# Quantized refrigerator for an atomic cloud

Wolfgang Niedenzu  Institut für Theoretische Physik, Universität Innsbruck,
Technikerstraße 21a, A-6020 Innsbruck, Austria<EMAIL_ADDRESS>Igor Mazets  Vienna Center for Quantum Science and Technology (VCQ),
Atominstitut, TU Wien, 1020 Vienna, Austria  Wolfgang Pauli Institute, c/o
Fakultät für Mathematik, Universität Wien, 1090 Vienna, Austria  Gershon
Kurizki  Department of Chemical Physics, Weizmann Institute of Science,
Rehovot 7610001, Israel  Fred Jendrzejewski  Heidelberg University, Kirchhoff-
Institut für Physik, Im Neuenheimer Feld 227, D-69120 Heidelberg, Germany
<EMAIL_ADDRESS>
(June 24, 2019)

###### Abstract

We propose to implement a quantized thermal machine based on a mixture of two
atomic species. One atomic species implements the working medium and the other
implements two (cold and hot) baths. We show that such a setup can be employed
for the refrigeration of a large bosonic cloud starting above and ending below
the condensation threshold. We analyze its operation in a regime conforming to
the quantized Otto cycle and discuss the prospects for continuous-cycle
operation, addressing the experimental as well as theoretical limitations.
Beyond its applicative significance, this setup has a potential for the study
of fundamental questions of quantum thermodynamics.

##  1 Introduction

Over the past two decades, extensive studies of thermal machines in the
quantum domain [1, 2, 3, 4, 5, 6, 7] have sought to reveal either
fundamentally new aspects of thermodynamics or unique quantum advantages
compared to their classical counterparts. Yet, despite the great progress
achieved, both theoretically [8, 9, 10, 11, 12, 13, 14, 15] and experimentally
[16, 17, 18, 19, 20, 21, 22], the potential of quantum thermal machines for
useful quantum technology applications only begins to unfold. In particular,
new insights into thermodynamics in the quantum domain may be obtained by
extension of these studies to hitherto unexplored quantum phenomena. In this
spirit, we suggest to realise thermal machines in cold atomic gases, aiming at
their improved refrigeration.

[width=]otto_fridge_plus_cyclev2

Figure 1: Quantized Otto cycle for refrigeration of the cold bath: a)
Schematic representation of the Otto cooling cycle. The working medium (green)
is located in a strongly confining harmonic potential. During the heat-
exchange strokes 1 and 3 it is coupled, sequentially to the cold bath (left,
blue) and the hot bath (right, red). The energy level spacing of the working
medium is adiabatically changed during the strokes 2 and 4. b) The quantized
Otto cycle in the energy–entropy plane of the working medium. The cycle is
transversed in the clockwise direction (refrigeration mode) between the end
points $A\rightarrow B\rightarrow C\rightarrow D\rightarrow A$.

Specifically, we propose to implement a quantized thermal machine, which is
based on a mixture of two cold atomic gases, and employ it as a refrigerator
for one of these species. The goal would be to cool the atomic gas starting
from the thermal regime and ending well within the quantum-degenerate regime.
The proposed cycle will be a fundamentally new avenue for cooling, only
limited by thermalization and not by coolant recoil or by atom loss, as do
existing schemes, such as laser cooling or evaporative cooling [23]. In a
simple implementation of the Otto cycle [24, 25, 26, 6, 27] (see Fig. 1) the
working medium (WM) is realised by one species that is alternately coupled to
spatially separated, hot and cold baths of another species. This allows for
progressive cooling of the cold bath at the expense of heat dumping into the
hot bath. Traversing the quantized Otto cycle several hundred times should
then allow us to reduce the temperature of the cold bath by an order of
magnitude and even cross the phase-transition threshold towards Bose–Einstein
condensation. The proposed cooling technique should also be applicable to
other types of baths, including strongly correlated Mott insulators or anti-
ferromagnetic phases, where cooling by existing techniques remains an
experimental challenge [28]. We note that in our refrigeration setup there is
a clear physical distinction between the WM and the two thermal baths (cf.
Fig. 1), contrary to the engine proposed in Ref. [29].

A multitude of diverse cooling techniques have revolutionized atomic physics
over the last thirty years. Conventional laser cooling that has founded the
field of ultracold atomic gases employs photon recoil for heat extraction from
the atomic cloud. However, the phase-space-density is fundamentally limited by
the recoil energy of the photon [23]. Degenerate quantum gases have been
realized by evaporative cooling, whereby the highest-energy atoms are
selectively removed from the cloud [30, 31]. These evaporative techniques are
fundamentally limited by the required spatial separation of regions possessing
high or low entropy. Such separation typically breaks down in the quantum
degenerate regime. None of these limitations should apply to the Otto cycle,
which we will study now.

##  2 Otto cycle for cold quantum gases

The four strokes of the quantized Otto cycle can be implemented as follows in
cold-atom setups. The working medium (WM) is formed by a tightly trapped
atomic gas, such that only two quantum states of the confined system are
accessible. Such a configuration arises naturally in highly focused optical
traps [32]. It can be modelled by the Hamiltonian
$H=\mathcal{E}\sigma_{+}\sigma_{-}$, where $\sigma_{-}$ ($\sigma_{+}$) are the
Pauli lowering (raising) operators in the effective two-level trap. The
spacing between the energy levels is varied by an external optical laser which
provides the work input required for refrigeration, analogous to a piston in
common thermodynamic machine cycles [1, 2, 24, 33, 34].

In the first stroke, the WM has energy-level spacing
$\mathcal{E}_{\mathrm{c}}$ and thermalizes through its interaction with a
thermal bath at temperature $T_{\mathrm{c}}$. Thermalization is ensured
through contact collisions between the atoms of the bath and the WM. Such
thermalization does not involve any radiative processes, which limit the
maximal rate of laser cooling. The proposed thermalization represents the
isochoric interaction with the cold bath of the Otto cycle (stroke 1 in Fig.
1). The cold bath is realized by an atomic species that differs from the WM
(Fig. 1a), modelled here as an ideal, uncondensed Bose gas. However, the same
cycle remains valid for more intricate baths, such as Mott insulators,
superfluids or fermionic (spin) baths. During this first stroke the WM
receives from the cold bath the heat
$Q_{\mathrm{c}}=\mathcal{E}_{\mathrm{c}}(\bar{n}_{\mathrm{c}}-\bar{n}_{\mathrm{h}})$,
where $\bar{n}_{i}$ ($i\in\\{\mathrm{c},\mathrm{h}\\}$) are the thermal
excitations of the two-level WM (see Appendix A).

In the second stroke, the WM is adiabatically decoupled from the cold bath and
its energy-level spacing is adiabatically raised to
$\mathcal{E}_{\mathrm{h}}>\mathcal{E}_{\mathrm{c}}$. This realizes the
isentropic compression stroke of the Otto cycle (stroke 2 in Fig. 1). The work
$W_{\mathrm{in}}=(\mathcal{E}_{\mathrm{h}}-\mathcal{E}_{\mathrm{c}})\bar{n}_{\mathrm{h}}$
which is needed for the WM compression is extracted from the classical optical
field which confines the WM. We here neglect the work necessary for the
compression of the atoms that remain in the ground state as it cancels with
the decompression in the last stroke of the cycle.

In the third stroke, the WM is coupled to the hot thermal bath at temperature
$T_{\mathrm{h}}>T_{\mathrm{c}}$, with which the WM thermalizes (Stroke 3 in
Fig. 1). This stroke realizes the second isochore in the Otto cycle in which
the WM deposits the heat
$Q_{\mathrm{h}}=\mathcal{E}_{\mathrm{h}}(\bar{n}_{\mathrm{h}}-\bar{n}_{\mathrm{c}})$
into the hot bath.

The fourth stroke that closes the cycle consists in adiabatic decoupling of
the WM from the hot bath and adiabatic decrease of the energy-level spacing
back to $\mathcal{E}_{\mathrm{c}}$ (stroke 4 in Fig. 1). This decompression of
the WM produces the work
$W_{\mathrm{out}}=(\mathcal{E}_{\mathrm{c}}-\mathcal{E}_{\mathrm{h}})\bar{n}_{\mathrm{c}}$
which is transferred to the optical field.

The Otto cycle acts as a refrigerator for the cold bath as long as
$Q_{\mathrm{c}}>0$ ($Q_{\mathrm{h}}<0$), which yields the cooling condition

$\bar{n}_{\mathrm{c}}>\bar{n}_{\mathrm{h}}\quad\Rightarrow\quad\frac{\mathcal{E}_{\mathrm{h}}}{T_{\mathrm{h}}}>\frac{\mathcal{E}_{\mathrm{c}}}{T_{\mathrm{c}}}.$
(1)

The theoretically-achievable minimal temperature of the cold bath is then
limited by condition (1), yielding

$T_{\mathrm{c}}^{\mathrm{min}}=\frac{\mathcal{E}_{\mathrm{c}}}{\mathcal{E}_{\mathrm{h}}}T_{\mathrm{h}}.$
(2)

The ratio of the two baths temperature is hence only limited by the tunability
of the energy-level spacing of the quantized WM. For an optically confined WM,
the energy-level spacing $\mathcal{E}$ is controlled by the square root of the
confining laser intensity $I$, $\mathcal{E}\propto\sqrt{I}$ [32]. Since
intensity ratios of $I_{\mathrm{h}}/I_{\mathrm{c}}\gtrsim 100$ are
experimentally controllable, it should be feasible to refrigerate the cold-
bath temperature by an order of magnitude.

##  3 Proposed experimental implementation

The quantized Otto refrigerator can be implemented by employing the following
state-of-the-art techniques: (i) The spatially separated hot and cold baths
can be created within two separate harmonic traps, as originally employed in
atomic tunneling junctions [35] or more recently in atomic circuits with
fermionic and bosonic baths [16, 36]. In the proposed setup (see Fig. 2) the
two baths must be completely separated such that no particle exchange or heat
flow can happen between them, except via the WM.

[width=]experimental_setup_v1

Figure 2: Envisaged implementation of the Otto cycle: The two baths are formed
by large clouds of an atomic species. The working medium is implemented by
another strongly confined atomic species. Alternate coupling of the working
medium to each bath (consecutively) is achieved by spatially moving the
working medium into the bath region.

(ii) The WM is formed by an atomic species, by means of a species-selective
optical lattice or tweezer. Such schemes have been employed for single bosonic
atoms immersed in a cold gas [37, 38] or for larger numbers of fermionic and
bosonic atoms immersed in a Bose–Einstein condensate (BEC) [39, 40].

(iii) Control of energy-level spacing and position of the WM can be
implemented by standard optical techniques [41, 42].

(iv) The temperature of each bath has to be measured independently through in-
situ measurements of the bath profile and density correlations [43, 44, 45].
Another intriguing approach would be to employ an impurity as a local
thermometer [38, 46, 47, 48], provided the thermalization between the bath and
the WM is well at hand [49].

##  4 Modeling the performance

In order to quantitatively evaluate the performance of the described cooling
technique, we have simulated the Otto cycle for realistic experimental
parameters [39, 38]. The simulation considers here bosonic baths formed by Cs
atoms and a WM formed by Rb atoms. Initially, both baths are thermal clouds at
temperature
$T_{\mathrm{c}}=T_{\mathrm{h}}=$1\text{\,}\mathrm{\SIUnitSymbolMicro K}$$.
Both Cs clouds are confined in a weak trap of frequency
$\omega_{\mathrm{T}}=2\pi\times$80\text{\,}\mathrm{Hz}$$ for the cold bath and
$\omega_{\mathrm{T}}=2\pi\times$150\text{\,}\mathrm{Hz}$$ for the hot bath.
The baths consist of $N_{\mathrm{at}}^{\mathrm{c}}=2\times 10^{5}$ and
$N_{\mathrm{at}}^{\mathrm{h}}=50\times 10^{5}$ atoms, respectively, and these
numbers are conserved throughout refrigeration. The initial temperatures of
both baths are well above the BEC critical temperature, namely, the bath are
initially thermal clouds. The critical temperatures for condensation of the
two baths are $T_{\mathrm{crit}}^{\mathrm{c}}\approx 395\,\mathrm{nK}$ and
$T_{\mathrm{crit}}^{\mathrm{h}}\approx 617\,\mathrm{nK}$, respectively, well
within the feasible refrigeration range.

After each cycle, the two bath temperatures are adapted as follows
($i\in\\{\mathrm{c},\mathrm{h}\\}$),

$T_{i}(n+1)=T_{i}(n)-\frac{N_{\mathrm{wm}}Q_{i}(n)}{C_{\mathrm{V}}(T_{i}(n),N_{\mathrm{at}}^{i})},$
(3)

where $N_{\mathrm{wm}}$ is the number of atoms in the WM. The explicit form of
the heat capacity $C_{\mathrm{V}}$ is given in Appendix B. The heat removed
from the cold bath in the $n$th cycle is $-N_{\mathrm{wm}}Q_{c}(n)$, where the
negative sign indicates the direction of the heat flow from the cold bath to
the WM.

[width=]FullTempEvolv

Figure 3: Temperatures of the two atomic clouds: Otto cooling of an initially
thermal atomic cloud that serves as a cold bath at temperature
$T_{\mathrm{c}}$ in the proposed refrigerator. The up- and down-ramps on the
energy-level spacing have here been terminated after 1000 cycles, consistently
with limitations on the total cooling time and tunability of the energy-level
spacing. See text for parameters and details.

The energy-level spacing of the WM, taken to consist of
$N_{\mathrm{wm}}=10^{4}$ particles, is initially set during the cold and hot
isochores to
$\mathcal{E}_{\mathrm{c}}/k_{\mathrm{B}}=$2\text{\,}\mathrm{\SIUnitSymbolMicro
K}$$ and
$\mathcal{E}_{\mathrm{h}}/k_{\mathrm{B}}=$4\text{\,}\mathrm{\SIUnitSymbolMicro
K}$$, respectively. This choice ensures that only the first excited level of
the WM has a non-negligible occupancy, so that heat transport between the two
baths can occur according to Eqs. (1) and (2). During the Otto cycle we have
ramped the energy-level spacings up and down in a linear fashion (see Fig. 3b)
such that the occupancy of the upper WM level is always a few percent. The up-
and down-ramps have been terminated after 1000 cycles, when the temperature of
the cold bath becomes comparable to the mode spacing in the cold bath,
$k_{\mathrm{B}}T_{\mathrm{c}}^{\mathrm{min}}\approx
7\hbar\omega_{\mathrm{T}}$. The reason for this choice is our wish to avoid
non-Markovian effects, which may become prominent as the number of occupied
bath modes becomes small (see below). We note that the minimal temperature is
calculated according to Eq. (2) from the final energy-level spacings and the
final temperature of the hot bath.

It can be seen in Fig. 3 that efficient cooling takes place for as many cycles
as needed to achieve the minimal allowed temperature. After approximately 500
cycles, the condensation threshold of the cold bath is crossed and efficient
cooling continues well into the deeply degenerate regime, down to $T\approx
0.1T_{\mathrm{crit}}^{\mathrm{c}}$, a regime that is hard to reach by other
methods [50]. Assuming a cycle time of roughly $10\text{\,}\mathrm{ms}$ we
estimate the cooling rate to be of the order of
$0.1\text{\,}\mathrm{n}\mathrm{K}\mathrm{/}\mathrm{m}\mathrm{s}$.

##  5 Model limitations

To reach our quantitative predictions we have made the following, assumptions:

(i) The WM interactions with the baths are assumed to be Markovian, i.e.,
devoid of memory effects such as back-propagation of excitations to their
origin. The Markovian assumption is unquestionable in the non-degenerate
regime. In the deeply degenerate regime, where only a few modes may be
excited, non-Markovian recurrences may be encountered [51]. Such non-
Markovianity is beyond the scope of the paper, but its effects may give rise
to a most fascinating experimental regime. It must, however, be noted that the
energy changes in the Otto cycle only depend on the four end points in Fig. 1b
and are thus independent of the dynamics that connects these points. We note
that this dependence of energy changes only on the end points holds since
energy is a system variable. Importantly, in this cycle the heat transfer
equals the energy change in the respective paths (since they are isochoric)
and hence is path-independent. Generally, this is not the case, e.g., in the
Diesel and Carnot cycles.

(ii) The discrete excitation spectrum of the bath has been neglected here,
i.e., the local density approximation has been adopted. This is an excellent
approximation for weak traps and high enough temperatures [52, 53] but it may
break down at extremely low temperatures, which is the range of possible non-
Markovian dynamics.

(iii) We have neglected interactions between the bath modes, which is an
excellent approximation even in the condensed regime, where the excitations
are well-described by non-interacting Bogoliubov modes [54, 51]. We note,
however, that these interactions must be included in a fully quantitative
study of the thermalization dynamics, which is beyond the scope of this paper.

(iv) The temperature of each bath has been assumed to be constant during each
stroke [cf. Eq. (3)] as we have a temperature change that is less than $1\%$
of the temperature of the bath itself. We can verify in Fig. 3 that this
approximation is well justified.

(v) For a simplified treatment of the WM, we have assumed that only two energy
levels are accessible. Such a situation can be naturally realized in an
optical trap, which is anisotropic and singles out the direction of
propagation of the laser [32].

(vi) Our treatment has assumed that thermal occupancy of higher modes can be
neglected, which means that the energy-level spacing is always larger than the
bath temperature. This assumption is valid for the chosen parameters in our
numerical treatment. Although optimization of heat extraction from the cold
bath may call for a WM which occupies a large number of modes, the entire
setup would then be well described by a classical (rather than quantized) Otto
cycle, similar to the description of Ref. [18].

(vii) The Otto cycle presumes an adiabatic change of the energy-level spacing
in the WM so as to avoid quantum friction [6]. For strokes that are longer
than $1\text{\,}\mathrm{ms}$ this is indeed the case. Strokes involving a
faster change of the trap frequency may be exploited in the future by
employing shortcuts to adiabaticity [55].

(viii) We have assumed weak coupling between the WM and the bath, consistently
with the Markovian approximation. This assumption is well established within
the context of the Fröhlich polaron, which has been extensively investigated
[56, 57, 58, 40]. Notably, we have verified that the dimensionless coupling
parameter, which describes the deformation of the bath density due to the
coupling with the WM, is always much smaller than unity. The weak-coupling
regime has also allowed us to neglect the energy cost of decoupling the WM
from the bath and the corresponding entropy changes.

(ix) Finally, thermalization between the baths and the WM during the isochores
has been ensured by considering time scales exceeding the relaxation time
[59], as discussed in more detail below. In the future we may explore how such
a thermal machine functions if thermalization is hampered (due to the
occurrence of non-thermal fixed points [60, 61, 62], the existence of dark
states [63], invariant subspaces [64] or many-body-localization [65]).

##  6 Experimental challenges

Having established the feasibility of the quantum gas refrigerator and its
conceivable limitations we now turn our attention to possible experimental
challenges. One challenge is the attainment of sufficiently fast
thermalization of the WM with the baths, relying on strong interspecies
interactions in Rb-Cs [37] or Na-K [66] mixtures. We can then estimate
thermalization rates based on the results of Ref. [39] that apply to the
proposed setup. With the assumed parameters, we obtain typical relaxation
times on the order of a $1\text{\,}\mathrm{ms}$, which are similar to the WM
adiabaticity time scale.

A fast enough relaxation is crucial in view of the finite lifetime of the
atomic clouds, which is typically of a few seconds, and residual heat effects.
The species-selective optical control of the WM has to be designed carefully;
such that it provides a strong confinement of the WM but does not perturb the
heat baths. For the proposed mixture of Rb-Cs the optical potential might be
tuned to $880\text{\,}\mathrm{nm}$, which is the tune-out wavelength of Cs
[67], such that it does not create a dipole potential for the baths. Still, a
confining potential with a waist of a few microns will cause an absorption
rate of a few $\mathrm{Hz}$ within the WM. The produced heat per particle and
per stroke is much smaller than the heat exchanged between the baths (Appendix
C) and therefore negligible.

The high density of atoms within the WM is unrealistic because of inelastic
processes that would degrade the WM. Hence, it would be important to decompose
the WM into a number of small and dilute WMs trapped in an optical lattice
potential as sketched in Fig. 2. Cooperative effects may play a major role in
this regime [68, 69, 70], a problem which has to be studied experimentally.

The physical decoupling of the WM from the bath is implicitly assumed to be
adiabatic. In the condensed regime the Bose-Einstein condensate is a
superfluid and motion below the critical velocity does not create any
excitations within the bath [71].

For the non-degenerate bath, on the other hand, we derive in Appendix D the
adiabatic decoupling condition according to which the WM has to move well
below the velocity $u_{\mathrm{a}}=\omega_{\mathrm{T}}\lambda_{\mathrm{dB}}$.
For our assumed parameters, this would imply typical cycles times in the order
of few tens $\text{\,}\mathrm{ms}$, such that only few strokes would be
possible. If the adiabatic decoupling condition is violated, the interaction
between the WM and bath is quenched in each cycle. The deposited energy is
then estimated to be $\Delta E=gN_{\mathrm{WM}}n_{\mathrm{bath}}$, which is
typically larger than the extracted heat. Thus the adiabatic decoupling
condition imposes severe restrictions on the Otto cycle operation in the non-
degenerate bath regime. These restrictions may however be overcome through
shortcuts to adiabaticity [72, 55, 73]. A fundamental solution to this problem
is a continuous cycle scheme, wherein coupling and decoupling are totally
absent, as we discuss in the next section.

##  7 Continuous cycles schemes

As an alternative to the Otto cycle, one may consider a different class of
thermodynamic cycles, namely, continuous cycles wherein both the heat baths as
well as the driving laser (the work reservoir) are simultaneously coupled to
the WM [33, 34, 74, 75, 3, 76, 77]. A possible setup is visualized in Fig. 4.

[width=]turbine_setup_v2

Figure 4: Continuous cycle schemes: The working medium is positioned in
between the two baths. The periodic modulation of the energy spacing
$\mathcal{E}_{0}(t)$ induces energy sidebands $\mathcal{E}_{\mathrm{h}}$ and
$\mathcal{E}_{\mathrm{c}}$. The spectra of the two baths have to be engineered
such that the working medium is stronger coupled at $\mathcal{E}_{\mathrm{h}}$
($\mathcal{E}_{\mathrm{c}}$) to the hot (cold) bath.

Instead of coupling the WM to the respective baths in two different
(temporarily-separated) isochoric strokes at respective WM energy-level
spacings $\mathcal{E}_{\mathrm{c}}$ and $\mathcal{E}_{\mathrm{h}}$, a spectral
separation of the two baths has been proposed, such that blue (red) sidebands
of the mean trap frequency $\omega_{\mathrm{T}}$ mainly couple to the hot
(cold) bath [34, 75], similarly to the Stokes and anti-Stokes sidebands in
laser cooling [78].

The salient advantage of such a continuous cycle is that neither (de)coupling
of the WM to or from the baths nor its translation between the baths are
necessary. Such a cycle is not restricted by the bottleneck of adiabatic
strokes and thus may be much faster than the Otto cycle and thereby
potentially increase the cooling power. On the other hand, the required
spectral separation of the two baths is non-trivial, yet it may be achieved
through internal-state manipulations of the baths confined in optical
lattices.

The theoretical description of continuous cycles, promising as they may be,
requires a more detailed knowledge of the concrete setup than the Otto cycle.
In particular, the Floquet analysis [79] of the Markovian master equation
governing the evolution of the WM in a thermal gas (derived in Ref. [80] for a
constant trap frequency) for a periodically-driven trap frequency may require
fine-tuning of the frequencies involved. Further, modulating the laser
intensity only varies the “spring constant” of the WM trap but not its mass,
which results in a non-diagonal modulation [6] with non-trivial time-
dependencies of the excitation and de-excitation operators, which are not
present in the previously-studied two-level atom case [34]. Finally, the
validity of the Markovian description (which is at the heart of Refs. [81, 34,
74, 75, 3, 76]) must be carefully re-analysed for such a cycle111First results
on a Markovian master equation below threshold for a constant trap frequency
can be found in the PhD thesis of Raphael Scelle [82].. Concrete predictions
require a detailed knowledge of the bath spectra at the respective sideband
frequencies. An experimental implementation of the continuous cycle would
hence provide crucial inputs to its theoretical descriptions.

##  8 Summary

We have adopted the harmonic Otto cycle of Ref. [6] to propose a novel
quantized refrigeration scheme for a cold bosonic-atom species, considered as
a trapped cold bath. The scheme is based on the coupling of this cold bath to
another trapped atomic species that acts as a working medium (WM), which
transfers heat from the cold bath to a hotter bath. The process is enabled by
the work invested in changing the energy-level spacing of the trapped WM by
modulating the intensity of the trapping field. Our analysis has shown the
feasibility of Otto-cycle refrigeration (especially when assisted by shortcuts
to adiabaticity) of a typical bosonic-atom (e.g., Cs) cloud from a thermal
regime to a deeply degenerate Bose–Einstein condensate while conserving the
number of atoms in the cloud. This task, shown here to be at hand, is
unfeasible by other means.

The proposed scheme promises to be a fundamentally new avenue for cooling as
it does not rely on radiative thermalization and is not recoil-limited.
Continuous versions of the proposed cycles are potentially even more
promising, because they are expected to be free of adverse non-adiabatic
effects [33, 34, 74, 3]. On the other hand, the continuous version requires
parameter choice and optimization that may be easier to achieve experimentally
than theoretically.

The minimal temperature achievable by the proposed refrigeration is an open
issue. The present minimum, Eq. (2), is only restricted by our technical
ability to ramp the confining laser intensity up and down, so that further
progress is expected in this respect. The more fundamental issue is the
ability to cool down to such low temperatures that the Markov and possibly the
Born approximations break down. The cooling speed may be affected at such
temperatures in ways still unknown. Related studies [83, 84, 85, 86] indicate
a plethora of fascinating effects in this non-Markovian domain.

The open fundamental issues outlined above suggest that the proposed
refrigerator may be a platform for studying principal aspects of quantum
thermodynamics in hitherto inaccessible regimes.

## Acknowledgements

We are grateful for fruitful discussions with David Gelbwaser-Klimovsky. W. N.
acknowledges support from an ESQ fellowship of the Austrian Academy of
Sciences (ÖAW). G. K. acknowledges support by the ISF. G. K, I. M. and F. J.
jointly acknowledge the DFG support through the project FOR 2724. I. M.
acknowledges support by the Wissenschafts- und TechnologieFonds (WWTF) project
No. MA16-066 (“SEQUEX”). F. J. acknowledges support by the DFG (Project-ID
377616843), the Excellence Initiative of the German federal government and the
state governments—funding line Institutional Strategy (Zukunftskonzept): DFG
project number ZUK49/Ü.

##  Appendix A Heat and work in the quantized Otto cycle

The respective changes in the energy of the WM during the four strokes of the
Otto cycle from Fig. 1 are [26]

$\displaystyle\Delta E_{1}$
$\displaystyle=\left\langle{H_{\mathrm{c}}}\right\rangle_{B}-\left\langle{H_{\mathrm{c}}}\right\rangle_{A}$
$\displaystyle=\mathrel{\mathop{:}}Q_{\mathrm{c}}$ (4a) $\displaystyle\Delta
E_{2}$
$\displaystyle=\left\langle{H_{\mathrm{h}}}\right\rangle_{C}-\left\langle{H_{\mathrm{c}}}\right\rangle_{B}$
$\displaystyle=\mathrel{\mathop{:}}W_{\mathrm{in}}$ (4b) $\displaystyle\Delta
E_{3}$
$\displaystyle=\left\langle{H_{\mathrm{h}}}\right\rangle_{D}-\left\langle{H_{\mathrm{h}}}\right\rangle_{C}$
$\displaystyle=\mathrel{\mathop{:}}Q_{\mathrm{h}}$ (4c) $\displaystyle\Delta
E_{4}$
$\displaystyle=\left\langle{H_{\mathrm{c}}}\right\rangle_{A}-\left\langle{H_{\mathrm{h}}}\right\rangle_{D}$
$\displaystyle=\mathrel{\mathop{:}}W_{\mathrm{out}}.$ (4d)

Here $H_{\mathrm{c,h}}=\mathcal{E}_{\mathrm{c,h}}\sigma_{+}\sigma_{-}$ are the
respective Hamiltonians of the WM during the cold and hot strokes with
$\mathcal{E}_{\mathrm{c}}\leq\mathcal{E}_{\mathrm{h}}$. The state of the WM
does not change during the adiabatic strokes ($B\rightarrow C$ and
$D\rightarrow A$), i.e., the ratio $\mathcal{E}/T$ remains constant.
Therefore, the thermal states at the four end points of the cycle are

$\displaystyle\rho_{A}$ $\displaystyle\equiv\rho_{D}$
$\displaystyle=\exp[-H_{\mathrm{h}}/(k_{\mathrm{B}}T_{\mathrm{h}})]/Z_{\mathrm{h}}$
$\displaystyle=\mathrel{\mathop{:}}\rho_{\mathrm{h}}$ (5a)
$\displaystyle\rho_{B}$ $\displaystyle\equiv\rho_{C}$
$\displaystyle=\exp[-H_{\mathrm{c}}/(k_{\mathrm{B}}T_{\mathrm{c}})]/Z_{\mathrm{c}}$
$\displaystyle=\mathrel{\mathop{:}}\rho_{\mathrm{c}}.$ (5b)

The work ($W$) and heat ($Q$) contributions (4) then evaluate to

$\displaystyle Q_{\mathrm{c}}$
$\displaystyle=\mathcal{E}_{\mathrm{c}}(\bar{n}_{\mathrm{c}}-\bar{n}_{\mathrm{h}})$
(6a) $\displaystyle W_{\mathrm{in}}$
$\displaystyle=(\mathcal{E}_{\mathrm{h}}-\mathcal{E}_{\mathrm{c}})\bar{n}_{\mathrm{c}}$
(6b) $\displaystyle Q_{\mathrm{h}}$
$\displaystyle=\mathcal{E}_{\mathrm{h}}(\bar{n}_{\mathrm{h}}-\bar{n}_{\mathrm{c}})$
(6c) $\displaystyle W_{\mathrm{out}}$
$\displaystyle=(\mathcal{E}_{\mathrm{c}}-\mathcal{E}_{\mathrm{h}})\bar{n}_{\mathrm{h}},$
(6d)

where ($i\in\\{\mathrm{c},\mathrm{h}\\}$)

$\bar{n}_{i}\mathrel{\mathop{:}}=\left\langle{\sigma_{+}\sigma_{-}}\right\rangle_{\rho_{i}}=\left[\exp\left[\mathcal{E}_{i}/(k_{\mathrm{B}}T_{i})\right]+1\right]^{-1}$
(7)

is the thermal occupation of the excited level of the two-level WM.

##  Appendix B Heat capacity of a Bose gas above and below threshold

The heat capacity of a BEC at constant volume in a three-dimensional harmonic
trap reads [23]

$C_{\mathrm{V}}=\begin{cases}3N_{\mathrm{at}}k_{\mathrm{B}}\left[1+\frac{\zeta(3)}{8}\left(\frac{T_{\mathrm{crit}}}{T}\right)^{3}\right]&T>T_{\mathrm{crit}}\\\
12\frac{\zeta(4)}{\zeta(3)}N_{\mathrm{at}}k_{\mathrm{B}}\left(\frac{T}{T_{\mathrm{crit}}}\right)^{3}&T<T_{\mathrm{crit}}\end{cases},$
(8)

where the critical temperature for condensation is [23]

$T_{\mathrm{crit}}=\frac{\hbar\omega_{\mathrm{T}}N_{\mathrm{at}}^{1/3}}{k_{\mathrm{B}}[\zeta(3)]^{1/3}}.$
(9)

Here $N_{\mathrm{at}}$ denotes the atom number in the bath,
$\omega_{\mathrm{T}}$ the trap frequency and $\zeta$ is the Riemann
$\zeta$-function.

##  Appendix C Heating due to spontaneous emission

The optical confinement of the WM induces spontaneous emission events during
which the WM atoms gain a recoil energy
$E_{\mathrm{R}}=h^{2}/(2m_{\mathrm{wm}}\lambda_{\mathrm{wm}}^{2})$, where $h$
is the Planck constant, $m_{\mathrm{wm}}$ the mass of the atoms and
$\lambda_{\mathrm{wm}}$ the wavelength of the emitted photons
($780\text{\,}\mathrm{nm}$ in the case of Rb and hence
$E_{\mathrm{R}}/k_{\mathrm{B}}\approx$180\text{\,}\mathrm{nK}$$). The produced
heat per atom in the WM can be estimated as
$Q_{\mathrm{sp}}=E_{\mathrm{R}}\gamma_{\mathrm{wm}}\tau_{\mathrm{stroke}}$,
where $\gamma_{\mathrm{wm}}$ is the rate of spontaneous emission (typically a
few $\mathrm{H}\mathrm{z}$) and $\tau_{\mathrm{stroke}}$ the duration of each
stroke (on the order of $\mathrm{ms}$). Therefore, we estimate
$Q_{\mathrm{sp}}k_{\mathrm{B}}\sim\textrm{a few }$\mathrm{nK}$$ per stroke,
which typically is much smaller than the exchanged heat in Eq. (6).

##  Appendix D Motion of the WM in the non-degenerate bath

Here we evaluate the effect of the non-adiabatic movement of a compact WM
through a bath. We assume that the bath to be composed of a non-degenerate and
(almost) ideal gas and strive to estimate the transferred energy
$\langle\Delta E\rangle$ per single atom. For the sake of simplicity, we
consider a 1D harmonic trap with the frequency $\omega_{\mathrm{T}}$. The
typical inverse length scale for an atom of mass $m$ is
$\alpha=\sqrt{m\omega_{\mathrm{T}}/\hbar}$. The energy spectrum is
$E_{n}=\hbar\omega_{\mathrm{T}}n$, $n=0,1,2,\dots$

Since we are interested in the case of small transferred energy, we apply the
lowest-order perturbative expression for the transition amplitude of an atom
being driven from the state $|n_{0}\rangle$ to the state $|n\rangle$ under the
time-dependent perturbation $V(z,t)$,

${\cal A}_{n_{0}\rightarrow
n}=-\frac{i}{\hbar}\int_{-\infty}^{\infty}dt\,e^{i(E_{n}-E_{n_{0}})t/\hbar}\langle
n|V|n_{0}\rangle.$ (10)

Then the transferred energy per atom is

$\langle\Delta E\rangle={\cal
Z}^{-1}\sum_{n_{0}=0}^{\infty}\sum_{n=0}^{\infty}(E_{n}-E_{n_{0}})e^{-\beta
E_{n_{0}}}|{\cal A}_{n_{0}\rightarrow n}|^{2},$ (11)

where $\beta$ is the inverse temperature and ${\cal
Z}=1/(1-e^{-\beta\hbar\omega_{\mathrm{T}}})$ the partition function. Slightly
reorganizing the summation, we see that

$\langle\Delta E\rangle={\cal
Z}^{-1}\sum_{n_{0}=0}^{\infty}\sum_{\,n=n_{0}+1}^{\infty}(E_{n}-E_{n_{0}})(e^{-\beta
E_{n_{0}}}-e^{-\beta E_{n}})|{\cal A}_{n_{0}\rightarrow n}|^{2}$ (12)

is always positive for a bath in thermal equilibrium.

For the purpose of evaluation of the non-adiabatic effects it is convenient to
approximate the potential of interaction of a bath atom with the working
medium. The contact interaction between the bath atoms and the working medium
reads $E=g_{\mathrm{IB}}\int\mathrm{d}z\leavevmode\nobreak\
n_{\mathrm{WM}}(z-ut)n_{\mathrm{th}}(z)$. For a very small working medium it
looks like a potential and the bath density is in its region, such that

$V(z,t)=V_{0}\delta(z-ut)\mbox{ with }V_{0}=g_{\mathrm{IB}}N_{\mathrm{wm}}$
(13)

Here we assume that the working medium passes through the bath at the constant
velocity $u$.

Now we readily calculate

${\cal A}_{n_{0}\rightarrow
n}=-\frac{iV_{0}}{u\hbar}\int_{-\infty}^{\infty}dz\,\psi_{n}^{*}(z)e^{i\omega_{\mathrm{T}}(n-{n_{0}})z/u}\psi_{n_{0}}(z),$
(14)

where $\psi_{n,\,n_{0}}$ are eigenfunctions of the harmonic oscillator.
Expressing the coordinate $z$ via the creation and annihilation operators
$\hat{a}^{\dagger}$ and $\hat{a}$, respectively, we recognize that ${\cal
A}_{n_{0}\rightarrow n}$ is proportional to the matrix element of the
displacement operator $\exp[i\lambda_{n}(\hat{a}^{\dagger}+\hat{a})]$ with
$\lambda_{n}=\omega_{\mathrm{T}}(n-n_{0})/(\sqrt{2}u\alpha)$ in the Fock
basis. These matrix elements are known, see, e.g., Ref. [87]. Since, according
to Eq. (12), $n>n_{0}$, we get

$\langle
n|e^{i\lambda_{n}(\hat{a}^{\dagger}+\hat{a})}|n\rangle=i^{n-n_{0}}e^{-\lambda_{n}^{2}/2}\lambda_{n}^{n-n_{0}}\sqrt{\frac{n_{0}!}{n!}}L_{n_{0}}^{n-n_{0}}(\lambda_{n}^{2}),$
(15)

where $L_{n_{0}}^{n-n_{0}}(x)$ is an associated Laguerre polynomial. This
reduces Eq. (12) to

$\langle\Delta
E\rangle=(1-e^{-\beta\hbar\omega_{\mathrm{T}}})\hbar\omega_{\mathrm{T}}\left(\frac{V_{0}}{\hbar
u}\right)^{2}\sum_{n_{0}=0}^{\infty}\sum_{l=1}^{\infty}e^{-\beta\hbar\omega_{\mathrm{T}}n_{0}}l(1-e^{-\beta\hbar\omega_{\mathrm{T}}l})\zeta^{2l}l^{2l}e^{-\zeta^{2}l^{2}}\frac{n_{0}!}{(n_{0}+l)!}[L_{n_{0}}^{l}(\zeta^{2}l^{2})]^{2},$
(16)

where $\zeta=\omega_{\mathrm{T}}/(\sqrt{2}\alpha u)$. The sum over $n_{0}$ in
Eq. (16) can be taken using the Hille–Hardy formula [88], thus yielding

$\langle\Delta E\rangle=2\hbar\omega_{\mathrm{T}}\left(\frac{V_{0}}{\hbar
u}\right)^{2}\sum_{l=1}^{\infty}l\sinh(\beta\hbar\omega_{\mathrm{T}}/2)\exp[-\zeta^{2}l^{2}\coth(\beta\hbar\omega_{\mathrm{T}}/2)]I_{l}[\zeta^{2}l^{2}/\sinh(\beta\hbar\omega_{\mathrm{T}}/2)],$
(17)

where $I_{l}(x)$ is the modified Bessel function.

We work in the high-temperature limit,

$\beta\hbar\omega_{\mathrm{T}}\ll 1.$ (18)

In this limit the adiabaticity condition requires that the velocity of the
working medium in the bath does not exceed a certain value,

$\displaystyle u\ll u_{\mathrm{a}}$
$\displaystyle\equiv\frac{\omega_{\mathrm{T}}}{\alpha}\sqrt{\frac{\beta\hbar\omega_{\mathrm{T}}}{8}}$
(19) $\displaystyle u_{\mathrm{a}}$
$\displaystyle=\omega_{\mathrm{T}}\lambda_{\mathrm{dB}}\mbox{ with
}\lambda_{\mathrm{dB}}=\sqrt{\frac{\beta\hbar^{2}}{8m}},$ (20)

where $\omega_{\mathrm{T}}$ sets the typical gap and and
$\lambda_{\mathrm{dB}}$ the typical distance. Note that condition (20) is much
more restrictive that the smallness of the oscillation period compared to the
passage time.

To derive this condition, we assume first a less stringent inequality,
$u\ll\omega_{\mathrm{T}}/(\alpha\sqrt{\beta\hbar\omega_{\mathrm{T}}})$, which
means that the time of the passage of the working medium across the bath of
the thermal radius $\sim 1/(\alpha\sqrt{\beta\hbar\omega_{\mathrm{T}}})$ is
much longer than the trap oscillation period. Then we can use the asymptotic
form $I_{l}(x)\approx e^{x}/\sqrt{2\pi x}$ for $x\rightarrow+\infty$. Equation
(17) is then reduced to

$\langle\Delta E\rangle=2\hbar\omega_{\mathrm{T}}\left(\frac{V_{0}}{\hbar
u}\right)^{2}\sum_{l=1}^{\infty}l\frac{\beta\hbar\omega_{\mathrm{T}}}{2}\exp\left(-\frac{\zeta^{2}l^{2}\beta\hbar\omega_{\mathrm{T}}}{4}\right)\frac{\sqrt{\beta\hbar\omega_{\mathrm{T}}}}{2\sqrt{\pi}\zeta
l},$ (21)

where condition (18) has been used. In the adiabatic limit given by Eq. (20)
we obtain

$\displaystyle\langle\Delta E\rangle$
$\displaystyle=\frac{2}{\sqrt{\pi}}\beta(\alpha
V_{0})^{2}\frac{u_{\mathrm{a}}}{u}\exp\left(-\frac{u_{\mathrm{a}}^{2}}{u^{2}}\right).$
(22)

Under the latter condition the decreasing exponent and the increasing modified
Bessel function almost compensate each other, hence the exponential
suppression of the heating rate requires further slowing down of the motion of
the working medium inside the bath. The strong exponential dependence (not a
power-law one) of the r.h.s. of Eq. (22) on the adiabaticity parameter
indicates that an adiabaticity condition similar to Eq. (20) should hold also
in 3D.

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11institutetext: Nishal Rai 22institutetext: 1Center for Astrophysics
Gravitation and Cosmology (CAGC), SRM University Sikkim, Upper Tadong, Sikkim,
India
2Department of Physics, SRM University Sikkim, Upper Tadong, Sikkim, India.
22email<EMAIL_ADDRESS>33institutetext: Eugenio Megías
44institutetext: 3 Departamento de Física Atómica, Molecular y Nuclear and
Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada,
Avenida de Fuente Nueva s/n, E-18071 Granada, Spain

# Holographic approach to anomalous transport in a massive $U(1)$ gauge theory

Nishal Rai1,2 and Eugenio Megías3

###### Abstract

In this study, we explore a massive $U(1)$ gauge holographic model with pure
gauge and mixed gauge-gravitational Chern-Simons terms. By considering the
full backreaction of the gauge field on the metric tensor, we explore the
vortical and energy transport sectors. Our findings for the chiral vortical
conductivity, $\sigma_{V}$, and the chiral magnetic/vortical conductivity of
energy current show that $\sigma^{\varepsilon}_{B}=\sigma^{\varepsilon}_{V}$.
Notably, we highlight a contribution to $\sigma_{V}$ induced by the mixed term
in the massive theory, which is absent in the massless case.

##  1 Introduction

AdS/CFT correspondence Maldacena:1997re ; Witten:1998qj  states that, in the
low energy limit, the large-$N_{c}$, ${\mathcal{N}}$ = 4 super Yang-Mills
field theory in four-dimensional space is equivalent to the type IIB string
theory in $AdS_{5}\times S^{5}$ space. It has become a very powerful
theoretical tool for studying strongly coupled systems. It has a wide
application in different fields of physics, such as condensed matter physics,
hydrodynamics and QCD Grieninger:2023myf ; Baggioli:2023tlc ; Morales-
Tejera:2024uzg ; Landsteiner:2022wap ; Rai:2023nxe ; Mukhopadhyay:2020tky ;
Rai:2019vrr ; Mukhopadhyay:2017ltk ; Mukhopadhyay:2017hgc ; Ammon:2021pyz ;
Grieninger:2021zik ; Ahn:2024ozz ; Chu:2024dti ; Landsteiner:2015lsa ;
Landsteiner:2012dm ; Megias:2013joa ; Newman:2005as ; Newman:2005hd ;
Megias:2016ery ; Landsteiner:2013aba  to name a few. Our study will be focused
on the application of the fluid/gravity correspondence in the context of
hydrodynamics, in particular the study of the anomalous transport in this kind
of systems.

Quantum chiral anomalies, arising in relativistic field theories of chiral
fermions beyond perturbation theory, play a crucial role in relativistic
hydrodynamics book1 ; book2 ; book3 ; Landsteiner:2012kd . Since the 1980s,
anomaly-induced transport mechanisms have been widely studied Vilenkin . Key
effects include the chiral magnetic effect, where a charge current is induced
parallel to an external magnetic field Fukushima:2008xe , and the chiral
vortical effect, where a vortex in a charged fluid generates a current
Kharzeev:2007tn ; Banerjee:2008th ; Son:2009tf ; Landsteiner:2011cp . These
effects may be detectable in heavy ion collisions at RHIC and LHC STAR:2009wot
and can lead to anomalous transport properties in materials like Weyl
semimetals Basar:2013iaa ; Landsteiner:2013sja . In this work, we explore and
investigate the chiral vortical effects in a massive $U(1)$ gauge holographic
model. To achieve this, we have accounted for the full backreaction of the
gauge field on the metric tensor and included a mixed gauge-gravitational
Chern-Simons term as well. To evaluate the transport coefficients we will be
using the Kubo formalism where the anomalous conductivities are given by the
following relations

$\begin{split}\sigma_{V}&=\lim_{k_{c}\rightarrow
0}\dfrac{i}{2k_{c}}\sum_{a,b}\epsilon_{abc}\langle
J^{a}T^{0b}\rangle|_{\omega=0}\,,\\\
\sigma_{V}^{\varepsilon}&=\lim_{k_{c}\rightarrow
0}\dfrac{i}{2k_{c}}\sum_{a,b}\epsilon_{abc}\langle
T^{0a}T^{0b}\rangle|_{\omega=0}\,,\end{split}$ (1)

where $\sigma_{V}$ is the chiral vortical conductivity, and
$\sigma^{\varepsilon}_{V}$ the chiral vortical conductivity of energy current,
respectively. The chiral magnetic conductivities for charge, $\sigma_{B}$, and
energy, $\sigma^{\varepsilon}_{B}$, current are given by

$\begin{split}\sigma_{B}&=\lim_{k_{c}\rightarrow
0}\dfrac{i}{2k_{c}}\sum_{a,b}\epsilon_{abc}\langle
J^{a}J^{b}\rangle|_{\omega=0}\,,\\\
\sigma_{B}^{\varepsilon}&=\lim_{k_{c}\rightarrow
0}\dfrac{i}{2k_{c}}\sum_{a,b}\epsilon_{abc}\langle
T^{0a}J^{b}\rangle|_{\omega=0}\,.\end{split}$ (2)

These correlators can be evaluated using the AdS/CFT dictionary
Landsteiner:2011iq ; Son:2002sd ; Herzog:2002pc .

##  2 Holographic model

The holographic action for a massive $U(1)$ gauge boson with pure gauge and a
mixed gauge-gravitational Chern-Simon term is given by Landsteiner:2011iq ;
Jimenez-Alba:2014iia ; Rai:2023nxe

$\displaystyle S$ $\displaystyle=$ $\displaystyle\dfrac{1}{16\pi G}\int
d^{5}x\sqrt{-g}\Big{[}R+2\Lambda-\dfrac{1}{4}F_{MN}F^{MN}$ (3)
$\displaystyle-\dfrac{m^{2}}{2}(A_{M}-\partial_{M}\theta)(A^{M}-\partial^{M}\theta)$
$\displaystyle+\epsilon^{MNPQR}(A_{M}-\partial_{N}\theta)\left(\dfrac{\kappa}{3}F_{NP}F_{QR}+\lambda
R^{A}\,{}_{BNP}R^{B}\,{}_{AQR}\right)\Big{]}$ $\displaystyle+S_{\rm GH}+S_{\rm
CSK}\,,$

where $S_{\rm GH}$ and $S_{\rm CSK}$ are the Gibbons-Hawking boundary term,
and a boundary term induced by the mixed gauge-gravitational anomaly,
respectively. $\theta$ is a field which ensures gauge invariance (up to gauge
anomalies), and thus the mass term enters in a consistent way. As it mentioned
in Klebanov:2002gr ; Gursoy:2014ela ; Casero:2007ae , the Stückelberg term
arises as the holographic realization of dynamical anomalies. For convenience
we will define a new field $B_{M}\equiv A_{M}-\partial_{M}\theta$, such that
$\theta$ does not appear explicitly in the equations of motion. One may check
that the action still remains invariant under gauge transformations. From now
onwards we will be working with the field $B_{\mu}$ instead of $A_{\mu}$.

###  2.1 Background solution

The ansatz for the background metric and fields is chosen as follows in
Fefferman-Graham coordinates Megias:2017czr ; Megias:2019djo ,

$\displaystyle ds^{2}$
$\displaystyle=-\dfrac{\ell^{2}}{\rho}g_{\tau\tau}(\rho)d\tau^{2}+\dfrac{\ell^{2}}{\rho}g_{xx}(\rho)d\vec{x}^{2}+\dfrac{\ell^{2}}{4\rho^{2}}d\rho^{2}\,,$
(4) $\displaystyle B_{\mu}dx^{\mu}$ $\displaystyle=B_{t}(\rho)dt\,,$

where the boundary lies at $\rho=0$ and the horizon at $\rho_{h}$, with
$g_{\tau\tau}(\rho_{h})=0$. Using this ansatz the equations of motion for the
background are given by

$\displaystyle
g_{xx}^{\prime\prime}(\rho)-\dfrac{g_{xx}^{\prime}(\rho)}{\rho}+\frac{1}{6\ell^{2}\rho}\dfrac{g_{xx}(\rho)}{g_{\tau\tau}(\rho)}\left(\dfrac{m^{2}\ell^{2}}{4}B_{t}(\rho)^{2}+\rho^{2}B_{t}^{\prime}(\rho)^{2}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
g_{\tau\tau}^{\prime}(\rho)\left(1-\rho\dfrac{g_{xx}^{\prime}(\rho)}{g_{xx}(\rho)}\right)+g_{\tau\tau}(\rho)\dfrac{g_{xx}^{\prime}(\rho)}{g_{xx}(\rho)}\left(3-\rho\dfrac{g_{xx}^{\prime}(\rho)}{g_{xx}(\rho)}\right)$
(5)
$\displaystyle-\frac{1}{3}\dfrac{\rho^{2}}{\ell^{2}}B_{t}^{\prime}(\rho)^{2}+\dfrac{1}{12}m^{2}B_{t}(\rho)^{2}$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
B_{t}^{\prime\prime}(\rho)+\frac{1}{2}\left(3\dfrac{g_{xx}^{\prime}(\rho)}{g_{xx}(\rho)}-\dfrac{g_{\tau\tau}^{\prime}(\rho)}{g_{\tau\tau}(\rho)}\right)B_{t}^{\prime}(\rho)-\dfrac{\ell^{2}m^{2}}{4\rho^{2}}B_{t}(\rho)$
$\displaystyle=$ $\displaystyle 0\,.$

We will solve numerically the above equations of motion (5) subjected to the
following boundary conditions

$B_{t}(\rho_{h})=0\,,\hskip 56.9055pt\lim_{\rho\to
0}\left(\rho^{\Delta/2}B_{t}(\rho)\right)=\mu_{5}\,,$ (6)

where $\mu_{5}$ corresponds to the source of the gauge field and $\Delta$
being the anomalous dimension of the dual current Jimenez-Alba:2014iia
related to the mass of the gauge field through the following relation
$m^{2}\ell^{2}=\Delta(\Delta+2)$. In addition to this, we demand the solution
to the metric tensor to be regular at the horizon.

It is possible to find an analytic solution of the background for the case
with $\mu_{5}=0$. This is given by

$g_{\tau\tau}(\rho)=\frac{1}{\rho_{h}^{2}}\frac{(\rho_{h}^{2}-\rho^{2})^{2}}{\rho_{h}^{2}+\rho^{2}}\,,\qquad
g_{xx}(\rho)=1+\frac{\rho^{2}}{\rho_{h}^{2}}\,,\qquad B_{t}(\rho)=0\,.$ (7)

We display in Fig. 1 (left) the numerical solution of the equations of motion
for $\mu_{5}=0$ (dots), and compare it with the corresponding analytical
solution (solid lines). We display also in the right panel the numerical
solution for $\mu_{5}=0.3$.

Figure 1: (color) Dependence of the background metric and gauge field with
$\rho$. We display the results for $g_{xx}(\rho)$ (blue), $g_{\tau\tau}(\rho)$
(orange), and $\rho^{\Delta/2}B_{t}(\rho)$ (green). Left panel: We have
considered $\mu_{5}=0$, with solid line being the analytic solution and the
dots the numerical one. Right panel: We have chosen $\Delta=0.1$ and
$\mu_{5}=0.3$.

##  3 Anomalous conductivity

To compute these correlators, we will follow the general procedure of the
perturbation of the fields on top of the background, and later use the AdS/CFT
dictionary to compute the correlators.

###  3.1 Fluctuations

Within this approach, we perturb the fields on top of the numerical solution
for the background displayed in Fig. 1. We will study the linear response of
the fluctuations by splitting the metric and gauge field into background and
linear perturbation components, i.e.

$g_{MN}=g_{MN}^{(0)}+\epsilon h_{MN},\hskip
56.9055ptB_{M}=B_{M}^{(0)}+\epsilon b_{M}\,,$ (8)

where the terms with (0) correspond to the background metric/field, and the
terms with $\epsilon$ the fluctuations of this metric/field. Then, we will
follow the general procedure of Fourier mode decomposition Amado:2011zx

$\displaystyle h_{MN}(\rho,x^{\mu})$ $\displaystyle=$
$\displaystyle\int\dfrac{d^{d}k}{(2\pi)^{d}}h_{MN}(\rho)e^{-i\omega
t+i\vec{k}.\vec{x}}\,,$ (9) $\displaystyle b_{M}(\rho,x^{\mu})$
$\displaystyle=$
$\displaystyle\int\dfrac{d^{d}k}{(2\pi)^{d}}b_{M}(\rho)e^{-i\omega
t+i\vec{k}.\vec{x}}\,.$ (10)

Without loss of generality, we consider perturbations with frequency $\omega$
and momentum $k$ in the $z$-direction. Since we aim to compute correlators at
zero frequency, we set the frequency-dependent parts to zero in the equations
and solve the system up to first order in $k$. In this limit, the fields
$h^{i}_{z}$ decouple from the system and become constant. Thus, we can write
the system of differential equations for the shear sector as follows

$\displaystyle
b_{i}^{\prime\prime}(\rho)+\frac{1}{2}\left(\frac{g_{xx}^{\prime}(\rho)}{g_{xx}(\rho)}+\frac{g_{\tau\tau}^{\prime}(\rho)}{g_{\tau\tau}(\rho)}\right)b_{i}^{\prime}(\rho)-\frac{\Delta(\Delta+2)}{4\rho^{2}}b_{i}(\rho)$
(11) $\displaystyle+\left(\frac{4i\kappa
k\epsilon_{ij}b_{j}(\rho)}{\sqrt{g_{xx}(\rho)g_{\tau\tau}(\rho)}}+\frac{g_{xx}(\rho)h^{i}{}_{t}^{\prime}(\rho)}{g_{\tau\tau}(\rho)}\right)B_{t}^{\prime}(\rho)+i\lambda
k\epsilon_{ij}h^{j}{}_{t}^{\prime}(\rho)\Omega(\rho)=0,$ $\displaystyle
h^{i}{}_{t}^{\prime\prime}(\rho)-\left(\dfrac{g_{\tau\tau}^{\prime}(\rho)}{2g_{\tau\tau}(\rho)}-\dfrac{5g_{xx}^{\prime}(\rho)}{2g_{xx}(\rho)}+\dfrac{1}{\rho}\right)h^{i}{}_{t}^{\prime}(\rho)+\dfrac{\rho
B_{t}^{\prime}(\rho)}{g_{xx}(\rho)}b_{i}^{\prime}(\rho)$
$\displaystyle+\dfrac{\Delta(\Delta+2)B_{t}(\rho)}{4\rho
g_{xx}(\rho)}b_{i}(\rho)+i\lambda k\epsilon_{ij}\Phi_{j}(\rho)=0\,,$ (12)

where $i,j=x,y$. The explicit expressions of the functions $\Omega(\rho)$ and
$\Phi_{j}(\rho)$ are given in Appendix A of Rai:2023nxe .

The asymptotic behaviour of the fields up to the first subleading term are
given by

$\displaystyle b_{i}(\rho)$ $\displaystyle=$ $\displaystyle
b_{i}^{(0)}\rho^{-\frac{\Delta}{2}}+b_{i}^{(1)}\rho^{\frac{\Delta}{2}+1}+\cdots\,,$
(13) $\displaystyle h_{t}^{i}(\rho)$ $\displaystyle=$ $\displaystyle
h_{t}^{i}{}^{(0)}+h_{t}^{i}{}^{(1)}\rho^{2}+\cdots\,,$ (14)

where the leading-order terms $b_{i}^{(0)}$ and $h_{t}^{i}{}^{(0)}$ play the
role of sources.

#### Conductivities

From the holographic description of the correlation functions, one can
evaluate the one-point functions as

$\langle J_{a}\rangle=\dfrac{\delta S_{\text{ren}}}{\delta
b_{a}^{(0)}}=-\dfrac{2}{16\pi G}(\Delta+1)b_{a}^{(1)}\,,\quad\langle
T_{0a}\rangle=\dfrac{\delta S_{\text{ren}}}{\delta
h_{t}^{a}{}^{(0)}}=\dfrac{1}{16\pi
G}\left(2h_{t}^{a}{}^{(0)}+h_{t}^{a}{}^{(1)}\right)\,,$ (15)

with $(a=x,y)$. $S_{\textrm{ren}}=S+S_{\textrm{ct}}$ is the renormalized
action, with $S$ the action given in Eq. (3) and $S_{\textrm{ct}}$ the
counterterm. The two-point functions can be obtained by taking the variation
of one-point function with respect to the corresponding source term and the
conductivities are evaluated using the Kubo formulae given in (1) and (2),
i.e.

$\displaystyle\sigma_{V}$ $\displaystyle=\lim_{k\to
0}\frac{1}{k}\textrm{Im}\langle
J_{x}T_{0y}\rangle\,,\qquad\sigma_{V}^{\varepsilon}=\lim_{k\to
0}\frac{1}{k}\textrm{Im}\langle T_{0x}T_{0y}\rangle\,,$ (16)
$\displaystyle\sigma_{B}$ $\displaystyle=\lim_{k\to
0}\frac{1}{k}\textrm{Im}\langle J_{x}J_{y}\rangle\,,\qquad\hskip
2.84544pt\sigma_{B}^{\varepsilon}=\lim_{k\to 0}\frac{1}{k}\textrm{Im}\langle
T_{0x}J_{y}\rangle\,.$ (17)

A comparison of the variation of the action under axial gauge transformations
with the consistent form of the anomaly for chiral fermions book1  allows us
to set the value of the anomaly coefficients to

$\kappa=-\frac{G}{2\pi}\,,\qquad\lambda=-\frac{G}{48\pi}\,.$ (18)

Figure 2: Upper panels: Plot of $\sigma_{V}$ (left) and
$\sigma^{\varepsilon}_{V}$ (right) vs $\Delta$. Lower panels: Plot of
$\sigma_{B}$ (left) and $\sigma^{\varepsilon}_{B}$ (right) vs $\Delta$. We
have considered $\mu_{5}=0.15$ in all the panels. We have divided the
conductivities by $|\kappa|$ to make them dimensionless.

In this regard we will be setting $G=2\pi$, so that $\kappa=-1$ and
$\lambda=-1/24$. In Fig. 2 we have plotted the anomalous conductivities vs
$\Delta$ (a parameter related to the mass of the gauge field). From this
figure, we observe that the chiral vortical conductivity and the chiral
magnetic conductivity for energy current are the same even at finite mass,
i.e., $\sigma_{V}=\sigma^{\varepsilon}_{B}$, and both increase with $\Delta$.
Additionally, we see that the chiral vortical conductivity of energy current,
$\sigma^{\varepsilon}_{V}$, decreases with $\Delta$, but the rate decreases
rapidly. Conversely, in the case of the chiral magnetic conductivity,
$\sigma_{B}$, it increases with $\Delta$ as shown in Fig. 2. In the limit of
vanishing mass, our results lead to $\frac{\sigma_{B}}{\mu_{5}|\kappa|}\simeq
16/3$, which exactly coincides with the results in Landsteiner:2011iq ;
Jimenez-Alba:2014iia  where $\alpha$ has been set to $\mu_{5}$ in both
references 111$\alpha$ corresponds to the asymptotic value of the gauge field
$A_{t}$ for $\rho\to 0$. In our case, we assume $\alpha=\mu_{5}$ for
$\Delta=0$..

##  4 Discussion

In this work we have studied the mass dependence of the anomalous
conductivities in a holographic model for a massive $U(1)$ gauge field,
incorporating both pure gauge and mixed gauge-gravitational anomaly terms. In
this model, the vortical sector was also accessible, and the conductivities
corresponding to this sector were also studied. We observed that the chiral
vortical conductivity, $\sigma_{V}$, and the chiral magnetic conductivity for
energy current, $\sigma^{\varepsilon}_{B}$, are equal and increase with
$\Delta$. An intriguing finding is the presence of contributions to
$\sigma_{B}$ induced by the mixed gauge-gravitational anomaly term in the
massive theory, which were absent in the massless case. Furthermore, the
conductivities $\sigma_{B}$, $\sigma_{V}$, and $\sigma^{\varepsilon}_{B}$ all
increase with $\Delta$, while the chiral vortical conductivity of energy
current, $\sigma^{\varepsilon}_{V}$, decreases with $\Delta$. We have
explicitly verified that all our numerical results for the conductivities at
finite mass converge to the known results at zero mass as $\Delta\to 0$.

## Acknowledgments

N.R. thanks the International Centre for Theoretical Sciences (ICTS) for the
support through the program - Field Theory and Turbulence (ICTS/FTT2023/12).
The works of N.R. and E.M. are supported by the project PID2020-114767GB-I00
and by the Ramón y Cajal Program under Grant RYC-2016-20678 funded by
MCIN/AEI/10.13039/501100011033 and by “FSE Investing in your future”, as well
as by the FEDER/Junta de Andalucía-Consejería de Economía y Conocimiento
2014–2020 Operational Program under Grant A-FQM-178-UGR18. The work of E.M. is
also supported by Junta de Andalucía under Grant FQM-225, and by the
“Prórrogas de Contratos Ramón y Cajal” Program of the University of Granada.

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