# Faraday imaging induced squeezing of a double-well Bose-Einstein condensate

Ebubechukwu O. Ilo-Okeke  New York University Shanghai, 1555 Century Ave,
Pudong, Shanghai 200122, China  Shinichi Sunami  Clarendon Laboratory,
University of Oxford, Oxford OX1 3PU, United Kingdom  Christopher J. Foot
Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, United Kingdom
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  National Institute of Informatics, 2-1-2 Hitotsubashi,
Chiyoda-ku, Tokyo 101-8430, Japan  Department of Physics, New York University,
New York, NY 10003, USA

###### Abstract

We examine how non-destructive measurements generate spin squeezing in an
atomic Bose-Einstein condensate confined in a double-well trap. The condensate
in each well is monitored using coherent light beams in a Mach-Zehnder
configuration that interacts with the atoms through a quantum nondemolition
Hamiltonian. We solve the dynamics of the light-atom system using an exact
wavefunction approach, in the presence of dephasing noise, which allows us to
examine arbitrary interaction times and a general initial state. We find that
monitoring the condensate at zero detection current and with identical
coherent light beams minimizes the backaction of the measurement on the atoms.
In the weak atom-light interaction regime, we find the mean spin direction is
relatively unaffected, while the variance of the spins is squeezed along the
axis coupled to the light. Additionally, squeezing persists in the presence of
tunneling and dephasing noise.

##  I Introduction

Squeezed states of quantum systems are a resource that have numerous
applications. They possess entanglement that can be used advantageously in
various quantum information applications Nielsen and Chuang (2000), quantum
metrology Giovannetti et al. (2011); Degen et al. (2017), precision
measurements Appel et al. (2009); D’Ariano et al. (2001), time keeping Jozsa
et al. (2000); Louchet-Chauvet et al. (2010); Kómár et al. (2014); Ilo-Okeke
et al. (2018), and quantum networks Elliot (2002); Hahn et al. (2019); Ilo-
Okeke et al. (2020); Nagele et al. (2020). Squeezed states Giovannetti et al.
(2011); Kitagawa and Ueda (1993); Ma et al. (2011) give phase sensitivities
that beat the standard quantum limit Wineland et al. (1992); Kitagawa and Ueda
(1993) set by the quantum noise of individual uncorrelated quantum particles
Wineland et al. (1994). Consequently, the use of entanglement and squeezing
for enhanced measurement and detection are becoming increasingly common 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); Degen et al. (2017), optical interferometry Dowling and Seshadreesan
(2015); Schnabel (2017), and gravitational wave detection Eberle et al.
(2010); Pitkin et al. (2011).

Realizing squeezed states in different systems is typically dependent on
generating quantum correlations or entanglement created by the nonlinear
interactions between the quantum particles. For instance, the Kerr effect Boyd
(2008); New (2011) is largely exploited in creating optical squeezed states
Slusher et al. (1985); Wu et al. (1986), and has been widely studied Scully
and Zubairy (1997); Loudon (2000); Gerry and Knight (2005) and applied in
various technologies Dowling and Seshadreesan (2015); Schnabel (2017); Dowling
(2008); Joana et al. (2016); Ishida et al. (2013). In atomic systems such as
Bose-Einstein condensates (BECs) Kitagawa and Ueda (1993); Choi and Bigelow
(2005); Estève et al. (2008); Böhi et al. (2009); Jing et al. (2019) and
trapped ions Leibfried et al. (2004); Ge et al. (2019), two-body spin
interactions Kitagawa and Ueda (1993) have been widely exploited for
generation of entanglement and squeezing.

Techniques such as quantum nondemolition (QND) measurement Takahashi et al.
(1999); Higbie et al. (2005); Kuzmich and Kennedy (2004); Meppelink et al.
(2010); Ilo-Okeke and Byrnes (2014, 2016) have also been used to realize
squeezing Appel et al. (2009); Schleier-Smith et al. (2010); Sewell et al.
(2012); Cox et al. (2016); Hosten et al. (2016) in atom systems. In this
approach, the phase shift acquired by the light pulse after passing through
the atomic samples is measured. The light probe has a large frequency detuning
from the atomic resonance transition to give very low photon scattering rates
making the measurement minimally-destructive. Another advantage of the
measurement technique is that the information about the atomic population is
contained in the phase of light making it available for collection and
readout. These features are utilized in minimally destructive measurement of
atom systems to produce entanglement between hyperfine levels of atoms Behbood
et al. (2014); Kong et al. (2020), and observe spin rotations of atoms placed
in an rf field below the standard quantum limit Kuzmich et al. (2000).

Recently, Vasilakis, Møller, Polzik, and co-workers used a stroboscopic back-
action-evading measurement Vasilakis et al. (2015); Møller et al. (2017) to
generate a squeezed state of an oscillator. The oscillator in this case
consisted of a polarized atomic ensemble. On applying a magnetic field
parallel to the polarization axis, the spin undergoes precession about the
applied field. A probe applied to the plane perpendicular to the magnetic
field axis couples to one of the spin components on that plane. By
manipulating the strength of atom-probe interactions, one can increase the
correlations between the atom and light. It is found that the times where the
oscillator strength is maximum, the noise of the measurement device is not
coupled to the oscillator Thorne et al. (1978); Braginsky et al. (1980). Hence
for measurements performed at these times, the normalized variance of the atom
operator appearing in the atom-light interaction Hamiltonian is squeezed.

Here we propose using the balanced detection scheme in a Mach-Zehnder
interferometer to generate a squeezed state of an oscillator as shown in Fig
1. This technique does not rely on performing measurements at the times where
the oscillator strength is maximum in order to generate squeezing. Rather, it
requires the optical coherent state of light in the two arms of the
interferometer to have the same amplitude and phase
$\lvert\alpha_{l}\rangle=\lvert\alpha_{r}\rangle$. For the case of zero
detection current ($n_{d}=n_{c}$), the measurement back-action is minimized,
and atom-light interactions modulate the BEC state. For sufficiently strong
atom-light coupling strength, squeezing appears.

To describe how the squeezing comes about, we consider an atomic BEC that is
prepared in the ground state of a double-well potential Milburn et al. (1997),
as shown in Fig 1. As such, atoms are completely delocalized between the two
wells. Working in a two-level approximation of the double-well trap, where the
eigenstates are symmetric and antisymmetric wavefunctions, we may form a
pseudospin. To produce squeezing, laser light is used to perform an
interferometric measurement on the atoms, as shown in Fig. 1, under the
conditions specified above. The measurement of the photons induces a
modification on the quantum state of the spins, while leaving its mean spin
direction unchanged. This is because the photon detection causes a random
evolution of atoms such that on the average the net motion of the mean spin
projection on the plane perpendicular to the mean spin direction is zero,
thereby leaving the mean spin direction unchanged. Additionally, the photon
detection constricts the random motion of the mean spin direction along the
spin axis that is coupled to light. The net effect is that the random motion
of the atoms is only able to shear the initial distribution in phase space,
with the resulting distribution having a reduced variance. The reduced
variance of the distribution caused by the detection of photons is the
squeezing, and manifests as a decrease in variance below $\sqrt{N}$ for the
spin component that is coupled to light.

We investigate and characterize the squeezed spin state generated in the
double-well trap using a QND measurement as described above. In the zero
tunneling limit, we find a simple analytic expression for the probability
density that confirms the spin state after photon detection is indeed
squeezed. We develop a more realistic model that includes both tunneling and
dephasing noise from the measurement and show that squeezing persists in the
presence of dephasing noise. One of the main features of our work is that we
use an exact wavefunction method to solve the dynamics of the quantum
nondemolition Hamiltonian. In many works relating to light generated
squeezing, the Holstein-Primakoff approximation is used for the atomic spins
Hald et al. (1999); Julsgaard et al. (2001); Vasilakis et al. (2015); Møller
et al. (2017). This is valid for highly spin polarized initial spins, and for
short evolution times, but at longer evolution times it loses validity Byrnes
et al. (2015). Our methods do not have this restriction and can be applied in
a more general setting. Additionally, experiments with stroboscopic
measurement Vasilakis et al. (2015); Sewell et al. (2012); Møller et al.
(2017) require exquisite engineering, precision and control in order to reduce
the noise of the pulsed laser light to below the required level, whereas the
continuous measurement as analysed here has less experimental parameters that
could introduce imperfection and noise.

The remainder of the paper is organized as follows. We start by developing the
theoretical framework for describing the tunneling of atoms in a double-well
potential and subsequent squeezing using the QND measurement, as well as
describing the framework for dephasing of the BEC in Sec. II. We develop an
exact wavefunction approach to explaining the origin of squeezing due to QND
measurement in Sec. III, and study the effect of tunneling on the squeezed
state in Sec. IV. Next in Sec. V, we make the model more realistic by
including the effects of dephasing on the squeezed state of atoms in the
double-well trap. We discuss how this scheme can be realized with existing
experimental technology, in Sec. VI. Finally, the summary and conclusions are
presented in Sec. VII.

Figure 1: A double-well trap containing atoms that are allowed to tunnel
between the wells. The atoms in each well are imaged using identical laser
light in the Mach-Zehnder interferometer configuration. The solid black line
is the double-well trap. The symmetric wavefunction is the ground state
$\lvert g\rangle$ wavefunction and the antisymmetric line is first excited
state $\lvert e\rangle$ belonging to the double-well trap, respectively.

##  II Double-well trap model and measurement of atoms

###  II.1 Tunneling in a double-well trap

Our model system is a many-body atomic Bose-Einstein condensate confined in a
double-well potential that can be made from magnetic Schumm et al. (2005);
Harte et al. (2018) or optical Shin et al. (2004) fields. This is illustrated
in Fig. 1. The many-body Hamiltonian governing the dynamics of the atoms in
double-well potential $V(r)$ is

$\displaystyle\hat{H}$ $\displaystyle=\int
d\mathbf{r}\hat{\Psi}^{\dagger}(\mathbf{r},t)\left[\frac{P^{2}}{2M}+V(\mathbf{r})\right]\hat{\Psi}(\mathbf{r},t)$
$\displaystyle+\frac{U_{0}}{2}\int
d\mathbf{r}\hat{\Psi}^{\dagger}(\mathbf{r},t)\left(\hat{\Psi}(\mathbf{r},t)\hat{\Psi}^{\dagger}(\mathbf{r},t)-1\right)\hat{\Psi}(\mathbf{r},t),$
(1)

where $U_{0}=4\pi\hbar^{2}a_{\mathrm{s}}/M$ is the two-body contact
interaction energy, $a_{s}$ is the s-wave scattering length, $M$ is the mass
of an atom, and $P$ is the momentum of an atom.

Assuming that the fluctuation about the mean atom number density $\langle
n(\mathbf{r})\rangle$ is small, we can perform a mean-field approximation and
neglect the quadratic noise terms. We approximate the contact interaction
Hamiltonian as

$\displaystyle\hat{H}_{\mathrm{int}}$ $\displaystyle=\frac{U_{0}}{2}\int
d\mathbf{r}\bigg{(}-\langle
n(\mathbf{r})\rangle^{2}+\hat{\Psi}^{\dagger}(\mathbf{r},t)\Big{[}2\langle
n(\mathbf{r})\rangle$
$\displaystyle-1)\Big{]}\hat{\Psi}(\mathbf{r},t)\bigg{)},$ (2)

where the first term merely adds a constant energy term, which can be ignored
so that the Hamiltonian of the system (II.1) becomes

$\displaystyle\hat{H}$ $\displaystyle=\int
d\mathbf{r}\hat{\Psi}^{\dagger}(\mathbf{r},t)\left[\frac{P^{2}}{2M}+V(\mathbf{r})+U_{0}\langle
n(\mathbf{r})\rangle\right]\hat{\Psi}(\mathbf{r},t).$ (3)

This single-particle Hamiltonian is of Gross-Pitaevskii form Dalfovo et al.
(1999); Leggett (2001).

We assume that only two states are allowed in the trap. Solving the Gross-
Pitaevskii equation for the double-well potential, we find the ground state
$\psi_{g}(\mathbf{r},t)$ is a symmetric state with energy $\hbar\omega_{g}$
and the first excited state $\psi_{e}(\mathbf{r},t)$ is the antisymmetric
state with energy $\hbar\omega_{e}$. Let the field operator
$\hat{\Psi}(\mathbf{r},t)$ be expanded in terms of the ground and excited
solutions,
$\hat{\Psi}(\mathbf{r},t)=\hat{b}_{g}\psi_{\mathrm{g}}(\mathbf{r},t)+\hat{b}_{e}\psi_{\mathrm{e}}(\mathbf{r},t)$,
where $\hat{b}_{g,e}$ is an annihilation operator. This acting on the vacuum
destroys it, $\hat{b}_{g,e}\lvert 0\rangle=0$. Due to orthogonality
conditions, the overlap vanishes, $\int
d\mathbf{r}\psi^{*}_{e}(\mathbf{r},t)\psi_{g}(\mathbf{r},t)=0$, and the
Hamiltonian density does not couple the ground and the excited states of the
macroscopic oscillator. Substituting for the field operator in (3) gives

$\displaystyle\hat{H}$
$\displaystyle=\frac{\hbar}{2}\left(\omega_{e}+\omega_{g}\right)\hat{N}+\hbar\Omega
J_{z},$ (4)

where
$\hat{N}=\hat{b}^{\dagger}_{g}\hat{b}_{g}+\hat{b}^{\dagger}_{e}\hat{b}_{e}$ is
the total particle number operator,
$J_{z}=(\hat{b}^{\dagger}_{e}\hat{b}_{e}-\hat{b}^{\dagger}_{g}\hat{b}_{g})/2$
measures the relative atom number in both the ground and excited states, and
$\Omega=\omega_{e}-\omega_{g}$ is the particle tunneling frequency. The other
spin operators are
$J_{x}=(\hat{b}^{\dagger}_{e}\hat{b}_{g}+\hat{b}^{\dagger}_{g}\hat{b}_{e})/2$,
$J_{y}=-i(\hat{b}^{\dagger}_{e}\hat{b}_{g}-\hat{b}^{\dagger}_{g}\hat{b}_{e})/2$.
In the following, we will neglect the constant energy term in (4) as it
contributes only a shift in the energy level.

The initial state of atoms is taken to be in a linear superposition of
symmetric and antisymmetric states as

$\lvert\alpha,\beta\rangle=\frac{1}{\sqrt{N!}}\left(\alpha\hat{b}_{e}^{\dagger}+\beta\hat{b}_{g}^{\dagger}\right)^{N}\lvert
0\rangle,$ (5)

where $\lvert 0\rangle$ is the vacuum state for the atom. The probability
$|\alpha|^{2}$ gives the fractional population of atoms found in the excited
state of the double-well potential, while $|\beta|^{2}$ gives the fractional
number of atoms found in the ground state of the double-well potential. These
probabilities satisfy $|\alpha|^{2}+|\beta|^{2}=1$. We may also write (5) in
terms of Bloch sphere angles Arecchi et al. (1972); Gross (2012),

$\lvert\theta,\phi\rangle=\frac{1}{\sqrt{N!}}\left(\sin(\theta/2)e^{-i\phi/2}\hat{b}_{e}^{\dagger}+\cos(\theta/2)e^{i\phi/2}\hat{b}_{g}^{\dagger}\right)^{N}\lvert
0\rangle.$ (6)

###  II.2 Light-atom interaction

In order to discuss the physics of the double-well it is convenient to
consider atoms localized in the left and right well, using the transformation
$\hat{b}_{g}=(\hat{b}_{l}+\hat{b}_{r})/\sqrt{2}$ and
$\hat{b}_{e}=(\hat{b}_{l}-\hat{b}_{r})/\sqrt{2}$. The atom operators $J_{x}$,
$J_{y}$ and $J_{z}$ defined in terms of the operators $b_{l}$ and $b_{r}$
become

$\displaystyle J_{x}$
$\displaystyle=\frac{\hat{b}_{l}^{\dagger}\hat{b}_{l}-\hat{b}^{\dagger}_{r}\hat{b}_{r}}{2},$
$\displaystyle J_{y}$
$\displaystyle=\frac{\hat{b}^{\dagger}_{l}\hat{b}_{r}-\hat{b}^{\dagger}_{r}\hat{b}_{l}}{2i},$
(7) $\displaystyle J_{z}$
$\displaystyle=\frac{b^{\dagger}_{l}\hat{b}_{r}+\hat{b}^{\dagger}_{r}\hat{b}_{l}}{2}.$

The total number operator
$\hat{N}=\hat{b}^{\dagger}_{l}\hat{b}_{l}+\hat{b}^{\dagger}_{r}\hat{b}_{r}$,
is a conserved quantity. In this basis, the operator $J_{x}$ gives the
relative atom number in each well. The operator $J_{z}$ gives the relative
atom number difference between the excited and ground states of the double-
well trap. For instance $\langle J_{z}\rangle$ being positive implies that
there are more atoms in the ground state of the double-well trap than in the
excited state and vice versa. Finally, $J_{y}$ gives the relative phase
between the condensates in the two wells.

We consider atoms in the double-well potential interacting with laser light
that is detuned from the atomic resonance transition Higbie et al. (2005);
Meppelink et al. (2010), as shown in Fig. 1. This set up can be described by
the QND Hamiltonian Julsgaard et al. (2001); Kuzmich and Kennedy (2004);
Behbood et al. (2014); Ilo-Okeke and Byrnes (2014). We write the total
Hamiltonian of the system including the light-matter interaction as

$H_{\mathrm{eff}}=\hbar\Omega J_{z}+2\hbar gJ_{x}S_{z},$ (8)

where $g$ is the atom-light coupling strength (Ilo-Okeke and Byrnes, 2014).
Writing the sum of atom-light interactions in both wells in terms of the
relative photon number
$S_{z}=(\hat{a}^{\dagger}_{l}\hat{a}_{l}-\hat{a}^{\dagger}_{r}\hat{a}_{r})/2$
that passes through the wells gives rise to the dependence of atom-light
interactions on the spins. The total number operator that introduces an
overall global phase has been ignored in writing $H_{\mathrm{eff}}$.

###  II.3 State evolution in the presence of ac Stark shift dephasing

Quantum states of matter such as the spin squeezed state that we generate are
usually fragile and are susceptible to decoherence Lone and Byrnes (2015) and
particle loss Li et al. (2008) caused by interactions with the environment.
For example, this may include imperfections in the experiment, interactions
among the particles, and the interactions with the probe. The statistical
fluctuations resulting from these interactions introduces an inherent
randomness in the quantum state of particles that interacts with the
environment. The randomness introduced by the environment leads to the
randomization of the phase of the particles that makes it impossible to
recover the coherence properties of the quantum state, even with
postselection. For example, photon scattering by atoms, which leads to a decay
in atomic polarization Grimm and Weidemller (1978), plays a role in the light
shift dephasing of BEC state. This is because the finite line width used in
calculating the ac Stark shift or light shift is also relevant in determining
the scattering rate. As such, the atomic operator $J_{x}$ coupling the atoms
to light as in (8) creates an ac Stark dephasing channel for the atoms. This
allows us to describe the effective dynamics of the coupled atom-light system
by a Markovian master equation Byrnes (2013); Lone and Byrnes (2015) for the
density operator $\rho$

$\displaystyle\frac{d\rho}{dt}=-\frac{i}{\hbar}\left[H_{\mathrm{eff}},\rho\right]+\gamma\left(J_{x}\rho
J_{x}-\frac{1}{2}\left\\{J_{x}^{2},\rho\right\\}\right),$ (9)

where $\gamma$ is the dephasing rate, and $H_{\mathrm{eff}}$ is as given in
(8).

At the end of the evolution of the density matrix, the light beams leaving the
trap are recombined as shown in Fig. 1. These photons are detected at the
detector $c$ and $d$, respectively, thereby collapsing the coherent state of
light. The full set of equations used to solve the master equation is given in
Appendix A. One of the techniques that we use is to expand the photons in the
coherent state basis, this avoids an expansion in the Fock basis which is much
more computationally demanding. This is possible because the Hamiltonian (8)
only couples to the light with photon number operators, which evolve the
coherent states by applying a phase. The state of the atomic BEC given that
$n_{c}$ and $n_{d}$ photons have been detected is $\rho_{n_{c},n_{d}}$ (A).
From here it is straightforward to calculate the averages of BEC operators as
$\langle A\rangle=\mathrm{Tr}\left[A\rho_{n_{c},n_{d}}\right]$. For instance,
the relative number of atoms in the wells given that $n_{c}$ and $n_{d}$ has
been detected may be inferred by the expectation value of $J_{x}$ defined as
$\langle
J_{x}\rangle_{n_{c},n_{d}}=\mathrm{Tr}\left[J_{x}\rho_{n_{c},n_{d}}\right]$.
Similarly, the variance is defined as

$\left(\Delta
J_{x}\right)^{2}_{n_{c},n_{d}}=\langle(J_{x})^{2}\rangle_{n_{c},n_{d}}-\langle
J_{x}\rangle^{2}_{n_{c},n_{d}},$ (10)

where
$\langle(J_{x})^{2}\rangle_{n_{c},n_{d}}=\mathrm{Tr}\left[J^{2}_{x}\rho_{n_{c},n_{d}}\right]$.

##  III Zero tunneling and zero decoherence limit

###  III.1 General wavefunction after measurement

In order to demonstrate how squeezing can be achieved using measurement, we
consider first the limit where the tunneling is negligible so that we set
$\Omega=0$ in (8). This limit corresponds to a separation between the well
minima that is very large in comparison to characteristic length of the
oscillator in each well. Coherent light is used to measure the BEC in the
double-well trap as shown in Fig. 1. The quantum state of 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

$\lvert\alpha_{s}\rangle=e^{-\frac{\lvert\alpha_{s}\rvert^{2}}{2}}e^{\alpha_{s}\hat{a}^{\dagger}_{s}}\lvert
0\rangle,$ (11)

$s=r,l$, $\lvert 0\rangle$ is the vacuum state. A generalized state of the BEC
is the $J_{x}$ Fock basis $\lvert k,N-k\rangle$, and is written as

$\lvert\Psi_{0}\rangle=\sum_{k}C_{k}\lvert k,N-k\rangle,$ (12)

where $\lvert k,N-k\rangle$ represents the state having $k$ atoms in the left
well and $N-k$ atoms in the right well,

$\lvert
k,N-k\rangle=\frac{(b_{l}^{\dagger})^{k}}{\sqrt{k!}}\frac{(b_{r}^{\dagger})^{N-k}}{\sqrt{(N-k)!}}\lvert
0\rangle\otimes\lvert 0\rangle.$ (13)

The evolution of the atom-light state is governed by the Schrödinger equation
$\lvert\Psi(t)\rangle=e^{-iH_{\mathrm{eff}}t/\hbar}\lvert\alpha_{l}\rangle\otimes\lvert\alpha_{r}\rangle\otimes\lvert\Psi_{0}\rangle$
and which can be solved exactly as

$\displaystyle\lvert\Psi(t)\rangle$
$\displaystyle=\lvert\alpha_{l}e^{-igtJ_{x}}\rangle\lvert\alpha_{r}e^{igtJ_{x}}\rangle\lvert\Psi_{0}\rangle,$
$\displaystyle=e^{-\frac{|\alpha_{l}|^{2}}{2}-\frac{|\alpha_{r}|^{2}}{2}}\sum_{k}C_{k}\exp\left(\alpha_{l}e^{-igt(k-N/2)}\hat{a}^{\dagger}_{l}\right)$
$\displaystyle\times\exp\left(\alpha_{r}e^{igt(k-N/2)}\hat{a}^{\dagger}_{r}\right)\lvert
0\rangle\otimes\lvert 0\rangle\otimes\lvert k,N-k\rangle,$ (14)

where in the second equality we have expanded the BEC state in Fock basis
(12). The phase of the photons encodes the information of the atoms. This
phase information carried by the photons can be accessed via interference of
the photons by passing the light through beamsplitter as shown in Fig. 1,
which is equivalent to the following transformation

$\hat{a}^{\dagger}_{l}=\frac{\hat{a}^{\dagger}_{c}+i\hat{a}^{\dagger}_{d}}{\sqrt{2}},\qquad\hat{a}^{\dagger}_{r}=\frac{i\hat{a}^{\dagger}_{c}+\hat{a}^{\dagger}_{d}}{\sqrt{2}}.$
(15)

The state after the beamsplitter becomes

$\displaystyle\lvert\psi_{\mathrm{BS}}\rangle$
$\displaystyle=e^{-\frac{|\alpha_{l}|^{2}}{2}-\frac{|\alpha_{r}|^{2}}{2}}\sum_{k}C_{k}\exp\left(\frac{\alpha_{c}(k)}{\sqrt{2}}\hat{a}^{\dagger}_{c}\right)$
$\displaystyle\times\exp\left(\frac{\alpha_{d}(k)}{\sqrt{2}}\hat{a}^{\dagger}_{d}\right)\lvert
0\rangle\otimes\lvert 0\rangle\otimes\lvert k,N-k\rangle,$ (16)

where

$\displaystyle\alpha_{c}(k)$
$\displaystyle=\left(\alpha_{l}+i\alpha_{r}\right)\cos[gt(k-N/2)]$
$\displaystyle-(i\alpha_{l}+\alpha_{r})\sin[gt(k-N/2)],$ (17)
$\displaystyle\alpha_{d}(k)$
$\displaystyle=(i\alpha_{l}+\alpha_{r})\cos[gt(k-N/2)]$
$\displaystyle+(\alpha_{l}+i\alpha_{r})\sin[gt(k-N/2)].$ (18)

After the beamsplitter, the photons collected in the detectors $c$ and $d$ are
counted. The probability $P(n_{c},n_{d})$ that $n_{c}$ and $n_{d}$ photons are
detected at the detectors $c$ and $d$, respectively, is obtained by projecting
the operator $\lvert n_{c},n_{d}\rangle\langle n_{c},n_{d}\rvert$ on the state
$\lvert\psi_{\mathrm{BS}}\rangle$,

$\displaystyle P(n_{c},n_{d})$ $\displaystyle=\sum_{k}\lvert\langle
n_{c},n_{d},k\lvert\psi_{\mathrm{BS}}\rangle\rvert^{2},$
$\displaystyle=\sum_{k}\lvert C_{k}\rvert^{2}\lvert
A_{n_{c},n_{d}}(k)\rvert^{2}.$ (19)

The state $\lvert\psi_{n_{c},n_{d}}\rangle$ after $n_{c}$ and $n_{d}$ photons
have been detected is

$\lvert\psi_{n_{c},n_{d}}\rangle=\frac{1}{\sqrt{P(n_{c},n_{d})}}\sum_{k}C_{k}A_{n_{c},n_{d}}(k)\lvert
k,N-k\rangle$ (20)

where

$A_{n_{c},n_{d}}(k)={e^{-\frac{|\alpha_{l}|^{2}}{2}-\frac{|\alpha_{r}|^{2}}{2}}}\frac{\left(\dfrac{\alpha_{c}(k)}{\sqrt{2}}\right)^{n_{c}}}{\sqrt{n_{c}!}}\frac{\left(\dfrac{\alpha_{d}(k)}{\sqrt{2}}\right)^{n_{d}}}{\sqrt{n_{d}!}}.$
(21)

Thus the effect of measurement is to modify the initial probability amplitude
of the BEC state, $C_{k}\rightarrow C_{k}A_{n_{c},n_{d}}(k)$.

In the limit $n_{d},\,n_{c}\gg 1$ and $gt>0$, the approximate state of the BEC
after the photon detection becomes

$\lvert\psi_{n_{c},n_{d}}^{\mathrm{approx}}\rangle\propto\sum_{k}C_{k}e^{-\frac{X_{0}}{4}(k-N/2-x_{0})^{2}}\lvert
k,N-k\rangle,$ (22)

where the peak and width of the measurement weighting function
$A_{n_{c},n_{d}}(k)$ are, respectively

$\displaystyle x_{0}$
$\displaystyle=\frac{1}{2gt}\left(\phi-\arcsin\left[\frac{|\alpha_{l}|^{2}+|\alpha_{r}|^{2}}{2|\alpha_{l}\alpha_{r}|}\frac{n_{c}-n_{d}}{n_{d}+n_{c}}\right]\right),$
(23) $\displaystyle X_{0}$
$\displaystyle=(gt)^{2}\left(\frac{n_{c}+n_{d}}{n_{c}n_{d}}\right)\Bigg{[}(n_{c}+n_{d})^{2}\left(\frac{2|\alpha_{l}\alpha_{r}|}{|\alpha_{l}|^{2}+|\alpha_{r}|^{2}}\right)^{2}$
$\displaystyle-(n_{d}-n_{c})^{2}\Bigg{]}.$ (24)

Detailed derivations of these equations above state are given in Appendix C.
From (22), we see that the factor $\lvert A_{n_{c},n_{d}}(k)\rvert$ (21)
modifies the original probability amplitude $C_{k}$ by suppressing the
amplitudes away from $k=N/2-x_{0}$.

This clearly shows that the measurement process induce squeezing effect for a
wide range of initial states. To illustrate the effect of measurement
$A_{n_{c},n_{d}}$, let us examine what happens to the probability amplitude of
the BEC being in the state $\lvert k,N-k\rangle$ after measurement of $n_{c}$
and $n_{d}$ photons. In the limit of no atom-light interaction with $gt=0$,
$A_{n_{c},n_{d}}(k)$ is independent of $k$, and the initial state of the BEC
is recovered. For a spin coherent state, the probability amplitude $C_{k}$ is
characterized by a distribution that is Gaussian with a width scaling as
$\sqrt{N}$. After the measurement with finite $gt$, the probability amplitude
is modified as $C_{k}A_{n_{c},n_{d}}(k)$. The state is squeezed if this new
probability amplitude $C_{k}A_{n_{c},n_{d}}(k)$ has a width that is smaller
than $\sqrt{N}$.

###  III.2 Example: initial spin coherent state

We now consider the specific case where the initial state of the atoms are
spin-polarized in the $J_{x}$-direction. The full calculations are shown in
Appendix C. The final approximate state for $n_{c},n_{d}\gg 1$ is

$\displaystyle\lvert\psi^{\mathrm{approx}}_{n_{c},n_{d}}\rangle\approx$
$\displaystyle\left(\frac{1}{2\pi\sigma}\right)^{1/4}\sum_{k}e^{-\frac{1}{4\sigma}\left(k-k_{0}\right)^{2}}$
$\displaystyle\times$ $\displaystyle
e^{ik\varphi}e^{i(n_{c}\phi_{c}(k)+n_{d}\phi_{d}(k))}\lvert k,N-k\rangle,$
(25)

where $\varphi$ is the initial relative phase between the atoms in the left
and right well, $\phi_{c}(k)$ is the argument of $\alpha_{c}(k)$, and
$\phi_{d}(k)$ is the argument of $\alpha_{d}(k)$. The general dependence of
the width $\sigma$ and peak $k_{0}$ of the modulating Gaussian in (III.2) on
detected number of photons $n_{c}$ and $n_{d}$ are derived in Appendix B. For
the case where $n_{d}=n_{c}$, the width $\sigma$ and the peak $k_{0}$ of the
magnitude of (III.2) are

$\displaystyle\sigma=$
$\displaystyle\frac{N\lvert\eta_{l}\eta_{r}\rvert^{2}}{1+8g^{2}t^{2}n_{c}N\lvert\eta_{l}\eta_{r}\rvert^{2}},$
(26) $\displaystyle k_{0}=$
$\displaystyle\frac{N|\eta_{l}|^{2}+4g^{2}t^{2}n_{c}N^{2}|\eta_{l}\eta_{r}|^{2}}{1+8g^{2}t^{2}n_{c}N|\eta_{l}\eta_{r}|^{2}},$
(27)

respectively, where $\eta_{l}=(\alpha+\beta)/\sqrt{2}$, and
$\eta_{r}=(\beta-\alpha)/\sqrt{2}$. It immediately follows from (III.2) that
the conditional probability density becomes

$\displaystyle P(k\lvert n_{c})$
$\displaystyle=\sqrt{\frac{1}{2\pi\sigma}}e^{-\frac{1}{2\sigma}\left(k-k_{0}\right)^{2}}.$
(28)

Clearly, the state (III.2) is squeezed since for $gt\neq 0$, the width of the
exponential term $\Delta k=\sqrt{\sigma}$ is smaller than
$\sqrt{N}\lvert\eta_{l}\eta_{r}\rvert$, the width (standard deviation) of the
spin coherent state.

Figure 2: The conditional probability of finding the atoms in the state
$\lvert k\rangle$. (a) The solid line is calculated numerically from (20)
while the dashed line is given by (28). (b) The numerically computed
probability density from (20) at large values of $gt$, $gt\sim 1/\sqrt{N}$ and
the dashed lines is given by (28). The parameters for the plots are $N=200$,
$\alpha_{l}=\alpha_{r}=\sqrt{20}$, $\alpha=0$, and $\beta=1$.

To illustrate this, (28) is plotted in Fig 2. It shows excellent agreement
with the exact calculation based on (20). In the presence of interaction, (28)
shows that the peak of the BEC’s probability distribution given that $n_{c}$
and $n_{d}$ photons have been detected is $k_{0}$. For $gt=0$, the BEC’s
probability distribution is that of a coherent state that has a peak at
$N\lvert\eta_{l}\rvert^{2}$ and a width $\Delta
k=\sqrt{N}\lvert\eta_{l}\eta_{r}\rvert$. With atom-light interactions $gt\neq
0$, the width of the probability density is modified as $\Delta
k=\sqrt{\sigma}$. Clearly this shows that the width of the distribution
decreases since the denominator of (26) would always be greater than unity if
$gt$ is not zero. In the small interaction regime shown in Fig 2(a), $gt\ll
1/\sqrt{N}$, the width decreases from that of a coherent state by an amount
$4g^{2}t^{2}\lvert\alpha_{r}\rvert^{2}N^{3/2}|\eta_{l}\eta_{r}|^{3}$. Since
the width of the BEC probability density distribution in a conditional
measurement of $n_{c}$ and $n_{d}$ photons is smaller than
$\sqrt{N}|\eta_{l}\eta_{r}|$, the BEC state is thus squeezed. For $gt\sim
1/\sqrt{N}$ the approximate probability (28) does not predict the many peaks
that emerge in this limit. Thus the approximate probability breaks down and is
no longer valid. The numerically computed results predict that the squeezing
is lost as the width of the probability density distribution is roughly
$\Delta k\sim N$. This is clearly supported by the results of Fig. 2(b) which
shows there is more than one peak in this limit which results in the
broadening of the width of the distribution. We remark that the inequality
$gt\ll 1/\sqrt{N}$ is an upper bound for the atom-light interaction strength
beyond which squeezing is being lost. It is possible to obtain a much lower
bound on atom-light interaction strength that would depend on the average
photon number used in the experiment.

The BEC state after measurement can also be visualized on a Bloch sphere using
the Husimi _Q_ function

$Q=\frac{N+1}{4\pi}\lvert\langle\theta,\phi\rvert\psi_{n_{c},n_{d}}\rangle\lvert^{2},$
(29)

where $\lvert\theta,\phi\rangle$ is the atomic coherent state (6). The results
of plotting (29) in Fig. 3 show that for $gt=0$, one obtains a distribution
that has equal width in $\theta$ and $\phi$, which is characteristic of a spin
coherent state. As the atom-light interaction strength $gt$ increases, $gt\sim
1/N$, the _Q_ -function distribution takes the shape of an ellipse that is
squeezed along _x_ -axis as shown at $gt=6/N$. For strong atom-light
interaction strength $gt\sim 1/\sqrt{N}$, the squeezing starts to degrade as
the ellipsoid shape is destroyed. There appears broader distribution around
the ellipse that distorts its shape, and the distribution starts to wrap
around the Bloch sphere. This can be linked to the many connected peaks that
emerge in this limit as shown in Fig 2(b). As result, the width of the
distribution broadens and is greater than $\sqrt{N}$.

Figure 3: The conditional Husimi _Q_ function. The parameters for the plots
are $N=200$, $\lvert\alpha_{l}\lvert=\lvert\alpha_{r}\rvert=\sqrt{20}$,
$\alpha=0$, and $\beta=1$. The parameter $gt$ is as shown in the figure. For
ease of visualization, the positive _z_ -axis is flipped and points downwards.

These calculations show that measurement can be used to create spin squeezed
states in an atomic BEC that is trapped in a double-well potential. Also, we
accounted for how the squeezing is lost due to strong measurement. In the
following section we model the system for the more general case, accounting
for the effects of dephasing and tunneling.

##  IV Tunneling with zero decoherence

In this section we calculate squeezing in the BEC state for measurements
performed on the condensates while tunneling between the wells is present but
without any dephasing. The resulting effect of the interactions on quantum
state of the BEC is then analyzed by studying its evolution on the Bloch
sphere. Similarly, the effects of measurement on the BEC state are analyzed
via the evolution of spin operators.

In the presence of tunneling alone without measurement, the initial state (5)
evolves within the mean field description as
$\lvert\psi(t)\rangle=e^{-i\hat{H}t/\hbar}\lvert\alpha,\beta\rangle$. Defining
the relative phase of the atom probability amplitudes
$\phi_{\alpha,\beta}=\phi_{\alpha}-\phi_{\beta}$, then the expectation values
of the spin operators take a simple form

$\displaystyle\langle\psi(t)\lvert J_{x}\rvert\psi(t)\rangle$
$\displaystyle=N\lvert\alpha\beta\rvert\cos(\Omega t-\phi_{\alpha,\beta}),$
$\displaystyle\langle\psi(t)\lvert J_{y}\rvert\psi(t)\rangle$
$\displaystyle=N\lvert\alpha\beta\rvert\sin(\Omega t-\phi_{\alpha,\beta}),$
(30) $\displaystyle\langle\psi(t)\rvert J_{z}\lvert\psi(t)\rangle$
$\displaystyle=N\frac{\lvert\alpha\rvert^{2}-\lvert\beta\rvert^{2}}{2}.$

The corresponding variances of the atomic spin operators are readily
calculated in a similar way

$\displaystyle(\Delta J_{x})^{2}=$
$\displaystyle\frac{N}{4}\left[1-4\lvert\alpha\beta\rvert^{2}\cos^{2}(\Omega
t-\phi_{\alpha,\beta})\right],$ $\displaystyle(\Delta J_{y})^{2}=$
$\displaystyle\frac{N}{4}\left[1-4\lvert\alpha\beta\rvert^{2}\sin^{2}(\Omega
t-\phi_{\alpha,\beta})\right],$ (31) $\displaystyle(\Delta J_{z})^{2}=$
$\displaystyle N\lvert\alpha\beta\rvert^{2}.$

The tunneling causes the atoms to oscillate between the two wells which on the
Bloch sphere is represented by a precession about the _z_ -axis with frequency
$\Omega$ as expressed in (IV). Since the squeezing occurs in the
$J_{x}$-direction, this means that the tunneling causes the state to rotate
with respect to the squeezing direction. As such, one strategy is to perform
measurement at specific times $t=n\pi/\Omega$, where the oscillator strength
is maximum, in order to generate squeezing Thorne et al. (1978); Braginsky et
al. (1980); Vasilakis et al. (2015). Alternatively, one could perform
continuous measurement on the oscillator and generate a squeezed state, for
sufficiently strong enough atom-light coupling strength, using the balanced
detection scheme described in Sec. III.2. The latter approach is what we study
in this section.

###  IV.1 Husimi _Q_ -Distribution

The effect of atom-light interactions on the BEC state in the presence of
tunneling without decoherence is examined by numerically calculating the
density matrix $\rho$ using (9), with the initial state
$\rho_{0}=\lvert\alpha,\beta\rangle\langle\alpha,\beta\rvert$, where
$\lvert\alpha,\beta\rangle$ is given in (5). At the end of evolution, the
photons are recombined using (15) and subsequently detected at the detectors
_c_ and _d_. The state $\rho_{n_{c},n_{d}}$ (A) resulting from the detection
of $n_{c}$ and $n_{d}$ photons is then used to calculate the Husimi _Q_
-function defined for a mixed state as

$Q_{n_{c},n_{d}}(\theta,\phi,t)=\frac{N+1}{4\pi}\langle\theta,\phi\lvert\rho_{n_{c},n_{d}}(t)\lvert\theta,\phi\rangle,$
(32)

where $\lvert\theta,\phi\rangle$ is a spin coherent state given in (6).

The results are shown in Fig. 4. The rows give the variations of the _Q_
-functions with time $\Omega t$, while the columns give the variation of the
state $\rho_{n_{c},n_{d}}$ with atom-light interactions $g/\Omega$. The first
row at $g=0$ confirms that state $\rho_{n_{c},n_{d}}$ is that of a spin
coherent state that is precessing about the _z_ -axis on a Bloch sphere. Due
to the slight offset from a perfectly _z_ -polarized state, the tunneling
causes coherent state to rotate around the _z_ -axis. Also we see from the
figure that weak atom-light interactions give rise to a slow build up of
squeezing of the BEC state. For instance at $\Omega t=4$, the BEC state is
still similar to that of a coherent state at $g=0.1\Omega/N$ compared to that
at $g=\Omega/N$ where the squeezing is already starting to degrade. At long
times, the squeezing of the BEC state arising from a weak atom-light
interactions is lost as shown for $\Omega t=60$ and $g=0.1\Omega/N$. This loss
occurs because of the squeezed spin state transitioning into other quantum
state as the interactions become stronger. We are able to probe this long
interaction time region because we use the exact wavefunction approach. This
is best seen in the case of strong atom-light interactions $g=\Omega/N$ where
the loss of squeezed state after $\Omega t=4$ leads to the emergence of two
spin coherent state at the north and south poles of the Bloch sphere for
$\Omega t=20$. Such a state is a Schrödinger cat state, which is a linear
superposition of two spin coherent state Dodonov et al. (1974); Byrnes and
Ilo-Okeke . Beyond the time $\Omega t=20$, the spin coherent state at the top
of the Bloch sphere splits into two and starts its motion back to the bottom
of the sphere where they again form a spin squeezed state and then a spin
coherent state (see also movies in the Supplementary Materials).

Figure 4: The conditional Husimi _Q_ -function. The parameters for the plots
are $N=30$, $\alpha=\sqrt{0.001}$, $\beta=\sqrt{0.999}$, $\Omega=\pi/4$ and
$\alpha_{l}=\alpha_{r}=2$. The parameters for the strength of the coupling $g$
and time $\Omega t$ are as shown in the figure. For ease of visualization, the
positive _z_ -axis is flipped and points downwards.

###  IV.2 Expectation values and variances

We now examine the spin averages and variances with the tunneling present. The
averages calculated at different atom-light coupling strength $g$ with the
dephasing parameter $\gamma$ set to zero is shown in Fig. 5 and Fig. 6. For
$g=0$, the expectations $J_{x}$ and $J_{y}$ execute harmonic oscillation with
a relative phase of $\pi/2$ between them. As a result, the mean spin
projection on the _x_ -_y_ plane traces a circle thereby confirming that the
atoms precess about the _z_ -axis on the Bloch sphere at frequency $\Omega$.
Physically this corresponds to the atoms shuttling between the two double-well
trap at frequency $\Omega$. Also, the relative atomic population in the
excited state remains fixed for all times. Since most of the atoms are
initialized to be dominantly in the ground state of the double-well, the
variance of $J_{z}$ remains fixed as predicted by (IV), as shown in Fig. 5.
However, the variances of $J_{x}$ and $J_{y}$ are large, i.e. around unity in
scaled units, see (IV).

Figure 5: The normalized conditional mean of $J_{x}$, $J_{y}$ and $J_{z}$ with
dephasing parameter $\gamma$ = 0. The atom-light interaction strengths are as
shown in the figure. The parameters for the figure are $N=30$, $\Omega=\pi/4$,
$\alpha=\sqrt{0.001}$, $\beta=\sqrt{0.999}$, $\alpha_{r}=\alpha_{l}=2$.

Turning on the measurement apparatus with coupling strength $g=0.1\Omega/N$ in
Fig. 5, the atoms execute sinusoidal oscillations between the wells with a
slightly modulated frequency. This can be understood from the fact that the
first order effect is to change the Hamiltonian to $H=\hbar\Omega J_{z}+\hbar
g\langle S_{x}\rangle J_{x}$ which changes the axis of the oscillations. The
amplitude of $\langle J_{x}\rangle$ oscillations decreases faster than the
amplitude of $\langle J_{y}\rangle$ oscillations as shown by the dashed lines
in Fig. 5. As a result, the circular motion traced by mean spin projection on
the _x_ -_y_ plane shrinks faster along _x_ than along _y_ giving rise to an
ellipsoidal trajectory. This squeezing effect is manifested in the variance of
$J_{x}$ which decreases below unity, while that of $J_{y}$ increases well
above unity as shown by the dashed line in Fig. 6. The variance of $J_{x}$
reaches its minimum value in the region where the amplitude of $\langle
J_{x}\rangle$ oscillations has its smallest value. Observing the variance to
be below unity indicates that the state of the oscillator is squeezed. At the
same time there is oscillation in both the _y_ -_z_ and _x_ -_z_ planes which
is absent in the case without measurement $g=0$. We also point out that there
are two time scales. One of them is the fast oscillation responsible for the
tunneling between the wells. In addition, there is modulation of fast
oscillations that is responsible for the squeezing effect. Comparing the
frequencies in the expectations of $J_{x}$ and $J_{y}$ at $g=0$ and
$g=0.1\Omega/N$, it is seen that atom-light interactions increases the period
of the fast time scale for $\Omega t>20$.

Increasing further the atom-light interaction strength to $g=\Omega/N$ as
represented by the dotted lines in Fig. 5 shows that the motion is no longer
periodic after a short time. However, the variance of $J_{x}$ is again well
below unity, while that of $J_{y}$ is well above unity within the short time
window as shown in Fig. 6. Notice that the squeezing of the state is more
easily seen with the variance of $J_{x}$ than in the _Q_ -function. Note that
the minimal width of a $Q$-function, even for an infinitely squeezed state, is
only one-half that of a spin coherent state.

Figure 6: The normalized conditional variance of $J_{x}$, $J_{y}$ and $J_{z}$
at dephasing parameter $\gamma$ = 0. The atom-light interaction strengths are
as shown in the figure. The dotted line at normalized conditional variance of
$J_{x}$ equal to zero is a guide for the eye. The parameters for the figure
are $N=30$, $\Omega=\pi/4$, $\alpha=\sqrt{0.001}$, $\beta=\sqrt{0.999}$,
$\alpha_{r}=\alpha_{l}=2$.

##  V Tunneling and dephasing effects

We now include the dephasing, and investigate how the state of the BEC is
impacted by visually inspecting changes in the plots of the _Q_ -function and
more quantitatively by analyzing the averages of the spin operators.

###  V.1 Husimi _Q_ -Distribution

We begin by calculating the Husimi $Q$-function given in (32). The density
matrix $\rho_{n_{c},n_{d}}$ is given in (A) with $\Omega\geq 0$ and
$\gamma\geq 0$. The results of the calculations are shown in Fig. 7. Moving
along the rows show variation of the _Q_ -functions for the state
$\rho_{n_{c},n_{d}}(t)$ with time, while moving along the column shows the
variation with dephasing rate $\gamma$. As time increases, there is
significant squeezing of state $\rho_{n_{c},n_{d}}(t)$. The state with weak
dephasing rate shows a slow build up to a noticeable amount of squeezing. At
longer times $\Omega t\geq 50$, the state can however lose its squeezing
depending on the dephasing rate $\gamma$. The onset for the loss of squeezing
happens earlier if the dephasing rate is very strong. For instance, for
$\gamma=3g$ the state loses its squeezing at $\Omega t=30$ while for
$\gamma=g$ the squeezing is still in tact at the same time. In the case of
dephasing rate $\gamma=0.1g$, the state exhibits strong squeezing at $\Omega
t=30$ which will begin to diminish at $\Omega t=50$.

###  V.2 Expectation values and variances

Figure 7: The conditional Husimi _Q_ -function. The parameters are $N=30$,
$\Omega=\pi/4$, $g=0.1\Omega/N$, $\alpha=\sqrt{0.001}$, $\beta=\sqrt{0.999}$,
and $\alpha_{l}=\alpha_{r}=2$. The dephasing rate $\gamma$ (row) and the
parameter proportional to time $\Omega t$ (columns) are as indicated. For ease
of visualization, the positive _z_ -axis is flipped and points downwards.

Fig. 8 and Fig. 9 show the spin expectation values and variances for the case
including dephasing, $\gamma\geq 0$. Additionally, Fig. 9(b) contains an
enlargement of the variance of $J_{x}$ around unity. One effect of dephasing
is to maintain the amplitude of the oscillations in $J_{y}$ while causing a
decrease in the oscillation amplitude of the $J_{x}$ expectation value. Hence
as the amplitude of $\langle J_{x}\rangle$ oscillations decrease and go
through its minimum point, the amplitude of $\langle J_{y}\rangle$
oscillations is roughly same as that at $\gamma=0$. Consequently, the mean
spin projection on the $x$-$y$ plane does not trace an ellipse until after the
minima of the $J_{x}$ amplitude has been passed and begins to increase. As
such there is a delay in the decrease of the variance of $J_{x}$ below unity
in the presence of dephasing compared to that when dephasing is absent
$\gamma=0$ as shown in Fig. 9(b). This effect becomes more noticeable for
large dephasing rate $\gamma\geq g$. As a result, the squeezing can persist in
the presence of dephasing but only within a certain time window. Similarly,
the minimum of the variance of $J_{x}$ is reached quite early for appreciable
dephasing rate. Hence, a strong squeezing effect can occur quite early when
the dephasing rate is appreciable while it happens much later if the dephasing
rate is weak. These results are in agreement with that already shown in Fig.
7. Also, note that the period of the fast oscillations is same for the
different strength of the dephasing value i.e dephasing does not affect the
effective tunneling period of the atoms.

Figure 8: The normalized conditional mean of $J_{x}$, $J_{y}$ and $J_{z}$ at
different values of dephasing parameter $\gamma$. The measurement strength
$g=0.1\Omega/N$ for all plots. The solid line is for $\gamma=0$, the dashed
line is for $\gamma=0.1g$, the dashed-dotted line is for $\gamma=g$ and the
dotted line is for $\gamma=3g$. The parameters for the figure are $N=30$,
$\Omega=\pi/4$, $\alpha=\sqrt{0.001}$, $\beta=\sqrt{0.999}$,
$\alpha_{r}=\alpha_{l}=2$.

##  VI Experimental realizations

Nondestructive measurement of atoms have been carried out in several
experiments Higbie et al. (2005); Meppelink et al. (2010). In these
realizations, the measurement is performed on atoms in a single-well trap,
where the single-site addressing is not required. Nondestructive measurement
of atoms in double well requires independent access to each site. The double-
well separation in cold-atom experiments is on the order of micrometers in
order to have reasonable control of the quantum tunneling between the wells
Schumm et al. (2005); Albiez et al. (2005). Diffraction limits the
experimentally feasible waist size of the probe laser beam to be of same order
corresponding to a few wavelengths. Hence it is difficult to obtain the
arrangement shown in Fig. 1.

However, for a spinor BEC in the double-well trap, it is possible to
separately access information about each site as described in Ref.Ilo-Okeke
and Byrnes (2014), where the effective Hamiltonian (8) can be recovered by
adding a weak microwave coupling between internal states Riedel et al. (2010).
In addition, the atoms in each well can be addressed separately by working in
the frequency domain where the scattered light from atoms in the left and
right wells are different to those of the incident light. This difference in
frequencies, where, the atoms in the left well are engineered to modulate the
incident light to lower frequency, while atoms in the right well absorb the
incident beam and emit at higher frequency or vice versa, can then be used to
distinguish the information from the atoms on the left well from those on the
right well. Such a technique can be implemented using Raman transitions
Kasevich and Chu (1991); Moler et al. (1992); Carraz et al. (2012); Giese et
al. (2013) between the ground state energies of the atom, and the modulated
output beam would be observed on a polarimeter.

Figure 9: The normalized conditional variance of $J_{x}$, $J_{y}$ and $J_{z}$
at different values of dephasing parameter $\gamma$. (a) Presented in this
column are the normalized variances of $J_{x}$, $J_{y}$ and $J_{z}$. The
dotted line at $4(\Delta J_{x})^{2}/N=0$ is a guide for the eye. (b) An
enlarged plot of the normalized variance of $J_{x}$ showing its behavior
around unity for different values of dephasing parameter. The solid line at
$4(\Delta J_{x})^{2}/N=1$ is guide for the eye. The parameters for the figures
are $N=30$, $\Omega=\pi/4$, $g=0.1\Omega/N$, $\alpha=\sqrt{0.001}$,
$\beta=\sqrt{0.999}$, and $\alpha_{r}=\alpha_{l}=2$.

Alternatively, an extension of the RF-dressed Voigt detection method Jammi et
al. (2018) in multiple-RF dressed double-well potentials Harte et al. (2018);
Barker et al. (2020) provides a convenient method of measurement explained in
Sec. II.3. In this method, the double-well potential is formed by a
combination of three RF dressing fields, and atoms at the two minima are on
resonance with maximum and minimum dressing RF components respectively. An
off-resonant, polarized probe laser beam passing through both trapped clouds
is phase modulated at the frequencies of the resonant dressing fields, and the
signal at each frequency can be detected with a balanced polarimeter Jammi et
al. (2018). The detection of the two frequency components at the dressing
field frequencies using two separate lock-in demodulations constitutes the
measurement equivalent to that shown Fig. 1.

##  VII Summary and Conclusions

We showed that minimally-destructive detection is a good tool for the creation
of a squeezed state of atoms in a double-well potential. To achieve the
minimally-destructive effect on the states of the atoms, the measurement is
performed with a balanced detection configuration. To understand the squeezing
effect, we first studied the system under the conditions of no tunneling and
no decoherence. Our main result in this case is (20), where a simple form of
the atomic wavefunction after measurement was derived. Here, it was found that
the measurement introduces an additional factor of $A_{n_{c},n_{d}}(k)$ into
the atomic wavefunction, which can be approximated as a Gaussian. Without
squeezing, the probability distribution of the quantum state of the atoms is
approximately a Gaussian with a width of roughly $\sqrt{N}$. Our scheme can
affect a reduction in the width of the probability distribution of the quantum
state of the atoms below unsqueezed value of $\sqrt{N}$ after conditional
detection of photons (26). One of the key features of our approach is that we
solve exactly the time evolution of the wavefunction under the QND
interaction. This contrasts to standard methods based on the Holstein-
Primakoff approximation which are limited to short interaction times and an
initial spin coherent state. Thus, we have been able to investigate the full
non-linear effect of the QND Hamiltonian, showing the generation of
oversqueezed states and exotic states such as Schrödinger cat states.

In the presence of tunneling, the squeezing effect was seen as a decrease in
the variance of the atom spin operator that was coupled to light.
Additionally, there is a modulation of the expectation values of the spin
operators arising from measurement which is absent when there is no tunneling
effect. We studied the effects of dephasing on the squeezed state due to the
presence of light using the master equation.These calculations showed that
dephasing has a negligible effect on squeezing when the atom-light interaction
strength is greater than the dephasing rate. Even in the strong dephasing
regime, where the dephasing strength is equal or greater than the measurement
strength, we found that squeezing persists.

We showed the overall effect of squeezing and dephasing on the quantum state
of the atoms by visualizing it on a Bloch sphere using the Husimi _Q_
-distribution. In the case without tunneling and dephasing, the initial mean
spin direction is not affected by the measurement. However, the measurement
shrinks the _Q_ -distribution by causing squeezing in the plane that is
perpendicular to the mean spin direction, and in particular shrinks only the
width of the _Q_ -distribution along the spin axis that is coupled to light.
In the presence of tunneling, the mean spin direction is relatively unaffected
by the measurement for small atom-light interaction strength. There are also
oscillations of the state about the mean spin direction, and for relatively
strong atom-light interactions, the width of the _Q_ -distribution along the
spin axis that is coupled to light decreases, which manifests in the squeezing
effect.

The squeezing effect observed in the presence of tunneling is a precursor to
formation of different quantum states of the BEC such as the Schrödinger cat-
state that is characterized by two _Q_ -distributions located on the opposite
(north and south) poles on the Bloch sphere. Due to the oscillations, the
state formation is actually reversible where the BEC state goes between the
cat-sate and the spin coherent state, with the spin squeezed state being one
of the intermediate states in the transition. This actually mimics the
collapse and revival of quantum state that have been observed in other
different oscillating systems like the Jaynes-Cummings model Gea-Banacloche
(1990); Bužek et al. (1992), and the Josephson junction Milburn et al. (1997);
Kirchmair et al. (2013). It is also analogous to the oscillatory effects
observed Byrnes (2013) in the two-axis squeezing of BEC where there is a
periodic evolution between a coherent state and a maximal entangled state. The
inclusion of the dephasing causes several other satellite _Q_ -distributions
to emerge. The splintered _Q_ -distributions are along the spin axis that is
coupled to light, a feature that is absent if the dephasing effect is zero.
Additionally, there is a broadening of the width of the _Q_ -distributions
along the spin axis coupled to light, linking the various _Q_ -distributions.
Hence, while squeezing is present at small atom-light interactions in the
presence of dephasing, all the features described previously about the _Q_
-function are not recovered, and squeezing is irreversibly lost in the strong
dephasing regime or at large atom-light interactions in the presence of
dephasing.

###### Acknowledgements.

T. B. is supported by the Shanghai Research Challenge Fund; New York
University Global Seed Grants for Collaborative Research; National Natural
Science Foundation of China (Grant No. 61571301); the Thousand Talents Program
for Distinguished Young Scholars (Grant No. D1210036A); and the NSFC Research
Fund for International Young Scientists (Grant No. 11650110425); NYU-ECNU
Institute of Physics at NYU Shanghai; and the Science and Technology
Commission of Shanghai Municipality (Grant No. 17ZR1443600). E. O. I. O.
acknowledges the Talented Young Scientists Program (NGA-16-001) supported by
the Ministry of Science and Technology of China. S. S. acknowledges Murata
scholarship foundation, Ezoe foundation and Daishin foundation. The authors in
Oxford carry out experimental work on cold atoms funded by EPSRC grant
EP/S013105/1.

##  Appendix A Solving The Master Equation

To solve for the evolution of the density matrix, we use the ansatz (García-
Ripoll et al., 2005; Hussain et al., 2014)

$\rho=\sum_{k,k^{\prime}}\rho_{kk^{\prime}}\lvert
k;\alpha_{k,l},\alpha_{k,r}\rangle\langle
k^{\prime};\alpha_{k^{\prime},l},\alpha_{k^{\prime},l}\rvert,$ (33)

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, (33) into (9) 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 $\langle\alpha_{\perp,k,s}\lvert\alpha_{k,s}\rangle=0$ (García-Ripoll et
al., 2005; Hussain et al., 2014) gives the equations for the evolution of
$\alpha_{k^{\prime},s}$ and $\alpha_{k,s}$ and they read

$\displaystyle\frac{d\alpha_{k,l}}{dt}$
$\displaystyle=-\frac{ig(2k-N)}{2}\alpha_{k,l},$
$\displaystyle\frac{d\alpha_{k,r}}{dt}$
$\displaystyle=\frac{ig(2k-N)}{2}\alpha_{k,r},$ (34)

whose solutions are

$\displaystyle\alpha_{k,l}(t)$
$\displaystyle=\alpha_{l}e^{-i(2k-N)\tfrac{gt}{2}},$
$\displaystyle\alpha_{k,r}(t)$
$\displaystyle=\alpha_{r}e^{i(2k-N)\tfrac{gt}{2}}.$ (35)

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 $\rho_{kk^{\prime}}$

$\displaystyle\frac{d\rho_{mm^{\prime}}}{dt}$
$\displaystyle=i\Omega\frac{\sqrt{m(N-m+1)}}{2}\langle\alpha_{m}\lvert\alpha_{m-1}\rangle\rho_{m-1,m^{\prime}}$
$\displaystyle+i\Omega\frac{\sqrt{(m+1)(N-m)}}{2}\langle\alpha_{m}\lvert\alpha_{m+1}\rangle\rho_{m+1,m^{\prime}}$
$\displaystyle-i\Omega\frac{\sqrt{m^{\prime}(N-m^{\prime}+1)}}{2}\langle\alpha_{m^{\prime}-1}\lvert\alpha_{m^{\prime}}\rangle\rho_{m,m^{\prime}-1}$
$\displaystyle-i\Omega\frac{\sqrt{(m^{\prime}+1)(N-m^{\prime})}}{2}\langle\alpha_{m^{\prime}+1}\lvert\alpha_{m^{\prime}}\rangle\rho_{m,m^{\prime}+1}$
$\displaystyle-\gamma(m-m^{\prime})\rho_{mm^{\prime}},$ (36)

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

$\displaystyle\langle\alpha_{m}\lvert\alpha_{m+1}\rangle$
$\displaystyle=e^{-(|\alpha_{l}|^{2}+|\alpha_{r}|^{2})}e^{|\alpha_{l}|^{2}e^{-igt}}e^{|\alpha_{r}|^{2}e^{igt}},$
$\displaystyle\langle\alpha_{m}\lvert\alpha_{m-1}\rangle$
$\displaystyle=e^{-(|\alpha_{l}|^{2}+|\alpha_{r}|^{2})}e^{|\alpha_{l}|^{2}e^{igt}}e^{|\alpha_{r}|^{2}e^{-igt}}.$
(37)

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

At the beamsplitter, photons that left each well are recombined as shown in
Fig. 1 and sorted into bins $c$ and $d$ according to (15). The state after
recombination at the beamsplitter becomes

$\displaystyle\rho_{\mathrm{BS}}(t)$
$\displaystyle=\sum_{k,k^{\prime}}\rho_{kk^{\prime}}(t)e^{-\left(\frac{\lvert\alpha_{k,l}\rvert^{2}}{2}+\frac{\lvert\alpha_{k,r}\rvert^{2}}{2}\right)}e^{\frac{\alpha_{k,l}+i\alpha_{k,r}}{\sqrt{2}}\hat{a}^{\dagger}_{c}}$
$\displaystyle\times
e^{\frac{i\alpha_{k,l}+\alpha_{k,r}}{\sqrt{2}}\hat{a}^{\dagger}_{d}}\lvert
k,0,0\rangle\langle k^{\prime},0,0\rvert
e^{-\left(\frac{\lvert\alpha_{k^{\prime},l}\rvert^{2}}{2}+\frac{\lvert\alpha_{k^{\prime},r}\rvert^{2}}{2}\right)}$
$\displaystyle\times
e^{\frac{\alpha^{*}_{k^{\prime},l}-i\alpha^{*}_{k^{\prime},r}}{\sqrt{2}}\hat{a}_{c}}e^{\frac{-i\alpha^{*}_{k^{\prime},l}+\alpha^{*}_{k^{\prime},r}}{\sqrt{2}}\hat{a}_{d}}.$
(38)

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,

$\displaystyle P(n_{c},n_{d})$
$\displaystyle=\mathrm{Tr}[\rho_{\mathrm{BS}}(t)\mathcal{P}],$
$\displaystyle=\frac{1}{n_{c}!}\frac{1}{n_{d}!}\sum_{k}\rho_{kk}(t)e^{-\left(\lvert\alpha_{k,l}\rvert^{2}+\lvert\alpha_{k,r}\rvert^{2}\right)}$
$\displaystyle\times\left(\frac{\lvert\alpha_{k,l}\rvert^{2}+i\alpha_{k,r}\alpha^{*}_{k,l}-i\alpha_{k,l}\alpha^{*}_{k,r}+\lvert\alpha_{k,r}\rvert^{2}}{2}\right)^{n_{c}}$
$\displaystyle\times\left(\frac{\lvert\alpha_{k,l}\rvert^{2}-i\alpha_{k,r}\alpha^{*}_{k,l}+i\alpha_{k,l}\alpha^{*}_{k,r}+\lvert\alpha_{k,r}\rvert^{2}}{2}\right)^{n_{d}}.$
(39)

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

$\displaystyle\rho_{n_{c},n_{d}}(t)$
$\displaystyle=\frac{1}{n_{c}!}\frac{1}{n_{d}!}\frac{1}{P(n_{c},n_{d})}\sum_{k,k^{\prime}}\rho_{kk^{\prime}}(t)\times$
$\displaystyle
e^{-\left(\frac{\lvert\alpha_{k,l}\rvert^{2}}{2}+\frac{\lvert\alpha_{k^{\prime},l}\rvert^{2}}{2}+\frac{\lvert\alpha_{k,r}\rvert^{2}}{2}\frac{\lvert\alpha_{k^{\prime},r}\rvert^{2}}{2}\right)}$
$\displaystyle\times\left(\frac{\alpha_{k,l}\alpha^{*}_{k^{\prime},l}+i\alpha_{k,r}\alpha^{*}_{k^{\prime},l}-i\alpha_{k,l}\alpha^{*}_{k^{\prime},r}+\alpha_{k,r}\alpha^{*}_{k^{\prime},r}}{2}\right)^{n_{c}}$
$\displaystyle\times\left(\frac{\alpha_{k,l}\alpha^{*}_{k^{\prime},l}-i\alpha_{k,r}\alpha^{*}_{k^{\prime},l}+i\alpha_{k,l}\alpha^{*}_{k^{\prime},r}+\alpha_{k,r}\alpha^{*}_{k^{\prime},r}}{2}\right)^{n_{d}}$
$\displaystyle\times\lvert k,N-k\rangle\langle k^{\prime},N-k^{\prime}\rvert.$
(40)

##  Appendix B Most Probable Outcome

Let us first consider that $gt=0$, which occurs when there are no atom-light
interactions. In this limit, the amplitudes $A_{n_{c}.n_{d}}(k)$ can be taken
outside the sum over $k$ and the contribution of the atoms to the probability
(III.1) sums up to unity, $\sum_{k}\lvert C_{k}\rvert^{2}=1$. Then, the
probability (III.1) of obtaining $n_{c}$ and $n_{d}$ photons after measurement
simplifies $P(n_{c},n_{d})=\sum_{k}\lvert C_{k}\rvert^{2}\lvert
A_{n_{c},n_{d}}(k)\rvert^{2}=\lvert A_{n_{c},n_{d}}(k)\rvert^{2}$. Using
(III.1) and (III.1),

$\displaystyle\lvert A_{n_{c},n_{d}}(k)\rvert^{2}=$ $\displaystyle
e^{-\lvert\alpha_{l}\rvert^{2}-\lvert\alpha_{r}\rvert^{2}}\frac{\left(\dfrac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}\right)^{n_{c}}}{n_{c}!}$
$\displaystyle\times$
$\displaystyle\frac{\left(\dfrac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{2}\right)^{n_{d}}}{n_{d}!}.$
(41)

Observe that $\lvert A_{n_{c},n_{d}}(k)\rvert^{2}$ can be written as a product
of two probabilities $P(n_{c})$ and $P(n_{d})$ as $\lvert
A_{n_{c},n_{d}}(k)\rvert^{2}=P(n_{c})P(n_{d})$ where

$P(n_{j})=e^{-\frac{|\alpha_{l}|^{2}+|\alpha_{r}|^{2}}{2}}\left(\frac{|\alpha_{l}|^{2}+|\alpha_{r}|^{2}}{2}\right)^{n_{j}}\frac{1}{n_{j}!},$
(42)

$j=c,d$. Suppose that $n_{c}$, $n_{d}\gg 1$ and using Stirling’s
approximation, one can write $P(n_{j})$ as

$P(n_{j})=\frac{1}{\sqrt{\pi\left(\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2}\right)}}\exp\left[-\frac{\left(n_{j}-\frac{\alpha_{r}\rvert^{2}-\lvert\alpha_{l}\lvert^{2}}{2}\right)^{2}}{\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2}}\right].$
(43)

It is then clear that the most probable outcome occurs for
$n_{d}=n_{c}=(\lvert\alpha_{r}\rvert^{2}+\lvert\alpha_{l}\rvert^{2})/2$, see
also Refs. Ilo-Okeke and Byrnes (2014, 2016) that proved this using a
different method.

For the number of photons $n_{c},n_{d}>1$ and using Stirling’s approximation,
the magnitude of the function $A_{n_{c},n_{d}}(k)$ becomes

$\displaystyle|A_{n_{c},n_{d}}(k)|=$
$\displaystyle\left(\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{n_{c}+n_{d}}\right)^{\frac{n_{c}+n_{d}}{2}}e^{\frac{n_{c}+n_{d}-\lvert\alpha_{l}\rvert^{2}-\lvert\alpha_{r}\rvert^{2}}{2}}$
$\displaystyle\times$
$\displaystyle\left(\frac{1}{4\pi^{2}n_{c}n_{d}}\right)^{1/4}e^{-\frac{X_{0}}{4}(k-N/2-x_{0})^{2}},$
(44)

where

$\displaystyle x_{0}$
$\displaystyle=\frac{1}{2gt}\left(\phi-\arcsin\left[\frac{|\alpha_{l}|^{2}+|\alpha_{r}|^{2}}{2|\alpha_{l}\alpha_{r}|}\frac{n_{c}-n_{d}}{n_{d}+n_{c}}\right]\right),$
(45) $\displaystyle X_{0}$
$\displaystyle=(gt)^{2}\left(\frac{n_{c}+n_{d}}{n_{c}n_{d}}\right)\Bigg{[}(n_{c}+n_{d})^{2}\left(\frac{2|\alpha_{l}\alpha_{r}|}{|\alpha_{l}|^{2}+|\alpha_{r}|^{2}}\right)^{2}$
$\displaystyle-(n_{d}-n_{c})^{2}\Bigg{]},$ (46)

and $\phi=\phi_{l}-\phi_{r}$ is the relative phase between the coherent light
beams on the left and right as shown in Fig. 1.

##  Appendix C The Approximate State of BEC After Photon Detection

Working in the _x_ -basis, the probability amplitude of the initial state
$\lvert\alpha,\beta\rangle$ (5)

$C_{k}=\sqrt{\frac{N!}{k!(N-k)!}}\left(\eta_{l}\right)^{k}\left(\eta_{r}\right)^{N-k},$
(47)

where $\eta_{l}=(\alpha+\beta)/\sqrt{2}$ and
$\eta_{r}=(\beta-\alpha)/\sqrt{2}$. For $N\gg 1$, the dominant contribution to
$C_{k}$ come from terms around $k=N\lvert\eta_{l}\rvert^{2}$ that has a width
$\Delta k\sim\sqrt{N}$. Using Stirling’s approximation, the magnitude of
$C_{k}$ becomes

$\lvert C_{k}\rvert\approx\left(\frac{1}{2\pi
N\lvert\eta_{l}\eta_{r}\rvert^{2}}\right)^{1/4}e^{-\frac{1}{4N\lvert\eta_{l}\rvert^{2}\lvert\eta_{r}\rvert^{2}}\left(k-N\lvert\eta_{l}\rvert^{2}\right)^{2}}.$
(48)

Hence the initial probability distribution $\lvert C_{k}\rvert^{2}$ of the
atoms in the state $\lvert\alpha,\beta\rangle$ can be approximated to
Gaussian.

The probability density $P(n_{c},n_{d})$ of detecting $n_{c},n_{d}$ photons in
a measurement (III.1) is easily calculated using $\lvert C_{k}\rvert$ (48),
and $\lvert A_{n_{c},n_{d}}(k)\rvert$ (B), and it evaluates to

$\displaystyle P(n_{c},n_{d})=$
$\displaystyle\sqrt{\frac{1}{4\pi^{2}n_{c}n_{d}(1+N\lvert\eta_{l}\eta_{r}\rvert^{2}X_{0})}}$
$\displaystyle\times$
$\displaystyle\left(\frac{\lvert\alpha_{l}\rvert^{2}+\lvert\alpha_{r}\rvert^{2}}{n_{c}+n_{d}}\right)^{n_{c}+n_{d}}e^{n_{c}+n_{d}-\lvert\alpha_{l}\rvert^{2}-\lvert\alpha_{r}\rvert^{2}}$
$\displaystyle\times$
$\displaystyle\exp\left[-\frac{X_{0}\left(x_{0}-\frac{N(\lvert\eta_{l}\rvert^{2}-\lvert\eta_{r}\rvert^{2})}{2}\right)^{2}}{2(1+N\lvert\eta_{l}\eta_{r}\rvert^{2}X_{0})}\right].$
(49)

The approximate state of BEC after detecting $n_{c}$ and $n_{d}$ photons using
(48), (B), and (C) in (20) becomes

$\displaystyle\lvert\psi^{\mathrm{approx}}_{n_{c},n_{d}}\rangle\approx$
$\displaystyle\left(\frac{1+N\lvert\eta_{l}\eta_{r}\rvert^{2}X_{0}}{2\pi
N\lvert\eta_{l}\eta_{r}\rvert^{2}}\right)^{1/4}$ $\displaystyle\times$
$\displaystyle\sum_{k}\exp\Bigg{[}-\frac{N|\eta_{l}\eta_{r}|^{2}X_{0}+1}{4N|\eta_{l}\eta_{r}|^{2}}$
$\displaystyle\times$
$\displaystyle\Bigg{(}k-\frac{N|\eta_{l}|^{2}+N|\eta_{l}\eta_{r}|^{2}X_{0}(N/2+x_{0})}{N|\eta_{l}\eta_{r}|^{2}X_{0}+1}\Bigg{)}^{2}\Bigg{]}$
$\displaystyle\times$ $\displaystyle
e^{ik\varphi}e^{i(n_{c}\phi_{c}(k)+n_{d}\phi_{d}(k))}\lvert k,N-k\rangle,$
(50)

where $\varphi$ is the initial relative phase of the BEC in left and right
well, $\phi_{c}(k)$ is the argument of $\alpha_{c}(k)$, and $\phi_{d}(k)$ is
the argument of $\alpha_{d}(k)$. It immediately follows that

$\displaystyle P(k\lvert n_{c},n_{d})$
$\displaystyle=\sqrt{\frac{N|\eta_{l}\eta_{r}|^{2}X_{0}+1}{2\pi
N|\eta_{l}\eta_{r}|^{2}}}\exp\Bigg{[}-\frac{N|\eta_{l}\eta_{r}|^{2}X_{0}+1}{2N|\eta_{l}\eta_{r}|^{2}}$
$\displaystyle\times\Bigg{(}k-\frac{N|\eta_{l}|^{2}+N|\eta_{l}\eta_{r}|^{2}X_{0}(N/2+x_{0})}{N|\eta_{l}\eta_{r}|^{2}X_{0}+1}\Bigg{)}^{2}\Bigg{]}.$
(51)

In order to simplify the expressions, we will work about the most probable
outcome, with $n_{d}=n_{c}$. We will use balanced detection scheme
$\alpha_{l}=\alpha_{r}$ (i.e. there is no relative phase between the light
beams). At these values, $x_{0}=0$ and $X_{0}=8g^{2}t^{2}n_{c}$. Substituting
these values in (22), (C) and (C) give the expression (III.2) in the main
text.

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