Anti-Dynamical Casimir Effect as a Resource for Work Extraction
AAnti-Dynamical Casimir Effect as a Resource for Work Extraction
A. V. Dodonov, ∗ D. Valente,
2, 3 and T. Werlang Institute of Physics and International Center for Physics,University of Brasilia, 70910-900, Brasilia, Federal District, Brazil Instituto de F´ısica, Universidade Federal de Mato Grosso, Cuiab´a MT, Brazil Laboratoire Pierre Aigrain (LPA), ´Ecole Normale Sup´erieure (ENS) - Paris,Centre National de la Recherche Scientifique (CNRS), 24 rue Lhomond, 75005 Paris, France
We consider the quantum Rabi model with external time modulation of the atomic frequency,which can be employed to create excitations from the vacuum state of the electromagnetic field asa consequence of the dynamical Casimir effect. Excitations can also be systematically subtractedfrom the atom-field system by suitably adjusting the modulation frequency, in the so-called anti-dynamical Casimir effect (ADCE). We evaluate the quantum thermodynamical work and show thata realistic out-of-equilibrium finite-time protocol harnessing ADCE allows for work extraction fromthe system, whose amount can be much bigger then the modulation amplitude, | W ADCE | (cid:29) (cid:126) (cid:15) Ω , incontrast to the case of very slow adiabatic modulations. We provide means to control work extractionin state-of-the-art experimental scenarios, where precise frequency adjustments or complete systemisolation may be difficult to attain. PACS numbers: 03.65-w, 42.50.Pq, 05.70-a
I. INTRODUCTION
The dynamical Casimir effect (DCE) consists in thegeneration of quanta from the initial vacuum (or anyother) state of some field due to time-dependent bound-ary conditions or varying material properties of somemacroscopic system (see [1–4] for reviews). In the major-ity of cases this corresponds to the generation of photonsdue to accelerated motion of a single mirror, a cavity wallor time-dependent dielectric permittivity/conductivity ofthe intracavity medium [5–11], although phonon gener-ation is also possible in such systems as Bose-Einsteincondensates [12, 13], quantum fluids of light [14] orlaser-cooled atomic gases [15]. In recent years, it hasbeen shown that a microscopic analogue of DCE can beachieved using a time-dependent quantum Rabi model[16–18] – a quantized single-mode electromagnetic fieldcoupled to a two-level atom (TLA) with time-dependentparameters. The photon generation occurs as a resultof time-variation of the transition frequency of the TLAor the atom-field coupling strength, while the macro-scopic boundary conditions for the field remain station-ary [19–21]. Ultimately, the photon creation relies onthe presence of the counter-rotating terms (CRT) in theRabi Hamiltonian (RH) [22], usually neglected underthe rotating wave approximation (RWA) [23, 24]. Thetime-dependent Rabi model can be implemented experi-mentally in the current circuit Quantum Electrodynam-ics (circuit QED) architecture [25–34] with artificial su-perconducting atoms coupled to microwave resonators,where DCE analogues were already observed both in asingle-mirror and a cavity configurations [35, 36]. How-ever, the DCE is not the only relevant effect that arises ∗ adodonov@fis.unb.br from the combination of the CRT and the temporal mod-ulation of the system parameters. It was predicted inRefs. [37–39] that by properly adjusting the modulationfrequency one can induce a coherent photon annihila-tion from non-vacuum states, in what became known asthe anti-dynamical Casimir effect (ADCE). Moreover, byletting the modulation frequency to slowly change withtime, effective Landau-Zener transitions may occur be-tween the dressed states of the time-independent RabiHamiltonian [40].The DCE and the ADCE involve, respectively, cre-ation and annihilation of excitations from some initialstate of the atom-field system. In order to implementthese physical processes an external agent is required tosupply or withdraw energy from the system by meansof appropriate modulation of parameters. However, tillnow there has been no clear relationship between thecreation (annihilation) of excitations and the amount ofenergy supplied (withdrawn) by the external agent. Inthis scenario, the following question can be stated: is itpossible to use the ADCE to extract work from the atom-field system ? In order to address this issue, one can usethe framework of quantum thermodynamics [41–44] - afield of physics seeking to establish a quantum versionfor thermodynamic principles and processes. Studies inquantum thermodynamics aim to introduce appropriatedefinitions of work and heat [45–51], the development ofquantum thermal machines [52–59], the analysis of theheat transport [60–65] and the validity of the second lawat the microscopic level [66–72], the study of the stochas-tic thermodynamics [73, 74] and the fluctuation theorems[75–80], just to name a few. An interesting result involv-ing the Rabi model in the context of the quantum ther-modynamics was presented in Ref. [81]. The authorshave shown that the CRT prevent the atom-field systemfrom reaching the absolute zero temperature, even in thelimit of an infinite number of cycles. a r X i v : . [ qu a n t - ph ] A p r In this paper we investigate the relationship betweenquantum work and the creation or annihilation of exci-tations in the Rabi model. Work extraction has alreadybeen investigated in the stationary regime of the Rabimodel [82], requiring ultra-strong couplings and hugeatom-field detunings. In our case, the time-dependentRabi model is shown to allow for finite-time work ex-traction even for perturbative modulation amplitudesand moderate atom-field detunings, without demandingultra-strong couplings. Work can be extracted either pe-riodically or steadily, depending on whether the modula-tion frequency is itself constant or time-varying.This paper is organized as follows. The theoreticalframework is presented in Sec. II. In Sec. II A the time-dependent RH is introduced, and the system dynamicsunder parametric modulation is elucidated. In Sec. II Bthe definitions of work and heat are delineated. Ourmain result is described in Sec. III: ADCE can be a re-source for work extraction. In Sec. III A we evidence howwork extraction is related to annihilation of system ex-citations driven by an out-of-equilibrium finite-time pro-tocol, implemented as a perturbative modulation of theatomic frequency. In Sec. III B we extend the methodfor multi-tone modulations, which are useful for realis-tic initial states of the system. In Sec. III C we em-ploy a time-dependent modulation frequency in order tosolve two issues, namely, the need for an asymptotic fi-nite work extraction and the challenge of finding a veryfine-tuned modulation frequency. Finally, in section III Dwe investigate how the presence of dissipation may reducethe amount of extracted work in actual implementations.Conclusions are presented in Sec. IV, and some formalanalytical derivations are summarized in the appendix A.
II. MODELA. Atom-Field Interaction
The atom-field interaction is described by the time-dependent Rabi Hamiltonian (we set (cid:126) = 1) [19–21, 83,84] H = ωa † a + Ω( t )2 σ z + g ( a + a † )( σ + + σ − ) , (1)where a ( a † ) is the cavity annihilation (creation) opera-tor, and σ + = | e (cid:105)(cid:104) g | and σ − = σ † + are the atomic lad-der operators. Here | g (cid:105) ( | e (cid:105) ) denotes the atomic ground(excited) state and σ z = | e (cid:105)(cid:104) e | − | g (cid:105)(cid:104) g | . The field num-ber states are denoted by | n (cid:105) , such that a † a | n (cid:105) = n | n (cid:105) . ω is the cavity frequency, g is the atom-field couplingstrength and Ω( t ) is the time-dependent atomic transi-tion frequency. The total average number of excitationsis given by N = Tr (cid:0) ρa † a (cid:1) + Tr ( ρ | e (cid:105)(cid:104) e | ), where ρ denotesthe atom-field density operator.We assume that the atomic frequency undergoes an external multi-tone modulation asΩ( t ) = Ω + (cid:88) k (cid:15) ( k )Ω sin( η ( k ) ( t ) t + φ ( k ) ) , (2)where η ( k ) ( t ) is the k - th modulation frequency and (cid:15) ( k )Ω is the k - th modulation amplitude. It is worth notingthat this particular choice for the modulation does notrestrict the generality of our results, since for the regimeconsidered here ( g (cid:28) ω, Ω) a weak modulation of anysystem parameter produces similar results [21, 37, 85].We suppose that the modulation frequency η ( k ) ( t ) mayalso slowly change as function of time. To uncover theeffects of temporal modulation on the system dynamics,we assume a perturbative regime characterized by (cid:15) ( k )Ω (cid:28) Ω and (cid:15) ( k )Ω (cid:46) g . Besides, for the validity of the single-mode approximation we require the inequality | ∆ − | (cid:28) ω ,where ∆ − = ω − Ω is the average field-atom detuning.In the following we restrict our attention to the dispersiveregime, g √ n max (cid:28) | ∆ − | /
2, where n max is the maximumnumber of system excitations.The time-dependent RH can be implemented in thecircuit QED architecture – area of research that investi-gates the interaction between the quantized Electromag-netic field confined in microwave resonators and super-conducting artificial atoms composed of Josephson junc-tions. Originally proposed in 2004 with the target of im-plementing the Jaynes-Cummings model in a highly con-trollable environment [86], this field has expanded enor-mously over the past ten years, and now it embraces aplenty of experimental architectures with different kindsof multi-level artificial atoms and sophisticated assem-blies of interconnected resonators and waveguides [25–28, 87–89]. However, all the setups have in common theattribute of strong atom-field coupling and the ability tocontrol in situ the system parameters (e.g., the atomictransition frequency or the coupling strength). The typi-cal parameters in current circuit QED architectures read[29, 34, 90–92]: ω/ π ∼ −
10 GHz, g/ω ∼ − − − , | ∆ − | /g ∼ − κ/ω ∼ γ/ω ∼ × − − × − , T r ∼ −
50 mK, where κ ( λ ) denotes de cavity (atom)damping rate and T r is the temperature. As will beshown in the following, these values are sufficient for therealization of the protocol proposed in this paper.For g/ω, (cid:15) Ω / Ω (cid:28) η ( t ) [19–21, 37–39, 84, 85]. By properly adjusting η ( t ), which depends on the initial state of the system, onecan resonantly couple a specific set of the dressed states(eigenstates) of the bare RH (see Appendix A for details).Below we resume three qualitatively different phenomenathat alter the total number of excitations: the DCE, theanti-Jaynes-Cummings behavior (AJC) and the ADCE.The DCE regime is characterized by the creation ofphoton pairs from the vacuum and occurs for the modula-tion frequency η DCE ≈ ω . For the initial zero-excitationstate | g, (cid:105) the number of excitations increases throughthe induced transitions between the states | g, (cid:105) , | g, (cid:105) , | g, (cid:105) , . . . , | g, k (cid:105) , where k depends on the values of ∆ − , g and (cid:15) Ω [37]. The population of the atomic excitedstate remains approximately unchanged in this scenario.(In reality, the transitions occur between the atom-fielddressed states, which in the dispersive regime can be ap-proximated as | g, n (cid:105) – see appendix A). On the otherhand, using the same initial state and η AJC ≈ ω + Ω , thedynamics consists of periodic transitions between the ap-proximate states | g, (cid:105) and | e, (cid:105) . So the increase in thenumber of quanta occurs due to the excitation of boththe atom and the cavity field. Such behavior is known asAJC regime [19, 37, 38, 93] or the blue-sideband transi-tion [83, 84].Both regimes presented so far are responsible for theincrease in the number of excitations. However, in Refs.[37–39, 85] the authors showed that the number of ex-citations can be reduced instead, in what they calledADCE. This effect consists of coherent annihilation oftwo system excitations due to the approximate transition | g, n (cid:105) ←→ | e, n − (cid:105) (for n ≥ η ADCE ≈ ω − Ω . For constant η the total number of system excitations presents a periodicbehavior, with the typical period of oscillation on the or-der of ∼ − g − for realistic experimental parameters.One way to get a reduction in the total number of ex-citations without a subsequent return to its initial valueis to use a time-dependent modulation frequency, η ( t ),swept across the expected resonance associated with theADCE [40]. A steep decrease in the number of excitationsis observed when η ( t ) ≈ η ADCE , and as η ( t ) moves awayfrom η ADCE the resonance condition is forfeit. As shownin appendix A this process can be viewed as an effec-tive Landau-Zener-Stueckelberg-Majorana problem [94–97], which asserts that the asymptotic transition betweenthe two involved states will be complete for sufficientlysmall | ˙ η | . Hence, it is possible to deterministically inducea steady decrease in the number of excitations. B. Quantum Thermodynamics
In the context of quantum thermodynamics, the inter-nal energy of a quantum system is the average energy U ( t ) = (cid:104) H ( t ) (cid:105) = Tr ( ρ ( t ) H ( t )), where H ( t ) is the sys-tem’s Hamiltonian and ρ ( t ) – its density operator. Thequantum version of the first law of thermodynamics reads[43, 52] ∆ U = W + Q, (3)where W is the work performed by an external agentand Q is the heat supplied to the system by its thermalenvironment. The quantum work W computed from time t i up to t f is related to the time variation of the system’sHamiltonian, W = (cid:90) t f t i Tr ( ρ ( t ) ∂ t H ( t )) dt, (4) while the heat Q is related to the time variation of thedensity operator Q = (cid:82) t f t i Tr ( ∂ t ρ ( t ) H ( t )) dt . For an iso-lated quantum system there is no heat exchange betweenthe system and its environment, Q = 0. Therefore, thevariation of the internal energy coincides with the workdone on the system ( W >
0) or extracted from the sys-tem (
W < t ) = Ω , the quantum work will always be equalto zero, even with the increase in the number of excita-tions caused by the counter-rotating terms. This resultshows that in order to extract work from the atom-fieldsystem it is necessary that the system’s Hamiltonian it-self evolves over time, driven by an external agent. III. RESULTS AND DISCUSSIONS
In this section we discuss the relationship between thecreation or annihilation of excitations and quantum workin the Rabi model. We shall show that generation of afinite number of excitations is accompanied by positivework performed on the system, while the coherent anni-hilation of quanta is accompanied by negative work of theorder ∼ ( − (cid:126) ω ), i.e., the energy is withdrawn by the ex-ternal agent. We shall also demonstrate that energy canbe added or withdrawn from the system without chang-ing the total number of excitation (with only infinitesi-mal changes in N ), however, the net amount of work is | W | < (cid:126) ω in this case. A. Single-tone modulations
We begin our analysis by investigating how the cre-ation of excitations from the zero-excitation state | g, (cid:105) affects the quantum work. The dynamics of the sys-tem was obtained through the numerical solution of theLiouville-von Neumann equation, ˙ ρ = − i [ H ( t ) , ρ ( t )],where H ( t ) is the time-dependent RH, Eq. (1). In Fig.1we plot the dynamics of the quantum work W ( t ) = (1 / (cid:90) t dt (cid:48) ˙Ω( t (cid:48) ) (cid:104) σ z ( t (cid:48) ) (cid:105) (5)and the total number of excitations N for the param-eters g/ω = 5 × − , ∆ − = 8 g = 0 . ω , φ = 0 and (cid:15) Ω = 0 . = 0 . ω . The DCE regime (Fig.1 - a1 anda2) was obtained using a single-tone modulation with η = 2 . ω . The AJC regime (Fig.1 - b1 and b2) wasobtained using η = 0 . + , where ∆ + = ω + Ω . Inboth regimes, the quantum work is predominantly pos-itive (it may take small negative values when N ≈ W = N (cid:126) ω − (cid:126) ∆ − P e , where P e = Tr ( ρ | e (cid:105)(cid:104) e | ) is theatomic excitation probability.Work can also be realized on the system without varia-tion of N . As the simplest example, consider the Jaynes-Cummings Hamiltonian (JCH) [23, 24], obtained fromEq. (1) by neglecting the counter-rotating terms. Thisapproximate model describes well the dynamics provided g (cid:28) ω, Ω and the modulation frequency is low, η (cid:28) ω .For JCH the total number of excitations is a constantof motion, and for the initial state | ϕ n, ± (cid:105) , which is theeigenstate of JCH with n excitations (see appendix A),the work reads W JC | η/ω → = ± ε Ω cos 2 θ n [sin ( ηt + φ ) − sin ( φ )] . (6)Therefore the work of magnitude | W JC | η/ω → | ≤ (cid:126) (cid:15) Ω canbe performed on or by the system without any change in N .Still under JCH, one can obtain a finite work, eitherpositive or negative, by setting the modulation frequencyto η ≈ | ∆ − | , thereby promoting the coupling betweenapproximate states | g, n (cid:105) and | e, n − (cid:105) in the dispersiveregime. This behavior is known as Jaynes-Cummingsor red-sideband regime [19, 83, 84, 93], and has recentlybeen implemented experimentally in circuit QED [33, 34].For instance, for the initial state | g, n (cid:105) a straightforwardenergetic reasoning predicts for the extremum amount ofwork W JC ≈ − (cid:126) ∆ − , so energy can be added or with-drawn from the system depending on the sign of ∆ − .This result holds under RH as well, as illustrated in Fig.1 - c1 and c2 for the initial state | g, (cid:105) and the modula-tion frequency η = 1 . − (all other parameters are aspreviously). We see that work is indeed extracted fromthe system, while the total number of excitations under-goes only infinitesimal changes due to the off-resonantcontribution of CRT.Because in the DCE and AJC regimes the maximalextracted work is W max > | W JC | , one can see that thecounter-rotating terms crucially contribute to the quan-tum work. Since the creation of excitations is related tothe performance of work on the system, the annihilationof excitations is expected to play a role in the extrac-tion of work. To investigate this point, we evaluate thedynamics of W ( t ) in the ADCE regime. For this, weadopted η = 1 . ω − Ω ) and the initial state | g, (cid:105) .Fig. 1- d1 and d2 show that the quantum work becomesnegative indeed, proving that the ADCE can be used toextract work from the atom-field system. Maximal workextraction occurs at times when the population of thestate | g, (cid:105) attain its minimum value due to the transferof population to the state | e, (cid:105) , that is, when N reachesits lowest value. We also see that in the ADCE regimeone can extract the energy ≈ (cid:126) ω from the system, com-pared to ≈ (cid:126) ω/ − − −
10 0 50 100 150024 5 10 15 20 250120 100 200 300123
NNN
DCE regime DCE regimeAJC regimeAJC regime ADCE regimeADCE regime (a1) (a2)(b1) (b2)(d1) (d2) W o r k N W o r k W o r k W o r k JC regime JC regime (c1) (c2) − − AJC regimeDCE regime (e1) (e2)
Figure 1. (color online) Quantum work and average totalnumber of excitations as a function of dimensionless time t/τ for the regimes: DCE (panels a1-a2), AJC (panels b1-b2), JC (panels c1-c2) and ADCE (panels d1-d2). Here τ − = g(cid:15) Ω / + with ∆ ± = ω ± Ω . For the DCE andAJC regimes the initial state is | g, (cid:105) ; for the JC and ADCEregimes it is | g, (cid:105) . Parameters: g/ω = 5 × − , ∆ − = 8 g , (cid:15) Ω = 0 . , φ = 0. The modulation frequencies are η DCE = 2 . ω , η AJC = 0 . + , η JC = 1 . − and η ADCE = 1 . ω − Ω ). Panels e1 and e2 show the dynam-ics of (cid:104) σ z (cid:105) over short timescales, exhibiting fast oscillationssynchronized with ˙Ω. In the DCE regime (panel e1) the oscil-lations are nonsinusoidal, but the characteristic period is still ≈ π/η . The main contribution of this paper to the field ofquantum thermodynamics is that it shows a finite-time,out-of-equilibrium realistic resource for work extractionfrom systems suitably described by the Rabi model. TheRabi model itself has already been investigated in thecontext of heat engines [82]. In that case, the so calledadiabatic regime was addressed, which restricts the rateof change in any system’s parameter to an infinitely slowpace, in order to keep it in equilibrium at all points ofthe cycle. As a consequence, the exchanged work is di-rectly proportional to the variation of the system’s en-ergy gap, since populations of the eigenstates remain con-stant during the work exchange protocol for an isolatedatom-field system. Therefore, a fair amount of work ex-traction requires huge variations of the gap, that meansequally huge changes in the atom-field detuning, as wellas an ultra-strong coupling regime, g ∼ ω . The out-of-equilibrium process we present here conveys a conceptu-ally different origin for work extraction: it comes from thetime-dependent gap variation that under resonance con-ditions induces the amplification of oscillations betweenthe eigenstates of the non-modulated system, thereby al-lowing the system to be driven to a lower-energy state.The quantum thermodynamics approach also leavesclear the interference nature associated with the genera-tion or annihilation of excitations in our protocol. As isclear from Eq. (5), the work averaged over a few peri-ods of oscillation of Ω would be zero unless (cid:104) σ z ( t ) (cid:105) alsoexhibits fast oscillations with frequencies of the order of η . This is precisely what one observes by making a zoomof (cid:104) σ z ( t ) (cid:105) corresponding to data in Figs. 1 - a1 and b1,as shown in Figs. 1 - e1 and e2. This confirms that afinite average amount of work can only be obtained whenthe oscillations of (cid:104) σ z ( t ) (cid:105) become synchronized with ˙Ω( t ),meaning that energy can be added or withdrawn fromthe system only under resonance conditions. Since themodulation frequency must also match the energy gapassociated to the transition between two system eigen-states, modulation frequencies of the order ∼ ω lead tolarger amount of added or extracted work than modula-tion frequencies of the order ∼ | ∆ − | . B. ADCE under multi-tone modulation
The initial state influences the maximum amount ofwork that can be extracted from the system by means ofADCE. As a realistic example, we consider the scenariowhere the field is initially in a thermal state and the atomis in the ground state, ρ (0) = | g (cid:105)(cid:104) g | ⊗ ρ fT . We recall thatthe thermal state is given by ρ fT = (cid:80) n p n | n (cid:105)(cid:104) n | , where p n = ¯ n n / (¯ n + 1) n +1 is the population of the state | n (cid:105) and¯ n is the average photon number related to temperature as T − = k B ω − ln( n ¯ n ) ( k B is the Boltzmann constant).Since the modulation frequency depends on the initialstate, we can adjust the value of η to select a partic-ular transition | g, n (cid:105) ←→ | e, n − (cid:105) for a given valueof n . Therefore, the amount of work extracted fromthe system depends on the initial populations of thestates | g, n (cid:105) and | e, n − (cid:105) . As shown in the appendixA, one is able to extract work from any initial stateof the form | g (cid:105)(cid:104) g | ⊗ ρ f provided the initial population P | g,n (cid:105) is larger than P | e,n − (cid:105) . To illustrate this point weplot in Fig. 2 - a the quantum work W as a functionof dimensionless time t/τ for two different modulationfrequencies: η (1) = 1 . ω − Ω ) (black curve) and η (2) = 1 . ω − Ω ) (red curve). We use the sameparameters as previously, ¯ n = 1 . φ (1 , = 0. Asdiscussed in section III A, the dynamics of the quantumwork in the ADCE regime presents a periodic behavioron large timescales. In Fig. 2 - a we adopt a time intervalthat corresponds to a single oscillation period. Frequency η (1) drives the transition | g, (cid:105) ←→ | e, (cid:105) , whereas fre-quency η (2) drives the transition | g, (cid:105) ←→ | e, (cid:105) . Sincethe initial population of state | g, (cid:105) is larger than the ini-tial population of state | g, (cid:105) , p ≈ . > p ≈ . − − − and single-tone modulation (a) two-tone modulation W o r k / (c) W o r k / − − (b) N − − (d)(e) single-tone modulation w i t h o u t d i ss i p a t i o n w i t h d i ss i p a t i o n single-tone modulation w i t h d i ss i p a t i o n w i t h o u t d i s s i p a t i o n Figure 2. (color online)
Panel a : Quantum work as a func-tion of dimensionless time t/τ for single-tone modulationswith { η (1) = 1 . ω − Ω ) , (cid:15) (1)Ω = 0 . } (black curve), { η (2) = 1 . ω − Ω ) , (cid:15) (2)Ω = 0 . } (red curve), andfor the two-tone modulation with the above frequencies andmodulation amplitudes (blue curve). All curves correspondto the ADCE regime and φ (1 , = 0. We used the initial state ρ (0) = | g (cid:105)(cid:104) g | ⊗ ρ fT , where ρ fT = (cid:80) n p n | n (cid:105)(cid:104) n | is a field thermalstate with p n = ¯ n n / (¯ n + 1) n +1 and ¯ n = 1 . τ m denotes the instant of time when theextracted work is maximum. Panel b:
Dynamics of quantumwork around the instant of time τ m for a single-tone modula-tion with frequency η (1) . Panel c:
Level diagram related tothe modulation-induced transitions between the approximateeigenstates of the time-independent RH for frequencies η (1) and η (2) . Panel d:
Quantum work for single-tone modulation η (1) in the presence of Markovian dissipation. The cavity andatom damping rates are κ = γ = 2 × − g , and the reser-voirs’ temperature is k B T r /ω = 0 . Panel e:
Behavior of N under unitary and dissipative dynamics. the modulation frequency η (1) rather than the frequency η (2) [98]. The instant of time τ m at which the work ex-traction is maximum depends on both the number ofphotons in the state | g, n (cid:105) and the modulation ampli-tude (cid:15) Ω : τ − m ∝ (cid:112) n ( n − n − (cid:15) ( k )Ω (see Appendix Afor details). Fig. 2 - b shows the dynamics of quan- − − (a) W o r k / time-dependent modulation frequency w i t h d i s s i p a t i o n w i t h o u t d i s s i p a t i o n time-dependent modulation frequency w i t h o u t d i ss i p a t i o n w i t h d i s s i p a t i o n N (b) − − (c) time-dependent modulation frequency with dissipation W o r k / Figure 3. (color online)
Panels a-b : Quantum work and N under unitary (black curves) and dissipative (green curves)evolution for aperiodic modulation with frequency η ( t ) = η (1) − λ + λ t , where η (1) = 1 . ω − Ω ), φ = 0 and λ = 1 . × − ω . The initial state and other parametersare as in Fig. 2. The dynamics resembles the typical Landau-Zener behavior for an effective TLA undergoing adiabatic fre-quency sweep. Panel c : Quantum work under dissipative dy-namics for two different choices of the modulation frequency: η ( t ) = η (1) − λ + λ t (red curve) and η ( t ) = η (1) − λ + λ t (blue curve). One can see that for the blue curve the workextraction is larger and faster, hence there is room for furtheroptimization of our protocol. tum work around the instant of time τ m . The quantumwork rapidly oscillates at time scale 1 /η (1) due to thefast oscillations of (cid:104) σ z (cid:105) , which are necessary to withdrawa finite amount of energy from the system (as discussedin section III A).In order to adjust the same τ m for both modulationfrequencies, η (1) and η (2) , the modulation amplitude as-sociated with frequency η (2) was set as (cid:15) (2)Ω = (cid:15) (1)Ω / (cid:15) (1)Ω = 0 . is the modulation amplitude as-sociated with frequency η (1) . This choice of modula-tion amplitudes is particularly convenient for employ-ing multi-tone modulations. The blue curve in Fig. 2- a describes the dynamics of the quantum work for atwo-tone modulation characterized by the atomic fre-quency Ω( t ) = Ω + (cid:80) k =1 , (cid:15) ( k )Ω sin( η ( k ) t ). In this case, the amount of work extracted is effectively affected byboth transitions: | g, (cid:105) ←→ | e, (cid:105) and | g, (cid:105) ←→ | e, (cid:105) .Hence, according to expression (5), the total amount ofwork extracted from the system is equal to the sum ofthe works extracted by each single-tone modulation indi-vidually. Therefore, in order to maximize the extractedwork, one needs to adjust the amplitudes (cid:15) ( k )Ω in such away that all the induced transitions have the same τ m . C. Effective Landau-Zener transitions
In the cases studied so far, we observed two featuresthat can possibly be regarded as issues. Firstly, thedynamics of the quantum work in the ADCE regimepresents a periodic behavior. The amount of work ex-tracted from the system will, then, be maximal aroundspecific instants of time ( t = nτ m with n = 1 , , . . . ), afterwhich it will return to zero. Secondly, the ADCE regimeis obtained for a very fine-tuned modulation frequency η , which must be ultimately found either numerically orexperimentally.We overcome these possible limitations by using atime-dependent modulation frequency η ( t ) [40]. As dis-cussed in section II A, when the modulation frequencyvaries over time, it can assume values close to the res-onance frequency η ADCE , that corresponds to the ADCEregime for a given initial state. When η ( t ) is close to η ADCE , the energy of the atom-field system will decreasedue to the work extraction. But, as η ( t ) moves awayfrom η ADCE , the resonance condition η ( t ) ≈ η ADCE is lost.Therefore, the external agent responsible for the atomicfrequency modulation will not be able to give back en-ergy to the system. To illustrate this point we choose η ( t ) = η (1) − λ + λ t , where λ is the transition ratebetween the states | g, (cid:105) and | e, (cid:105) . The initial state isthe local thermal state ρ (0) = (cid:80) n p n | g, n (cid:105)(cid:104) g, n | , and allother parameters are as in section III B. In this case, η (1) = 1 . ω − Ω ) and λ = 1 . × − ω .In Fig. 3 - a we plot the quantum work as a functionof dimensionless time t/τ . The transition | g, (cid:105) −→ | e, (cid:105) begins at t b ≈ /λ ≈ τ , corresponding to η ( t b ) = η (1) − λ , instead of η ( t b ) = η (1) one would expectnaively. Such discrepancy can be explained after rigor-ous derivation of the effective Hamiltonian, which reads H eff = V ( t )( | g, (cid:105)(cid:104) g, | − | e, (cid:105)(cid:104) e, | ) / λ | g, (cid:105)(cid:104) e, | + h.c. ), where V ( t ) = 2 λ t − λ (see appendix A). Indeed,for t ≈ t b one obtains V ( t b ) ≈
0, corresponding to the ex-pected resonance condition. Fortunately, we do not haveto solve analytically the time evolution according to theHamiltonian H eff , since this was made independently in1932 by Landau, Zener, Stueckelberg and Majorana , inwhat became known as Landau-Zener problem [94–97].As shown in the appendix A, for sufficiently slow ˙ η thetransition from | g, (cid:105) to | e, (cid:105) is almost complete, evenfor finite duration of the process [40]. Therefore, thework extracted from the system tends to a steady value W ≈ − . (cid:126) ω . Note that this steady value is practicallyequal to the maximum extracted work using the time-independent modulation frequency η = η (1) , shown inFig. 2 - a (black curve). D. Account of dissipation
For actual experimental implementation of our pro-posal it is necessary to take into account the interac-tion between the system and its environments. Forthe quantum Rabi model with moderate coupling rates, g/ω (cid:46) − , one can use the Markovian master equationin the dressed picture, which was rigorously deduced fromthe first principles in [83]. We did solve it numericallyand verified that for the parameters of Figs. 1 and 2 theresults are almost indistinguishable from the predictionsof a much simpler ‘standard master equation’ of Quan-tum Optics: ˙ ρ = − i [ H ( t ) , ρ ( t )] + L ( ρ ). The Liouvilliansuperoperator L ( ρ ) reads [38, 40, 83, 86] L ( ρ ) = γ (1 + n a ) D [ σ − ] ρ + γn a D [ σ + ] ρ + κ (1 + n c ) D [ a ] ρ + κn c D [ a † ] ρ, (7)where D [ c ] ρ ≡ (2 cρc † − c † cρ − ρc † c ), κ ( γ ) is the de-cay rate of the cavity (atom) and n c = [exp( β (cid:126) ω ) − − ( n a = [exp( β (cid:126) Ω ) − − ) is the mean number of ther-mal excitations in the cavity (atom). Here β = k B T r and T r is the common temperature of both reservoirs,adjusted so that n c = 0 .
05. We use the same param-eters as previously, which give n a = 0 .
19 for the pos-itive detuning adopted in this paper. For the dissipa-tive rates we assume the state-of-the-art values in circuitQED: κ = γ = 2 × − g .We studied the influence of dissipation on the quantumwork and average number of excitations in the ADCEregime for the initial state ρ (0) = | g (cid:105)(cid:104) g |⊗ ρ fT with ¯ n = 1 . W and N for a single-tone modulation that drives the transition | g, (cid:105) ←→ | e, (cid:105) . For initial times the amount of ex-tracted work is approximately equal to the one obtainedin the lossless case, although its absolute value decreasesas time goes on. The average number of excitations de-creases with exponential envelope superposed with smalloscillations due to the periodic transitions between | g, (cid:105) and | e, (cid:105) , which are resolvable for initial times.The aperiodic regime with time-dependent η is alsofeasible in the presence of dissipation, as illustrated inFig. 3. The behavior of W and N for the modulationfrequency η ( t ) = η (1) − λ + λ t is illustrated in Fig.3 - a and b. The qualitative behavior of quantum workis similar in the unitary and dissipative cases, althoughthe amount of extracted work is roughly 50% smaller dueto the losses. The behavior of N is also affected by theADCE: instead of an exponential decay expected for puredamping, N exhibits an accentuated decrease around t ∼ τ , which is precisely where the LZ transition takesplace.We finally note that in the presence of dissipation it isadvantageous to decrease the duration of the frequency sweep in the effective LZ process, so that the transitionoccurs at earlier times while the population of the state | g, (cid:105) is as high as possible. The downside of such a dras-tic measure is that the probability of complete populationtransfer from | g, (cid:105) to | e, (cid:105) is lowered and the Landau-Zener formula (A4) loses its validity. Yet a compromisecan be found in order to optimize the work extraction.An example is shown in Fig. 3 - c for the modulation fre-quency η ( t ) = η (1) − λ + λ t (blue curve): the LZ tran-sition takes place earlier than in the previous case, andthe amount of extracted works is slightly higher. Thisresults demonstrate that our protocol can be optimizedfor maximum work extraction in realistic scenarios. IV. CONCLUSIONS
We have established for the first time the direct rela-tionship between the quantum work and generation orannihilation of excitations in the quantum Rabi modelwith time-modulated atomic frequency. Our results arevalid in the dispersive regime of light-matter interaction( | ∆ − | ∼ g ), moderate coupling strengths ( g/ω ∼ . (cid:15) Ω / Ω ∼ . | W max | (cid:46) (cid:126) | ∆ − | < (cid:126) ω in this case.Even in this number-conserving scenario, the out-of-equilibrium protocol we propose outperforms the veryslow adiabatic modulations, i.e., | W max | ∼ . (cid:126) ω > (cid:126) (cid:15) Ω = 0 . (cid:126) ω .Much greater amount of energy can be transferred be-tween the system and the external agent by harness-ing the ‘counter-rotating’ (or ‘anti-Jaynes-Cummings’)terms. We showed that the generation of excitationsis accompanied by positive work, as in the dynamicalCasimir and anti-Jaynes-Cummings effects. On the otherhand, the work becomes negative in the so called anti-dynamical Casimir effect (ADCE), when one pair of exci-tations is coherently annihilated under appropriate mod-ulation frequency. For ADCE the maximum amount ofextracted work is | W ADCE | ∼ (cid:126) ω (cid:29) (cid:126) (cid:15) Ω = 0 . (cid:126) ω , andstrongly depends on the initial state. We also explainedwhy the generation and annihilation of excitations arealways accompanied by fast low-amplitude oscillationsof the atomic population inversion: the attainment of afinite amount of work requires the synchronization of os-cillations of (cid:104) σ z ( t ) (cid:105) and the atomic transition frequencyΩ( t ).We extended our results to realistic initial states in cir-cuit QED, arguing that multi-tone modulations are moreeffective in extracting energy from the system when thefield is prepared in a thermal equilibrium state. Addition-ally, we have shown how effective Landau-Zener transi-tions between the system dressed-states may be employedfor obtaining aperiodic work extraction, which also solvesthe problem of an extremely fine-tuned adjustment of themodulation frequency. Lastly, we carried out numericalsimulations to assess the feasibility of our proposal inrealistic circuit QED setup subject to atomic and cav-ity dampings, demonstrating that periodic and aperiodicwork extraction is still possible, although in a smalleramount compared to the lossless case. We hope these re-sults will find applications in out-of-equilibrium quantumthermal machines of finite-time cycles. Appendix A: Analytical approach
In this appendix we briefly describe how to obtain theresonant modulation frequencies and the correspondingtransition rates mentioned in the main text. We expandthe state corresponding to the Hamiltonian (1) as | ψ ( t ) (cid:105) = (cid:88) l A l ( t ) e − itE l | R l (cid:105) , where E l and | R l (cid:105) are the eigenvalues and eigenstates( dressed states ) of the time-independent Rabi Hamilto-nian H ≡ H [Ω( t ) = Ω ], where l increases with energy.The probability amplitudes obey the differential equa-tions i ˙ A j = (cid:88) l (cid:80) k ε ( k )Ω sin (cid:0) η ( k ) ( t ) t + φ ( k ) (cid:1) (cid:104) R j | σ z | R l (cid:105)× e − it ( E l − E j ) A l . (A1)Therefore modulation frequency η ( k ) = | E l − E j | mayinduce a resonant coupling between the amplitudes A j and A l with transition rate ∝ |(cid:104) R j | σ z | R l (cid:105)| .To diagonalize H we perform the unitary transforma-tion U = exp (cid:2) Λ( aσ − − a † σ + ) + ξ ( a − a † ) σ z (cid:3) , whereΛ = g / ∆ + , ξ = g Λ / ω , ∆ ± = ω ± Ω . To the firstorder Λ we obtain U † H U = ( ω + δ + σ z ) n + Ω + δ + σ z + g ( aσ + + a † σ − ) , where δ ± = g / ∆ ± . Hence we find the approximateeigenvalues and eigenstates: E = − (Ω + δ + ) / | R (cid:105) = U | g, (cid:105) , E m, ± = ωm − ω + δ + ± (cid:113) (∆ − − δ + m ) + 4 g m | R m, − (cid:105) = U (cos θ m | g, m (cid:105) − sin θ m | e, m − (cid:105) ) | R m, + (cid:105) = U (sin θ m | g, m (cid:105) + cos θ m | e, m − (cid:105) ) , where m ≥ θ m = ∆ − − δ + m + (cid:113) (∆ − − δ + m ) + 4 g m g √ m . The Jaynes-Cummings eigenvalues and eigenstates (de-noted as | ϕ n, ± (cid:105) ) are obtained simply by setting U = 1and δ + = 0 in the above formulae. To compute Eq.(6), we must note that (cid:104) σ z ( t ) (cid:105) = ± (cos θ m − sin θ m ) = ± cos 2 θ m for the initial state | ϕ n, ± (cid:105) .In the following we restrict our attention to the dis-persive regime, | ∆ − | / (cid:29) g √ n max , where n max is themaximum number of system excitations and we assumethe condition | ∆ − | (cid:28) ω . The approximate eigenenergiesthen read E m, D ≈ ( ω + δ − − δ + ) m − αm + E E m, −D ≈ ( ω − δ − + δ + ) m + αm − ∆ − + E , where D ≡ ∆ − / | ∆ − | = ± α = g / ∆ − . To the firstorder in g/ ∆ − the dressed states are | R (cid:105) ≈ | g, (cid:105) , | R m, D (cid:105) ≈ | g, m (cid:105) + g √ m ∆ − | e, m − (cid:105)| R m, −D (cid:105) ≈ | e, m − (cid:105) − g √ m ∆ − | g, m (cid:105) . (A2) AJC regime . For the initial state | g, (cid:105) and the mod-ulation frequency η ( k ) = ∆ + − δ − − δ + ) + 4 α − ν ( t )one can neglect the rapidly oscillating terms in (A1) andobtain the effective Hamiltonian [40]˜ H = E | R (cid:105)(cid:104) R | + E , −D | R , −D (cid:105)(cid:104) R , −D | +( λe − it ( E , −D − E − ν ( t )) | R , −D (cid:105)(cid:104) R | + h.c. ) λ = i D g + ε ( k )Ω exp( − iφ ( k ) ) . Performing the time-dependent unitary transformation U = exp (cid:26) − i (cid:20)(cid:18) E + ν ( t )2 (cid:19) t | R (cid:105)(cid:104) R | + (cid:18) E , −D − ν ( t )2 (cid:19) t | R , −D (cid:105)(cid:104) R , −D | (cid:21)(cid:27) we get the final effective Hamiltonian H f = ˜ ν ( t )2 ( | R , −D (cid:105)(cid:104) R , −D | − | R (cid:105)(cid:104) R | )+ ( λ | R , −D (cid:105)(cid:104) R | + h.c. ) (A3)˜ ν ( t ) ≡ ν ( t ) + ˙ ν ( t ) t . This is the standard Hamiltonian for a two-level sys-tem with a time-varying energy splitting and constantoff-diagonal coupling. For ν ( t ) = 0 the dynamics con-sists of periodic oscillation between the states | R (cid:105) and | R , −D (cid:105) , which approximately corresponds to the transi-tion | g, (cid:105) ↔ | e, (cid:105) . On the other hand, if ν ( t ) = | ξ | t and t is varied from −∞ to + ∞ one can use the well knownformula for the Landau-Zener transition [94–97]. If for t = −∞ the initial state was | R (cid:105) , then for t → + ∞ theprobability of occupancy of the state | R , −D (cid:105) is P = 1 − exp (cid:18) − π | λ | | ξ | (cid:19) . (A4)Therefore if | ˙ ν | (cid:46) | λ | one can accomplish an almostcomplete transition from | R (cid:105) to | R , −D (cid:105) . As shown in[40] similar conclusion holds even if ν ( t ) varies within afinite interval around ν ( t ) = 0. DCE regime . For the modulation frequency η ( k ) =2 ω +2 ( δ − − δ + ) − α − ν ( t ) we obtain the effective DCEHamiltonian H f = n max (cid:88) m =0 (cid:18) ˜ ν ( t )2 m − αm ( m − (cid:19) | R m, D (cid:105)(cid:104) R m, D | + n max (cid:88) m =0 ( λ m | R m +2 , D (cid:105)(cid:104) R m, D | + h.c. ) (A5) λ m = − iδ − + (cid:112) ( m + 1) ( m + 2) ε ( k )Ω exp( − iφ ( k ) ) . where n max is the maximum number of excitationsallowed by the dispersive approximation and we de-note | R , D (cid:105) ≡ | R (cid:105) . As shown in [37] the Kerr term αm ( m −
2) in (A5) is responsible for non-periodiccollapse-revival behavior of (cid:104) n (cid:105) . JC Regime . For the modulation frequency η ( k ) J = | ∆ − − δ + J | + 2 | δ − | J − | α | J − ν , with J ≥
1, weobtain the effective Hamiltonian (see [40] for the validityrange) H f = D ˜ ν ( t )2 ( | R J, D (cid:105)(cid:104) R J, D | − | R J, −D (cid:105)(cid:104) R J, −D | )+( λ J | R J, D (cid:105)(cid:104) R J, −D | + h.c. ) (A6) λ J = − ig − √ Jε ( k )Ω exp( −D iφ ( k ) ) . (A7)This corresponds roughly to the transition between thestates | g, J (cid:105) and | e, J − (cid:105) , reliant only on the rotatingterms in the Hamiltonian. ADCE Regime . For the modulation frequency η ( k ) J =3 ω − Ω + 2 ( δ − − δ + ) ( J − − α (cid:0) J − J + 2 (cid:1) − ν ( t ),where J ≥
3, we obtain the effective Hamiltonian H f = ˜ ν ( t )2 ( | R J, D (cid:105)(cid:104) R J, D | − | R J − , −D (cid:105)(cid:104) R J − , −D | )+ ( λ J | R J, D (cid:105)(cid:104) R J − , −D | + h.c. ) (A8) λ J = − i D gδ − + ∆ − (cid:112) J ( J −
1) ( J − ε ( k )Ω exp( − iφ ( k ) ) . (A9)A more rigorous calculation performed in [37] resulted inthe expression λ (cid:48) J = λ J (cid:18) ω − ∆ − ω + ∆ − ω + ∆ − ω (cid:19) instead of (A9). For the parameters of this paper, ∆ − =0 . ω , we obtain λ (cid:48) J /λ J = 14 / ≈ .
93, so the differencebetween λ J and λ (cid:48) J is insignificant and becomes smalleras the ratio | ∆ − | /ω decreases.ADCE corresponds to the induced coupling betweenthe dressed states | R J, D (cid:105) and | R J − , −D (cid:105) , or approxi-mately | g, J (cid:105) and | e, J − (cid:105) . In particular, for ν = 0we can solve the unitary dynamics for any initial stateby writing the density matrix in the dressed basis ρ ( t ) = (cid:88) S , T ∞ (cid:88) n,l =0 ρ S , T n,l ( t ) | R n, S (cid:105)(cid:104) R l, T | If initially the off-diagonal elements ρ S , −S n − ,n (0) are zero,as occurs for the thermal state considered in this work,the solution for the affected dressed states reads ρ D , D J,J = ρ D , D J,J (0) cos ( | λ J | t ) + ρ −D , −D J − ,J − (0) sin ( | λ J | t ) ρ −D , −D J − ,J − = ρ −D , −D J − ,J − (0) cos ( | λ J | t ) + ρ D , D J,J (0) sin ( | λ J | t ) , while other probabilities remain unaltered. Therefore,one is able to annihilate two system excitations provided ρ D , D J,J (0) > ρ −D , −D J − ,J − (0), or approximately when the ini-tial population of | g, J (cid:105) is bigger than | e, J − (cid:105) . ACKNOWLEDGMENTS
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