# Cavity optomechanical detection of persistent currents and solitons in a
bosonic ring condensate

Nalinikanta Pradhan  Department of Physics, Indian Institute of Technology,
Guwahati 781039, Assam, India  Pardeep Kumar  Max Planck Institute for the
Science of Light, Staudtstraße 2, 91058 Erlangen, Germany  Rina Kanamoto
Department of Physics, Meiji University, Kawasaki, Kanagawa 214-8571, Japan
Tarak Nath Dey  Department of Physics, Indian Institute of Technology,
Guwahati 781039, Assam, India  M. Bhattacharya  School of Physics and
Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive,
Rochester, New York 14623, USA  Pankaj Kumar Mishra  Department of Physics,
Indian Institute of Technology, Guwahati 781039, Assam, India

###### Abstract

We present numerical simulations of the cavity optomechanical detection of
persistent currents and bright solitons in an atomic Bose-Einstein condensate
confined in a ring trap. This work describes a novel technique that measures
condensate rotation in situ, in real-time, and with minimal destruction, in
contrast to currently used methods, all of which destroy the condensate
completely. For weakly repulsive inter-atomic interactions, the analysis of
persistent currents extends our previous few-mode treatment of the condensate
[P. Kumar et al. Phys. Rev. Lett. 127, 113601 (2021)] to a stochastic Gross-
Pitaevskii simulation. For weakly attractive atomic interactions, we present
the first analysis of optomechanical detection of matter-wave soliton motion.
We provide optical cavity transmission spectra containing signatures of the
condensate rotation, sensitivity as a function of the system response
frequency, and atomic density profiles quantifying the effect of the
measurement backaction on the condensate. We treat the atoms at a mean-field
level and the optical field classically, account for damping and noise in both
degrees of freedom, and investigate the linear as well as nonlinear response
of the configuration. Our results are consequential for the characterization
of rotating matter waves in studies of atomtronics, superfluid hydrodynamics,
and matter-wave soliton interferometry.

##  I Introduction

An atomic Bose-Einstein Condensate (BEC) confined in a ring potential exhibits
superflow, i.e., transport without dissipation [1, 2, 3]. It is therefore a
natural platform for studying superfluid hydro-dynamical phenomena such as
quantized persistent currents [4], phase slips [5, 6], excitations [7], two-
component rotation [8, 9], hysteresis [10, 11], and shock waves [12]; a
versatile enabler for applications such as matter-wave interferometry [13,
14], atomtronic circuits [15, 16, 17, 18], and gyroscopy [19, 20]; and a
convenient simulator of topological excitations [21, 22, 23], early universe
cosmology [24], and time crystals [25].

Inspired by the experimental activity in the field, a large number of
theoretical proposals have been put forward, based on the BEC-in-a-ring
system, characterizing plain wave to soliton transitions [26], self-trapping
[27], simulated Hawking radiation [28], the Berry phase [29], qubits for
computation [30], critical velocities [31, 32], superflow decay [33, 34, 35,
36], phonons [37], rotating lattices [38], rotation sensing [39], gauge fields
[40], matter-wave interference [41], double-ring geometries [42, 43, 44], etc.

In all these studies, knowledge of the condensate rotation is an important
consideration. At present, all demonstrated methods of detecting such rotation
in ring BECs are destructive of the condensate [45]. Due to issues related to
optical resolution, the methods typically also require time-of-flight
expansion of the atoms, making in situ measurements difficult. A theoretical
proposal exists based on atom counting for a minimally destructive measurement
of the condensate rotation [46].

Recently, our group suggested a method for detecting condensate rotation in
real time, in situ and with minimal destruction to the condensate [47]. This
method proposed to use the techniques of cavity optomechanics, a discipline
that addresses the coupling of mechanical motion to electromagnetic fields
confined in resonators [48]. Probably the best-known optomechanical device in
existence is the Laser Interferometer Gravitational-Wave Observatory (LIGO),
which detected the gravitational waves predicted by Einstein’s theory of
general relativity [49], an accomplishment recognized by a Nobel prize.

Cavity optomechanics is now a mature field that is capable of supporting the
sensitive detection of any physical variable that actuates the mechanical
motion coupling to the electromagnetic fields in the cavity. Thus, cavity
optomechanical principles have been employed to construct accelerometers [50],
magnetometers [51], thermometers [52], mass [53] and force [54] sensors, etc.
In our previous proposal, which considered a rotating BEC in a cavity, it was
shown that the resulting sensitivity of BEC rotation measurement was three
orders of magnitude better than demonstrated hitherto [47]. This conclusion
regarding the detection of a persistent current was based on a few-mode
approximation for the condensate.

In the present work, we consider a BEC confined in a ring trap and interacting
with an optical cavity mode carrying orbital angular momentum (OAM) [47]. This
may be regarded as the rotational analog of a BEC with a linear degree of
mechanical freedom combined with a standing wave optical cavity lattice in an
optomechanical context [55].

For weak repulsive atomic interactions, we extend the previous two-mode
characterization of the condensate to a mean field, i.e., Gross-Pitaevskii,
treatment. Our method allows us to confirm the basic results of the two-mode
treatment regarding the rotation detection of a persistent current, to
investigate the modifications resulting from taking the full condensate
dynamics into account, and to quantify the effect of measurement backaction on
the condensate. It also allows us to consider the detection of a superposition
of persistent current states in the condensate.

We also investigate, for the first time, the case of weak attractive atomic
interactions [56] \- which results in a bright-soliton ground state in the
ring condensate - in the optomechanical context. Such solitons are of great
interest e.g., to rotation sensing and matter-wave interferometry [57, 58, 59,
60, 61, 62, 63]. However, a soliton is not amenable to a few-mode
optomechanical treatment, due to the large number of matter-wave OAM states
contributing to the condensate dynamics. Our numerical simulations make this
case tractable, extracting, as in the case of the persistent currents, cavity
transmission spectra with signatures of soliton rotation, the sensitivity of
the measurement as a function of system response frequency, and atomic density
profiles showing the effect of the measurement on the condensate. In all
simulations, the matter is treated at the mean-field level, light is treated
classically, and noise arising from both optical as well as matter-wave fields
are taken into account.

This paper is organized as follows. In Section II the theoretical model and
details of the numerical simulation are presented. In Sections III.1 and III.2
we provide the dynamics, OAM content, optical spectra, measurement
sensitivity, and condensate density fidelity for the persistent current and
bright soliton detection, respectively. The conclusions are presented in
Section IV.

##  II Theoretical model and details of numerical simulation

In this section, we present the theoretical model for the configuration of
interest, shown in Fig. 1, i.e., a BEC confined in a one-dimensional ring trap
coupled to a cavity using Laguerre-Gauss beams [47].

Figure 1: A schematic setup for the BEC with winding number $L_{p}$ rotating
in a ring trap around the axis of the Fabry-Perot cavity. The red beam
represents the Laguerre-Gauss modes with the orbital angular momentum of $\pm
l\hbar$ used to probe the BEC rotation. The output signal $a_{out}$ is the
field transmitted from the cavity.

The dynamical equations governing the system are given by [64, 65, 66, 47]

$\displaystyle(i-\Gamma)\frac{d\psi}{d\tau}=\biggl{[}-\frac{d^{2}}{d\phi^{2}}$
$\displaystyle+\frac{U_{0}}{\omega_{\beta}}|\alpha(\tau)|^{2}\cos^{2}\left({l\phi}\right)$
(1)
$\displaystyle-\mu+2\pi\frac{\chi}{N}|\psi|^{2}\biggr{]}\psi+\xi(\phi,\tau),$

and

$\displaystyle i\frac{d\alpha}{d\tau}=\biggl{\\{}-\biggl{[}\Delta_{c}$
$\displaystyle-
U_{0}\langle\cos^{2}\left(l\phi\right)\rangle_{\tau}+i\frac{\gamma_{0}}{2}\biggr{]}\alpha$
(2)
$\displaystyle+i\eta\biggr{\\}}\omega_{\beta}^{-1}+i\sqrt{\gamma_{0}}\omega_{\beta}^{-1}\alpha_{in}(\tau).$

In the above, Eq. (1) is the stochastic Gross-Pitaevskii equation, where
$\psi\equiv\psi(\phi,\tau)$ represents the microscopic wave function of the
condensate with $\phi$ the angular variable along the ring, and $\tau$ the
scaled time, to be defined below. The wave function obeys, at any time, the
normalization condition

$\int_{0}^{2\pi}|\psi(\phi,\tau)|^{2}d\phi=N,$

where $N$ is the number of atoms in the condensate. In order to obtain the
dimensionless Eq. (1), energy and time have been scaled using the quantities

$\hbar\omega_{\beta}=\frac{\hbar^{2}}{2mR^{2}}\,\,\mathrm{and}\,\,\tau=\omega_{\beta}t,$
(3)

respectively, where $m$ is the atomic mass and $R$ is the radius of the ring-
shaped trap.

The first term inside the square bracket on the right-hand side of Eq. (1)
stands for the kinetic energy of the atoms due to their rotational motion. The
second term in the bracket represents the optical lattice potential created
with a superposition of two Laguerre-Gauss beams having orbital angular
momenta $\pm l\hbar$ respectively, with $U_{0}=g_{0}^{2}/\Delta_{a}$, where
$g_{0}$ is the single photon-single atom coupling and $\Delta_{a}$ is the
detuning of the driving laser from the atomic resonance. The third term in the
bracket corresponds to the chemical potential $\mu$ of the condensate, which
is corrected by $\Delta\mu$ at each time step ($\Delta\tau$) as [67]

$\Delta\mu=(\Delta\tau)^{-1}\ln{\left[\int\lvert\psi(\phi,\tau)\rvert^{2}d\phi/\int|\psi(\phi,\tau+\Delta\tau)\rvert^{2}d\phi\right]},$

to conserve the normalization of the condensate in the presence of the
dissipation

$\Gamma=\frac{\omega_{m}}{\omega_{\beta}},$ (4)

set by the lifetime $\omega_{m}^{-1}$ of the persistent currents [1, 47]. The
fourth term inside the bracket represents the scaled atomic interaction

$\chi=\frac{gN}{2\pi\hbar\omega_{\beta}}.$ (5)

Here

$g=\frac{2\hbar\omega_{\rho}a_{s}}{R},$ (6)

with $a_{s}$ the $s$-wave atomic scattering length and $\omega_{\rho}$ the
harmonic trap frequency along the radial direction [47]. Thermal noise $\xi$,
with zero mean and correlations provided below, has been added to the
condensate in accordance with fluctuation-dissipation theory [68].

The dynamics of the complex intracavity coherent field amplitude $\alpha$ is
described by Eq. (2). In our simulations, we have treated $\alpha$ as a
classical quantity as this approximation has been shown to be adequate for
similar setups experimentally. For example, in [55] although bistability is
observed at intracavity photon numbers below unity $(|\alpha|^{2}\lesssim 1)$,
the corresponding experimental data is very well described using a classical
theory for the optical field [69]. As explained in [70], this is due to the
fact that in the ‘bad cavity’ limit [48], where the condensate mechanical
(i.e., sidemode) oscillation frequencies are lower than the cavity linewidth,
the number of photons passing through the cavity during one mechanical period
is much larger than one. The quantum fluctuations in the photon number thus
have a negligible effect on the dynamics of the condensate density
modulations. In our simulations below, we have ensured that the bad cavity
limit always applies.

In the first term inside the square bracket on the right-hand side of Eq. (2),
$\Delta_{c}$ signifies the detuning of the driving field frequency from the
cavity resonance frequency $\omega_{c}$. The second term represents the
coupling between the light mode and condensate, where the expectation value of
the light potential $\cos^{2}\left(l\phi\right)$ taken with respect to the
condensate wave function $\psi(\phi,\tau)$

$\langle\cos^{2}\left(l\phi\right)\rangle_{\tau}=\int_{0}^{2\pi}\left|\psi\left(\phi,\tau\right)\right|^{2}\cos^{2}\left(l\phi\right)d\phi,$
(7)

is a time-dependent quantity. In the third term, $\gamma_{0}$ is the energy
decay rate of the cavity field. The last term inside the curly braces
represents the laser drive with pump rate
$\eta=\sqrt{P_{in}\gamma_{0}/\hbar\omega_{c}}$, where $P_{in}$ is the input
optical power. The last term on the right-hand side of Eq.(2) signifies the
optical shot noise present in the system. The thermal and optical fluctuations
each have zero mean and their correlations are given by [71, 47]

$\displaystyle\langle\xi(\phi,\tau)\xi^{*}(\phi^{\prime},\tau^{\prime})\rangle$
$\displaystyle=2\Gamma T\delta(\phi-\phi^{\prime})\delta(\tau-\tau^{\prime}),$
(8)
$\displaystyle\langle\alpha_{in}(\tau)\alpha_{in}^{*}(\tau^{\prime})\rangle$
$\displaystyle=\omega_{\beta}\delta(\tau-\tau^{\prime}),$ (9)

where $T$ is the non-dimensionalized temperature in units of
$k_{B}/{(\hslash\omega_{\beta})}$, with $k_{B}$ being the Boltzmann constant.
For the numerical simulation of these stochastic equations, the noise terms
are modeled as follows

$\displaystyle\xi(\phi,\tau)$ $\displaystyle=\sqrt{2\Gamma T/(d\phi
d\tau)}\mathcal{N}(0,1,N_{\phi})\mathcal{N}(0,1,N_{\phi}),$ (10)
$\displaystyle\alpha_{in}(\tau)$
$\displaystyle=\sqrt{\omega_{\beta}/d\tau}\mathcal{N}(0,1,1),$ (11)

where $\mathcal{N}(0,1,N_{\phi})$ is a normally distributed random variable
with zero mean and unit variance, where the third argument in
$\mathcal{N}(0,1,N_{\phi})$ refers to the size of the array containing the
random numbers and $N_{\phi}$ is the number of grid points in the $\phi$
direction. We have considered $N_{\phi}=1024$ for all the simulation runs.

To attain the dynamics of the persistent current, we have considered the
initial state as a plane wave, and then we evolve the system in real-time
using the coupled BEC - cavity equations and the RK4 scheme [72]. A different
approach is taken for the case of soliton. Initially, we prepare a localized
state with a Gaussian density profile, and then we evolve it in imaginary time
using the Strang splitting Fourier method [73] to reach a soliton as its
ground state in the presence of an optical lattice as well as atomic
interactions. The resulting ground state is then used as the initial state for
the subsequent real-time evolution using the RK4 scheme. All the results in
the paper have been presented for single realizations of a BEC, and each
realization has been averaged over $5$ $\mu$s. We have adopted the time step
$d\tau=10^{-7}$ for all the simulation runs.

##  III Results

###  III.1 Persistent current

####  III.1.1 Rotational eigenstate

In this section, we present the dynamics accompanying the detection of a
persistent current in the ring BEC. Such currents can exist for macroscopic
times as metastable flow states of the condensate with atoms that weakly repel
each other [1]. The basic idea is for the circular optical lattice to act as a
probe of the angular momentum, and hence the winding number, of the condensate
[47]. For low intra-cavity photon number, the matter wave Bragg diffracts from
the weak optical lattice. This results, in the first order, in two additional
OAM states (sidemodes), which modulate the condensate density. These
modulations add sidebands to the optical modes, which can subsequently be
detected in the cavity transmission.

Figure 2: (a) Angular profile of the condensate density per particle for a
persistent current rotational eigenstate (b) OAM state content of the
condensate. Parameters used here are $\Gamma=0.0001$, $T=10$ nK $\times
k_{B}/{(\hslash\omega_{\beta})}$ , $g/\hbar\simeq 2\pi\times 0.02$ Hz,
$L_{p}=1$, $l=10$, $N=10^{4}$, $\tilde{\Delta}=-2\pi\times 173$ Hz,
$U_{0}=2\pi\times 212$ Hz, $\gamma_{0}=2\pi\times 2$ MHz, $P_{in}=0.2$ pW,
$\omega_{c}=2\pi\times 10^{15}$ Hz, $m=23$ amu, and $R=12$ $\mu$m.
Figure 3: Power spectra of the output phase quadrature of the cavity field as
a function of the system response frequency for $L_{p}=1$ (blue) and $L_{p}=2$
(red). The vertical dashed lines (grey and green) correspond to the analytical
predictions for the side modes of $L_{p}=1$ and $L_{p}=2$ respectively,
including atomic interactions, made earlier [47]. Other parameters are same as
mentioned in Fig. 2.
Figure 4: Persistent current eigenstate rotation measurement sensitivity
$\zeta$ [Eq. (14)] as a function of the system response frequency $\omega$.
Here $G=2\pi\times 7.5$ kHz and $|\alpha_{s}|^{2}=0.096$, which corresponds to
$P_{in}=0.2$ pW. Other parameters are the same as for Fig. 2.

In our simulation, for which we have used 23Na atoms [1], a phase gradient is
imprinted initially on the condensate, in order to impart a winding number
$L_{p}$ to it. The resulting persistent current then gets coupled to the
angular optical lattice, which displays $2l$ interference maxima along the
ring on which the BEC is trapped. For all our simulations related to the
persistent current, we consider an initial state for the condensate wave
function of the form

$\psi(\phi)=\sqrt{\frac{N}{2\pi}}e^{iL_{p}\phi},$ (12)

which corresponds to an eigenstate of condensate rotation in the absence of
the optical lattice.

Figure 5: Variation of the fidelity [Eq. (15)] of the ring condensate density
for a persistent current with time. The red dashed line is the guide to see
the fidelity value when the density profile of the condensate at different
times are in phase with the initial state. The set of parameters are same as
those in Fig. 2.

The resulting condensate density ($\psi$) obtained from the Eqs. (1)-(2),
modulated by the presence of the condensate sidemodes created by the optical
lattice, is shown in Fig. 2(a). The OAM content of the modulated condensate
density ($\lvert\tilde{\psi}\rvert^{2}$), where $\tilde{\psi}$ is the Fourier
amplitude of the condensate density ($\psi$), is shown in Fig. 2(b), which
displays the first-order peaks, resulting from matter-wave Bragg diffraction,
at $L_{p}\pm 2l$. The figure, which accounts for the full Gross-Pitaevskii
condensate dynamics, implies that only three OAM modes are dominant and
therefore provides justification for the few-mode model proposed earlier.
[47].

Figure 6: (a) Angular profile of the condensate density per particle for a
persistent current superposition [Eq. (16)] with $L_{p1}=3,L_{p2}=4$ (b) OAM
state content of the condensate. Here $P_{in}=1$ pW and the other parameters
are the same as in Fig. 2.

In Fig. 3 we show the phase quadrature of the resulting cavity transmission
spectrum [47],

$S(\omega)=\left|\mathrm{Im}\left[\alpha_{out}(\omega)\right]\right|^{2},$
(13)

where $\alpha_{out}$ is the output field, transmitted from the cavity and it
is related to the input field into the cavity, through the input-output
relation $\alpha_{out}=-\alpha_{in}+\sqrt{\gamma_{0}}\alpha$ [48]. The
vertical dashed lines for $L_{p}=1$ correspond to the sidemode frequencies
$\omega_{c,d}$ defined in the Supplementary Material of [47], see Eq.S33.
These definitions account for the atomic interactions. The agreement between
the vertical lines and the peak locations show that the Gross-Pitaevskii
simulation retains the results predicted by the few-mode theory presented
earlier. It also shows that our classical treatment of $\alpha$, the optical
field, reproduces the results of [47], which treated the optical field quantum
mechanically. It can also be seen that the peaks for $L_{p}=2$ are spectrally
distinct from those for $L_{p}=1$. Thus our method can reliably distinguish
between neighboring values of condensate winding number $L_{p}$. We have also
analyzed the effect of high cavity power drive $P_{in}$ on the power spectra
of the output phase of the cavity and find that increasing the $P_{in}$ leads
to the deviation of the different peaks of $S(\omega)$ from the analytical
results as well as generation of the other OAM modes at higher frequencies as
shown in the Fig. A.2.

Figure 7: Persistent current superposition (a) Power spectrum of the output
phase quadrature of the cavity field as a function of the system response
frequency. The vertical dashed lines correspond to the analytical predictions
for the side modes corresponding to $L_{p1}=3,L_{p2}=4$. (b) Rotation
measurement sensitivity. The parameters used are the same as in Fig. 6.
Figure 8: Variation of the fidelity [Eq. (15)] of the condensate density with
time, resulting from the superposition of two persistent currents. The set of
parameters are same as in Fig. 6. Inset shows the fidelity from the initial
time to $100$ ms.

We quantify the performance of our scheme by using the sensitivity of the
rotation measurement, defined as

$\zeta=\frac{S(\omega)}{\partial
S(\omega)/\partial\Lambda}\times\sqrt{t_{meas}},$ (14)

where $t_{meas}^{-1}=8(\alpha_{s}G)^{2}/\gamma_{0}$ is the optomechanical
measurement rate in the bad cavity limit, $G=U_{0}\sqrt{N}/2\sqrt{2}$ [47],
and $\alpha_{s}$ is the steady state of the cavity field [47]. The sensitivity
of Eq. (14), obtained from a fit to the spectrum [Eq. (13)], is displayed in
Fig. 4, and matches the result from [47] quite well.

To quantify to what extent the measurement backaction affects the condensate,
we display the fidelity $F(t)$, i.e. the position-averaged autocorrelation
function of the condensate density

$F(t)=\int_{0}^{2\pi}\left[\psi^{*}(\phi,t)\psi(\phi,0)\right]^{2}d\phi,$ (15)

as a function of the unscaled time $t$ in Fig. 5. As can be seen, the density
fidelity stays close to unity for small times. For macroscopic times, it
decays mainly due to the damping ($\Gamma$) and noise ($\xi$) of the
persistent current. This indicates that the measurement backaction on the
condensate is small. Certainly, unlike existing techniques, the measurement
does not completely destroy the condensate.

####  III.1.2 Two state superposition

In this section, we investigate the dynamics of the condensate prepared in a
superposition state of two different winding numbers i.e. $L_{p1}\neq L_{p2}$.
These states could be of interest in the context of quantum information
processing, matter wave interferometry, as well as studies of mesoscopic
quantum mechanics and [74, 75]. We start with the initial state

$\psi(\phi)=\sqrt{\frac{N}{4\pi}}\left(e^{iL_{p1}\phi}+e^{iL_{p2}\phi}\right).$
(16)

Fig. 6(a) shows the condensate density resulting from the superposition of two
persistent currents. The modulation in the condensate density is relatively
high due to the high input optical power required to observe condensate
rotation. The OAM content of the corresponding state is shown in Fig. 6(b).
The increased complexity in the OAM content of the state is due to the
interference between the two persistent currents, which introduces additional
modulations to the system. Despite these complicated modulations, sidebands
corresponding to $L_{p1}$ and $L_{p2}$ can be observed in the phase quadrature
of the resulting cavity transmission spectrum, which is presented in Fig.
7(a).

The sensitivity of the rotation measurement is shown in Fig. 7(b) and is
comparable to that of the measurement of the rotational eigenstate [Fig. 4].
Variation of the fidelity of the condensate density with time is shown in Fig.
8. In this case, due to the interference between multiple sidemodes, the
fidelity of the superposition measurement is at best around $0.5$.

###  III.2 Bright soliton

In this section, we present the dynamics accompanying the detection of a
bright soliton in the ring BEC. Such solitons can be sustained by the
condensates, in which the atoms weakly attract each other [76, 77, 78, 79, 80,
81, 82, 83, 84, 85]. In our simulations, we imprint a density and phase
modulation on a uniform condensate of 7Li atoms [83] and this leads to a
bright soliton rotating on the ring and carrying a winding number $L_{p}$
(e.g. see Eq. (43) of [26]). For this situation, we consider the initial state
as

$\psi(\phi)=\sqrt{\frac{N}{\sqrt{\pi}}}e^{-\phi^{2}/2}e^{iL_{p}\phi}.$ (17)

Figure 9: (a) Temporal evolution of the moving soliton density profile (b) OAM
content of the soliton. Here $N=6000$, $a_{s}=-27.6a_{0}$, where $a_{0}$ is
the Bohr radius, $m=7.01$ amu, $L_{p}=1$, and $P_{in}=0.4$ pW, and all other
parameters are same as in Fig. 2.
Figure 10: (a) Power spectrum of the output phase quadrature of the cavity
field as a function of the system response frequency for a soliton. The
vertical dashed lines (grey and green) correspond to the analytical
predictions for the side modes of $L_{p}=1$ and $L_{p}=2$ respectively. (b)
Variation of rotation measurement sensitivity $\zeta$ with the system response
frequency $\omega$ for $L_{p}=1$. The set of parameters used here are the same
as in Fig. 9.

For $L_{p}=1$, the temporal evolution of the soliton density profile is shown
in Fig. 9(a). As can be seen, the spatial profile of the soliton stays close
to its initial shape as it moves in the ring. The OAM distribution of the
soliton, when it has interacted with the optical lattice, is shown in Fig.
9(b).

The resulting cavity transmission spectrum used to detect the winding number
of the soliton is shown in Fig. 10(a). As can be seen, the $L_{p}=1$ and
$L_{p}=2$ peaks are resolvable, indicating that our method can distinguish
between neighboring winding numbers for the soliton. Remarkably, the
analytical predictions for the spectral locations of the sidemode peaks from
the analytical treatment of the persistent current case [47] agree quite well
with the full Gross-Pitaevskii treatment of the soliton. This can be seen from
the coincidence of the numerically obtained side mode peaks and the vertical
dashed lines in Fig. 10(a).

Figure 11: Fidelity (Eq. 15) of the soliton density profile versus time for
$P_{in}=0.4$ pW. The red dashed line is the guide to the fidelity value when
the solitons at different times are in phase with the initial state. The
remaining set of parameters are the same as in Fig. 9.

The corresponding measurement sensitivity is shown in Fig. 10(b). As can be
seen, the benefits of high measurement sensitivities in our optomechanical
scheme carry over from the persistent current case (where it was three orders
of magnitude better than demonstrated previously [47]) to the case of the
bright soliton. The fidelity of the density profile is shown in Fig. 11 and
remains close to unity for macroscopic times. The decay in fidelity is largely
due to the dissipation ($\Gamma$) and noise ($\xi$) in the system, and not so
much due to measurement backaction. Hence our scheme represents a minimally
destructive measurement of the motion of a bright soliton in a ring BEC. For
high input powers results the appearance of more modes accompanied with the
noise in the condensate as shown in the Fig. A.2.

##  IV Summary and Conclusions

Using a stochastic mean-field Gross-Pitaevskii formalism for modeling atoms in
a BEC in a ring trap, and a classical approximation for the optical mode, we
have demonstrated that cavity optomechanics can make real-time, in situ, and
minimally destructive measurements of both persistent currents as well as
bright solitons. In support of our conclusions, we have presented numerical
simulations of cavity transmission spectra, measurement sensitivities as a
function of the system response frequency, and the fidelity of condensate
density profiles.

Our numerical simulations have verified and extended the analytical model
proposed by us earlier. Remarkably, the previously analytically-found
locations of the peaks in the cavity transmission crucial for determining
condensate rotation agree well with the numerical results for both persistent
current states as well as solitons.

We expect our findings to be of interest to studies of superfluid
hydrodynamics, atomtronics, and soliton interferometry. The technique we have
presented could be extended to other systems such as polariton ring
condensates [86].

##  V Acknowledgments

We thank the International Centre for Theoretical Sciences, Bengaluru, where
this work was initiated, for hosting us. M.B. would like to thank the Air
Force Office of Scientific Research (FA9550-23-1-0259) for support. R.K.
acknowledges support from JSPS KAKENHI Grant No. JP21K03421. We also
gratefully acknowledge our supercomputing facility Param-Ishan (IITG), where
all the simulation runs were performed.

##  Appendix A Noise for high input power

In this appendix, we present the effect of increasing the cavity drive power
$P_{in}$ and thus making a stronger measurement of the persistent current and
bright soliton rotation.

In Fig. A.1, we show the power spectra of the output phase quadrature of the
cavity field in the frequency domain for three different input powers, namely,
$P_{in}=0.5$ pW, $1$ pW and $2$ pW for the persistent current. We find that
increasing the power (from the left column to the right) results in nonlinear
behavior, as discussed below.

Density profiles of the soliton have been shown in Figs. A.2(a)-(c). We can
see that at low powers $P_{in}$ the density profile is only slightly
modulated, while it is quite heavily modulated at high $P_{in}$. Thus, as
expected, the measurement backaction increases with cavity power. However, in
the regime of the powers presented, the soliton does not break up as a result
of interaction with the probe lattice, and thus the measurement is not fully
destructive.

The OAM content of the soliton for the corresponding powers have been shown in
Figs. A.2(d)-(f). The spectra are displayed in Figs. A.2(g)-(i), the peaks
labeled by the winding number $L_{p}$. As can be seen, use of higher optical
powers results in additional peaks and more noise. Fianlly the sensitivities
are shown in Figs. A.2(j)-(l), showing that they are comparable to the low
power version.

Figure A.1: Power spectrum of the output phase quadrature of the cavity field
as a function of the system response frequency for different input powers, (a)
$P_{in}=0.5$ pW, (b) $P_{in}=1$ pW, and (c) $P_{in}=2$ pW. Vertical dashed
lines indicate the analytical predictions of [47]. All other parameters are
the same as in Fig. 2.

Figure A.2:  (a)-(c)Temporal evolution of the moving soliton density profile
(d)-(f) OAM states of the condensate occupied by the soliton (g)-(i) the power
spectrum of the imaginary part of cavity field versus response frequency
(j)-(l) soliton rotation measurement sensitivity as a function of system
response frequency for $P_{in}=0.7$ pW, $1$ pW, and $2$ pW, respectively. Here
$G=2\pi\times 5.8$ kHz and $|\alpha_{s}|^{2}=0.33,0.48,0.96$ for the above
input power values. The remaining set of parameters are the same as in Fig. 9.

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# On the Feynman path integral formulation of the Bell-Clauser-Horne-Shimony-
Holt inequality in Quantum Field Theory

G. Peruzzo1, S. P. Sorella2,

1Instituto de Física, Universidade Federal Fluminense,
Campus da Praia Vermelha, Av. Litorânea s/n,
24210-346, Niterói, RJ, Brasil
2UERJ – Universidade do Estado do Rio de Janeiro,
Instituto de Física – Departamento de Física Teórica – Rua São Francisco
Xavier 524,
20550-013, Maracanã, Rio de Janeiro, Brasil
gperuzzofisica<EMAIL_ADDRESS>
###### Abstract

By employing a free scalar Quantum Field Theory model previously introduced
[1], we attempt at formulating the Bell-CHSH inequality within the Feynman
path integral. This possibility relies on the observation that the Bell-CHSH
inequality exhibits a natural extension to Quantum Field Theory in such a way
that it is compatible with the time ordering $T$. By treating the Feynman
propagator as a distribution and by introducing a suitable localizing set of
compact support smooth test functions, we work out the path integral setup for
the Bell-CHSH inequality, recovering the same results of the canonical
quantization.

##  1 Introduction

The study of the Bell-Clauser-Horne-Shimony-Holt [2, 3, 4, 5, 6, 7] inequality
is one of the corner stone of the physics of the entanglement, as documented
by the large literature available on the subject in Quantum Mechanics.

From the theoretical point of view of relativistic Quantum Field Theory, it
seems fair to state that the study of the Bell-CHSH inequality is yet to be
considered at its beginning. Let us mention that the topic has a great
phenomenological interest in view of the future experiments at LHC, see [8]
and refs. therein.

The study of the Bell-CHSH inequality in Quantum Field Theory started with the
pioneering work of [9, 10, 11, 12, 13] who, making use Algebraic Quantum Field
Theory [14], showed that even free fields lead to a violation of the Bell-CHSH
inequality. This result highlights the strength of entanglement in Quantum
Field Theory [15]. Though, many aspects remain still to be unraveled. Let us
quote, for example, the general treatment of interacting Quantum Field
Theories as well as the construction of a $BRST$ invariant setup for the Bell-
CHSH inequality in Abelian and non-Abelian gauge theories.

More recently, following [9, 10, 11, 12, 13], we have constructed an explicit
Quantum Field Theory model built out by means of a massive free scalar field
and of a suitable Bell-CHSH operator exhibiting a violation of the Bell-CHSH
inequality at the quantum level [1]. The results obtained in [1] relied on the
use of the canonical quantization.

The aim of the present work is that of pursuing the investigation of the Bell-
CHSH inequality in Quantum Field Theory. More precisely, we shall attempt at
formulating the Bell-CHSH inequality within the framework of the Feynman path
integral, a topic which, to our knowledge, has not yet been addressed.
Needless to say, the path integral formulation will enable us to study the
Bell-CHSH inequality for interacting field theories by employing the
dictionary of the Feynman diagrams, including the BRST invariant formulation
of Abelian and non-Abelian gauge theories.

Several issues arise when trying to achieve the path integral formulation of
the Bell-CHSH inequality. Willing to present them briefly, we might start to
mention that the Feynman path integral is intrinsically related to the
chronological time ordering $T$. A second issue concerns the complex character
of the Feynman propagator, i.e.

$\Delta_{F}(x-x^{\prime})=\int\frac{d^{4}p}{(2\pi)^{4}}\frac{e^{-ip(x-x^{\prime})}}{p^{2}-m^{2}+i\varepsilon}\neq(\Delta_{F}(x-x^{\prime}))^{\dagger}\;.$
(1)

Both aspects have to be properly addressed when comparing the Hermitian
expression of the Bell-CHSH correlator obtained via canonical quantization
with the corresponding quantum correlator evaluated with the Feynman path
integral.

As we shall see in the following, these issues can be faced by making use of
smeared fields, namely

$\varphi(f)=\int_{\Omega}d^{4}x\;\varphi(x)f(x)\;,$ (2)

where $f(x)$ is a test function with compact support, $\Omega$, belonging to
the space ${\cal C}_{0}^{\infty}(\mathbb{R}^{4})$, i.e. to the space of smooth
infinitely differentiable functions decreasing as well as their derivatives
faster than any power of $(x)\in\mathbb{R}^{4}$ in any direction [14]. As it
apparent from (2), the introduction of the test function $f(x)$ has the effect
of localizing the field $\varphi(x)$ in the region $\Omega$. Working with
smeared fields has many advantages. First, the use of a suitable set of test
functions will enable us to introduce a Bell-CHSH correlator compatible with
the time ordering $T$, a basic requirement in order to have a path integral
formulation. Secondly, the analytic properties of the Fourier transform of
test functions belonging to ${\cal C}_{0}^{\infty}(\mathbb{R}^{4})$, see [16],
allow to handle the Feynman $i\varepsilon$ prescription by the usual Cauchy
theorem, so as to recover exactly the result of the canonical setup.

Moreover, besides the pure mathematical aspects related to the introduction of
the test functions, we would like to point out that, in the case of the study
of the violation of the Bell-CHSH inequality, the smearing procedure acquires
a rather clear and simple physical meaning. Looking at the details of one of
the most recent experiments [17], one realizes that issues like the so-called
causality loophole, i.e. the effective experimental implementation of the
space-like separation between the two polirazers, namely Alice and Bob’s
devices, is very carefully handled. Both polarizers are randomly rotating
while the pair of entangled photons emitted by the source is flying towards
them, so that it turns out to be impossible for the photons to communicate
each other about the direction in which their respective polarization is being
measured. This very sophisticated setup has the practical effect of closing
the causality loophole. Willing thus to achieve a relativistic Quantum Field
Theory framework for the Bell-CHSH inequality, it seems to us very helpful
employing a clear localization procedure which stays as close as possible with
the experiments. This is precisely the role played by the introduction of the
test functions, namely: allowing for a well defined localization procedure in
space-time.

The paper is organized as follows. In Section (2) we elaborate on the
comparison between the Bell-CHSH operator of Quantum Mechanics and that of
Quantum Field Theory, pointing out the very basic requirement of compatibility
with the time ordering $T$. This section contains a detailed construction of
the Quantum Field Theory operators entering the Bell-CHSH correlator. Here, we
shall rely on a free scalar field model and on a set of field operators
slightly different from those employed in [1]. Though, similarly to [1], the
model exhibits a small violation of the Bell-CHSH inequality. Section (3) is
devoted to the Feynman path integral formulation of the Bell-CHSH inequality
by establishing the equivalence with the canonical formalism. In (4) we
collect our conclusion.

Overall, for the benefit of the reader, we attempted at presenting the various
topics in a self-contained way.

##  2 The Bell-CHSH inequality in Quantum Mechanics and in relativistic
Quantum Field Theory: smearing and compatibility with the time ordering $T$

###  2.1 The Bell-CHSH inequality in Quantum Mechanics: a short reminder

Let us begin by reminding the construction of the Bell-CHSH operator in
Quantum Mechanics, as presented in Quantum Mechanics textbooks, see for
example [18, 19, 20]. One starts by introducing a two spin $1/2$ operator

${\cal
C}_{CHSH}=\left[\left({\vec{\alpha}}\cdot{\vec{\sigma}}_{A}+{\vec{\alpha}^{\prime}}\cdot{\vec{\sigma}}_{A}\right)\otimes{\vec{\beta}}\cdot{\vec{\sigma}}_{B}+\left({\vec{\alpha}}\cdot{\vec{\sigma}}_{A}-{\vec{\alpha}^{\prime}}\cdot{\vec{\sigma}}_{A}\right)\otimes{\vec{\beta}^{\prime}}\cdot{\vec{\sigma}}_{B}\right]\;,$
(3)

where $(A,B)$ refer to Alice and Bob, $\vec{\sigma}$ are the spin $1/2$ Pauli
matrices and
$({\vec{\alpha}},{\vec{\alpha}^{\prime}},{\vec{\beta}},{\vec{\beta}^{\prime}})$
are four arbitrary unit vectors.111Due to
$\sigma_{i}\sigma_{j}=\delta_{ij}+i\varepsilon_{ijk}\sigma_{k}$, it follows
that $({\vec{n}}\cdot{\vec{\sigma}})^{2}=1$ for any unit vector $|\vec{n}|=1$.
The operator (3) has the renowned form

${\cal C}_{CHSH}=(A+A^{\prime})B+(A-A^{\prime})B^{\prime}\;.$ (4)

with $(A,A^{\prime})$ and $(B,B^{\prime})$ denoting the Alice and Bob spin
operators

$A={\vec{\alpha}}\cdot{\vec{\sigma}}_{A}\;,\qquad
A^{\prime}={\vec{\alpha}^{\prime}}\cdot{\vec{\sigma}}_{A}\;,\qquad
B={\vec{\beta}}\cdot{\vec{\sigma}}_{B}\;,\qquad
B^{\prime}={\vec{\beta}^{\prime}}\cdot{\vec{\sigma}}_{B}$ (5)

fulfilling the following commutation relations

$\left[A,B\right]=0\;,\qquad\left[A,B^{\prime}\right]=0\;,\qquad\left[A^{\prime},B\right]=0\;,\qquad\left[A^{\prime},B^{\prime}\right]=0\;.$
(6)

Moreover, $(A,A^{\prime})$ and $(B,B^{\prime})$ are all Hermitian, with
eigenvalues ${\pm 1}$.

On the basis of the so-called local realism of hidden variables [21], one
expects that

$|{\cal C}_{CHSH}|\leq 2\;,$ (7)

for any possible choice of the unit vectors
$({\vec{\alpha}},{\vec{\alpha}^{\prime}},{\vec{\beta}},{\vec{\beta}^{\prime}})$.

Nevertheless, it turns out that this inequality is violated by Quantum
Mechanics, due to entanglement. In fact, when evaluating the Bell-CHSH
correlator in Quantum Mechanics, i.e. $\langle\psi|{\cal
C}_{CHSH}|\psi\rangle$, where $|\psi\rangle$ is an entangled state as, for
example, the Bell singlet, one gets

$|\langle\psi|{\cal
C}_{CHSH}|\psi\rangle|=2\sqrt{2}\;,\qquad|\psi\rangle=\frac{|+\rangle_{A}\otimes|-\rangle_{B}-|-\rangle_{A}\otimes|+\rangle_{B}}{\sqrt{2}}\;.$
(8)

The bound $2\sqrt{2}$ is known as Tsirelson’s bound [22, 23], providing the
maximum violation of the $CHSH$ inequality (7). The experiments carried out
over the last decades, see [17] and refs therein, have largely confirmed the
violation of the Bell-CHSH inequality, being in very good agreement with the
bound $2\sqrt{2}$.

###  2.2 Construction of the Bell-CHSH in Quantum field theory: localization
and compatibility with the time ordering $T$

####  2.2.1 Basic features of the canonical quantization

In order to address the issue of the construction of the analogue of the Bell-
CHSH operator (3) in Quantum Field Theory, it is useful to recall here a few
basic properties of the canonical quantization of a free massive scalar field
[14]:

${\cal
L}=\frac{1}{2}\left(\partial^{\mu}\varphi\partial_{\mu}\varphi-m^{2}\varphi^{2}\right)\;.$
(9)

Expanding $\varphi$ in terms of annihiliation and creation operators, one gets

$\varphi(t,{\vec{x}})=\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega(k,m)}\left(e^{-ikx}a_{k}+e^{ikx}a^{\dagger}_{k}\right)\;,\qquad
k^{0}=\omega(k,m)=\sqrt{{\vec{k}}^{2}+m^{2}}\;,$ (10)

where

$[a_{k},a^{\dagger}_{q}]=(2\pi)^{3}2\omega(k,m)\delta^{3}({\vec{k}-\vec{q}})\;,\qquad[a_{k},a_{q}]=0\;,\qquad[a^{\dagger}_{k},a^{\dagger}_{q}]=0\;,$
(11)

are the canonical commutation relations. A quick computation shows that

$\left[\varphi(x),\varphi(y)\right]=i\Delta_{\textrm{PJ}}(x-y)=0\;\qquad{\rm
for}\;\;\;(x-y)^{2}<0\;,$ (12)

where $\Delta_{\textrm{PJ}}(x-y)$ is the Lorentz invariant causal Pauli-Jordan
function, encoding the principle of relativistic causality

  * •

$\Delta_{\textrm{PJ}}(x-y)=\frac{1}{i}\int\frac{d^{4}k}{(2\pi)^{3}}(\theta(k^{0})-\theta(-k^{0}))\delta(k^{2}-m^{2})e^{-ik(x-y)}\;,$
(13)

  * •

$\Delta_{\textrm{PJ}}(x-y)=-\Delta_{\textrm{{PJ}}}(y-x)\;,\qquad(\partial^{2}_{x}+m^{2})\Delta_{\textrm{PJ}}(x-y)=0\;,$
(14)

  * •

$\Delta_{\textrm{PJ}}(x-y)=\left(\frac{\theta(x^{0}-y^{0})-\theta(y^{0}-x^{0})}{2\pi}\right)\left(-\delta((x-y)^{2})+m\frac{\theta((x-y)^{2})J_{1}(m\sqrt{(x-y)^{2}})}{2\sqrt{(x-y)^{2}}}\right)\;,$
(15)

where $J_{1}$ is the Bessel function.

It is known that expression (10) is a too singular object, being in fact an
operator valued distribution [14]. To give a well defined meaning to eq.(10),
one introduces the smeared field

$\varphi(h)=\int d^{4}x\;\varphi(x)h(x)\;,$ (16)

where $h(x)$ is a test function belonging to the space of compactly supported
smooth functions ${\cal C}_{0}^{\infty}(\mathbb{R}^{4})$. The support of
$h(x)$, $supp_{h}$, is the region in which the test function $h(x)$ is non-
vanishing. Moving to the Fourier space

${\hat{h}}(p)=\int d^{4}x\;e^{ipx}h(x)\;,$ (17)

expression (16) becomes

$\varphi(h)=\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega(k,m)}\left({\hat{h}}^{*}(\omega(k,m),{\vec{k}})a_{k}+{\hat{h}}(\omega(k,m),{\vec{k}})a^{\dagger}_{k}\right)=a_{h}+a^{\dagger}_{h}\;,$
(18)

where $(a_{h},a^{\dagger}_{h})$ stand for

$a_{h}=\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega(k,m)}{\hat{h}}^{*}(\omega(k,m),{\vec{k}})a_{k}\;,\qquad
a^{\dagger}_{h}=\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega(k,m)}{\hat{h}}(\omega(k,m),{\vec{k}})a^{\dagger}_{k}\;.$
(19)

One sees that the smearing procedure has turned the too singular object
$\varphi(x)$, eq.(10), into an operator acting on the Hilbert space of the
system, eq.(18). When rewritten in terms of the operators
$(a_{f},a^{\dagger}_{g})$, the canonical commutation relations (11) read

$\left[a_{h},a^{\dagger}_{h^{\prime}}\right]=\langle h|h^{\prime}\rangle\;,$
(20)

where $\langle h|h^{\prime}\rangle$ denotes the Lorentz invariant scalar
product between the test functions $h$ and $h^{\prime}$. i.e.

$\langle
h|h^{\prime}\rangle=\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega(k,m)}{\hat{h}}^{*}(\omega(k,m),{\vec{k}}){\hat{h}^{\prime}}(\omega(k,m),{\vec{k}})=\int\frac{d^{4}{\vec{k}}}{(2\pi)^{4}}2\pi\;\theta(k^{0})\delta(k^{2}-m^{2}){\hat{h}}^{*}(k){\hat{h}}^{\prime}(k)\;.$
(21)

The scalar product (21) can be recast in configuration space. Taking the
Fourier transform, one has

$\langle h|h^{\prime}\rangle=\int d^{4}xd^{4}x^{\prime}\;h(x){\cal
D}(x-x^{\prime})h^{\prime}(x^{\prime})\;,$ (22)

where ${\cal D}(x-x^{\prime})$ is the so-called Wightman function

${\cal D}(x-x^{\prime})=\langle
0|\varphi(x)\varphi(x^{\prime})|0\rangle=\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega(k,m)}e^{-ik(x-x^{\prime})}\;,\qquad
k^{0}=\omega(k,m)\;.$ (23)

which can be decomposed as

${\cal
D}(x-x^{\prime})=\frac{i}{2}\Delta_{\textrm{PJ}}(x-x^{\prime})+H(x-x^{\prime})\;,$
(24)

where $H(x-x^{\prime})=H(x^{\prime}-x)$ is the real symmetric quantity [24]

$H(x-x^{\prime})=\frac{1}{2}\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega(k,m)}\left(e^{-ik(x-x^{\prime})}+e^{ik(x-x^{\prime})}\right)\qquad
k^{0}=\omega(k,m)\;.$ (25)

Finally, the commutation relation (12) can be expressed in terms of smeared
fields as

$\left[\varphi(h),\varphi(h^{\prime})\right]=i\Delta_{\textrm{PJ}}(h,h^{\prime})$
(26)

where $h$, $h^{\prime}$ are test functions and

$\Delta_{\textrm{PJ}}(h,h^{\prime})=\int
d^{4}x\;d^{4}x^{\prime}h(x)\Delta_{\textrm{PJ}}(x-x^{\prime})h^{\prime}(x^{\prime}).$
(27)

Therefore, the causality condition in terms of smeared fields becomes

$\left[\varphi(h),\varphi(h^{\prime})\right]=0,$ (28)

if $supp_{h}$ and $supp_{h^{\prime}}$ are space-like.

####  2.2.2 Weyl operators

For further use, let us present here the so-called Weyl operators, which will
be the building blocks for the construction of the Bell-CHSH operator. The
Weyl operators are bounded unitary operators built out by exponentiating the
smeared field, namely

${\cal A}_{h}=e^{i{\varphi}(h)}\;,$ (29)

where ${\varphi}(h)$ is the smeared field defined in eq.(16). Making use of
the following relation

$e^{A}\;e^{B}=\;e^{A+B+\frac{1}{2}[A,B]}\;,$ (30)

valid for two operators $(A,B)$ commuting with $[A,B]$, one immediately checks
that the Weyl operators give rise to the following algebraic structure

$\displaystyle{\cal A}_{h}\;{\cal A}_{h}^{\prime}$ $\displaystyle=$
$\displaystyle e^{-\frac{1}{2}[{\varphi}(h),{\varphi}(h^{\prime})]}\;{\cal
A}_{(h+h^{\prime})}=e^{-\frac{i}{2}\Delta_{\textrm{PJ}}(h,h^{\prime})}\;{\cal
A}_{(h+h^{\prime})}$ $\displaystyle{\cal A}^{\dagger}_{h}$ $\displaystyle=$
$\displaystyle{\cal A}_{(-h)}\;,$ (31)

where $\Delta_{\textrm{PJ}}(h,h^{\prime})$ is defined in eq.(27). Also, using
the canonical commutation relations written in the form (20), for the vacuum
expectation value of ${\cal A}_{h}$, one gets

$\langle 0|\;{\cal A}_{h}\;|0\rangle=\;e^{-\frac{1}{2}{\lVert
h\rVert}^{2}}\;,$ (32)

where the vacuum state $|0\rangle$ is the Fock vacuum: $a_{k}|0\rangle=0$ for
all modes $k$.

####  2.2.3 Construction of the Bell-CHSH operator and compatibility with the
time ordering $T$

We are now ready to face the issue of the construction of the Bell-CHSH
operator in Quantum Field Theory. We follow here the setup outlined in [9, 10,
11, 12, 13] and introduce the notion of eligibility. A set of four field
operators $(A,A^{\prime})$ and $(B,B^{\prime})$ are called eligible for the
Bell-CHSH inequality if:

  * •

they are all Hermitian

$A=A^{\dagger}\;,\qquad A^{\prime}=A^{\prime\dagger}\;,\qquad
B=B^{\dagger}\;,\qquad B^{\prime}=B^{\prime\dagger}\;,$ (33)

  * •

obey the condition

$(A,A^{\prime})\;{\rm and\;}(B,B^{\prime})\;{\rm
are\;bounded\;operators,\;taking\;values\;in\;the\;interval\;}[-1,1]\;,$ (34)

  * •

Alice’s operators $(A,A^{\prime})$ commute with Bob’s operators
$(B,B^{\prime})$, namely

$\left[A,B\right]=0\;,\qquad\left[A,B^{\prime}\right]=0\;,\qquad\left[A^{\prime},B\right]=0\;,\qquad\left[A^{\prime},B^{\prime}\right]=0\;.$
(35)

When implemented in Quantum Field Theory, the above requirements give rise to
a rather restricted class of operators, built out essentially as suitable
combinations of Weyl operators. Following the construction outlined in [1], an
example of such operators can be given by

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{\left(e^{ia\varphi(f)}\;e^{ib\varphi(f^{\prime})}+e^{-ib\varphi(f^{\prime})}\;e^{-ia\varphi(f)}\right)}{2}=A^{\dagger}\;,$
$\displaystyle A^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\left(e^{ia^{\prime}\varphi(f)}\;e^{ib\varphi(f^{\prime})}+e^{-ib\varphi(f^{\prime})}\;e^{-ia^{\prime}\varphi(f)}\right)}{2}=A^{\prime\dagger}\;,$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{\left(e^{ia\varphi(g)}\;e^{ib\varphi(g^{\prime})}+e^{-ib\varphi(g^{\prime})}\;e^{-ia\varphi(g)}\right)}{2}=B^{\dagger}\;,$
$\displaystyle B^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\left(e^{ia\varphi(g)}\;e^{ib^{\prime}\varphi(g^{\prime})}+e^{-ib^{\prime}\varphi(g^{\prime})}\;e^{-ia\varphi(g)}\right)}{2}=B^{\prime\dagger}\;,$
(36)

where $(a,a^{\prime})$ and $(b,b^{\prime})$ are arbitrary real parameters
which play a role akin to that of the four vectors
$({\vec{\alpha}},{\vec{\alpha}^{\prime}},{\vec{\beta}},{\vec{\beta}^{\prime}})$
of the quantum mechanical operator (3). In addition, $(f,f^{\prime})$ and
$(g,g^{\prime})$ are two pairs of ${\cal C}_{0}^{\infty}(\mathbb{R}^{4})$ test
functions attached, respectively, to Alice’s operators, $(A,A^{\prime})$ and
to Bob’s operators, $(B,B^{\prime})$.

As one easily sees, all operators $(A,A^{\prime},B,B^{\prime})$ in (36) are
Hermitian. Notice that, from the Weyl algebra, eq.(31), the operator $A$ can
be rewritten as

$A=\frac{\left(e^{-\frac{iab}{2}\Delta_{\textrm{PJ}}(f,f^{\prime})}\;{\cal
A}_{(af+bf^{\prime})}+e^{\frac{iab}{2}\Delta_{\textrm{PJ}}(f,f^{\prime})}\;{\cal
A}_{(-af-bf^{\prime})}\right)}{2}\;,$ (37)

so that, using the triangle inequality for the norm operator222We remind that
the norm of an operator $O$ acting on a Hilbert space ${\cal H}$ is defined
as:  $\parallel O\parallel=\\{sup\left(\frac{\parallel Ox\parallel}{\parallel
x\parallel}\right)\;,\;\;\forall x\in\cal{H}\\}\;.$ (38) , it follows that

$\parallel A\parallel\leq\frac{1}{2}\left(\parallel{\cal
A}_{(af+bf^{\prime})}\parallel+\parallel{\cal A}_{(-af-
bf^{\prime})}\parallel\right)\leq 1\;.$ (39)

Equations (36), (39) show that the first two properties, eqs.(33),(34), are
fulfilled by the four operators defind in (36). From now on, expression (37)
will be taken as the definition of the operator $A$, as well as for operators
$A^{\prime}$, $B$ and $B^{\prime}$.

Let us focus now on eq.(35). Its fulfillment requires a well specified
localization property of both Alice and Bob operators in space-time. More
precisley, relying on the relativistic causality, eq.(28), one is led to
demand that the supports of Alice’s test functions $(f,f^{\prime})$ belong to
a space-time region $\Omega_{A}$ which is space-like with respect to the
region $\Omega_{B}$ containing the supports of Bob’s test functions
$(g,g^{\prime})$, i.e.

$(supp_{(f,f^{\prime})})\;\;\;\;{\rm space-
like\;with\;respect\;to}\;\;\;\;(supp_{(g,g^{\prime})})\;,$ (40)

see Fig.(1).

Figure 1: Location of the labs of Alice and Bob in a two-dimensional spacetime
diagram.

These considerations make clear the key role of the relativistic causality
when analysing the Bell-CHSH inequality in Quantum Field Theory. The use of
smeared fields and of the test functions acquires a clear physical meaning:
$(f,f^{\prime})$ and $(g,g^{\prime})$ act as space-time localizers for Alice’s
and Bob’s operators, implementing in a practical way the fundamental principle
of relativistic causality.

Furthermore, the demand of space-like separation between Alice and Bob has a
relevant consequence for the time ordering $T$. In fact, if $O_{1}(x)$ and
$O_{2}(y)$ are two field operators and $(x-y)^{2}<0$, it follows that

$[O_{1}(x),O_{2}(y)]=0\;,\qquad(x-y)^{2}<0\;,$ (41)

so that the $T$ product reduces to the identity, i.e.

$T\left(O_{1}(x)O_{2}(y)\right)=\theta(x^{0}-y^{0})O_{1}(x)O_{2}(y)+\theta(y^{0}-x^{0})O_{2}(y)O_{1}(x)=O_{1}(x)O_{2}(y)\;.$
(42)

An immediate consequence of all this is that the Bell-CHSH combination is, by
construction, left invariant by the time ordering $T$, namely

$T\left((A+A^{\prime})B+(A-A^{\prime})B^{\prime})\right)=(A+A^{\prime})B+(A-A^{\prime})B^{\prime}\;.$
(43)

Although equation (43) looks a simple consequence of the requirements
(33),(34),(35), it seems fair to state that it expresses a deep property of
the Bell-CHSH particular combination. It paves the Feynman path integral
formulation.

###  2.3 Example of the violation of the Bell-CHSH inequality in free Quantum
Field Theory

Having constructed the set of four eligible operators $(A,A^{\prime})$,
$(B,B^{\prime})$, eq.(36), we can proceed with the investigation of the
violation of the Bell-CHSH inequality. Introducing the Hermitian field
operator

${\cal
C}^{aa^{\prime}bb^{\prime}}_{ff^{\prime}gg^{\prime}}=(A+A^{\prime})B+(A-A^{\prime})B^{\prime}\;,$
(44)

we shall say, in agreement with [9, 10, 11, 12, 13], that the Bell-CHSH
inequality is violated at the qunatum level in the vacuum if

$|\langle 0|{\cal
C}^{aa^{\prime}bb^{\prime}}_{ff^{\prime}gg^{\prime}}|0\rangle|>2\;.$ (45)

Using the algebra of the Weyl operators, for the correlation function $\langle
0|{\cal C}^{aa^{\prime}bb^{\prime}}_{ff^{\prime}gg^{\prime}}|0\rangle$ we get

$\displaystyle\langle 0|{\cal
C}^{aa^{\prime}bb^{\prime}}_{ff^{\prime}gg^{\prime}}|0\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(e^{\frac{1}{2}\parallel
af+bf^{\prime}+ag+bg^{\prime}\parallel^{2}}\cos\left(\frac{ab}{2}(\Delta_{PJ}^{ff^{\prime}}+\Delta_{PJ}^{gg^{\prime}})\right)+e^{\frac{1}{2}\parallel
af+bf^{\prime}-ag-
bg^{\prime}\parallel^{2}}\cos\left(\frac{ab}{2}(\Delta_{PJ}^{ff^{\prime}}-\Delta_{PJ}^{gg^{\prime}})\right)\right)$
$\displaystyle+$ $\displaystyle\frac{1}{2}\left(e^{\frac{1}{2}\parallel
a^{\prime}f+bf^{\prime}+a^{\prime}g+bg^{\prime}\parallel^{2}}\cos\left(\frac{a^{\prime}b}{2}(\Delta_{PJ}^{ff^{\prime}}+\Delta_{PJ}^{gg^{\prime}})\right)+e^{\frac{1}{2}\parallel
a^{\prime}f+bf^{\prime}-a^{\prime}g-bg^{\prime}\parallel^{2}}\cos\left(\frac{a^{\prime}b}{2}(\Delta_{PJ}^{ff^{\prime}}-\Delta_{PJ}^{gg^{\prime}})\right)\right)$
$\displaystyle+$ $\displaystyle\frac{1}{2}\left(e^{\frac{1}{2}\parallel
af+b^{\prime}f^{\prime}+ag+b^{\prime}g^{\prime}\parallel^{2}}\cos\left(\frac{ab^{\prime}}{2}(\Delta_{PJ}^{ff^{\prime}}+\Delta_{PJ}^{gg^{\prime}})\right)+e^{\frac{1}{2}\parallel
af+b^{\prime}f^{\prime}-ag-b^{\prime}g^{\prime}\parallel^{2}}\cos\left(\frac{ab^{\prime}}{2}(\Delta_{PJ}^{ff^{\prime}}-\Delta_{PJ}^{gg^{\prime}})\right)\right)$
$\displaystyle+$ $\displaystyle\frac{1}{2}\left(e^{\frac{1}{2}\parallel
a^{\prime}f+b^{\prime}f^{\prime}+a^{\prime}g+b^{\prime}g^{\prime}\parallel^{2}}\cos\left(\frac{a^{\prime}b^{\prime}}{2}(\Delta_{PJ}^{ff^{\prime}}+\Delta_{PJ}^{gg^{\prime}})\right)+e^{\frac{1}{2}\parallel
a^{\prime}f+b^{\prime}f^{\prime}-a^{\prime}g-b^{\prime}g^{\prime}\parallel^{2}}\cos\left(\frac{a^{\prime}b}{2}(\Delta_{PJ}^{ff^{\prime}}-\Delta_{PJ}^{gg^{\prime}})\right)\right)\;,$

where $\Delta_{PJ}^{ff^{\prime}}$ stands for

$\Delta_{PJ}^{ff^{\prime}}=\Delta_{PJ}(f,f^{\prime})\;,$ (47)

as given in eq.(27).

In order to specify the scalar product between the test functions
$(f,f^{\prime})$, $(g,g^{\prime})$, we employ the same parametrization of the
original work [10]333See Proposition 3.1:

$\displaystyle\parallel f\parallel^{2}$ $\displaystyle=$
$\displaystyle\parallel f^{\prime}\parallel^{2}=\parallel
g\parallel^{2}=\parallel
g^{\prime}\parallel^{2}=\frac{1+\lambda^{2}}{1-\lambda^{2}}\;,$
$\displaystyle\langle f|f^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\langle g|g^{\prime}\rangle=i\;,$ $\displaystyle\langle
f|g\rangle$ $\displaystyle=$ $\displaystyle-\langle
f^{\prime}|g^{\prime}\rangle=\frac{2\lambda}{1-\lambda^{2}}\;,$
$\displaystyle\langle f|g^{\prime}\rangle$ $\displaystyle=$ $\displaystyle
0=\langle f^{\prime}|g\rangle\;,$ (48)

where $\lambda$ is a real parameter, $0<\lambda<1$.

As a consequence, expression (LABEL:qcf) becomes

$\displaystyle\langle 0|{\cal
C}^{aa^{\prime}bb^{\prime}}_{ff^{\prime}gg^{\prime}}|0\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(e^{-\left(a^{2}\frac{1+\lambda}{1-\lambda}+b^{2}\frac{1-\lambda}{1+\lambda}\right)}\cos(2ab)+e^{-\left(a^{\prime
2}\frac{1+\lambda}{1-\lambda}+b^{2}\frac{1-\lambda}{1+\lambda}\right)}\cos(2a^{\prime}b)\right)$
(49) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(e^{-\left(a^{2}\frac{1+\lambda}{1-\lambda}+b^{\prime
2}\frac{1-\lambda}{1+\lambda}\right)}\cos(2ab^{\prime})-e^{-\left(a^{\prime
2}\frac{1+\lambda}{1-\lambda}+b^{\prime
2}\frac{1-\lambda}{1+\lambda}\right)}\cos(2a^{\prime}b)\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left(e^{-\left(a^{2}\frac{1-\lambda}{1+\lambda}+b^{2}\frac{1+\lambda}{1-\lambda}\right)}+e^{-\left(a^{\prime
2}\frac{1-\lambda}{1+\lambda}+b^{2}\frac{1+\lambda}{1-\lambda}\right)}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left(e^{-\left(a^{2}\frac{1-\lambda}{1+\lambda}+b^{\prime
2}\frac{1+\lambda}{1-\lambda}\right)}-e^{-\left(a^{\prime
2}\frac{1-\lambda}{1+\lambda}+b^{\prime
2}\frac{1+\lambda}{1-\lambda}\right)}\right)\;.$

In Fig.(2) we present the graph of $\langle 0|{\cal
C}^{aa^{\prime}bb^{\prime}}_{ff^{\prime}gg^{\prime}}|0\rangle\equiv\langle{\cal
C}^{aa^{\prime}bb^{\prime}}_{ff^{\prime}gg^{\prime}}\rangle$ (cyan surface) as
a function of $a^{\prime}$ and $b^{\prime}$, for $a=b=0,001$ and
$\lambda=0,75$. Also, to easily see the violation we plot the plane $z=2$
(orange surface). In the figure on the right, a section of the graph is shown
along the plane $a^{\prime}=0.2$. As one can see, there is a region where the
CHSH inequality is violated, i.e. $\langle{\cal
C}^{aa^{\prime}bb^{\prime}}_{ff^{\prime}gg^{\prime}}\rangle>2$.

Figure 2:  $CHSH$ correlator for $\lambda=0,75$.
Behavior of the $CHSH$ correlator $\langle{\cal
C}^{aa^{\prime}bb^{\prime}}_{(ff^{\prime}gg^{\prime})}\rangle$, cyan surface,
for $a=b=0,001$ and $\lambda=0,75$. To observe the violation more easily, we
have also plotted the plane z=2, corresponding to the orange surface. The blue
line in the right hand side figure shows the behavior of $\langle{\cal
C}^{aa^{\prime}bb^{\prime}}_{(ff^{\prime}gg^{\prime})}\rangle$ for
$a=b=0,001$, $\lambda=0,75$ and $a^{\prime}=0,2$.

##  3 Feynman path integral formulation of the Bell-CHSH inequality

Relying on the results and observations of the previous sections, let us
discuss now the Feynman path integral formulation of the Bell-CHSH inequality.
To that end, let us start by introducing the generating functional ${\cal
Z}(j)$ of the time ordered correlation functions:

${\cal Z}(j)=\frac{\int[D\varphi]\;e^{i\left(S(\varphi)+\int
d^{4}xj\varphi\right)}}{\int[D\varphi]\;e^{i\left(S(\varphi)\right)}}=e^{-\frac{i}{2}\int
d^{4}xd^{4}yj(x)\Delta_{F}(x-y)j(y)}\;,$ (50)

where $\Delta_{F}(x-y)$ is the Feynman propagator, i.e.

$\Delta_{F}(x-y)=\int\frac{d^{4}p}{(2\pi)^{4}}\frac{e^{-ip(x-y)}}{p^{2}-m^{2}+i\varepsilon}\;.$
(51)

It is useful to remind here the expression of $\Delta_{F}(x-y)$ in
configuration space. Using the same notations of [25], $\Delta_{F}(x-y)$ can
be written as

$\Delta_{F}(x-y)=\frac{1}{2}(\theta(x^{0}-y^{0})-\theta(y^{0}-x^{0}))\Delta_{PJ}(x-y)-iH(x-y)\;,$
(52)

where $\Delta_{PJ}$ is the Pauli-Jordan function, eq.(13), and $H$ is the
symmetric expression of eq.(25). Explicitly:

$\displaystyle\Delta_{F}(x-y)$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\delta((x-y)^{2})+\frac{m\theta((x-y)^{2})}{8\pi\sqrt{(x-y)^{2}}}\left(J_{1}(m\sqrt{(x-y)^{2}})-iN_{1}(m\sqrt{(x-y)^{2}})\right)$
(53) $\displaystyle-$
$\displaystyle\frac{im\theta(-(x-y)^{2})}{4\pi^{2}\sqrt{-(x-y)^{2}}}K_{1}(m\sqrt{-(x-y)^{2}})\;,$

where, from the second line, one observes the well known non-causal behavior
of the Feynman propagator. In the above expression, $J_{1}$ is the Bessel
function, $N_{1}$ the Neumann function and $K_{1}$ the modified Bessel
function.

To achieve the equivalence between the path integral and the canonical
formalism we evaluate the correlation functions of two Weyl operators in both
cases, the aim being that of showing that

$\langle e^{i\varphi(h)}\;e^{i\varphi(h^{\prime})}\rangle_{Feyn}=\langle
0|e^{i\varphi(h)}\;e^{i\varphi(h^{\prime})}|0\rangle_{can}\;,$ (54)

where the supports of the two test functions $(h,h^{\prime})\in{\cal
C}_{0}^{\infty}(\mathbb{R}^{4})$ are taken as located in the positive half-
plane $t>0$, as in Fig.(1), and are space-like, i.e.

$(supp_{(h)})\;\;\;\;{\rm space-
like\;with\;respect\;to}\;\;\;\;(supp_{(h^{\prime})})\;,$ (55)

a feature which, as already underlined, reduces the chronological ordering $T$
to unity.

The computation of the left hand side of eq.(54) is easily done with the help
of (50), namely

${\cal Z}(j)=\langle e^{i\varphi(j)}\rangle_{Feyn}\;.$ (56)

Therefore, setting

$j=h+h^{\prime}\;,$ (57)

it follows that

$\langle
e^{i\varphi(h)}\;e^{i\varphi(h^{\prime})}\rangle_{Feyn}=e^{-\frac{i}{2}\Delta_{F}(h+h^{\prime},h+h^{\prime})}=e^{-\frac{i}{2}\left(\Delta_{F}(h,h)+2\Delta_{F}(h,h^{\prime})+\Delta_{F}(h^{\prime},h^{\prime})\right)}\;,$
(58)

where
$(\Delta_{F}(h,h),\Delta_{F}(h,h^{\prime}),\Delta_{F}(h^{\prime},h^{\prime}))$
denote the smeared expressions

$\displaystyle\Delta_{F}(h,h^{\prime})$ $\displaystyle=$ $\displaystyle\int
d^{4}xd^{4}y\;h(x)\Delta_{F}(x-y)h^{\prime}(y)\;,$ (59)
$\displaystyle\Delta_{F}(h,h)$ $\displaystyle=$ $\displaystyle\int
d^{4}xd^{4}y\;h(x)\Delta_{F}(x-y)h(y)\;,\qquad\Delta_{F}(h^{\prime},h^{\prime})=\int
d^{4}xd^{4}y\;h^{\prime}(x)\Delta_{F}(x-y)h^{\prime}(y).\;.$ (60)

One sees that eq.(58) demands the evaluation of two kinds of smeared
expressions involving the Feynman propagator. Let us first consider expression
(59), where the smearing is done with respect to two different test functions:
$(h,h^{\prime})$. Reminding that the supports of $h$ and $h^{\prime}$ are
space-like, eq.(55), one can rely directly on expression (52), from which one
realizes that the Pauli-Jordan term $\Delta_{PJ}(x-y)$ does not contribute
since it vanishes for space-like separations. Thus,

$\Delta_{F}(h,h^{\prime})=-iH(h,h^{\prime})=-i\int
d^{4}xd^{4}y\;h(x)H(x-y)h^{\prime}(y)\;.$ (61)

Though, owing to the general definition of the scalar product of test
functions in terms of Wightman two point function, eqs.(22),(23),(24), it
follows that

$\Delta_{F}(h,h^{\prime})=-i\langle h|h^{\prime}\rangle\;.$ (62)

Let us now focus on the expressions $\Delta_{F}(h,h)$ and
$\Delta_{F}(h^{\prime},h^{\prime})$ in eq.(60). These quantities require a
different handling, as the Feynman propagator is smeared exactly over the same
support. We proceed by moving to the Fourier space, i.e.

$\Delta_{F}(h,h)=\int\frac{d^{4}p}{(2\pi)^{4}}\;\frac{1}{p^{2}-m^{2}+i\varepsilon}|h(p_{0},\vec{p})|^{2}\;,$
(63)

where $h(p_{0},\vec{p})$ is the Fourier transform of $h(x)$

$h(p_{0},\vec{p})=\int d^{x}\;e^{ipx}h(x)\;.$ (64)

As is well known [16], being $h(x)$ a Schwartz type test function, its Fourier
transform displays an exponential decay at larhe |p|. Moreover, since
$h(x)\in{\cal C}_{0}^{\infty}(\mathbb{R}^{4})$ and its support is located in
the positive half-plane $t>0$, Fig.(1), it follows that $h(p_{0},\vec{p})$ can
be analytically continued to an entire function in the complex $p_{0}$ plane,
decaying very fast for large values of $Im(p_{0})$ in the positive complex
$p_{0}$ half-plane.

Let us illustrate this relevant property with a simple one-dimensional example
taken from Chapter II of [16]. Consider the function $f(x)\in{\cal
C}_{0}^{\infty}(\mathbb{R})$ defined as

$f(x)=\begin{cases}{\cal
C}e^{-\frac{1}{(x-a)^{2}(x-b)^{2}}}&\quad\text{if}\;x\in[a,b]\;,a,b>0\\\
0&\quad\text{if}\;x\notin[a,b]\;,\end{cases}$ (65)

where ${\cal C}$ is a normalization factor. The function $f(x)$ is a smooth
function, infinitely differentiable, which is non-vanishing only in the
interval $x\in[a,b]$. For the Fourier transform, we have

${\hat{f}}(p)=\int_{-\infty}^{\infty}dx\;e^{ipx}f(x)\;,$ (66)

Moreover, since $f(x)$ has compact support, the integral (66) becomes

${\hat{f}}(p)=\int_{a}^{b}dx\;e^{ipx}f(x)\;,$ (67)

and can be analytically continued to an entire function in the complex plane

$z=p+i\tau\;,\qquad{\hat{f}}(z)=\int_{a}^{b}dx\;e^{ipx}e^{-\tau x}f(x)\;,$
(68)

Notice that the analytic continuation of ${\hat{f}}(p)$ to an entire function
is possible thanks to the fact that $f(x)$ has compact support, so that the
integral (68) does exist for all values of $\tau$. Of corse ${\hat{f}}(z)$
decays very fast to zero when $Im(p)=\tau\rightarrow\infty$:

$\lim_{\tau\rightarrow\infty}{\hat{f}}(p+i\tau)=0\;.$ (69)

Therefore, as a consequence of these properties, we can evaluate expression
(63) by employing the residue Cauchy theorem in the complex $p_{0}$ plane, by
closing the contour to infinity in the upper positive imaginary half plane,
getting nothing but

$\Delta_{F}(h,h)=-i\parallel h\parallel^{2}\;.$ (70)

Collecting everything, it turns out that

$\langle
e^{i\varphi(h)}\;e^{i\varphi(h^{\prime})}\rangle_{Feyn}=e^{\frac{i}{2}\Delta_{F}(h+h^{\prime})}=e^{-\frac{\parallel
h+h^{\prime}\parallel^{2}}{2}}\;,$ (71)

which is exactly the result obtained from the canonical quantization, eq.(32).
Finally, we get

$\langle e^{i\varphi(h)}\;e^{i\varphi(h^{\prime})}\rangle_{Feyn}=\langle
0|e^{i\varphi(h)}\;e^{i\varphi(h^{\prime})}|0\rangle_{can}\;,$ (72)

showing thus the equivalence between the Feynman path integral and the
canonical quantization for the Bell-CHSH inequality.

##  4 Conclusion

In this work we have pursued the study of the Bell-CHSH inequality in
relativistic Quantum Field Theory by implementing its formulation within the
Feynman path integral. Both canonical quantization and functional integral
yield the same expression for the correlation function of the Weyl operators
entering the Bell-CHSH inequality.

This feature relies on the observation that, by construction, the Bell-CHSH
combination is compatible with the fundamental principle of relativistic
causality, as required by demanding that Alice and Bob be space-like
separated. Moreover, the localization of Alice anf Bob in space-time can be
given a precise mathematical formulation by employing a suitable set of smoorh
test functions with compact support, which act alike localizers for the
bounded operators entering the Bell-CHSH inequality, which turns out to be
compatible with the time ordering $T$, a key property for the path integral
formulation.

We strenghten that the Feynman path integral formulation of the Bell-CHSH
inequality opens the door to many applications such as:

  * •

treatment of interacting field theories by employing the usual dictionary of
Feynman diagrams

  * •

study of the Bell-CHSH inequality in Abelian and non-Abelian gauge theories in
a manifest BRST invariant setting, through the use of the Faddeev-Popov BRST
invariant action. In this regard, we refer to [1], where the BRST invariant
formulation for the Wyl operators of Yang-Mills theories in presence of Higgs
fields has been outlined.

  * •

Finally, the path integral formulation might enable us to estimate possible
non-perturbative contributions to the Bell-CHSH inequality stemming from the
existence of soliton sectors of the theory under investigation.

## Acknowledgements

The authors would like to thank the Brazilian agencies CNPq and FAPERJ for
financial support. This study was financed in part by the Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior–Brasil (CAPES) –Finance Code 001.
S.P. Sorella is a level $1$ CNPq researcher under the contract 301030/2019-7.

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