††thanks: The indicated authors are joint first authors††thanks: The indicated
authors are joint first authors

# Optical and atomic decoherence in entangled atomic ensembles generated by
quantum nondemolition measurements

Shuai Gao  Joint International Research Laboratory of Information Display and
Visualization, School of Electronic Science and Engineering, Southeast
University, Nanjing 210096, China  State Key Laboratory of Precision
Spectroscopy, School of Physical and Material Sciences, East China Normal
University, Shanghai 200062, China  Shuang Li  State Key Laboratory of
Precision Spectroscopy, School of Physical and Material Sciences, East China
Normal University, Shanghai 200062, China  Manish Chaudhary  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  Matthew
Prest  New York University Shanghai, 1555 Century Ave, Pudong, Shanghai
200122, China  Ebubechukwu O. Ilo-Okeke  New York University Shanghai, 1555
Century Ave, Pudong, Shanghai 200122, China  Department of Physics, School of
Science, Federal University of Technology, P. M. B. 1526, Owerri, Imo state
460001, Nigeria  Valentin Ivannikov  New York University Shanghai, 1555
Century Ave, Pudong, Shanghai 200122, China  Tim Byrnes<EMAIL_ADDRESS>New York University Shanghai, 1555 Century Ave, Pudong, Shanghai 200122, China
State Key Laboratory of Precision Spectroscopy, School of Physical and
Material Sciences, East China Normal University, Shanghai 200062, 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 study the effects of decoherence in the form of optical phase diffusion,
photon loss and gain, and atomic dephasing in entangled atomic ensembles
produced via quantum nondemolition (QND) measurements. For the optical
decoherence channels, we use the technique of integration within ordered
operators (IWOP) to obtain the Kraus operators that describe the decoherence.
We analyze the effect of different decoherence channels on a variety of
quantities such as the variances of the spin operators, entanglement and
correlation criteria, logarithmic negativity, and the Bell-CHSH inequality. We
generally find a smooth decay of correlations and entanglement in the presence
of decoherence. We find that various quantities retain showing non-classical
properties under all three types of decoherence, in the short interaction time
range. Our results show that such QND measurements are one of the most
promising methods for entanglement generation between two Bose-Einstein
condensates.

##  I Introduction

Quantum entanglement [1, 2, 3, 4] is one of the central concepts in quantum
mechanics, and in quantum information science is considered to be an
indispensable resource to perform tasks that exceed the performance of
classical physics [5, 6, 7, 8, 9, 10, 11, 12, 13]. Quantum entanglement has
been generally associated with the microscopic world, but with the progress of
technology, quantum entanglement has been observed increasingly in the
macroscopic world [14, 15, 16, 17] Spin squeezing is a prime example of a
practical application of such macroscopic entanglement [18, 19, 20, 21], where
the quantum noise of an observable is suppressed below the standard quantum
limit [22, 14, 15, 23, 24, 25]. Optical systems were the first system where
such squeezed states were realized in experiment [26, 27, 28]. One well-
studied way that such squeezed states can be produced is through nonlinear
effects such as the Kerr effect [29, 30]. There are however other ways in
which squeezing can be generated, including quantum nondemolition (QND)
measurements [31], adiabatic transitions [32], state-dependent forces [33],
Rydberg excitations [34], splitting a single squeezed BEC [35], cavity QED
[17, 36, 32, 37, 38]. Squeezed states have also been obtained in some other
systems such as atomic ensembles [39, 40]. Using QND measurements squeezed
states have been generated in atomic ensembles [41, 42, 43, 44, 45]. There are
some other systems like mechanical resonators where squeezing has also been
experimentally realized [46, 47, 48, 49, 50, 51].

Squeezed states are a type of multipartite entangled state [52] where the
collective degrees of freedom, rather than the individual identity of the
underlying particles is the quantity of interest. In the context of Bell
correlations, however, the identity and the location of the particules is a
crucial property and underlies quantification of the non-local correlations
using the Clauser-Horne-Shimony-Holt (CHSH) inequality[53]. For multipartite
states, the two-mode squeezed state is one of the most important type of
squeezed states which exhibit Einstein-Podolsky-Rosen (EPR) correlations. Such
states can also be produced in other systems in addition to optical systems,
where the first demonstrations of atomic ensembles were pioneered by Polzik
and co-workers [54, 55, 56, 57, 58, 59, 60]. In these experiments, two
spatially separated Cesium gas clouds [61] were entangled using QND
measurements [62, 63, 64, 58, 65, 66]. Applications of the entanglement, such
as quantum teleportation for continuous variables [67] was successfully
realized [68, 54, 69]. For Bose-Einstein condensates (BECs), the creation of
many-particle entanglement in single BECs has been studied in various contexts
[70, 71, 72, 73]. Entanglement in different spatial regions of a single BEC
has been observed [14, 15, 16, 17]. However, the entanglement between two
spatially distinct BECs has not been realized in experiments to date. Numerous
entangling schemes between BECs have been proposed, ranging from methods using
adiabatic transitions [32], state-dependent forces [33], Rydberg excitations
[34], splitting a single squeezed BEC [35], and cavity QED [17, 36, 32, 37,
38]. Generating entanglement is elementary to various quantum information
tasks based on atomic ensembles, such as quantum teleportation [74, 75],
remote state preparation, clock synchronization [76, 77], and quantum
computing [78, 79, 38].

In this paper, we calculate the effects of decoherence in QND measurement-
based entanglement generation between atomic ensembles [80, 81, 82].
Previously, a protocol generating entanglement between BECs based on QND
measurement induced entanglement was theoretically analyzed [80]. The key
difference of this theory of QND measurements in comparison to past theories
[23, 56, 59, 83, 84] is that the QND dynamics is exactly solved such that
Holstein-Primakoff (HP) approximation is not used, and non-Gaussian effects
can be examined. Another difference is the geometry of the light that is used
(see Fig. 1) where a Mach-Zehnder configuration is used, instead of the
sequential configuration of [85, 56]. This is advantageous in the case where
there are many ensembles, as the entanglement can be generated in a scalable
all-to-all configuration. We investigate the effect of optical phase
diffusion, loss and gain of the light, and atomic dephasing on the entangled
state that is generated by the QND scheme. For the optical decoherence
channels, we use the integral within the ordered operator (IWOP) approach [86,
87] to analytically evaluate the decoherence effects. This allows for a direct
derivation of the Kraus operators for the decoherence channels, giving a
powerful way of evaluating the effects. We note that we have previously
performed a study of decoherence on QND-induced entangled states of BECs [81].
This work differs from our previous work in that we analyze phase diffusion
and incoherent gain which was not done before. For the atomic dephasing, we
also analyze several different quantities that were not calculated before,
such as the violation of CHSH inequality [88, 89].

The structure of this paper is as follows. Sec. II briefly reviews the QND
entangling protocol, defining the physical model and describing the basic
system we are working on. Sec. III examines the effect of optical phase
diffusion on the optical modes involved in the QND measurement. In Sec. IV,
the effects of photonic loss and gain are examined. In Sec. V, we analyze the
effects of atomic dephasing and show its effect on the Wineland squeezing
parameter and the Bell-CHSH inequality. Finally, the results are then
summarized in Sec. VI.

Figure 1:  Experimental scheme for generating entanglement between two atomic
ensembles via QND measurements. A coherent light pulse $|\alpha\rangle$ enters
the first beam splitter of a Mach-Zehnder interferometer. Each atomic ensemble
or BEC has two relevant internal energy states that are populated by atoms,
with corresponding bosonic annihilation operators $g_{j},e_{j}$, where
$j\in\\{1,2\\}$ labels each atomic ensemble or BEC. The optical mode is split
into two modes labeled by $a_{j}$, $j\in\\{1,2\\}$ after the first beam
splitter. Then each optical mode interacts with the atoms via the QND
Hamiltonian. After the second beam splitter, the photons are detected with
counts $n_{c}$ and $n_{d}$.

##  II QND Entangling scheme

In this section, we briefly summarize the theory for QND-induced entanglement
generation between two atomic ensembles or BECs [80].

###  II.1 QND Measurement Model and Hamiltonian

We first give a brief description of the QND entangling scheme as shown in
Fig. 1. Two atomic ensembles or BECs are placed in well-separated traps such
as separate magnetic traps on an atom chip or two optical dipole traps [90,
91, 38]. The atoms have two internal energy states that are populated. For
instance, a suitable choice of the two internal states for ${}^{87}\text{Rb}$
could be two hyperfine ground states ($F=1,m_{F}=-1$ and $F=2,m_{F}=+1$) of an
atom. These have bosonic annihilation operators $g_{j},e_{j}$ respectively,
where $j\in\\{1,2\\}$ is used to distinguish the two atomic ensembles or BECs.

We introduce the collective spin using the Schwinger boson operators, which
are defined by the form as follows

$\displaystyle S^{x}_{j}$
$\displaystyle=e^{{\dagger}}_{j}g_{j}+g^{{\dagger}}_{j}e_{j},$ $\displaystyle
S^{y}_{j}$ $\displaystyle=-ie^{{\dagger}}_{j}g_{j}+ig^{{\dagger}}_{j}e_{j},$
$\displaystyle S^{z}_{j}$
$\displaystyle=e^{{\dagger}}_{j}e_{j}-g^{{\dagger}}_{j}g_{j}.$ (1)

The commutation relations for Schwinger boson operators are
$[S^{l},S^{m}]=2i\epsilon_{lmn}S^{n}$, where $\epsilon_{lmn}$ is the
completely anti-symmetric Levi-Civita tensor with $l,m,n\in\\{x,y,z\\}$. The
spin operators (II.1) most directly describe atoms within a BEC, where the
atoms are indistinguishable bosons. The algebra of spin operators for an
ensemble of distinguishable atoms is equivalent as long as the state of the
system remains in the subspace that is symmetric under particle interchange
[40]. Hence we will use the above formalism to equally describe an atomic
ensemble as well as a BEC.

We define the spin coherent state of the atomic ensemble or BECs to be

$\displaystyle|\theta,\phi\rangle\rangle_{j}$
$\displaystyle\equiv\frac{1}{\sqrt{N!}}\left(e^{\dagger}_{j}\cos\frac{\theta}{2}+e^{i\phi}g^{\dagger}_{j}\sin\frac{\theta}{2}\right)^{N}|\text{vac}\rangle$
$\displaystyle=\sum_{k=0}^{N}\sqrt{\binom{N}{k}}\cos^{k}(\frac{\theta}{2})\sin^{N-k}(\frac{\theta}{2})e^{i(N-k)\phi}|k\rangle_{j}.$
(2)

The direction of the polarization is given by the angles $0\leq\theta\leq\pi$,
$-\pi\leq\phi\leq\pi$ on the Bloch sphere. The initial state of atoms is taken
to be spins polarized in the $x$-direction

$\displaystyle|\frac{\pi}{2},0\rangle\rangle_{1}|\frac{\pi}{2},0\rangle\rangle_{2}.$
(3)

Meanwhile, the Fock states on the $j$th atomic ensemble are defined as

$\displaystyle{|k\rangle}_{j}=\frac{\left(e_{j}^{\dagger}\right)^{k}\left(g_{j}^{\dagger}\right)^{N-k}}{\sqrt{k!(N-k)!}}|\operatorname{vac}\rangle.$
(4)

The initial state of optical mode $b$ is a coherent light that could be
defined as

$|\alpha\rangle_{b}\equiv e^{-|\alpha|^{2}/2}e^{\alpha
b^{\dagger}}|\text{vac}\rangle.$ (5)

The coherent light enters the first beam splitter and is split into two modes
labeled by $a_{j}$, where $j\in\left\\{1,2\right\\}$, evolving to the state

$\displaystyle|\frac{\alpha}{\sqrt{2}}\rangle_{1}|\frac{\alpha}{\sqrt{2}}\rangle_{2}.$
(6)

Meanwhile, the initial state of the atomic ensembles is prepared in the
$S^{x}$ direction

$\displaystyle|\Psi_{0}\rangle$
$\displaystyle=\left.|\frac{\pi}{2},0\rangle\rangle_{1}\right.\left.|\frac{\pi}{2},0\rangle\rangle_{2}\right.$
$\displaystyle=|k_{x}=N\rangle_{1}|k_{x}=N\rangle_{2}.$ (7)

Each optical mode interacts with a BEC that is separated in two arms of a
Mach-Zehnder interferometer via the QND Hamiltonian

$\displaystyle H=\frac{\hbar\Omega}{2}\left(S_{1}^{z}-S_{2}^{z}\right)J^{z}.$
(8)

This entangles the light and the atomic spin degrees of freedom.

###  II.2 Entangled wavefunction after QND measurement

Evolving the initial states (3) and (6) with the Hamiltonian (8), and evolving
the photons through the second beamsplitter gives the state [80]

$\displaystyle|\psi(\tau)\rangle=$
$\displaystyle\frac{1}{2^{N}}\sum_{k_{1},k_{2}=0}^{N}\sqrt{\binom{N}{k_{1}}\binom{N}{k_{2}}}|k_{1},k_{2}\rangle$
$\displaystyle\times\left|\alpha\cos\left(k_{1}-k_{2}\right)\tau\right\rangle_{c}\left|-i\alpha\sin\left(k_{1}-k_{2}\right)\tau\right\rangle_{d}.$
(9)

Then projecting the light on Fock states gives rise to a modified atomic
wavefunction [80]

$\displaystyle|{\psi}_{n_{c}n_{d}}(\tau)\rangle$
$\displaystyle=\frac{1}{\sqrt{\cal
N}}\sum_{k_{1},k_{2}=0}^{N}\sqrt{{{N}\choose{k_{1}}}{{N}\choose{k_{2}}}}$
$\displaystyle\times
C_{n_{c}n_{d}}^{\alpha}[(k_{1}-k_{2})\tau]|k_{1},k_{2}\rangle.$ (10)

Here the dimensionless time is defined by $\tau=\Omega t$. The $C$-function is
defined by

$\displaystyle
C_{n_{c}n_{d}}^{\alpha}(\chi)\equiv\frac{\alpha^{n_{c}+n_{d}}e^{-|\alpha|^{2}/2}}{\sqrt{n_{c}!n_{d}!}}\cos^{n_{c}}\chi\sin^{n_{d}}\chi,$
(11)

and the normalization factor ${\cal N}$ is

$\displaystyle{\cal
N}=\sum_{k_{1},k_{2}=0}^{N}{{N}\choose{k_{1}}}{{N}\choose{k_{2}}}|C_{n_{c}n_{d}}[(k_{1}-k_{2})\tau]|^{2}.$
(12)

The most likely photon counts that are measured are distributed around
$n_{c}+n_{d}\approx|\alpha|^{2}$, due to photon number conservation. The
$C$-function takes the form of a Gaussian and can be approximated in the short
interaction time regime as [92]

$\displaystyle
C_{n_{c}n_{d}}(\chi)\propto\exp\left(-\frac{[|\chi|-\frac{1}{2}\arccos(\frac{n_{c}-n_{d}}{n_{c}+n_{d}})]^{2}}{2\sigma_{n_{c}n_{d}}^{2}}\right),$
(13)

where

$\displaystyle\sigma_{n_{c}n_{d}}\approx\frac{1}{\sqrt{n_{c}+n_{d}}}.$ (14)

For outcomes $n_{c}\gg n_{d}$ and large photon numbers $|\alpha|^{2}\gg 1$,
the state is then well approximated by

$\displaystyle|{\psi}_{n_{c}\gg n_{d}}(\tau)\rangle\approx\left(\frac{4}{\pi
N}\right)^{1/4}\sum_{k=0}^{N}e^{-\frac{2}{N}\left(k-\frac{N}{2}\right)^{2}}|k\rangle|k\rangle.$
(15)

This is an entangled state due to the correlations in Fock states that are
present.

##  III Optical phase diffusion

The first type of decoherence that we will analyze is phase diffusion for the
optical modes. During the QND scheme as shown in Fig. 1, the optical modes may
undergo phase diffusion. The state prior to photon detection thus becomes a
mixed state, which results in a source of decoherence for the final atomic
wavefunction. We will use the IWOP technique to obtain the exact density
matrix after applying the phase diffusion channel. We illustrate the effects
of the decoherence on various quantities such as logarithmic negativity,
variances of spin correlators, probability distributions, correlation-based
criteria, and Bell-CHSH inequalities.

###  III.1 Phase diffusion master equation

An optical mode $c$ undergoing phase diffusion is described by the master
equation [93, 94]

$\displaystyle\frac{d}{dt}\rho=-\kappa\left(c^{\dagger}c\rho+\rho
cc^{\dagger}-c\rho c^{\dagger}-c^{\dagger}\rho c\right),$ (16)

where $\kappa$ is the phase diffusion rate. We use the methods of Ref. [95] to
exactly solve the master equations. This is done by transforming the density
operators into ordinary functional equations by using the representation of
thermal entanglement states. Exact solutions to the master equations are
obtained using the technique of integration within ordered operators (IWOP).
The density matrix including the effects of phase diffusion is given as [96]

$\displaystyle\rho(t)=\sum_{m,n=0}^{\infty}M_{m,n}^{\text{PD}}(t)\rho(0){M_{m,n}^{\text{PD}}}^{\dagger}(t)$
(17)

where the Kraus operators for phase diffusion are defined as

$\displaystyle M_{m,n}^{\text{PD}}(t)=\sqrt{\frac{1}{m!n!}\frac{(\kappa
t)^{m+n}}{(\kappa t+1)^{m+n+1}}}c^{\dagger m}\left(\frac{1}{1+\kappa
t}\right)^{c^{\dagger}c}c^{n}.$ (18)

Here, $\rho(0)$ is the initial density matrix without decoherence.

In our case, phase diffusion is applied at the end of the QND entanglement
preparation process. We note that due to the two modes $c,d$ there are two
sets of Kraus operators that need to be applied. The initial density matrix in
our case is therefore

$\displaystyle\rho_{0}$ $\displaystyle=|\psi(\tau)\rangle\langle\psi(\tau)|,$
(19)

where $|\psi(\tau)\rangle$ is the state as defined in (9). Applying the phase
diffusion and performing the photonic measurements gives the density matrix

$\displaystyle\rho_{\text{PD}}=$ $\displaystyle\langle n_{c}|\langle
n_{d}|\sum_{m,n,m^{\prime},n^{\prime}=0}^{\infty}M_{m,n}^{\text{PD},c}(t)M_{m^{\prime},n^{\prime}}^{\text{PD},d}(t)$
$\displaystyle\times\rho_{0}{M_{m^{\prime},n^{\prime}}^{\text{PD},d}}^{\dagger}(t){M_{m,n}^{\text{PD},c}}^{\dagger}(t)|n_{c}\rangle|n_{d}\rangle,$
(20)

where we have labeled a mode on the Kraus operator with a superscript. The
explicit evaluated expression is given in Appendix A. The above expression
assumes that the time evolution of the QND interaction $\tau$ is equal to the
time that phase diffusion occurs for in (16). The rate at which the QND
interaction occurs $\Omega$ and the dephasing rate $\kappa$ can be adjusted
according to the ratio $\tilde{\kappa}=\kappa/\Omega$. Using this density
matrix we calculate various quantities of interest as below.

###  III.2 Effect of decoherence on various quantities

####  III.2.1 Entanglement

We first examine the effect of phase diffusion on the entanglement between the
atomic ensembles. We use logarithmic negativity to quantify the entanglement,
which is an entanglement monotone for mixed states [97, 98]

$\displaystyle E=\log_{2}\left\|\rho^{T_{2}}\right\|,$ (21)

where $\rho^{T_{2}}$ is partial transpose on the second BEC. We calculate the
normalized logarithmic negativity, in relation to the maximum value that is
possible between two $N+1$ dimensional systems

$\displaystyle E_{\max}=\log_{2}(N+1).$ (22)

In Fig. 2(a), we show the dependence of the entanglement for various values of
the dimensionless phase diffusion rate $\tilde{\kappa}$. We observe that for
small values of $\tilde{\kappa}\leq 0.1$, the entanglement is relatively
unaffected for all interaction times. The curve shows a characteristic
“devil’s crevasse” form as observed in previous works [99, 85]. With
increasing $\tilde{\kappa}$, the entanglement shows an exponential decay, and
the sharp dips are smoothed out. The entanglement rises sharply corresponding
to the generation of entanglement using the QND scheme within the dephasing
time. For longer times, the entanglement is increasingly affected by the
dephasing, due to the longer times that the decoherence has to degrade the
state. Beyond $\tilde{\kappa}\geq 10$, the devil’s crevasse structure is no
longer visible. We note that there are no decoherence effects at times
$\tau=n\pi$, due to such times having no entanglement even in the decoherence-
free case. This does not necessarily mean that the state is unaffected since
the entanglement is only one aspect of the state and may still affect the
state.

Figure 2: Entanglement as quantified by the normalized logarithmic negativity
(21) for the state (9) in the presence of different kinds of decoherence
channels. (a) Entanglement in the phase diffusion channel versus time with
decoherence $\tilde{\kappa}=\kappa/\Omega$ as marked. (b) Entanglement for the
photonic loss/gain channel versus time with decoherence rates as marked. We
set the photonic gain rate to be $g/\Omega=0.1$ and show various photon loss
rates $\tilde{\gamma}=\gamma/\Omega$. The parameters used in (a) are
$N=10,n_{c}=20,n_{d}=0,\alpha=\sqrt{20}$ while
$N=10,n_{c}=100,n_{d}=0,\alpha=10$ for other plots.

####  III.2.2 Probability distribution

We next consider the probability distribution of measuring state (20) in
various bases

$\displaystyle p_{l}(k_{1},k_{2})=\langle
k_{1},k_{2}|^{(l)}\rho|k_{1},k_{2}\rangle^{(l)}.$ (23)

where we use the notation

$\displaystyle|k_{1},k_{2}\rangle^{(l)}=|k_{1}\rangle^{(l)}\otimes|k_{2}\rangle^{(l)}$
(24)

for $l\in\\{x,y,z\\}$. The Fock states in the $x$ and $y$ bases are defined as

$\displaystyle|k\rangle^{(x)}$ $\displaystyle=e^{-iS^{y}\pi/4}|k\rangle^{(z)}$
$\displaystyle|k\rangle^{(y)}$
$\displaystyle=e^{-iS^{z}\pi/4}e^{-iS^{y}\pi/4}|k\rangle^{(z)}.$ (25)

where the Fock state $|k\rangle^{(z)}$ is the same as that defined in (4). The
Fock states in the $x$ and $y$-basis can be written in terms of the $z$-basis
Fock states using the relations given in Appendix B.

Figure 3: Probability distributions after measuring state (10) with
decoherence in various bases, as defined in (23). (a) The phase diffusion
channel with dissipative coefficient $\tilde{\kappa}\in\\{0,1,20\\}$ from top
to bottom, for the state (20). (b) The photonic loss/gain channel with loss
coefficient $\tilde{\gamma}\in\\{0.2,1,10\\}$ from top to bottom, and
$g/\Omega=0.1$. The parameters for all plots are
$N=10,n_{c}=100,n_{d}=0,\tau=0.1$.

Figure 3(a) shows the probabilities with phase diffusion channel for various
amounts of decoherence $\tilde{\kappa}$ at times $\tau=0.1$, corresponding to
times where two-spin squeezing is visible. For the decoherence-free case, we
see a pattern of correlations for $l=x,z$ and anti-correlations for $l=y$. As
the dephasing is increased, the pattern of correlations and anti-correlations
are modified. The spin correlations in the $z$-direction are decreased, and
the diagonal probability distribution is modified to circular distribution
which exhibits no correlations. However, the correlations do not become
completely removed even for very large amounts of phase diffusion, such as
$\tilde{\kappa}=20$. Interestingly, the $y$-correlations appear to be improved
under phase diffusion, where the width of the anti-correlations is reduced. We
confirm this effect in the variances in the following section. For the
$x$-correlations, from the probability distribution, there is little visible
change under the phase diffusion, except for the removal of interference
fringes. Overall, we observe that even under very strong phase diffusion, the
correlations remain relatively unaffected for the probability distributions
that are shown in Fig. 3(a). This is likely due to the relatively short
interaction time that is examined, where the type of entangled state is
relatively robust in the presence of decoherence [99].

####  III.2.3 Variances

Variances of spin correlators $S^{x}_{1}-S^{x}_{2}$, $S^{y}_{1}+S^{y}_{2}$,
and $S^{z}_{1}-S^{z}_{2}$ under the phase diffusion channel are shown as a
function of interaction time in Fig. 4. We choose these observables since they
are expected to show squeezing effects. Comparing the $\tilde{\kappa}=1$ with
the decoherence-free case of $\tilde{\kappa}=0$, we observe that there is very
little difference for all spin correlators at most evolution times. The
largest differences arise at times $\tau=\pi$, where the state involves
entangled Schrodinger cat states [81], which are particularly susceptible to
decoherence. The variance of $S_{1}^{x}-S_{2}^{x}$ remains large, whereas this
drops to zero in the decoherence-free case. Similarly, the variance of
$S_{1}^{y}+S_{2}^{y}$ in the decoherence-free case reaches a large value at
$\tau=\pi$ but is reduced when phase diffusion is introduced. Meanwhile, the
variance of $S_{1}^{z}-S_{2}^{z}$ is relatively affected at all times when
comparing the $\tilde{\kappa}=1$ and decoherence-free cases. However, as the
phase diffusion rate is further increased, generally the variance of
$S_{1}^{z}-S_{2}^{z}$ is increased, which can be understood as the loss of
correlations. Meanwhile, the $S_{1}^{x}-S_{2}^{x}$ and $S_{1}^{y}+S_{2}^{y}$
correlations are relatively unaffected with a further increase of phase
diffusion beyond $\tilde{\kappa}=1$.

Figure 4: The variance of the state (20) with phase diffusion decoherence for
the operators $S_{1}^{x}-S_{2}^{x},S_{1}^{y}+S_{2}^{y}$ and
$S_{1}^{z}-S_{2}^{z}$. Variances are plotted for (a)(c)(e) long time scales;
(d)(e)(f) zoomed into short time scales. The parameters are
$N=10,n_{c}=100,n_{d}=0,\alpha=\sqrt{n_{c}}=10$.

####  III.2.4 Correlation-based criteria

While logarithmic negativity is a well-established method for detecting
entanglement, in an experimental setting it requires full tomography of the
quantum state which has a large overhead and may not be practically feasible
in many cases. A preferable approach is to detect entanglement with low-order
spin expectation values, which are more readily measured and are relatively
insensitive to imperfect spin measurements. Here we describe and calculate
three correlation-based methods to detect entanglement and EPR steering.

The first correlation-based entanglement criterion is the Wineland squeezing
parameter

$\displaystyle\xi$ $\displaystyle=\frac{{2(\Delta
S_{\bot}^{2})}_{\text{min}}}{\left|\left\langle S\right\rangle\right|}$
$\displaystyle\geq 1\hskip 28.45274pt\text{(separable)}$ (26)

where $S_{\bot}$ is any component perpendicular to the average total spin and
$\Delta S_{\bot}^{2}$ is minimum fluctuation in the direction of the vertical
component. The coefficient $\xi$ corresponds to the degree of squeezing. A
smaller value means a better degree of squeezing. The inequality (26) is
derived assuming a separable state, hence $\xi<1$ signals the presence of
entanglement.

In a previous study [80], it was found that the Hofmann-Takeuchi criterion
[100] is one of the most effective spin correlation-based detection methods
for entanglement. In our case, it is defined as

$\displaystyle{\cal{C}}_{\text{ent }}$
$\displaystyle\equiv\frac{\operatorname{Var}\left(S_{1}^{x}-S_{2}^{x}\right)+\operatorname{Var}\left(S_{1}^{y}+S_{2}^{y}\right)+\operatorname{Var}\left(S_{1}^{z}-S_{2}^{z}\right)}{4N}$
$\displaystyle\geq 1.\hskip 28.45274pt\text{(separable)}$ (27)

Again, the inequality (27) is derived assuming separable states, hence
detecting ${\cal{C}}_{\text{ent }}<1$ signals the presence of entanglement.

Finally, EPR steering is a subclass of entangled states where one party can
nonlocally affect the other party’s state through measurements [101, 102,
103]. A correlated-based inequality that detects EPR steering is [104]

$\displaystyle{\cal{C}}_{\text{steer }}^{1\rightarrow 2}$
$\displaystyle\equiv\frac{\operatorname{Var}\left(S_{1}^{y}+S_{2}^{y}\right)\operatorname{Var}\left(S_{1}^{z}-S_{2}^{z}\right)}{\left\langle
S_{1}^{x}\right\rangle^{2}}$ $\displaystyle\geq 1\hskip 28.45274pt\text{(un-
steerable)}$ (28)

A violation of the inequality (28) implies the existence of EPR steering from
BEC 1 to BEC 2.

In Fig.5, we compare the above three different correlation-based criteria.
Fig.5(a) shows Wineland squeezing parameter in the region where entanglement
is shown. The time region of sustained squeezing for which $\xi<1$ is reduced
as the phase diffusion rate $\tilde{\kappa}$ is increased. This is consistent
with the logarithmic negativity calculations of Fig. 2(a), where the phase
diffusion has the effect of reducing entanglement. Now turning to the Hofmann-
Takeuchi criterion, we find it can detect entanglement for a wide range of
times except for some particular points where ${\cal{C}}_{\text{ent }}\approx
1$. As with the Wineland squeezing criterion, generally, the degree of
violation of the inequality (27) is reduced with an increasing phase diffusion
rate. It however remains a powerful method to detect entanglement, and in
comparison to the Wineland squeezing criterion, it detects entanglement in a
much wider range.

Finally, for the EPR steering criterion, the effect of phase diffusion is to
decrease the region where violation of the inequality (28) occurs. As would be
expected, for EPR steering it is generally more difficult to find a violating
region, which we attribute to the fact that EPR steerable states are a more
specialized class of entangled states. Beyond $\tilde{\kappa}>10$, the
criterion fails to detect any entangled states, while the Wineland squeezing
and the Hofmann-Takeuchi criterion continue to detect entangled states.

Figure 5: Detection of entanglement and EPR-steering using correlation-based
criteria. (a) The Wineland squeezing criterion (26) for the state (20) with
phase diffusion decoherence. (b) The Hofmann-Takeuchi criterion (27) for the
state (20) with phase diffusion decoherence. (c) The EPR steering criterion
(28) for the state (20) with phase diffusion decoherence. The parameters used
here are $N=10,n_{c}=100,n_{d}=0,\alpha=\sqrt{n_{c}+n_{d}}=10$. The
dissipative coefficient is as marked. We set $\Omega=1$ such that the
decoherence rate is in units of $\Omega$, and the time units are $1/\Omega$.

####  III.2.5 Bell-CHSH inequality

Figure 6: Bell-CHSH correlations for the state (10) with two types of optical
deocherence. (a) Time dependence of (29) for phase diffused state (20) for
$N=5$ and $\theta_{B}=0.37$. (b) Optimal values of (29) concerning $\tau$ and
$\theta_{B}$ for various $N$ with phase diffused state (20). (c) Time
dependence of (29) for the state (34) which has undergone photon loss and
gain. (d) Optimal values of (29) concerning $\tau$ and $\theta_{B}$ for
various $N$ under photon loss and gain (34). Shaded regions indicate the
regions where there is a violation of the CHSH inequality. The common
parameters are $n_{c}=100,n_{d}=0,\alpha=10$.

In Ref.[105], it was shown that the Bell-CHSH inequality can be violated using
the state (10). The form of the CHSH inequality that was used, reads as

$\displaystyle\mathcal{C}_{\text{CHSH}}\equiv$
$\displaystyle\Big{|}\left\langle
M_{1}^{(1)}M_{2}^{(1)}\right\rangle+\left\langle
M_{1}^{(1)}M_{2}^{(2)}\right\rangle$ $\displaystyle-\left\langle
M_{1}^{(2)}M_{2}^{(1)}\right\rangle+\left\langle
M_{1}^{(2)}M_{2}^{(2)}\right\rangle\Big{|}$ $\displaystyle\leq$ $\displaystyle
2\hskip 28.45274pt\text{(local HV)},$ (29)

which is valid for a local hidden variable (HV) theory and

$\displaystyle M_{1}^{(1)}$ $\displaystyle=\text{sgn}(S^{z}_{1})$
$\displaystyle M_{1}^{(2)}$
$\displaystyle=\text{sgn}(S^{z}_{1}\cos\theta_{B}+S^{y}_{1}\sin\theta_{B})$
$\displaystyle M_{2}^{(1)}$
$\displaystyle=\text{sgn}(S^{z}_{2}\cos\frac{\theta_{B}}{2}+S^{y}_{2}\sin\frac{\theta_{B}}{2})$
$\displaystyle M_{2}^{(2)}$
$\displaystyle=\text{sgn}(S^{z}_{2}\cos\frac{\theta_{B}}{2}-S^{y}_{2}\sin\frac{\theta_{B}}{2}).$
(30)

Here, $\text{sgn}(x)$ takes the sign of the eigenvalue of the operator. Using
the sign of spin operators is an experimentally viable observable since it is
largely insensitive to atom number fluctuations [106]. The superscripts
(1),(2) are the two measurement choices that can be made on the two atomic
ensembles. The angle $\theta_{B}$ must be optimized to find the largest
violation and was found to be well-approximated by the empirical relation
$\theta_{B}\approx(3.2/N+1.7/N^{2})/(1+2.1/N)$ [105].

In Ref. [105], the Bell-CHSH inequality was studied without considering
decoherence effects. Figure 6(a) shows the time dependence of the left-hand
side of the criterion (29) with phase channel decoherence for $N=5$. The
shaded area indicates the classical region of Bell-CHSH inequality.
Remarkably, we see very little change in the values of
$\mathcal{C}_{\text{CHSH}}$, and above a particular interaction time, CHSH
violations are seen despite the presence of strong phase diffusion of the
optical modes. In Fig. 6(b), the optimal values of (29) are as a function of
$1/N$. We observe the violations occur for all $N$ in the presence of strong
phase diffusion. We again find that decoherence has little effect on the
optimal Bell-CHSH inequality. While this is a positive from an experimental
context, we note that for large ensemble sizes the level of violation is
reduced, hence an increased precision of $\mathcal{C}_{\text{CHSH}}$ will be
required to observe any violation.

##  IV Photonic Loss and Gain

We now consider a loss and gain channel for the photons during the QND
measurement induced entanglement generation as described in Sec. II.2.
Physically, the loss of photons may occur during the transmission of the light
through the Mach-Zehnder interferometer and is potentially a major source of
decoherence to the scheme. Photonic gain may also occur where ambient
incoherent photons may enter the modes, also acting as a source of
decoherence.

###  IV.1 Photon loss and gain master equation

The master equation for photonic loss and gain is written as [107, 108, 109]

$\displaystyle\frac{d\rho}{dt}=\gamma\left(2c\rho
c^{\dagger}-c^{\dagger}c\rho-\rho c^{\dagger}c\right)+g\left(2c^{\dagger}\rho
c-cc^{\dagger}\rho-\rho cc^{\dagger}\right),$ (31)

where $\gamma$ is the photon loss rate and $g$ is the photon gain rate.
Solving this master equation using the IWOP technique can be written in terms
of Kraus operators as [110, 111]

$\displaystyle\rho(t)$
$\displaystyle=\sum_{p,q=0}^{\infty}M_{p,q}^{\text{LG}}(t)\rho(0){M_{p,q}^{\text{LG}}}^{\dagger}(t)$
(32)

where

$\displaystyle M_{p,q}^{\text{LG}}(t)$
$\displaystyle=\sqrt{\frac{\gamma^{p}g^{q}T_{1}^{p+q}T_{3}}{p!q!T_{2}^{2q}}}e^{c^{\dagger}c\ln
T_{2}}c^{\dagger q}c^{p}$ $\displaystyle T_{1}$
$\displaystyle=\frac{1-e^{-2(\gamma-g)t}}{\gamma-ge^{-2(\gamma-g)t}}$
$\displaystyle T_{2}$ $\displaystyle=\frac{(\gamma-g)e^{-(\gamma-g)t}}{\gamma-
ge^{-2(\gamma-g)t}}$ $\displaystyle T_{3}$
$\displaystyle=\frac{\gamma-g}{\gamma-ge^{-2(\gamma-g)t}}=1-gT_{1}.$ (33)

The above expression reduces to the standard form of the Kraus operator for
loss [81] when $g=0$, where the probability of the photon loss is
$1-e^{-2\gamma t}$.

Using the above Kraus operator we may derive the effect of photonic loss and
gain on the state by applying it to (19)

$\displaystyle\rho_{\text{LG}}=$ $\displaystyle\langle n_{c}|\langle
n_{d}|\sum_{p,q,p^{\prime},q^{\prime}=0}^{\infty}M_{p,q}^{\text{LG},c}(t)M_{p^{\prime},q^{\prime}}^{\text{LG},d}(t)$
$\displaystyle\times\rho_{0}{M_{p^{\prime},q^{\prime}}^{\text{LG},d}}^{\dagger}(t){M_{p,q}^{\text{LG},c}}^{\dagger}(t)|n_{c}\rangle|n_{d}\rangle.$
(34)

The evaluated expression is given in Appendix C. As with phase diffusion, the
above expression assumes that the time evolution of the QND interaction $\tau$
is equal to the time that phase diffusion occurs for in (16). We note that
this is different to the assumption made in Ref. [81], where the probability
of photon loss was set to be a constant, and hence was a time-independent
quantity. The rate at which the QND interaction occurs $\Omega$ and the loss
rate $\gamma$ can be adjusted according to the ratios
$\tilde{\gamma}=\gamma/\Omega$ and $\tilde{g}=g/\Omega$. Using this density
matrix we calculate various quantities of interest as below.

###  IV.2 Effect of decoherence on various quantities

####  IV.2.1 Entanglement

In Fig. 2(b), we show the logarithmic negativity for a fixed photon gain rate
$\tilde{g}$ and various rates of loss $\tilde{\gamma}$. We choose such a
regime since it is the most physically relevant regime, where the dominant
decoherence mechanism is photon loss. The constant photon gain accounts for
the possibility that there is an incoherent source of photons entering the
interferometer, but we assume that this can be suitably controlled such that
is a relatively smaller effect. We again see the characteristic devil’s
crevasse structure to the entanglement as the time is varied. Generally, the
amount of entanglement decreases with longer times due to the assumption here
that the time of evolving the photonic loss and gain master equation is equal
to the QND interaction time. This results in a probability of the photon loss
that exponentially approaches 1, such that the entanglement exponentially
reduces with time.

The results shown in Fig. 2(b) are qualitatively different from what was shown
in Ref. [81], where there was little difference between the ideal entanglement
curves with and without photon loss. We attribute this to the effectively
larger photon loss that our current calculations include, compared to Ref.
[81]. In Ref. [81], the largest photon loss probability that was considered
was 0.95. In comparison, for example, the photon loss probability for $\xi=1$
at $\tau=\pi$ is 0.998. Once the photonic population is entirely removed
through photon loss, it is clear that the entangling scheme cannot work since
the state (9) no longer involves any photons and the $C$-function (11) no
longer produces an amplitude on the states. We also note that the dependence
of the phase diffusion plots in Fig. 2(a), are somewhat different from the
photon loss and gain dependence, which tends to maintain high levels of
entanglement followed by a sharp drop off. This is in contrast with phase
diffusion has the effect of decreasing the overall level of entanglement. We
again interpret this as the QND measurement being relatively robust in the
presence of photon loss until all the photons are removed, at which point the
protocol no longer is capable of generating entanglement.

####  IV.2.2 Probability distribution

Fig. 3(b) shows the probability distribution (23) for the state (34). The
ideal case without any decoherence is shown in the top row of Fig. 3(a). We
see that there is very little difference between the ideal case and the case
with photon loss $\tilde{\gamma}=0.2,1$ and gain $\tilde{g}=0.1$. At very
large loss rates $\tilde{\gamma}=10$, some difference starts to develop in the
$x$ correlations and $y$ anti-correlations but the $z$-correlations remain
visually the same. As with the phase diffusion case, the $y$ anti-correlations
tend to improve slightly for the larger photon loss cases. We do not have a
good interpretation of this effect, but this appears to be a feature of both
types of photonic decoherence that the $y$ anti-correlations are improved.

####  IV.2.3 Variances

Fig. 7(a)(c)(e) shows the variances of the state (34) for the operators
$S_{1}^{x}-S_{2}^{x},S_{1}^{y}+S_{2}^{y}$, and $S_{1}^{z}-S_{2}^{z}$
respectively for the loss and gain channel. We again consider the case with
fixed incoherent photon gain and vary the loss rate. We see that for the $x$\-
and $y$-basis correlators, there is relatively little change to the dependence
despite relatively large amounts of loss present. Some of the small structure
in the variations is removed for $\tilde{\kappa}=10$, but this is a small
effect with respect to the overall variance. For the $z$-basis correlators,
for early times, all values of the variance remain similar, and squeezing is
observed. For longer times, the variance becomes more affected by the loss,
and at $\tilde{\kappa}=10$ the variance returns to its initial value. This can
be attributed due to the fact that the states at longer evolution times tend
to be more sensitive to the decoherence such that it minimizes the
correlations.

Figure 7: Variances and correlation-based criteria for the QND entangled state
under photon loss and gain. (a)(b)(c) Expectation values
$S_{1}^{x}-S_{2}^{x},S_{1}^{y}+S_{2}^{y}$ and $S_{1}^{z}-S_{2}^{z}$; (d)
Wineland squeezing; (e) Hofmann-Takeuchi criterion; (f) EPR steering
criterion. The parameters are $N=10,n_{c}=100,n_{d}=0,\alpha=\sqrt{n_{c}}=10$.
We set $\Omega=1$ such that $\tilde{\kappa}$ and $g$ is in units of $\Omega$,
and the time units are $1/\Omega$.

####  IV.2.4 Entanglement criteria and Bell-CHSH inequality

In Fig.7(b)(d)(f), we show the three different correlation-based criteria of
the Wineland squeezing parameter (26), Hofmann-Takeuchi criterion (27), and
the EPR steering criterion (28). For the Wineland squeezing parameter, we find
a generally similar behavior to phase diffusion, where the region where
entanglement is detected tends to reduce with increasing photon loss.
Generally, the criterion only detects entanglement in the short time regime
where the initial squeezing of states occurs with respect to the spin
variances. Fig.7(d) shows the Hofmann-Takeuchi criterion, which tends to
follow the time dependence of the $S_{1}^{z}-S_{2}^{z}$ variance. This is due
to the fact that variances of $S_{1}^{x}-S_{2}^{x}$ and $S_{1}^{y}+S_{2}^{y}$
show a perfect symmetry, which means that the sum
$\text{Var}(S_{1}^{x}-S_{2}^{x})+\text{Var}(S_{1}^{y}+S_{2}^{y})$ is a
constant. We find that this is again a powerful way of detecting entanglement,
showing that entanglement is present in the system for nearly all times for
most values of $\tilde{\gamma}$. The EPR steering criterion in Fig.7(f) shows
again a similar trend where the increased photon loss causes a more limited
region where EPR steering is detected. Figure.6(c) shows the Bell-CHSH
criterion (29) as a function of time including the effects of photon loss and
gain. Again we see that there is great robustness in observing a violation,
even in the presence of strong photon loss. In Fig.6(d) we see the dependence
of the level of violation as a function of $1/N$. We again see that for all
$N$ values a Bell violation can be observed and there is very little
difference between the levels of violation including photon loss and gain and
the ideal case. The main difficulty in seeing such correlations will then be
in observing the Bell’s violation for larger atomic ensembles where the level
of violation approaches the classical bound of ${\cal C}_{\text{CHSH}}=2$.

##  V Atomic Dephasing

We now examine a third channel of decoherence, atomic dephasing, which
directly acts on the atoms to remove coherence in the atomic state. This may
physically arise due to noise sources that the atoms may experience,
originating from technical noise in the atomic traps, or due to photonic
scattering due to the QND Hamiltonian (8). In a previous study [81], the
effect of $S^{z}$ dephasing on the state (10) was examined, where quantities
such as the expectation values, variances, entanglement criteria, and
distribution were investigated. However, several quantities such as the
Wineland criterion on Bell-CHSH inequality were not examined. We discuss some
of the quantities that were not considered in Ref. [81] here, to analyze the
impact of decoherence.

###  V.1 Dephasing master equation

The master equation for atomic dephasing reads

$\displaystyle\frac{d\rho}{dt}=-\Gamma\sum_{n=1}^{2}\left[\left(S_{n}^{z}\right)^{2}\rho-2S_{n}^{z}\rho
S_{n}^{z}+\rho\left(S_{n}^{z}\right)^{2}\right],$ (35)

where $\Gamma$ is the dephasing rate and it has been assumed to be the same
for both atomic ensembles. Solving the master equation (35) exactly, the
density matrix elements evolve according to

$\displaystyle\rho_{\text{D}}=$
$\displaystyle\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}}|k_{1},k_{2}\rangle\langle
k_{1}^{\prime},k_{2}^{\prime}|\langle
k_{1},k_{2}|\psi_{n_{c}n_{d}}(\tau)\rangle$
$\displaystyle\times\langle\psi_{n_{c}n_{d}}(\tau)|k_{1}^{\prime},k_{2}^{\prime}\rangle
e^{-2\tilde{\Gamma}\tau[(k_{1}-k_{1}^{\prime})^{2}+(k_{2}-k_{2}^{\prime})^{2}]},$
(36)

where the atomic wavefunction after the photonic measurement is given by (10).
As before, we set the dephasing time to be equal to the QND interaction time.
The relative strengths of the dephasing and the interaction can be adjusted by
the dimensionless ratio $\tilde{\Gamma}=\Gamma/\Omega$.

###  V.2 Effect of decoherence on various quantities

####  V.2.1 Wineland squeezing criteria

Figure 8(a) shows the Wineland squeezing parameter (26) as a function of time
$\tau$ for different decoherence factors $\tilde{\Gamma}$. For small dephasing
rates, the Wineland squeezing parameter detects entanglement (shaded region)
in the short time regime. In comparison to the Hofmann-Takeuchi and EPR
steering criterion as calculated in Ref. [81], the region is reduced,
indicating that this is a less sensitive detector of entanglement. As was
found in Ref. [81], entanglement is present for all times $0<\tau<\pi$ for
dephasing rates $\Gamma=0,0.01,0.1$ as detected by logarithmic negativity. The
various entanglement witnesses are only a sufficient condition for
entanglement, and a lack of violation of the criterion does not indicate that
entanglement is not present. For small dephasing rates, there is an optimal
interaction time $\tau_{\text{opt}}$ which minimizes the Wineland squeezing
parameter (Fig. 8(b)). This shows generally a linear relationship with $1/N$.
In Fig. 8(c), the relationship between optimal Wineland squeezing parameter
$\xi_{opt}=\xi(\tau_{\text{opt}})$ with atom numbers $1/N$ are shown for
various dephasing rates. We observe that $\xi_{opt}$ remains in the
entanglement detection region for the dephasing rates $\tilde{\Gamma}=0,0.01$.
For $\tilde{\Gamma}=0.1$, $\xi_{\text{opt}}=1$ for larger values of $N$ and
hence no entanglement is detected. For this regime, the dephasing is strong
enough to remove the minimum as seen in Fig. 8(a). Hence we see that it
becomes more difficult to detect entanglement using the Wineland squeezing
criterion for large ensembles and larger dephasing rates. Hence as an
entanglement detection tool, other methods such as the Hoffman-Takeuchi
criterion appear to be a better choice.

Figure 8: The Wineland squeezing parameter for the state (36) with
decoherence. (a) Wineland squeezing parameter versus QND interaction time
$\tau$ with decoherence rates $\tilde{\Gamma}$ as marked. The atom number is
$N=20$. (b) The optimal squeezing time $\tau_{\text{opt}}$ determined by
minimizing $\xi(\tau)$ plotted as a function of $1/N$. A fit of the optimal
times for $\Gamma=0$ is given by
$\tau_{\text{opt}}^{(\tilde{\Gamma}=0)}=0.104+\frac{0.0413}{N}$. (c) The
optimal Wineland squeezing $\xi(\tau_{\text{opt}})$ versus $1/N$ for different
decoherence rates $\tilde{\Gamma}$. For all plots, we choose parameters
$n_{c}=100,n_{d}=0,\alpha=10$.

####  V.2.2 Bell-CHSH inequality

In Fig. 9(a), we calculate the left-hand side of the Bell-CHSH inequality (29)
for the state (36) as a function of the QND interaction time with different
dephasing rates. For each dephasing rate, there is a slightly different
dependence with $\tau$, where larger dephasing rates tend to diminish the
violation for longer times. Fig. 9(b) shows the optimized value of (29) as a
function of $N$. We see that the Bell violations are observed for all $N$ even
in the presence of dephasing. The surprisingly robust nature of the Bell
violations in the presence of dephasing is a positive sign that such
correlations may be observable in experimental situations. We note however
that the violations tend to diminish for larger ensembles so that for larger
systems, such that alternative criteria such as the Hoffman-Takeuchi criterion
may be more sensitive detectors of entanglement.

Figure 9: Bell-CHSH correlations for the state (36) with atomic dephasing. (a)
Time dependence of (29) for different decoherence ratios as marked. Here we
choose parameters $\theta_{B}=0.251,N=11$. (b) Optimal values of (29) with
respect to $\tau$ and $\theta_{B}$ for various $N$. Common parameters
throughout are $n_{c}=100,n_{d}=0,\alpha=10$.

##  VI Conclusion

We have studied the effect of three types of decoherence (optical phase
diffusion, photon loss and gain, and atomic dephasing) on entangled states
produced by QND measurements in atomic ensembles. Generally, we find a similar
conclusion to Ref. [81] where the states are relatively robust for all types
of decoherence in the short time regime. For example, in the entanglement
plots of Fig. 2, even for very large decoherence rates there is a significant
amount of entanglement in the short time regime but decays for longer times.
The correlations, as shown in Figs. 3,4,7 are all robust in the short time
regime but can degrade at longer interaction and decoherence times. As such,
correlation-based detection of entanglement and EPR steering tend to be
robustly observed in the short time regime as seen in Figs. 5,7,8.
Interestingly, Bell correlations are also very robust under various types of
decoherence, as seen in Figs. 6,9. The robustness of Bell’s violation can be
attributed to the fact that the optimal interaction time occurs for the short
time regime. The main issue here with experimental observation is that for
larger $N$ the level of violation diminishes, making its detection more
challenging.

Our results suggest that producing entangled states based on QND measurements
should be one of the most experimentally viable methods in the context of
BECs. As mentioned in the introduction, while such methods are well
established for entanglement generation in hot atomic ensembles [56, 31],
there is no corresponding experiment that has been realized for BECs. While
there have been numerous approaches that have been theoretically investigated
for entanglement generation in BECs already [33, 34, 35, 17, 36, 32, 37, 38],
so far none have been realized experimentally. The QND approach allows for a
robust and versatile method to generate such entanglement. The theory that has
been developed in [80] and summarized in this paper allows for the
investigation beyond the short time regime, to produce more exotic types of
entanglement. The generation of such highly entangled many-body states is of
fundamental interest to the macroscopic nature of such states, and also has
been shown to have applications in quantum information [74, 75, 76, 77, 78,
79].

## Acknowledgments

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.

## Data availability statement

The data that supports the findings of this study are available within the
article (and any supplementary material).

##  Appendix A Density matrix for optical phase diffusion

In this section, we give details of the explicit expression for the density
matrix including optical phase diffusion. Starting from the initial state
(19), we evaluate

$\displaystyle\rho_{\text{PD}}=\langle n_{c}|\langle
n_{d}|\sum_{m,n,m^{\prime},n^{\prime}=0}^{\infty}M_{m,n}^{\text{PD},c}(t)M_{m^{\prime},n^{\prime}}^{\text{PD},d}(t)$
$\displaystyle\times\rho_{0}{M_{m^{\prime},n^{\prime}}^{\text{PD},d}}^{\dagger}(t){M_{m,n}^{\text{PD},c}}^{\dagger}(t)|n_{c}\rangle|n_{d}\rangle$
$\displaystyle=\frac{1}{4^{N}}\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}=0}^{N}\sqrt{\binom{N}{k_{1}}\binom{N}{k_{2}}\binom{N}{k_{1}^{\prime}}\binom{N}{k_{2}^{\prime}}}z^{2}(\tau)$
$\displaystyle\times
e^{-|\alpha|^{2}}e^{u(\tau){|\alpha|}^{2}\cos[(k_{1}-k_{2}-k_{1}^{\prime}+k_{2}^{\prime})\tau]}$
$\displaystyle\times
z^{2(n_{c}-m+n_{d}-m^{\prime})}(\tau)\sum_{m=0}^{n_{c}}\sum_{m^{\prime}=0}^{n_{d}}\frac{u^{m+m^{\prime}}(\tau)}{m!m^{\prime}!}$
$\displaystyle\times\left\\{\alpha^{2}\cos[(k_{1}-k_{2})\tau]\cos[(k_{1}^{\prime}-k_{2}^{\prime})\tau]\right\\}^{n_{c}-m}$
$\displaystyle\times\left\\{\alpha^{2}\sin[(k_{1}-k_{2})\tau]\sin[(k_{1}^{\prime}-k_{2}^{\prime})\tau]\right\\}^{n_{d}-m^{\prime}}$
$\displaystyle\times\frac{n_{c}!n_{d}!}{[(n_{c}-m)!]^{2}[(n_{d}-m^{\prime})!]^{2}}|k_{1}\rangle|k_{2}\rangle\langle
k_{1}^{\prime}|\langle k_{2}^{\prime}|$
$\displaystyle=\frac{1}{4^{N}}\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}=0}^{N}\sqrt{\binom{N}{k_{1}}\binom{N}{k_{2}}\binom{N}{k_{1}^{\prime}}\binom{N}{k_{2}^{\prime}}}$
$\displaystyle\times\frac{[-u(\tau)]^{n_{c}}U\left(-{n_{c}},1,-\frac{{n_{c}}\cos[({k_{1}}-{k_{2}})\tau]\cos[({k_{1}^{\prime}}-{k_{2}^{\prime}})\tau]z^{2}(\tau)}{u(\tau)}\right)}{n_{c}!}$
$\displaystyle\times\frac{[-u(\tau)]^{n_{d}}U\left(-{n_{d}},1,-\frac{{n_{d}}\sin[({k_{1}}-{k_{2}})\tau]\sin[({k_{1}^{\prime}}-{k_{2}^{\prime}})\tau]z^{2}(\tau)}{u(\tau)}\right)}{n_{d}!}$
$\displaystyle\times
z^{2}(\tau)D[(k_{1}-k_{2}-k_{1}^{\prime}+k_{2}^{\prime})\tau]|k_{1}\rangle|k_{2}\rangle\langle
k_{1}^{\prime}|\langle k_{2}^{\prime}|.$ (37)

Here

$\displaystyle z\left(\tau\right)$
$\displaystyle=\frac{1}{\tilde{\kappa}\tau+1}$ $\displaystyle u(\tau)$
$\displaystyle=\frac{\tilde{\kappa}\tau}{\tilde{\kappa}\tau+1}$ $\displaystyle
D(\chi)$ $\displaystyle=e^{-|\alpha|^{2}[1-u(\tau)\cos(\chi)]}$ (38)

and $U$ is the confluent hyper-geometric function of the second kind.

##  Appendix B Fock state matrix elements

The matrix elements of the $S^{y}$ rotation is given by [40]

$\displaystyle\langle
k|e^{-iS^{y}\theta/2}|k^{\prime}\rangle=\sqrt{k^{\prime}!(N-k^{\prime})!k!(N-k)!}$
$\displaystyle\times\sum_{n}\frac{(-1)^{n}\cos^{k-k^{\prime}+N-2n}(\theta/2)\sin^{2n+k^{\prime}-k}(\theta/2)}{(k-n)!(N-k^{\prime}-n)!n!(k^{\prime}-k+n)!},$
(39)

where $|k\rangle$ are the eigenstates of $S^{z}$. The matrix elements of
$S^{x}$ are accordingly given by

$\displaystyle\langle
k|e^{-iS^{x}\theta/2}|k^{\prime}\rangle=i^{k^{\prime}-k}\langle
k|e^{-iS^{y}\theta/2}|k^{\prime}\rangle,$ (40)

using the fact $S^{x}=e^{-iS^{z}\pi/4}S^{y}e^{iS^{z}\pi/4}$.

##  Appendix C Density matrix for photon loss and gain

In this section, we give the explicit expression of the density matrix
including photonic gain and loss. Starting from the initial state (19), we
evaluate

$\displaystyle\rho_{\text{LG}}=\langle n_{c}|\langle
n_{d}|\sum_{p,q,p^{\prime},q^{\prime}=0}^{\infty}M_{p,q}^{\text{LG}}(c)M_{p^{\prime},q^{\prime}}^{\text{LG}}(d)$
$\displaystyle\times\rho_{0}{M_{p^{\prime},q^{\prime}}^{\text{LG}}}^{\dagger}(d){M_{p,q}^{\text{LG}}}^{\dagger}(c)|n_{c}\rangle|n_{d}\rangle$
$\displaystyle=\frac{1}{4^{N}}\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}=0}^{N}\sqrt{\binom{N}{k_{1}}\binom{N}{k_{2}}\binom{N}{k_{1}^{\prime}}\binom{N}{k_{2}^{\prime}}}$
$\displaystyle{T_{3}}^{2}R[(k_{1}-k_{2}-k_{1}^{\prime}+k_{2}^{\prime})\tau]\sum_{p=0}^{n_{c}}\sum_{q=0}^{n_{d}}\frac{(gT_{1})^{p+q}}{p!q!}$
$\displaystyle\times\\{\alpha^{2}\cos[(k_{1}-k_{2})\tau]\cos[(k_{1}^{\prime}-k_{2}^{\prime})\tau]\\}^{n_{c}-p}$
$\displaystyle\times\\{\alpha^{2}\sin[(k_{1}-k_{2})\tau]\sin[(k_{1}^{\prime}-k_{2}^{\prime})\tau]\\}^{n_{d}-q}$
$\displaystyle\times{T_{2}}^{2(n_{c}-p+n_{d}-q)}\frac{n_{c}!}{{[(n_{c}-p)!]}^{2}}\frac{n_{d}!}{{[(n_{d}-q)!]}^{2}}|k_{1}\rangle|k_{2}\rangle\langle
k_{1}^{\prime}|\langle k_{2}^{\prime}|$
$\displaystyle=\frac{1}{4^{N}}\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}=0}^{N}\sqrt{\binom{N}{k_{1}}\binom{N}{k_{2}}\binom{N}{k_{1}^{\prime}}\binom{N}{k_{2}^{\prime}}}$
$\displaystyle\frac{(-g{T_{1}})^{{n_{c}}}U\left(-{n_{c}},1,-\frac{\alpha^{2}\cos[({k_{1}}-{k_{2}})\tau]\cos[({k_{1}^{\prime}}-{k_{2}^{\prime}})\tau]{T_{2}}^{2}}{g{T_{1}}}\right)}{{n_{c}}!}$
$\displaystyle\frac{(-g{T_{1}})^{{n_{d}}}U\left(-{n_{d}},1,-\frac{\alpha^{2}\sin[({k_{1}}-{k_{2}})\tau]\sin[({k_{1}^{\prime}}-{k_{2}^{\prime}})\tau]{T_{2}}^{2}}{g{T_{1}}}\right)}{{n_{d}}!}$
$\displaystyle\times{T_{3}}^{2}R[(k_{1}-k_{2}-k_{1}^{\prime}+k_{2}^{\prime})\tau]|k_{1}\rangle|k_{2}\rangle\langle
k_{1}^{\prime}|\langle k_{2}^{\prime}|,$ (41)

where $R(\chi)=e^{-|\alpha|^{2}[1-\gamma T_{1}\cos(\chi)]}$.

## References

  * Horodecki et al. [2009] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
  * Togan et al. [2010] E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. Dutt, A. S. Srensen, P. R. Hemmer, and A. S. Zibrov, Nature 466, 730 (2010).
  * Jeong [2005] H. Jeong, Phys. Rev. A 72, 034305 (2005).
  * Paternostro et al. [2007] M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, Physical review letters 99, p.250401.1 (2007).
  * Ma et al. [2011] J. Ma, X. Wang, C. Sun, and F. Nori, Physics Reports 509, 89 (2011).
  * Schleich et al. [2016] W. Schleich, K. Ranade, C. Anton, M. Arndt, M. Aspelmeyer, M. Bayer, G. Berg, T. Calarco, H. Fuchs, E. Giacobino, et al., Applied Physics B: Lasers and Optics 122 (2016).
  * Rozema et al. [2014a] L. A. Rozema, D. H. Mahler, R. Blume-Kohout, and A. M. Steinberg, Phys.rev.x 4 (2014a).
  * Gerry and Knight [2004] C. Gerry and P. Knight, _Introductory Quantum Optics_ (Cambridge University Press, 2004).
  * Zhang and Duan [2014] Z. Zhang and L. M. Duan, New Journal of Physics 16, 103037 (2014).
  * Rozema et al. [2014b] L. A. Rozema, D. H. Mahler, R. Blume-Kohout, and A. M. Steinberg, Physical Review X 4, 041025 (2014b).
  * Buluta and Nori [2009] I. Buluta and F. Nori, Science 326, 108 (2009).
  * Georgescu et al. [2014] I. M. Georgescu, S. Ashhab, and F. Nori, Reviews of Modern Physics 86, 153 (2014).
  * Byrnes et al. [2007] T. Byrnes, P. Recher, N. Y. Kim, S. Utsunomiya, and Y. Yamamoto, Physical review letters 99, 016405 (2007).
  * Kunkel et al. [2018] P. Kunkel, M. Prüfer, H. Strobel, D. Linnemann, A. Frölian, T. Gasenzer, M. Gärttner, and M. K. Oberthaler, Science 360, 413 (2018).
  * Fadel et al. [2018] M. Fadel, T. Zibold, B. Décamps, and P. Treutlein, Science 360, 409 (2018).
  * Lange et al. [2018] K. Lange, J. Peise, B. Lücke, I. Kruse, G. Vitagliano, I. Apellaniz, M. Kleinmann, G. Tóth, and C. Klempt, Science 360, 416 (2018).
  * Li et al. [2013] L. Li, Y. O. Dudin, and A. Kuzmich, Nature 498, 466 (2013).
  * Stoler [1971] D. Stoler, Phys. Rev. D 4, 1925 (1971).
  * Walls and D. [1983] Walls and F. D., Nature 306, 141 (1983).
  * Kitagawa and Ueda [1993] M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).
  * Vasilakis et al. [2014] G. Vasilakis, H. Shen, K. Jensen, M. Balabas, D. Salart, B. Chen, and E. S. Polzik (2014).
  * Sørensen et al. [2001] A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Nature 409, 63–66 (2001).
  * Kuzmich et al. [2000] A. Kuzmich, L. Mandel, and N. Bigelow, Physical Review Letters 85, 1594 (2000).
  * Esteve et al. [2008] J. Esteve, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, Nature 455, 1216 (2008).
  * You et al. [2017] C. You, S. Adhikari, Y. Chi, M. L. LaBorde, C. T. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, et al., Journal of Optics 19, 124002 (2017).
  * Slusher et al. [1985] R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, Physical Review Letters 55, 2409 (1985).
  * Breitenbach et al. [1997] G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997).
  * Wu et al. [1986] L.-A. Wu, H. Kimble, J. Hall, and H. Wu, Physical review letters 57, 2520 (1986).
  * New [2011] G. New, _Introduction to nonlinear optics_ (Cambridge University Press, 2011).
  * Gerry and Grobe [1994] C. C. Gerry and R. Grobe, Phys. Rev. A 49, 2033 (1994).
  * Hammerer et al. [2010a] K. Hammerer, A. S. Sørensen, and E. S. Polzik, Reviews of Modern Physics 82, 1041 (2010a).
  * Ortiz et al. [2018] S. Ortiz, Y. Song, J. Wu, and T. Byrnes, Phys. Rev. A 98 (2018).
  * Treutlein et al. [2006] P. Treutlein, T. W. Hänsch, J. Reichel, A. Negretti, M. A. Cirone, and T. Calarco, Physical Review A 74, 022312 (2006).
  * Idlas et al. [2016] S. Idlas, L. Domenzain, R. Spreeuw, and T. Byrnes, Physical Review A 93, 022319 (2016).
  * Jing et al. [2019a] Y. Jing, M. Fadel, V. Ivannikov, and T. Byrnes, New J. Phys. 21, 093038 (2019a).
  * Rosseau et al. [2014] D. Rosseau, Q. Ha, and T. Byrnes, Physical Review A 90 (2014).
  * Hussain et al. [2014] M. I. Hussain, E. O. Ilo-Okeke, and T. Byrnes, Physical Review A 89, 053607 (2014).
  * Abdelrahman et al. [2014] A. Abdelrahman, T. Mukai, H. Häffner, and T. Byrnes, Optics express 22, 3501 (2014).
  * Gross [2012a] C. Gross, Journal of Physics B: Atomic, Molecular and Optical Physics 45, 103001 (2012a).
  * Byrnes and Ilo-Okeke [2021] T. Byrnes and E. O. Ilo-Okeke, _Quantum atom optics: Theory and applications to quantum technology_ (Cambridge university press, 2021).
  * Windpassinger et al. [2008] P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L. G. Alzar, J. Appel, J. H. Müller, N. Kjærgaard, and E. S. Polzik, Phys. Rev. Lett. 100, 103601 (2008).
  * Leroux et al. [2009] I. D. Leroux, M. H. Schleier-Smith, and V. Vuleti (2009).
  * Sewell et al. [2012] R. Sewell, M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood, and M. Mitchell, Physical review letters 109, 253605 (2012).
  * Hosten et al. [2016] O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich, Science 352, 1552 (2016).
  * Cox et al. [2016] K. C. Cox, G. P. Greve, J. M. Weiner, and J. K. Thompson, Phys. Rev. Lett. 116, 093602 (2016).
  * Yang et al. [2018] C. Yang, L. Zhang, and W. Zhang, EPL 122 (2018).
  * Le Coq et al. [2020] Y. Le Coq, K. Molmer, and S. Seidelin, Modern Physics Letters B 34 (2020).
  * Vanner et al. [2011] M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, Proceedings of the National Academy of Sciences of the United States of America 108, 16182 (2011).
  * Qing-Hong et al. [2018] L. Qing-Hong, Y. Yang, L. Hong-Zhen, and Z. Nan-Run, Acta Physica Sinica 67 (2018).
  * Asjad et al. [2016] M. Asjad, S. Zippilli, and D. Vitali, Physical Review A 93 (2016).
  * Purdy et al. [2013] T. P. Purdy, P. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, Physical Review X 3 (2013).
  * Radhakrishnan et al. [2020] C. Radhakrishnan, M. Laurière, and T. Byrnes, Physical Review Letters 124, 110401 (2020).
  * Clauser et al. [1969a] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Physical review letters 23, 880 (1969a).
  * Krauter et al. [2011] H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, Physical review letters 107, 080503 (2011).
  * Hammerer et al. [2010b] K. Hammerer, A. S. Sørensen, and E. S. Polzik, Rev. Mod. Phys. 82, 1041 (2010b).
  * Julsgaard et al. [2001] B. Julsgaard, A. Kozhekin, and E. S. Polzik, Nature 413, 400 (2001).
  * Kong et al. [2020] J. Kong, R. Jiménez-Martínez, C. Troullinou, V. G. Lucivero, G. Tóth, and M. W. Mitchell, 11, 2415 (2020).
  * Kuzmich and Kennedy [2004] A. Kuzmich and T. A. B. Kennedy, Phys. Rev. Lett. 92, 030407 (2004).
  * Duan et al. [2000] L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, Phys. Rev. Lett. 85, 5643 (2000).
  * Kuzmich and Polzik [2000] A. Kuzmich and E. S. Polzik, Phys. Rev. Lett. 85, 5639 (2000).
  * Krauter et al. [2013] H. Krauter, D. Salart, C. Muschik, J. M. Petersen, H. Shen, T. Fernholz, and E. S. Polzik, Nature Physics 9, 400 (2013).
  * Takahashi et al. [1999] Y. Takahashi, K. Honda, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, Physical Review A 60 (1999).
  * Meppelink et al. [2009] R. Meppelink, R. A. Rozendaal, S. B. Koller, J. M. Vogels, and P. Straten, Physics 81, 1532 (2009).
  * Higbie et al. [2005] J. M. Higbie, L. E. Sadler, S. Inouye, A. P. Chikkatur, S. R. Leslie, K. L. Moore, V. Savalli, and D. M. Stamper-Kurn, Phys. Rev. Lett. 95, 050401 (2005).
  * Ilo-Okeke and Byrnes [2016] E. O. Ilo-Okeke and T. Byrnes, Phys. Rev. A 94, 013617 (2016).
  * Ilo-Okeke and Byrnes [2014] E. O. Ilo-Okeke and T. Byrnes, Physical review letters 112, 233602 (2014).
  * Braunstein and Van Loock [2005] S. L. Braunstein and P. Van Loock, Reviews of modern physics 77, 513 (2005).
  * Pfaff et al. [2014] W. Pfaff, B. J. Hensen, H. Bernien, S. B. van Dam, M. S. Blok, T. H. Taminiau, M. J. Tiggelman, R. N. Schouten, M. Markham, D. J. Twitchen, et al., Science 345, 532 (2014).
  * Sherson et al. [2008] J. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, and E. S. Polzik, Journal of Physics B: Atomic, Molecular and Optical Physics 41, 223001 (2008).
  * Gross [2012b] C. Gross, Journal of Physics B: Atomic, Molecular and Optical Physics 45, 103001 (2012b).
  * Sorensen et al. [2001] A. Sorensen, L. M. Duan, J. I. Cirac, and P. Zoller, Nature 409, 63 (2001).
  * Augusto et al. [2014] Augusto, Smerzi, Markus, K., Oberthaler, Daniel, Iinnemann, David, B., and Hume, Science 345, 424 (2014).
  * Sørensen and Mølmer [2001] A. S. Sørensen and K. Mølmer, Phys. Rev. Lett. 86, 4431 (2001).
  * Pyrkov and Byrnes [2014a] A. N. Pyrkov and T. Byrnes, New Journal of Physics 16, 073038 (2014a).
  * Pyrkov and Byrnes [2014b] A. N. Pyrkov and T. Byrnes, Physical Review A 90, 062336 (2014b).
  * Ilo-Okeke et al. [2018] E. O. Ilo-Okeke, L. Tessler, J. P. Dowling, and T. Byrnes, npj Quantum Information 4, 1 (2018).
  * Chaudhary et al. [2021] M. Chaudhary, M. Fadel, E. O. Ilo-Okeke, A. N. Pyrkov, V. Ivannikov, and T. Byrnes, Phys. Rev. A 103, 062417 (2021).
  * Byrnes et al. [2012] T. Byrnes, K. Wen, and Y. Yamamoto, Physical Review A 85, 040306 (2012).
  * Byrnes et al. [2015] T. Byrnes, D. Rosseau, M. Khosla, A. Pyrkov, A. Thomasen, T. Mukai, S. Koyama, A. Abdelrahman, and E. Ilo-Okeke, Optics Communications 337, 102 (2015).
  * Aristizabal-Zuluaga et al. [2021] J. E. Aristizabal-Zuluaga, I. Skobleva, L. Richter, Y. Ji, Y. Mao, M. Kondappan, V. Ivannikov, and T. Byrnes, Journal of Physics B: Atomic, Molecular and Optical Physics 54 (2021).
  * Gao et al. [2022] S. Gao, E. O. Ilo-Okeke, Y. Mao, M. Kondappan, J. E. Aristizabal-Zuluaga, V. Ivannikov, and T. Byrnes, Journal of Physics B: Atomic, Molecular and Optical Physics 55, 195501 (2022).
  * Chaudhary et al. [2022] M. Chaudhary, Y. Mao, M. Kondappan, A. S. P. Paz, V. Ivannikov, and T. Byrnes, Phys. Rev. A 105, 022443 (2022).
  * Serafin et al. [2021] A. Serafin, M. Fadel, P. Treutlein, and A. Sinatra, Physical Review Letters 127, 013601 (2021).
  * Tsang and Caves [2012] M. Tsang and C. M. Caves, Physical Review X 2, 031016 (2012).
  * Pettersson and Byrnes [2017] O. Pettersson and T. Byrnes, Physical Review A 95, 043817 (2017).
  * Wang et al. [2013] Z. Wang, X. Meng, and H.-Y. Fan, Journal of Physics A: Mathematical and Theoretical 46 (2013).
  * Hu et al. [2010] L.-y. Hu, X.-x. Xu, Z.-s. Wang, and X.-f. Xu, Phys. Rev. A 82, 043842 (2010).
  * Clauser et al. [1969b] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969b).
  * Coladangelo [2018] A. Coladangelo, Phys. Rev. A 98, 052115 (2018).
  * Reichel and Vuletic [2011] J. Reichel and V. Vuletic, _Atom chips_ (John Wiley & Sons, 2011).
  * Whitlock and Gerritsma [2009] S. Whitlock and R. Gerritsma, New Journal of Physics 11, 023021 (2009).
  * Kondappan et al. [2022] M. Kondappan, M. Chaudhary, E. O. Ilo-Okeke, V. Ivannikov, and T. Byrnes, arXiv preprint arXiv:2210.06923 (2022).
  * Liu et al. [2014] T.-K. Liu, C.-J. Shan, J.-B. Liu, and H.-Y. Fan, Chinese Physics B 23, 030303 (2014).
  * Carmichael [2007] H. Carmichael, _Statistical Methods in Quantum Optics 2: Non-Classical Fields_ , vol. 2008 (Springer Berlin Heidelberg, 2007).
  * Hong-yi and Yue [1996] F. Hong-yi and F. Yue, Phys. Rev. A 54, 958 (1996).
  * Fan and Hu [2008a] H.-y. Fan and L.-y. Hu, Modern Physics Letters B 22, 2435 (2008a).
  * Plenio [2005] M. B. Plenio, Phys. Rev. Lett. 95, 090503 (2005).
  * Vidal and Werner [2002] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002).
  * Byrnes [2013] T. Byrnes, Physical Review A 88, 023609 (2013).
  * Hofmann and Takeuchi [2003] H. F. Hofmann and S. Takeuchi, Phys. Rev. A 68, 032103 (2003).
  * Wiseman et al. [2007] H. Wiseman, S. Jones, and D. Andrew, in _APS Division of Atomic, Molecular and Optical Physics Meeting Abstracts_ (2007), vol. 38, pp. R1–126.
  * Sun et al. [2018] K. Sun, X. J. Ye, Y. Xiao, X. Y. Xu, Y. C. Wu, J. S. Xu, J. L. Chen, C. F. Li, and G. C. Guo, npj Quantum Information (2018).
  * Ma et al. [2019] Z.-H. Ma, J. Cui, Z. Cao, S.-M. Fei, V. Vedral, T. Byrnes, and C. Radhakrishnan, EPL (Europhysics Letters) 125, 50005 (2019).
  * Reid et al. [2009] M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, Rev. Mod. Phys. 81, 1727 (2009).
  * Asitzabal-Zuluana et al. [2020] J. Asitzabal-Zuluana, I. Skobleva, L. Richter, Y. Ji, Y. Mao, M. Kondappan, V. Ivannikov, and T. Byrnes, 409 (2020).
  * Jing et al. [2019b] Y. Jing, M. Fadel, V. Ivannikov, and T. Byrnes, New Journal of Physics 21, 093038 (2019b).
  * Agarwal [1971] G. S. Agarwal, Phys. Rev. A 4, 739 (1971).
  * Daniel and Milburn [1989] D. J. Daniel and G. J. Milburn, Phys. Rev. A 39, 4628 (1989).
  * Kim and Bužek [1992] M. S. Kim and V. Bužek, Phys. Rev. A 46, 4239 (1992).
  * Fan and Hu [2008b] H.-y. Fan and L.-y. Hu, Modern Physics Letters B 22, 2435 (2008b).
  * hua Chen and yi Fan [2013] J. hua Chen and H. yi Fan, Annals of Physics 334, 272 (2013).