi))/(1+4\sigma\Omega^{2}t^{2}\kappa^{2})$,
and $q^{\prime}$ is obtained by replacing $x_{0}$ in $q$ with $x^{\prime}_{0}$
(C). In analysis of the probability amplitude in Sec. V.1, the second term of
(D) is exponentially small for $n_{c}=n_{d}$, and $n_{c}+n_{d}-2a=0)$. Hence
the second term is neglected in the analysis of Sec. V.1. Notice that the
upper limit in summing over $n$ is fixed by the total number of atoms in the
double well trap.

In Ref. Ilo-Okeke et al. (2021), we decided that we will work within the
balanced detection configuration, $\alpha_{l}=\alpha_{r}$ and the most
probable outcome
$n_{c},\,n_{d}=(\lvert\alpha_{r}|^{2}+\lvert\alpha_{l}\rvert^{2})/2$. The
justification for these choices becomes obvious since
$n_{c}+n_{d}-2a=(n_{c}+n_{d})(\rvert\alpha_{r}\lvert^{2}-\rvert\alpha_{l}\lvert^{2})/(\rvert\alpha_{r}\lvert^{2}+\rvert\alpha_{l}\lvert^{2})=0$
in the balanced detection configuration. Also, in the regime $\lvert
n_{c}-n_{d}\rvert\ll n_{c}+n_{d}$, $x_{0}\approx 0$, and hence $q=0$. Thus
these conditions combine to limit the back-action of the measurement on
quantum state of the atoms. Hence at balanced detection, the probability
amplitude simplifies. The amplitude at $x^{\prime}_{0}$ is exponentially small
and is neglected. Provided that
$\frac{2\sigma\Omega^{2}t^{2}\kappa^{2}}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\ll
1$, the sum over $n$ converges rapidly. Taking these into account one obtain
the form of the probability density (V.1) written in the main text.

##  Appendix E Evaluating Expectations Of Atomic Operators

Here we evaluate various expectation value of atom operator that are used in
Sec. V.3 for finding the expectations of $\hat{X}$ and $\hat{P}$, and their
variances. We begin by evaluating the probability amplitude $\langle
n\lvert\psi_{n_{c},n_{d}}\rangle$ and expectations of the form $\langle
n\lvert A\rvert\psi_{n_{c},n_{d}}\rangle$, where
$A=b^{\dagger}_{e},\,b_{e},\,(b^{\dagger}_{e})^{2}$. Since the
$x^{\prime}_{0}$ term of the Fourier coefficient (C) is exponentially small in
the region of interest, it is neglected in the following calculations. Acting
the operator $b^{\dagger}_{e}$ on the state $\lvert\psi_{n_{c},n_{d}}\rangle$
(V.2) and finding its expectation in the atom subspace gives

$\displaystyle\langle n\rvert b^{\dagger}_{e}\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\alpha_{r}\right)^{n_{c}+n_{d}}\left(\frac{n_{c}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}$
$\displaystyle\times\left(\frac{n_{d}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}\frac{\left(i\Omega
t\kappa\right)^{n-1}}{(n-1)!}$
$\displaystyle\sqrt{\frac{n!}{P(n_{c},n_{d})}}\sqrt{\frac{1}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times\Bigg{\\{}e^{i[b-a(x_{0}-\phi)]}e^{-\frac{2i\sigma\Omega^{2}t^{2}\kappa^{2}(x_{0}-\phi)(n_{c}+n_{d}-2a)}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times
e^{-\frac{\Omega^{2}t^{2}\kappa^{2}(n_{c}+n_{d}-2a)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}e^{-\frac{\sigma(x_{0}-\phi)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}$
$\displaystyle\times\sum_{k=0}^{\frac{n-1}{2}}\frac{(n-1)!}{k!(n-1-2k)!}\left(\frac{2\sigma}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\right)^{k}$
$\displaystyle\times\left(n_{c}+n_{d}-2(a+q)\right)^{n-1-2k}\Bigg{\\}}$ (70)

where
$q=\sigma(2\Omega^{2}t^{2}\kappa^{2}(n_{c}+n_{d}-2a)-i(x_{0}-\phi))/(1+4\sigma\Omega^{2}t^{2}\kappa^{2})$.
For $n_{c},n_{d}=(\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2})/2$,
and $\lvert n_{c}-n_{d}\rvert\ll n_{c}+n_{d}$ the above simplifies to

$\displaystyle\langle n\rvert b^{\dagger}_{e}\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\alpha_{r}\right)^{n_{c}+n_{d}}\left(\frac{n_{c}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}$
$\displaystyle\times\left(\frac{n_{d}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}\sqrt{\frac{1}{P(n_{c},n_{d})}}$
(71)
$\displaystyle\times\sqrt{\frac{1}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}\left(\frac{2\sigma}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\right)^{\frac{n-1}{2}}$
$\displaystyle\times e^{ib}\left(i\Omega
t\kappa\right)^{n-1}\frac{\sqrt{n}!}{\left(\dfrac{n-1}{2}\right)!},\quad\mathrm{for}\,n\,\mathrm{odd},$
$\displaystyle\langle n\rvert b^{\dagger}_{e}\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=0,\quad\mathrm{for}\,n\,\mathrm{even}.$ (72)

Similarly, following same recipe for $b^{\dagger}_{e}$, the expectation value
of the destruction operator $b_{e}$ in the atom subspace becomes

$\displaystyle\langle n\rvert b_{e}\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\alpha_{r}\right)^{n_{c}+n_{d}}\left(\frac{n_{c}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}$
$\displaystyle\times\left(\frac{n_{d}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}\frac{\left(i\Omega
t\kappa\right)^{n+1}}{\sqrt{n!}}$
$\displaystyle\sqrt{\frac{1}{P(n_{c},n_{d})}}\sqrt{\frac{1}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times\Bigg{\\{}e^{i[b-a(x_{0}-\phi)]}e^{-\frac{2i\sigma\Omega^{2}t^{2}\kappa^{2}(x_{0}-\phi)(n_{c}+n_{d}-2a)}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times
e^{-\frac{\Omega^{2}t^{2}\kappa^{2}(n_{c}+n_{d}-2a)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}e^{-\frac{\sigma(x_{0}-\phi)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}$
$\displaystyle\times\sum_{k=0}^{\frac{n+1}{2}}\frac{(n+1)!}{k!(n+1-2k)!}\left(\frac{2\sigma}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\right)^{k}$
$\displaystyle\times\left(n_{c}+n_{d}-2(a+q)\right)^{n+1-2k}\Bigg{\\}}$ (73)

For $n_{c},n_{d}=(\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2})/2$,
and $\lvert n_{c}-n_{d}\rvert\ll n_{c}+n_{d}$ the above simplifies to

$\displaystyle\langle n\rvert b_{e}\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\alpha_{r}\right)^{n_{c}+n_{d}}\left(\frac{n_{c}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}$
$\displaystyle\times\left(\frac{n_{d}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}\sqrt{\frac{1}{P(n_{c},n_{d})}}$
(74)
$\displaystyle\times\sqrt{\frac{1}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}\left(\frac{2\sigma}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\right)^{\frac{n+1}{2}}$
$\displaystyle\times e^{ib}\frac{\left(i\Omega
t\kappa\right)^{n+1}}{\sqrt{n!}}\frac{(n+1)!}{\left(\dfrac{n+1}{2}\right)!},\quad\mathrm{for}\,n\,\mathrm{odd},$
$\displaystyle\langle n\rvert b_{e}\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=0,\quad\mathrm{for}\,n\,\mathrm{even}.$ (75)

Also, the products of operators that will be used in finding the variance may
be calculated in the same way. For instance, the expectation of operator
$(b^{\dagger}_{e})^{2}$ in the atom subspace is

$\displaystyle\langle
n\rvert(b^{\dagger}_{e})^{2}\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\alpha_{r}\right)^{n_{c}+n_{d}}\left(\frac{n_{c}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}$
$\displaystyle\times\left(\frac{n_{d}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}\frac{\left(i\Omega
t\kappa\right)^{n-2}}{(n-2)!}\sqrt{n!}$
$\displaystyle\sqrt{\frac{1}{P(n_{c},n_{d})}}\sqrt{\frac{1}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times\Bigg{\\{}e^{i[b-a(x_{0}-\phi)]}e^{-\frac{2i\sigma\Omega^{2}t^{2}\kappa^{2}(x_{0}-\phi)(n_{c}+n_{d}-2a)}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times
e^{-\frac{\Omega^{2}t^{2}\kappa^{2}(n_{c}+n_{d}-2a)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}e^{-\frac{\sigma(x_{0}-\phi)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}$
$\displaystyle\times\sum_{k=0}^{\frac{n-2}{2}}\frac{(n-2)!}{k!(n-2-2k)!}\left(\frac{2\sigma}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\right)^{k}$
$\displaystyle\times\left(n_{c}+n_{d}-2(a+q)\right)^{n-2-2k}\Bigg{\\}}$ (76)

For $n_{c},n_{d}=(\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2})/2$,
$\lvert n_{c}-n_{d}\rvert\ll n_{c}+n_{d}$ the above simplifies to

$\displaystyle\langle
n\rvert(b^{\dagger}_{e})^{2}\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\alpha_{r}\right)^{n_{c}+n_{d}}\left(\frac{n_{c}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}$
$\displaystyle\times\left(\frac{n_{d}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}\sqrt{\frac{1}{P(n_{c},n_{d})}}$
(77)
$\displaystyle\times\sqrt{\frac{1}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}\left(\frac{2\sigma}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\right)^{\frac{n-2}{2}}$
$\displaystyle\times e^{ib}\left(i\Omega
t\kappa\right)^{n-2}\frac{\sqrt{n!}}{\left(\dfrac{n-2}{2}\right)!},\quad\mathrm{for}\,n\,\mathrm{even},$
$\displaystyle\langle
n\rvert(b^{\dagger}_{e})^{2}\lvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=0,\quad\mathrm{for}\,n\,\mathrm{odd}.$ (78)

The remainder of operators $b^{2}_{e}$, $b^{\dagger}_{e}b_{e}$ may calculated
following same procedure. Note that the expectations of
$(b^{\dagger}_{e})^{2}$ and $b^{2}_{e}$ are equal
$\langle\psi_{n_{c},n_{d}}\lvert
b_{e}^{2}\rvert\psi_{n_{c},n_{d}}\rangle=\langle\psi_{n_{c},n_{d}}\lvert(b^{\dagger}_{e})^{2}\rvert\psi_{n_{c},n_{d}}\rangle$

Finally for completeness, we calculate the probability amplitude $\langle
n\rvert\psi_{n_{c},n_{d}}\rangle$, see also Sec. V.1. Projecting the state
$\lvert n\rangle$ having $n$ atoms on the state (V.2),
$\lvert\psi_{n_{c},n_{d}}\rangle$, given that $n_{c}$ and $n_{d}$ has been
measured gives

$\displaystyle\langle n\rvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\alpha_{r}\right)^{n_{c}+n_{d}}\left(\frac{n_{c}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}$
$\displaystyle\times\left(\frac{n_{d}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}\frac{\left(i\Omega
t\kappa\right)^{n}}{\sqrt{n!}}\sqrt{\frac{1}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times\Bigg{\\{}e^{i[b-a(x_{0}-\phi)]}e^{-\frac{2i\sigma\Omega^{2}t^{2}\kappa^{2}(x_{0}-\phi)(n_{c}+n_{d}-2a)}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}$
$\displaystyle\times
e^{-\frac{\Omega^{2}t^{2}\kappa^{2}(n_{c}+n_{d}-2a)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}e^{-\frac{\sigma(x_{0}-\phi)^{2}}{2(1+4\sigma\Omega^{2}t^{2}\kappa^{2})}}$
$\displaystyle\times\sum_{k=0}^{\frac{n}{2}}\frac{n!}{k!(n-2k)!}\left(\frac{2\sigma}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\right)^{k}$
$\displaystyle\times\left(n_{c}+n_{d}-2(a+q)\right)^{n-2k}$ (79)

For $n_{c},n_{d}=(\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2})/2$,
$\lvert n_{c}-n_{d}\rvert\ll n_{c}+n_{d}$ the above simplifies to

$\displaystyle\langle n\rvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=\frac{e^{-\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}}}{\sqrt{n_{c}!}\sqrt{n_{d}!}}\left(\alpha_{r}\right)^{n_{c}+n_{d}}\left(\frac{n_{c}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{c}}{2}}$
$\displaystyle\times\left(\frac{n_{d}(1+\eta^{2})}{n_{c}+n_{d}}\right)^{\frac{n_{d}}{2}}\sqrt{\frac{1}{P(n_{c},n_{d})}}$
(80)
$\displaystyle\times\sqrt{\frac{1}{1+4\sigma\Omega^{2}t^{2}\kappa^{2}}}\left(\frac{2\sigma}{1+4\sigma\Omega^{2}t{{}^{2}\kappa^{2}}}\right)^{\frac{n}{2}}$
$\displaystyle\times e^{ib}\left(i\Omega
t\kappa\right)^{n}\frac{\sqrt{n!}}{\left(\dfrac{n}{2}\right)!},\quad\mathrm{for}\,n\,\mathrm{even},$
$\displaystyle\langle n\rvert\psi_{n_{c},n_{d}}\rangle$
$\displaystyle=0,\quad\mathrm{for}\,n\,\mathrm{odd}.$ (81)

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