Quantized refrigerator for an atomic cloud
Wolfgang Niedenzu1 , Igor Mazets2,3 , Gershon Kurizki4 , and Fred Jendrzejewski5
1 Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria
2 Vienna Center for Quantum Science and Technology (VCQ), Atominstitut, TU Wien, 1020 Vienna, Austria
3 Wolfgang Pauli Institute, Universität Wien, 1090 Vienna, Austria
4 Department of Chemical Physics, Weizmann Institute of Science, Rehovot 7610001, Israel
5 Heidelberg University, Kirchhoff-Institut für Physik, Im Neuenheimer Feld 227, D-69120 Heidelberg, Germany

June 24, 2019

Entropy of theworking medium

arXiv:1812.08474v3 [quant-ph] 24 Jun 2019

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

1 Introduction
Over the past two decades, extensive studies of thermal machines in the quantum domain [1–7] havesought to reveal either fundamentally new aspects ofthermodynamics or unique quantum advantages compared to their classical counterparts. Yet, despite thegreat progress achieved, both theoretically [8–15] andexperimentally [16–22], 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.
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 andending well within the quantum-degenerate regime.
The proposed cycle will be a fundamentally new avenue for cooling, only limited by thermalization and
Wolfgang Niedenzu: Wolfgang.Niedenzu@uibk.ac.at
Fred Jendrzejewski: fnj@kip.uni-heidelberg.de

Accepted in Quantum 2019-06-13, click title to verify

Figure 1: Quantized Otto cycle for refrigeration of thecold 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 theworking 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 → B → C → D → A.

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 [6, 24–27] (see Fig. 1) the working medium (WM)
is realised by one species that is alternately coupled
to spatially separated, hot and cold baths of anotherspecies. This allows for progressive cooling of thecold 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 thresholdtowards Bose–Einstein condensation. The proposedcooling technique should also be applicable to othertypes of baths, including strongly correlated Mott insulators or anti-ferromagnetic phases, where cooling1

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 fieldof ultracold atomic gases employs photon recoil forheat extraction from the atomic cloud. However, thephase-space-density is fundamentally limited by therecoil 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 evaporativetechniques are fundamentally limited by the required
spatial separation of regions possessing high or low
entropy. Such separation typically breaks down inthe 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 theconfined system are accessible. Such a configurationarises naturally in highly focused optical traps [32].
It can be modelled by the Hamiltonian H = Eσ+ σ− ,where σ− (σ+ ) 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 forrefrigeration, analogous to a piston in common thermodynamic machine cycles [1, 2, 24, 33, 34].
In the first stroke, the WM has energy-level spacing Ec and thermalizes through its interaction with athermal bath at temperature Tc . Thermalization is
ensured through contact collisions between the atomsof 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 thecold bath of the Otto cycle (stroke 1 in Fig. 1). The
cold bath is realized by an atomic species that differsfrom 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 coldbath the heat Qc = Ec (n̄c − n̄h ), where n̄i (i ∈ {c, h})
are the thermal excitations of the two-level WM (seeAppendix A).
In the second stroke, the WM is adiabatically decoupled from the cold bath and its energy-level spacing is adiabatically raised to Eh > Ec . This realizes
Accepted in Quantum 2019-06-13, click title to verify

the isentropic compression stroke of the Otto cycle(stroke 2 in Fig. 1). The work Win = (Eh − Ec )n̄h
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 ofthe cycle.
In the third stroke, the WM is coupled to the hotthermal bath at temperature Th > Tc , 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 Qh = Eh (n̄h − n̄c ) into the
hot bath.
The fourth stroke that closes the cycle consists inadiabatic decoupling of the WM from the hot bathand adiabatic decrease of the energy-level spacing
back to Ec (stroke 4 in Fig. 1). This decompression of
the WM produces the work Wout = (Ec − Eh )n̄c which
is transferred to the optical field.
The Otto cycle acts as a refrigerator for the coldbath as long as Qc > 0 (Qh < 0), which yields thecooling conditionn̄c > n̄h

⇒

Eh
Ec
>
.
Th
Tc

(1)

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

Ec
Th .
Eh

(2)

The ratio of the two baths temperature is hence only
limited by the tunability of the energy-level spacingof the quantized WM. For an optically confined WM,
the energy-level spacing E is controlled by the√squareroot of the confining laser intensity I, E ∝ I [32].
Since intensity ratios of Ih /Ic & 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 becreated within two separate harmonic traps, as originally employed in atomic tunneling junctions [35] ormore recently in atomic circuits with fermionic andbosonic baths [16, 36]. In the proposed setup (seeFig. 2) the two baths must be completely separated
such that no particle exchange or heat flow can happen between them, except via the WM.
(ii) The WM is formed by an atomic species, bymeans of a species-selective optical lattice or tweezer.
Such schemes have been employed for single bosonicatoms immersed in a cold gas [37, 38] or for larger2

hot batha)working medium
in moving trap

T [nK]

cold bath
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 spatiallymoving the working medium into the bath region.

numbers of fermionic and bosonic atoms immersed ina 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 ofthe bath profile and density correlations [43–45]. Another intriguing approach would be to employ an impurity as a local thermometer [38, 46–48], provided
the thermalization between the bath and the WM iswell at hand [49].

4 Modeling the performance
In order to quantitatively evaluate the performanceof the described cooling technique, we have simulated the Otto cycle for realistic experimental parameters [38, 39]. The simulation considers here bosonicbaths formed by Cs atoms and a WM formed by Rb
atoms. Initially, both baths are thermal clouds attemperature Tc = Th = 1 µK. Both Cs clouds are confined in a weak trap of frequency ωT = 2π × 80 Hz forthe cold bath and ωT = 2π × 150 Hz for the hot bath.
chThe baths consist of Nat= 2 × 105 and Nat= 50 × 105
atoms, respectively, and these numbers are conserved
throughout refrigeration. The initial temperatures ofboth baths are well above the BEC critical temperature, namely, the bath are initially thermal clouds.
The critical temperatures for condensation of the twochbaths are Tcrit≈ 395 nK and Tcrit≈ 617 nK, respectively, well within the feasible refrigeration range.
After each cycle, the two bath temperatures are
adapted as follows (i ∈ {c, h}),Ti (n + 1) = Ti (n) −Nwm Qi (n),i )CV (Ti (n), Nat(3)

where Nwm is the number of atoms in the WM. The
Accepted in Quantum 2019-06-13, click title to verify/kB [nK]

1000
750 Thcrit
500 Tccrit
250 minTc
00200b) 4000

Th

Tc
400

600

800

1000 1200
h

3000
2000
1000
0

c

0

200

400

600 800
cycles1000 1200

Figure 3: Temperatures of the two atomic clouds: Otto
cooling of an initially thermal atomic cloud that serves asa cold bath at temperature Tc in the proposed refrigerator.
The up- and down-ramps on the energy-level spacing havehere been terminated after 1000 cycles, consistently withlimitations on the total cooling time and tunability of theenergy-level spacing. See text for parameters and details.

explicit form of the heat capacity CV is given in Appendix B. The heat removed from the cold bath in
the nth cycle is −Nwm Qc (n), where the negative sign
indicates the direction of the heat flow from the coldbath to the WM.
The energy-level spacing of the WM, taken to consist of Nwm = 104 particles, is initially set during
the cold and hot isochores to Ec /kB = 2 µK andEh /kB = 4 µK, respectively. This choice ensures that
only the first excited level of the WM has a nonnegligible 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 energylevel spacings up and down in a linear fashion (seeFig. 3b) such that the occupancy of the upper WM
level is always a few percent. The up- and downramps have been terminated after 1000 cycles, when
the temperature of the cold bath becomes comparable
to the mode spacing in the cold bath, kB Tcmin ≈ 7~ωT .
The reason for this choice is our wish to avoid nonMarkovian 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 energylevel spacings and the final temperature of the hotbath.
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 iscrossed and efficient cooling continues well into thecdeeply degenerate regime, down to T ≈ 0.1Tcrit, a3

regime that is hard to reach by other methods [50].
Assuming a cycle time of roughly 10 ms we estimate
the cooling rate to be of the order of 0.1 nK/ms.

5 Model limitations
To reach our quantitative predictions we have madethe 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 thatthe 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 theenergy change in the respective paths (since they areisochoric) 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 bathhas 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 nonMarkovian dynamics.
(iii) We have neglected interactions between thebath modes, which is an excellent approximationeven in the condensed regime, where the excitations are well-described by non-interacting Bogoliubov modes [51, 54]. We note, however, that theseinteractions 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 haveassumed that only two energy levels are accessible.
Such a situation can be naturally realized in an opticaltrap, which is anisotropic and singles out the directionof propagation of the laser [32].
(vi) Our treatment has assumed that thermal occupancy of higher modes can be neglected, which meansthat the energy-level spacing is always larger than thebath temperature. This assumption is valid for the
chosen parameters in our numerical treatment. AlAccepted in Quantum 2019-06-13, click title to verify

though optimization of heat extraction from the coldbath may call for a WM which occupies a large number of modes, the entire setup would then be welldescribed by a classical (rather than quantized) Otto
cycle, similar to the description of Ref. [18].
(vii) The Otto cycle presumes an adiabatic changeof the energy-level spacing in the WM so as to avoid
quantum friction [6]. For strokes that are longer than
1 ms this is indeed the case. Strokes involving a fasterchange 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 establishedwithin the context of the Fröhlich polaron, which has
been extensively investigated [40, 56–58]. Notably,
we have verified that the dimensionless coupling parameter, which describes the deformation of the bathdensity 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 andthe 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 occurrenceof non-thermal fixed points [60–62], the existence ofdark states [63], invariant subspaces [64] or manybody-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 onstrong interspecies interactions in Rb-Cs [37] or NaK [66] mixtures. We can then estimate thermalization
rates based on the results of Ref. [39] that apply tothe proposed setup. With the assumed parameters,
we obtain typical relaxation times on the order of a
1 ms, which are similar to the WM adiabaticity timescale.
A fast enough relaxation is crucial in view of the finite lifetime of the atomic clouds, which is typically ofa few seconds, and residual heat effects. The speciesselective optical control of the WM has to be designed
carefully; such that it provides a strong confinementof the WM but does not perturb the heat baths. For
the proposed mixture of Rb-Cs the optical potential
might be tuned to 880 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
4

rate of a few Hz within the WM. The produced heatper particle and per stroke is much smaller than theheat exchanged between the baths (Appendix C) andtherefore 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 inFig. 2. Cooperative effects may play a major role inthis regime [68–70], a problem which has to be studied
experimentally.
The physical decoupling of the WM from the bath isimplicitly 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 ua = ωT λdB . For our assumed parameters, this would imply typical cycles times in theorder of few tens ms, such that only few strokes would
be possible. If the adiabatic decoupling condition isviolated, the interaction between the WM and bath isquenched in each cycle. The deposited energy is thenestimated to be ∆E = gNWM nbath , which is typicallylarger than the extracted heat. Thus the adiabatic
decoupling condition imposes severe restrictions on
the Otto cycle operation in the non-degenerate bathregime. These restrictions may however be overcome
through shortcuts to adiabaticity [55, 72, 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 [3, 33, 34, 74–77]. A possible
setup is visualized in Fig. 4.
Instead of coupling the WM to the respective
baths in two different (temporarily-separated) isochoric strokes at respective WM energy-level spacings
Ec and Eh , a spectral separation of the two baths
has been proposed, such that blue (red) sidebandsof the mean trap frequency ωT mainly couple to thehot (cold) bath [34, 75], similarly to the Stokes andanti-Stokes sidebands in laser cooling [78].
The salient advantage of such a continuous cycleis that neither (de)coupling of the WM to or fromthe baths nor its translation between the baths arenecessary. Such a cycle is not restricted by the bottleneck of adiabatic strokes and thus may be much fasterthan the Otto cycle and thereby potentially increase
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Figure 4: Continuous cycle schemes: The working medium
is positioned in between the two baths. The periodic modulation of the energy spacing E0 (t) induces energy sidebands Eh
and Ec . The spectra of the two baths have to be engineeredsuch that the working medium is stronger coupled at Eh (Ec )to the hot (cold) bath.

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 theWM 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 excitationand 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. [3, 34, 74–76, 81]) must be
carefully re-analysed for such a cycle1 . 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 SummaryWe have adopted the harmonic Otto cycle of Ref. [6]to propose a novel quantized refrigeration scheme fora 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 thatacts 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
1 First results on a Markovian master equation below threshold for a constant trap frequency can be found in the PhD thesisof Raphael Scelle [82].

5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 athermal regime to a deeply degenerate Bose–Einsteincondensate while conserving the number of atoms in
the cloud. This task, shown here to be at hand, isunfeasible 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 potentiallyeven more promising, because they are expected to be
free of adverse non-adiabatic effects [3, 33, 34, 74]. On
the other hand, the continuous version requires parameter choice and optimization that may be easierto 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 thisrespect. The more fundamental issue is the ability tocool 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 [83–86]
indicate a plethora of fascinating effects in this nonMarkovian 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 Academyof 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.
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.

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
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are [26]∆E1 = hHc iB − hHc iA =: Qc(4a)∆E2 = hHh iC − hHc iB =: Win(4b)∆E3 = hHh iD − hHh iC =: Qh(4c)∆E4 = hHc iA − hHh iD =: Wout .

(4d)Here Hc,h = Ec,h σ+ σ− are the respective Hamiltonians of the WM during the cold and hot strokes withEc ≤ Eh . The state of the WM does not change during
the adiabatic strokes (B → C and D → A), i.e., theratio E/T remains constant. Therefore, the thermal
states at the four end points of the cycle areρA ≡ ρD = exp[−Hh /(kB Th )]/Zh =: ρh(5a)ρB ≡ ρC = exp[−Hc /(kB Tc )]/Zc =: ρc .

(5b)

The work (W ) and heat (Q) contributions (4) thenevaluate to
Qc = Ec (n̄c − n̄h )(6a)Win = (Eh − Ec )n̄c(6b)Qh = Eh (n̄h − n̄c )(6c)Wout = (Ec − Eh )n̄h ,(6d)where (i ∈ {c, h})n̄i := hσ+ σ− iρi = [exp [Ei /(kB Ti )] + 1]

−1

(7)

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

B Heat capacity of a Bose gas aboveand below threshold
The heat capacity of a BEC at constant volume in athree-dimensional harmonic trap reads [23]h iTcrit 3
3Nat kB 1 + ζ(3)T > Tcrit
8
T
3

CV =
, (8)
T
12 ζ(4) N k
T < Tcritζ(3) at B Tcrit
where the critical temperature for condensation is [23]
1/3

Tcrit =

~ωT Nat.
kB [ζ(3)]1/3(9)Here Nat denotes the atom number in the bath, ωT
the trap frequency and ζ is the Riemann ζ-function.

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 ER = h2 /(2mwm λ2wm ), where h is
the Planck constant, mwm the mass of the atoms and6λwm the wavelength of the emitted photons (780 nm
in the case of Rb and hence ER /kB ≈ 180 nK). The
produced heat per atom in the WM can be estimatedas Qsp = ER γwm τstroke , where γwm is the rate of spontaneous emission (typically a few Hz) and τstroke theduration of each stroke (on the order of ms). Therefore, we estimate Qsp kB ∼ a few nK per stroke, which
typically is much smaller than the exchanged heat in
Eq. (6).

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 transferredenergy h∆Ei per single atom. For the sake of simplicity, we consider
p a 1D harmonic trap with the frequency ωT .
The typical inverse length scale for an atom of mass m is α = mωT /~. The energy spectrum is En = ~ωT n,
n = 0, 1, 2, . . .
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 |n0 i to the state |ni under thetime-dependent perturbation V (z, t),Zi ∞dt ei(En −En0 )t/~ hn|V |n0 i.
(10)
An0 →n = −~ −∞Then the transferred energy per atom is∞ X∞Xh∆Ei = Z −1

(En − En0 )e−βEn0 |An0 →n |2 ,(11)

n0 =0 n=0

where β is the inverse temperature and Z = 1/(1 − e−β~ωT ) the partition function. Slightly reorganizing thesummation, we see thath∆Ei = Z −1

∞
X

∞
X

(En − En0 )(e−βEn0 − e−βEn )|An0 →n |2(12)

n0 =0 n=n0 +1

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 ofinteraction of a bath atom with theR working medium. The contact interaction between the bath atoms andthe working medium reads E = gIB dz nWM (z − ut)nth (z). For a very small working medium it looks like apotential and the bath density is in its region, such that
V (z, t) = V0 δ(z − ut) with V0 = gIB Nwm(13)

Here we assume that the working medium passes through the bath at the constant velocity u.
Now we readily calculateZ
iV0 ∞An0 →n = −dz ψn∗ (z)eiωT (n−n0 )z/u ψn0 (z),
u~ −∞

(14)

where ψn, n0 are eigenfunctions of the harmonic oscillator. Expressing the coordinate z via the creation andannihilation operators â† and â, respectively, we recognize that An0 →n√is proportional to the matrix elementof the displacement operator exp[iλn (â† + â)] with λn = ωT (n − n0 )/( 2uα) in the Fock basis. These matrix
elements are known, see, e.g., Ref. [87]. Since, according to Eq. (12), n > n0 , we getrn0 ! n−n0 2
iλn (â† +â)n−n0 −λ2n /2 n−n0
|ni = ieλn(λn ),
(15)
hn|e
L
n! n0
0
where Ln−n(x) is an associated Laguerre polynomial. This reduces Eq. (12) ton0h∆Ei = (1 − e−β~ωT )~ωT



V0
~u

2 X
∞ X
∞
n0 =0 l=1

Accepted in Quantum 2019-06-13, click title to verify

2 2e−β~ωT n0 l(1 − e−β~ωT l )ζ 2l l2l e−ζ ln0 !
[Ll (ζ 2 l2 )]2 ,(n0 + l)! n0

(16)

7

√
where ζ = ωT /( 2αu). The sum over n0 in Eq. (16) can be taken using the Hille–Hardy formula [88], thus
yieldingh∆Ei = 2~ωT



V0
~u

2 X
∞

l sinh(β~ωT /2) exp[−ζ 2 l2 coth(β~ωT /2)]Il [ζ 2 l2 / sinh(β~ωT /2)],

(17)

l=1

where Il (x) is the modified Bessel function.
We work in the high-temperature limit,
β~ωT  1.

(18)

In this limit the adiabaticity condition requires that the velocity of the working medium in the bath does notexceed a certain value,r
ωT β~ωT(19)u  ua ≡α8r
β~2ua = ωT λdB with λdB =
,
(20)
8m
where ωT sets the typical gap and and λdB the typical distance. Note that condition (20) is much more restrictivethat the smallness of the oscillation period compared to the passage time.
√means thatTo derive this condition, we assume first a less stringent inequality, u  ωT /(α β~ωT ), which√
the time of the passage of the working medium across the bath of the thermal radius ∼ 1/(αβ~ωT ) is much√
longer than the trap oscillation period. Then we can use the asymptotic form Il (x) ≈ ex / 2πx for x → +∞.
Equation (17) is then reduced toh∆Ei = 2~ωT



V0
~u

2 X
∞
l=1

 22
√
β~ωT
ζ l β~ωT
β~ω
√ T,
l
exp −
2
4
2 πζl

where condition (18) has been used. In the adiabatic limit given by Eq. (20) we obtain
 2
u
2
2 ua
exp − a2 .
h∆Ei = √ β(αV0 )
u
u
π

(21)

(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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