Speeding up antidynamical Casimir effect with nonstationary qutrits
SSpeeding up antidynamical Casimir effect with nonstationary qutrits
A. V. Dodonov, ∗ J. J. D´ıaz-Guevara, A. Napoli,
3, 4 and B. Militello
3, 4 Institute of Physics and International Centre for Condensed Matter Physics,University of Brasilia, 70910-900, Brasilia, Federal District, Brazil Departamento de F´ısica, Universidad de Guadalajara,Revoluci´on 1500, Guadalajara, Jalisco 44420, Mexico Dipartimento di Fisica e Chimica, Universit`a degli Studi di Palermo, Via Archirafi 36, I-90123 Palermo, Italy I.N.F.N. Sezione di Catania
The antidynamical Casimir effect (ADCE) is a term coined to designate the coherent annihilationof excitations due to resonant external perturbation of system parameters, allowing for extractionof quantum work from nonvacuum states of some field. Originally proposed for a two-level atom(qubit) coupled to a single cavity mode in the context of nonstationary quantum Rabi model, itsuffered from very low transition rate and correspondingly narrow resonance linewidth. In this paperwe show analytically and numerically that the ADCE rate can be increased by at least one orderof magnitude by replacing the qubit by an artificial three-level atom (qutrit) in a properly chosenconfiguration. For the cavity thermal state we demonstrate that the dynamics of the average photonnumber and atomic excitation is completely different from the qubit’s case, while the behavior ofthe total number of excitations is qualitatively similar yet significantly faster.
PACS numbers: 42.50.Pq, 42.50.Ct, 42.50.Hz, 32.80-t, 03.65.Yz
I. INTRODUCTION
The broad term dynamical Casimir effect (DCE) refersto the generation of excitations of some field (Electro-magnetic, in the majority of cases) due to time-dependentboundary conditions, such as changes in the geometry ormaterial properties of the system [1–4] (see [5, 6] for re-views; see also [7–9] for the related problem of a particlein a wall with moving boundaries). In the so called cavityDCE one considers nonadiabatic (periodic or not) mod-ulation of the cavity natural frequency by an externalagent, investigating the accumulation of intracavity pho-tons or the photon emission outside the cavity [1, 10–12]. The additional interaction of the cavity field witha stationary ‘detector’ during the modulation (harmonicoscillator, few-level atom or a set of two-level atoms inthe simplest examples) may dramatically alter the pho-ton generation dynamics, for instance, altering the fieldstatistics, shifting the resonance frequency and inhibitingthe photon growth [13–19] (see [20] for a short review).Moreover, the degree of excitation of the detector variesaccording to the regime of parameters, and entanglementcan be created between the cavity field and the detector,or between the set of atoms coupled to the field [21–25].Over the past ten years a new path has attracted at-tention of the community working on nonstationary phe-nomena in cavity Quantum Electrodynamics (QED). In-stead of changing the cavity frequency, different stud-ies suggested the parametric modulation of the ‘detec-tor’ instead, promoting it from a passive to an activeagent responsible for both the generation and detectionof photons [26–35]. Beside eliminating the inconvenience ∗ Electronic address: adodonov@fis.unb.br of time-dependent Fock states of the field associated totime-varying cavity frequency [10], this scheme makes fulluse of the counter-rotating terms in the light–matter in-teraction Hamiltonian and does not require the inclusionof additional parametric down-conversion terms in theformalism [26, 31, 34, 35]. Moreover, it benefits fromrecent advances in the coherent control and readout ofmicroscopic few-level quantum devices developed in therealm of the circuit QED for applications in QuantumInformation Processing (see [36] for a recent review).The area of circuit QED investigates the interaction ofartificial superconducting atoms, formed by a sophisti-cated array of Josephson Junctions, and the Electromag-netic field confined in increasingly complex microwaveresonators, ranging from waveguide resonators or 3D cav-ities [37–41]. The advances in engineering allowed forimplementation of multi-level atoms, with controllabletransition frequencies and coupling strengths, that caninteract with multiple cavities and other atoms controlledindependently [32, 36, 38, 42–48]. Moreover, circuit QEDallows for unprecedented atom–field coupling strength,in what became known as ultrastrong and deep strongcoupling regimes [49–52]. In the context of DCE, theexquisite control over the parameters of the Hamilto-nian allows for multi-tone multi-parameter modulations[26, 53–55], while quantum optimal control strategies canbe used to enhance the desired effects [56].Photon generation is not the only phenomenon in-duced by parametric modulations in circuit QED. It wasshown recently that the counter-rotating terms can alsobe employed to annihilate excitations of the Electromag-netic field from nonvacuum initial states, in what be-came known as antidynamical Casimir effect (ADCE)[34]. This effect was predicted in the context of the quan-tum Rabi model, which describes the interaction of thecavity field with a two-level atom [57–59], and consists in a r X i v : . [ qu a n t - ph ] J u l the coherent annihilation of three photons accompaniedby the excitation of the far-detuned atom [60, 61] (fourphotons could be annihilated by employing a two-tonemodulation [54]) . Thus an amount of energy (cid:46) (cid:126) ω could be extracted from the system due to resonant per-turbation of some parameter, where ω is the cavity fre-quency [55]. However, in the more accessible regimeof weak atom–field interaction (beneath the ultrastrongcoupling regime) the associated transition rate is quitesmall, so the modulation frequency must be finely tunedand the dissipation strongly affects the behavior [54, 55].In this paper we uncover that the ADCE rate canbe enhanced by almost two orders of magnitude by em-ploying artificial three-level atoms (qutrits) in the stan-dard ladder configuration and weak coupling regime [39].We obtain closed approximate description of the uni-tary dynamics when one or more atomic parameters un-dergo a low-amplitude multi-tone external perturbation,and assess the advantages and disadvantages of differ-ent regimes of parameters for the initial thermal state ofthe cavity field. We also discuss eventual complicationsthat qutrits bring into the problem, such as adjustmentof atomic energy levels with respect to the cavity fre-quency and two-tone driving with management of themodulation phases. Nevertheless it is argued that thesubstantial gain in the ADCE rate compensates for theadditional technical issues.This paper is organized as follows. In Sec. II we defineour problem and derive the general mathematical for-malism to obtain approximate expressions for the systemdynamics in the dressed-states basis. In Sec. III we dis-cuss three specific configurations of the qutrit for whichthe overall behavior is most easily inferred: the double-resonant, dispersive and mixed regimes. In Sec. IV weidentify the regimes of parameters and the transitionsfor which excitations can be annihilated from the cavitythermal state, assuming that the atom was initially inthe lowest energy state. In Sec. V we evaluate analyt-ically the transition rates associated to ADCE betweendifferent dressed states and compare our predictions tothe exact numerical solution of the Schr¨odinger equation,demonstrating that the ADCE rate can undergo almost50-fold increase compared to the qubit’s case while theamount of annihilated excitations is roughly the same.Our conclusions are summarized in Sec. VI. II. MATHEMATICAL FORMALISM
We consider a three-level artificial atom (qutrit) inter-acting with a single cavity mode of constant frequency ω , as described by the Hamiltonian (we set (cid:126) = 1)ˆ H = ω ˆ n + (cid:88) k =1 E k ˆ σ k,k + (cid:88) k =0 G k (ˆ a + ˆ a † )(ˆ σ k +1 ,k + ˆ σ k,k +1 ) . (1)ˆ a (ˆ a † ) is the cavity annihilation (creation) operator andˆ n = ˆ a † ˆ a is the photon number operator. The atomic eigenenergies are E ≡ , E and E , with the corre-sponding states denoted as | (cid:105) , | (cid:105) , | (cid:105) ; the atomic oper-ators read ˆ σ k,j ≡ | k (cid:105)(cid:104) j | . The parameters G k ( k = 0 , {| k (cid:105) , | k + (cid:105)} mediated by the cavity field.We assume that all the atomic parameters can be mod-ulated externally as E k ( t ) ≡ E ,k + ε E,k f E,k ( t ) , G k ( t ) ≡ G ,k + ε G,k f G,k ( t ) , where { ε E,k , ε
G,k } are the modulation depths and { E ,k , G ,k } are the corresponding bare values. The di-mensionless functions f l ( t ) = (cid:88) j w ( j ) l sin (cid:16) η ( j ) t + φ ( j ) l (cid:17) (2)represent the externally prescribed modulation, wherethe collective index l denotes { E ; k = 1 , } or { G ; k =0 , } . Constants 0 ≤ w ( j ) l ≤ φ ( j ) l are the weightand the phase corresponding to the harmonic modulationof l with frequency η ( j ) , and the index j runs over all theimposed frequencies (in this paper at most 2-tone mod-ulations will be examined). We normalize the weights sothat (cid:80) j w ( j ) l = 1 for any set l , so that ε l characterizescompletely the modulation strength (in our examples weshall set w ( j ) l = 1 and φ ( j ) l = 0 unless stated otherwise).To obtain a closed analytical description we firstrewrite the Hamiltonian as ˆ H = ˆ H + ˆ H c , whereˆ H = ω ˆ n + (cid:88) k =0 (cid:2) E ,k ˆ σ k,k + G ,k (ˆ a ˆ σ k +1 ,k + ˆ a † ˆ σ k,k +1 ) (cid:3) (3)is the bare Hamiltonian in the absence of modulationand counter-rotating terms (to shorten the formulas wedefined formally G , = ε G, = 0). For the realistic weakcoupling regime ( G , , G , (cid:28) ω ) we expand the wave-function corresponding to the total Hamiltonian ˆ H as | ψ ( t ) (cid:105) = ∞ (cid:88) n =0 (cid:88) S ( n ) e − itλ n, S A n, S ( t ) | ϕ n, S (cid:105) , (4)where λ n, S and | ϕ n, S (cid:105) are the n -excitations eigenvaluesand eigenstates ( dressed states ) of the Hamiltonian ˆ H and the index S labels different states with a fixed num-ber of excitations n , which is the quantum number as-sociated to the operator ˆ N = ˆ n + | (cid:105)(cid:104) | + 2 | (cid:105)(cid:104) | . Asshown in Sec.III, the range of values of S depends on n ,and we denote such degeneration with g ( n ). Moreover,the number of excitations in the subspace coincides withthe number of photons of the state having the atom inits ground ( | , n (cid:105) ).Following the approach detailed in [31, 61] we pro-pose a change of variables that maps each group of g ( m )variables A m, T into another set b m, T , so that A m, T = (cid:80) T (cid:48) α T T (cid:48) b m, T (cid:48) . In particular, we consider the followingtransformation: A m, T = e i Φ m, T ( t ) (cid:8) e − itν m, T b m, T ( t ) (5) − i (cid:88) S ( m ) (cid:54) = T e − itν m, S b m, S ( t ) × (cid:88) (cid:48) j (cid:88) k =0 (cid:88) L = E,G Υ L,k,jm, T , S × (cid:88) r = ± e riφ ( j ) L,k e it ( λ m, T − λ m, S + rη ( j ) ) − λ m, T − λ m, S + rη ( j ) (cid:41) Φ m, T ( t ) = (cid:88) j (cid:88) k =0 (cid:88) L = E,G Υ L,k,jm, T , T η ( j ) (6) × (cid:104) cos( η ( j ) t + φ ( j ) L,k ) − cos φ ( j ) L,k (cid:105) , where we divided the sum in two parts: (cid:80) (cid:48) j runs over‘fast’ frequencies η ( j (cid:48) ) ∼ λ m +2 , S − λ m, T and (cid:80) (cid:48)(cid:48) j runsover the ‘slow’ ones η ( j (cid:48)(cid:48) ) ∼ | λ m, S − λ m, T | . The smallfrequency shift ν m, T will be given in Eq. (13) and weintroduced constant coefficients ( k = 0 , , E,k,jm, T , S ≡ ε E,k w ( j ) E,k (cid:104) ϕ m, T | ˆ σ k,k | ϕ m, S (cid:105) (7)Υ G,k,jm, T , S ≡ ε G,k w ( j ) G,k (cid:104) ϕ m, T | (ˆ a ˆ σ k +1 ,k + ˆ a † ˆ σ k,k +1 ) | ϕ m, S (cid:105) . (8)After substituting A m, T into the Schr¨odinger equa-tion and systematically eliminating the rapidly oscillat-ing terms via Rotating Wave Approximation (RWA) [31],to the first order in ε E,k and ε G,k we obtain the approx-imate differential equation for the effective probabilityamplitude˙ b m, T = (cid:88) S ( m ) (cid:54) = T ς m, T , S e it (˜ λ m, T − ˜ λ m, S ) b m, S (9)+ (cid:88) (cid:48)(cid:48) j (cid:88) S ( m ) (cid:54) = T Ξ ( j ) m, T , S e it(cid:36) m, T , S ( | ˜ λ m, T − ˜ λ m, S |− η ( j ) ) b m, S + (cid:88) (cid:48) j (cid:88) S ( m +2) Θ ( j ) m +2 , T , S e − it (˜ λ m +2 , S − ˜ λ m, T − η ( j ) ) b m +2 , S − (cid:88) S ( m − Θ ( j ) ∗ m, S , T e it (˜ λ m, T − ˜ λ m − , S − η ( j ) ) b m − , S . The time-independent transition rates between thedressed states are ς m, T , S = i (cid:88) k,l =0 G ,k G ,l (cid:88) R ( m +2) Λ k,m +2 , T , R Λ l,m +2 , S , R λ m +2 , R − λ m, S − (cid:88) R ( m − Λ k,m, R , T Λ l,m, R , S λ m, S − λ m − , R Ξ ( j ) m, T , S = (cid:36) m, T , S (cid:88) k =0 (cid:88) L = E,G Υ L,k,jm, T , S e − i(cid:36) m, T , S φ ( j ) L,k Θ ( j ) m +2 , T , S = (cid:88) k =0 G ,k (cid:40) − ε ( j ) G,k Λ k,m +2 , T , S G ,k (10)+ (cid:88) l =0 (cid:88) L = E,G (cid:88) R ( m +2) Λ k,m +2 , T , R Υ L,l,jm +2 , R , S e iφ ( j ) L,l λ m +2 , R − λ m +2 , S + η ( j ) − (cid:88) R ( m ) Λ k,m +2 , R , S Υ L,l,jm, T , R e iφ ( j ) L,l λ m, T − λ m, R + η ( j ) Λ k,m +2 , T , S = (cid:104) ϕ m, T | ˆ a ˆ σ k,k +1 | ϕ m +2 , S (cid:105) . (11)Here (cid:36) m, T , S ≡ sign(˜ λ m, T − ˜ λ m, S ) and we introducedthe complex modulation depth ε ( j ) l ≡ ε l w ( j ) l exp( iφ ( j ) l ).Moreover, we defined the corrected eigenfrequencies˜ λ m, T ≡ λ m, T + ν m, T + ∆ ν, (12)where the correction due to counter-rotating terms reads ν m, T = (cid:88) S ( m − (cid:16)(cid:80) k =0 G ,k Λ k,m, S , T (cid:17) λ m, T − λ m − , S (13) − (cid:88) S ( m +2) (cid:16)(cid:80) k =0 G ,k Λ k,m +2 , T , S (cid:17) λ m +2 , S − λ m, T and ∆ ν denotes the neglected contributionssmaller than ν m, T and the terms of the order ∼ (Υ L,k,jm, T , S ) /ω , ( ε G,k Λ k,m, S , T ) /ω .Throughout the derivation of the formula (9) we haveassumed the constraints | λ m, T − λ m, S | , | Υ L,k,jm, T , S | , (cid:12)(cid:12)(cid:12)(cid:12) G ,k Λ l,m, S , T λ m +2 , T − λ m, S (cid:12)(cid:12)(cid:12)(cid:12) G ,l (cid:28) ω (14) G ,k | Λ k,m +2 , S , T | (cid:46) ω . Under these approximations we have | A m, T | ≈ | b m, T | , sofrom Eq. (9) one can easily infer the evolution of popu-lations of the dressed states. Besides, the generalizationof our method for N -level atoms and second-order effectsis straightforward [61].It is worth noting that the occurrence of ADCE is es-sentially governed by the transition rates Θ ( j ) m, T , S thatcouple states belonging to subspaces with different num-bers of excitations. Of course the whole dynamics is de-termined also by the transitions occurring inside eachsubspace, but the annihilation of (two) excitations is pos-sible only in the presence of non negligible Θ-terms. III. ANALYTICAL REGIMES
We shall confine ourselves to three different regimes ofparameters when the dressed states have simple analyti-cal expressions. With the aid of these formulas we shallbe able to evaluate analytically the coefficients Θ ( j ) m +2 , T , S in the section IV.The ground state of ˆ H is | ϕ (cid:105) = | , (cid:105) and the corre-sponding eigenenergy is λ = 0. In this paper we denote | k , n (cid:105) ≡ | k (cid:105) atom ⊗ | n (cid:105) field , where k stands for the atomiclevel and n stands for the Fock state. Moreover, we definethe bare atomic transition frequencies asΩ = E , − E , ≡ ω − ∆ Ω = E , − E , ≡ ω − ∆ , where ∆ and ∆ are the bare detunings. A. Two-level atom (2L)
We include this case ( G , = 0) to compare the ad-vantages and disadvantages of using qutrits instead ofqubits. The exact expressions for m ≥ λ m, ± D = ω m − ∆ ± D β m | ϕ m, ± D (cid:105) = 1 √ β m (cid:104)(cid:112) β m, ± | , m (cid:105) ± D (cid:112) β m, ∓ | , m − (cid:105) (cid:105) , (16)where β m = (cid:113) ∆ + 4 G , m , β m, ± = ( β m ± | ∆ | ) / detuning symbol D = +1 for ∆ ≥ D = − < m = 1; the dressedstates with m ≥ B. Double-resonant regime (RR)
When both G , and G , are nonzero, first we considerthe special case when ∆ = − ∆ , so that we have thedouble-resonance Ω = E , − E , = 2 ω . The exactformulas read (for m ≥ λ m, = mω , λ m, ± D = mω ± D(cid:37) m, ∓ (17) | ϕ m, (cid:105) = N − m, (cid:2) − G , √ m − | , m (cid:105) + √ mG , | , m − (cid:105) (cid:3) | ϕ m, ± D (cid:105) = N − m, ∓ (cid:2) √ mG , | , m (cid:105) ± D(cid:37) m, ∓ | , m − (cid:105) + √ m − G , | , m − (cid:105) (cid:3) , where we defined (cid:37) m = (cid:113) ∆ / mG , + ( m − G , (cid:37) m, ± = (cid:37) m ± | ∆ | / , (cid:37) m, = (cid:113) mG , + ( m − G , N m, = (cid:37) m, , N m, ± = (cid:112) (cid:37) m (cid:37) m, ± . For example, if G , ∼ G , and | ∆ | (cid:29) G , √ n for allrelevant values of n we have approximately | ϕ m, − D (cid:105) ∼| , m − (cid:105) , | ϕ m,D (cid:105) ∼ ( | , m (cid:105) + | , m − (cid:105) ) / √
2, while for | ∆ | (cid:28) G , , G , (near the atom–field resonance) we get | ϕ m, ± D (cid:105) ∼ ( | , m (cid:105) ± √ | , m − (cid:105) + | , m − (cid:105) ) / C. Dispersive regime (DR)
Now we assume that both the atomic transition fre-quencies are far-detuned from the cavity frequency | ∆ | , | ∆ | , | ∆ + ∆ | (cid:29) G , √ m, G , √ m − . (18)From the perturbation theory we obtain to the 4thorder in G , / ∆ and G , / ∆ λ m, = mω + δ m (cid:34) G , ( m − (∆ + ∆ ) − G , m ∆ (cid:35) | ϕ m, (cid:105) = N − m, (cid:20) | , m (cid:105) + ρ m, G , √ m ∆ | , m − (cid:105) + r m, G , G , (cid:112) m ( m − (∆ + ∆ ) | , m − (cid:105) (cid:35) λ m, = mω − ∆ − [ δ m − δ ( m − × (cid:34) − G , m ∆ − G , ( m − (cid:35) | ϕ m, (cid:105) = N − m, (cid:20) | , m − (cid:105) − ρ m, G , √ m ∆ | , n (cid:105) + r m, G , √ m − | , m − (cid:105) (cid:21) λ m, = mω − ∆ − ∆ − δ ( m − × (cid:34) G , m ∆ (∆ + ∆ ) − G , ( m − (cid:35) | ϕ m, (cid:105) = N − m, (cid:20) | , m − (cid:105) − ρ m, G , √ m − | , m − (cid:105) + r m, G , G , (cid:112) m ( m − (∆ + ∆ ) | , m (cid:105) (cid:35) , where we defined the dispersive shifts δ ≡ G , / ∆ and δ ≡ G , / ∆ . We adopted an intuitive notation in whichthe second index in | ϕ m, S (cid:105) represents the most probableatomic state in a given dressed state (for example, in theexpansion of | ϕ m, (cid:105) the bare state | , m (cid:105) appears with thehighest weight). The parameters ρ m, S , r m, S and N m, S are equal to 1 to the first order in G , / ∆ , G , / ∆ andare summarized in [62]. D. Mixed regime (MR)
In the mixed regime we assume ∆ = 0 and | ∆ | (cid:29) G , √ n, G , √ n − , (19)i. e., the atomic transition | (cid:105) → | (cid:105) is resonant withthe cavity mode, while the transition | (cid:105) → | (cid:105) is far-detuned. To the second order in G , / ∆ we obtain λ m, = mω + ∆ G , m ∆ − G , ( m − | ϕ m, (cid:105) = N − m, (cid:8) G , √ m − ρ m, | , m − (cid:105) + ρ m, ∆ | , m − (cid:105) + | , m (cid:105)} λ m, ± D = mω − D (cid:0) | ∆ | ∓ G , √ m −
1+ 12 G , m | ∆ | ∓ G , √ m − (cid:33) | ϕ m, ± D (cid:105) = N − m, ± { (1 − r m, ± ) | , m − (cid:105)± D (1 + r m, ± ) | , m − (cid:105) + ρ m, ± | , m (cid:105)} , where we defined ρ m, ± = G , √ mG , √ m − ∓ | ∆ | , ρ m, = G , √ m ∆ − G , ( m − r m, ± = 14 G , mG , √ m − G , √ m − ∓ | ∆ | ) N m, = (cid:113) ρ n, (cid:2) ∆ + ( m − G , (cid:3) N m, ± = (cid:113) r m, ± + ρ m, ± . IV. ADCE
Our goal is to study the coherent annihilation of sys-tem excitations from the initial separable state ˆ ρ = | (cid:105)(cid:104) | ⊗ ˆ ρ th , where ˆ ρ th = (cid:80) ∞ m =0 ρ m | m (cid:105)(cid:104) m | is the cav-ity thermal state with ρ m = ¯ n m / (¯ n + 1) m +1 . Here¯ n = (cid:0) e ωβ − (cid:1) − is the average initial photon number, β − = k B T , T is the absolute temperature and k B isthe Boltzmann’s constant. From Eq. (10) it is clear thatsuch process can be implemented via transition of theform | ϕ m, T (cid:105) → | ϕ m − , S (cid:105) when the modulation frequencyis η ( res ) = ˜ λ m, T − ˜ λ m − , S . So first we must determinethe dressed states for which the initial population of thestate | ϕ m, T (cid:105) , denoted as P m, T , is larger than P m − , S . We assume a small integer m (for the sake of illustrationwe choose m = 4, although the overall behavior is simi-lar for other values of m ) and set the realistic parameters G , = 6 × − ω and ¯ n = 1 .
5. We verified numericallythat when G , is of the same order of G , the exactvalue of G , does not affect qualitatively the results, soin this paper we set G , = 1 . G , . See [62] for an illus-tration of the quantitative differences in the results when G , = G , or G , = 0 . G , . ( 0 , 1 ) d ( D , - D ) ( 1 , 2 ) d ( D , - D ) m ( D , - D ) r ( 0 , - D ) r ( 0 , D ) m (0,- D ) m ( 0 , 2 ) d P(4, T ,S) | D | / G Figure 1: (color online) Difference of initial populations P ( m, T , S ) ≡ P m, T − P m − , S for m = 4 and differentregimes as function of the absolute value of the detuning ∆ .Regimes: 2-level atom (2L), double-resonant regime (r), dis-persive regime (d) and mixed regime (m). Only the states forwhich P ( m, T , S ) > T , S ) areindicated alongside the curves. (Here G , = 1 . G , .) In Fig. 1 we plot the initial population difference P ( m, T , S ) ≡ P m, T − P m − , S as function of | ∆ | for m = 4. Only positive values of P (4 , T , S ) are plottedand the values ( T , S ) are indicated next to the curves,where the index stands for 2-level (2L), double-resonant(r), dispersive (d) and mixed (m) regimes. In the dis-persive and mixed regimes we assume | ∆ | /G , ≥ = 6 G , sign (∆ )in the dispersive regime so that | ∆ + ∆ | never ap-proaches zero, as required by the inequality (18). Onecan see that large detuning | ∆ | favors the implementa-tion of ADCE; the transitions (1 , d and ( D, − D ) m arenot particularly useful since the population differencesare always small and are inversely proportional to the de-tuning. As already known, for a qubit the ADCE relieson the transition ( D, − D ) L . From Fig. 1 we discoverthat for a qutrit we have the following candidates forthe realization of ADCE; ( D, − D ) r and (0 , − D ) r in thedouble-resonant regime; (0 , d and (0 , d in the disper-sive regime; (0 , D ) m and (0 , − D ) m in the mixed regime.Now we are in position to evaluate the ADCE rate indifferent regimes according to Eq. (10). For the tran- - 1 0 - 8 - 6 - 4 - 2 2 4 6 8 1 0 - 5 - 4 - 3 ( 0 , 1 ) d ( 0 , - D ) m ( 0 , D ) m ( 1 , 2 ) d ( 0 , 2 ) d ( 0 , 2 ) d ( 1 , 2 ) d ( 0 , D ) m ( 0 , - D ) m ( D , - D ) m ( D , - D ) m ( D , - D ) ( 0 , - D ) r ( D , - D ) r | Q T , S | / w D / G - 1 0 - 8 - 6 - 4 - 2 2 4 6 8 1 0 - 5 - 4 - 3 ( b ) (0,1) d (0 ,1 ) d ( 1 , 2 ) d ( 1 , 2 ) d ( 0 , 2 ) d ( 0 , 2 ) d (0, D )m ,(0,- D )m (0, D ) m ,(0,- D ) m ( D , - D ) m ( D , - D ) m ( D , - D ) ( 0 , - D ) r ( D , - D ) r | Q T , S | / w D / G ( a ) Figure 2: (color online) a) Transition rate for ADCE in differ-ent regimes as function of ∆ /G , for m = 4 and modulationof E . b) Same as (a) but for the simultaneous modulation of E and E with the same frequency. In the dispersive regime(d) we set ∆ = 6 G , sign(∆ ). In the mixed regime (m)the lines (0 , D ) and (0 , − D ) are very close, so for the sakeof compactness they are not discerned separately. We do notshow the transition rate near | ∆ | = 0, since all the popula-tion differences P ( m, T , S ) are negative in this case. Noticethe increment by at least one order of magnitude of the tran-sition rate in the double-resonant regime (r) [compared to thequbit’s case (2L)] for | ∆ | (cid:29) G , . sition | ϕ m, T (cid:105) → | ϕ m − , S (cid:105) [denoted as ( T , S )] we evalu-ate analytically Θ m, T , S under the resonant modulationfrequency η (res) = ˜ λ m, T − ˜ λ m − , S . In Fig. 2a we plotthe dimensionless transition rate | Θ m, T , S | /ω for m = 4assuming the harmonic modulation of E with perturba-tive amplitude ε E, = 5 × − Ω . We disregard theregion near ∆ = 0 since P ( m, T , S ) < D, − D ) m is substantially higher than for thequbit, however this transition is not useful for ADCE dueto small population difference P ( m, D, − D ). We alsonote that in the dispersive regime one can induce thetransition | ϕ m, (cid:105) → | ϕ m − , (cid:105) for modulation frequency η (res) ≈ ω − Ω , that corresponds approximately tothe four-photon transition | , m (cid:105) → | , m − (cid:105) . Howeverthe associated transition rate is even smaller than theADCE rate for a qubit, hindering practical applicationsof such process.In the dispersive regime the transition rate and thepopulation difference for the process | ϕ m, (cid:105) → | ϕ m − , (cid:105) [denoted as (0 , d in the figures] is roughly the same asthe process | ϕ m,D (cid:105) → | ϕ m − , − D (cid:105) for a qubit [denotedas ( D, − D ) L ]. Therefore, the behavior of multi-levelatoms with respect to ADCE is similar to the one for aqubit, provided all the transitions are far detuned from the cavity frequency. Moreover, for the mixed regime andlarge detuning | ∆ | the population differences P ( m, , D )and P ( m, , − D ) are roughly the same as for the qubit,while the transition rates are several times larger, so theimplementation of ADCE would be facilitated.The main finding of the paper is the observation thatin the double-resonant regime the ADCE rate is at leastone order of magnitude larger than for the qubit, andthe difference increases for larger | ∆ | , as can be seenfrom Fig. 2a. Besides, in this regime the populationdifferences P ( m, D, − D ) and P ( m, , − D ) also increaseproportionally to | ∆ | , achieving sufficiently large valuesfor | ∆ | ∼ G , (see Fig. 1). Thus, it seems that onecould speed up ADCE by at least one order of magnitudeusing three-level atoms in the double-resonant configura-tion instead of qubits, provided the detuning | ∆ | is largeenough.In real circuit QED setups it might be tricky to modu-late only one parameter at a time, while keeping the otherparameters constant. So in figure 2b we consider the si-multaneous modulation of E and E (with the samemodulation frequency η (res) = ˜ λ m, T − ˜ λ m − , S ) assumingparameters ε E, = 5 × − Ω , ε E, = 5 × − Ω , φ E, = 0 and φ E, = π . Conveniently the ADCE transi-tion rates increase even more when compared to an iso-lated modulation of either E or E .In [62] we illustrate in details the transition rates andthe population differences for different values of G , andisolated modulations of E , G and G . It is found thatthe modulation of G does not speed up significantly thetransition rate in comparison to a qubit, whereas themodulation of E or G does increase the transition ratein the double-resonant regime by at least one order ofmagnitude. We also verified that under the simultaneousmodulation of all the parameters ( E , E , G and G ) thetotal transition rate is still substantially higher than for aqubit, provided the phases are properly adjusted. Hence,the simultaneous modulation of several parameters is notan issue from the experimental point of view, providedone can manage the phases φ ( j ) l corresponding to differentmodulation components. V. NUMERICAL VERIFICATION
Now we proceed to the numerical verification of thephenomenon predicted in the previous section, namely,the enhancement of the ADCE rate in the double-resonant regime. We solved numerically the Schr¨odingerequation for the Hamiltonian (1) using the initial lo-cal thermal state ˆ ρ = | (cid:105)(cid:104) | ⊗ ˆ ρ th and parameters m = 4, G , = 6 × − ω , G , = 1 . G , , ¯ n = 1 . = − ∆ = − G , . One downside of using thedouble-resonant regime for qutrits is clear from Fig. 1:both the populations differences (0 , − D ) r and ( D, − D ) r ,involved in the ADCE, are roughly twice smaller than thepopulation difference ( D, − D ) L for the qubit. Hence,considering the connection between ADCE and quantum n p h G t ( x 1 0 ) n a t G t ( x 1 0 ) n t o t G t ( x 1 0 ) n p h G t ( x 1 0 ) n a t G t ( x 1 0 ) n t o t G t ( x 1 0 ) n p h G t ( x 1 0 ) n a t G t ( x 1 0 ) n t o t G t ( x 1 0 ) ( a )( b )( c ) Figure 3: (color online) Exact numerical dynamics of ADCEobtained for the Hamiltonian (1) and the initial local thermalstate ˆ ρ in the double-resonant regime. a) 2-level atom andharmonic modulation of E . b) 3-level atom and 2-tone mod-ulation of E . c) 3-level atom and 2-tone double-modulationof E and E . Notice that in all cases the amount of annihi-lated excitations n tot is roughly the same, while the durationof the process in (c) is roughly 40 times smaller than in (a). thermodynamic processes recently analyzed in Ref.[55],we can say that the work extraction would be half smallerif one used qutrits instead of qubits. This nuisancecan be readily surpassed by employing 2-tone modula-tion with frequencies η (1) = ˜ λ m, − ˜ λ m − , − D and η (2) =˜ λ m,D − ˜ λ m − , − D that drives simultaneously the transi-tions | ϕ m, (cid:105) → | ϕ m − , − D (cid:105) and | ϕ m,D (cid:105) → | ϕ m − , − D (cid:105) .In figure 3a we illustrate the dynamics of the aver-age photon number n ph = (cid:104) ˆ n (cid:105) , the average number ofatomic excitations n at = (cid:104) (cid:80) k =1 k ˆ σ k,k (cid:105) and the total av-erage number of excitations n tot = n ph + n at for a qubit(setting momentarily G = 0) with modulation depth ε E, = 5 × − Ω . We observe the sinusoidal oscillationof n ph , n at and n tot with typical period τ ≈ × G − , .The coherent annihilation of excitations does take place,but since the initial population of the state | ϕ ,D (cid:105) was P ,D ≈ × − , the average number of annihilated exci-tations is ∼ P m,D ≈ .
1, in agreement with the numer-ical data.In figure 3b we consider the qutrit under 2-tone mod-
P ( 4 , D ) G t ( x 1 0 ) P ( 3 , D ) G t ( x 1 0 ) P ( 3 , 0 ) P ( 4 , 0 ) G t ( x 1 0 ) P ( 2 , - D ) P ( 3 , - D ) Figure 4: (color online) Dynamics of populations of relevantdressed states for the 2-tone double-modulation of E and E analyzed in Fig. 3c. There is a coherent transfer ofpopulations from the states | ϕ ,D (cid:105) and | ϕ , (cid:105) to the state | ϕ , − D (cid:105) . Moreover, one observes periodic oscillations betweenthe dressed states | ϕ k,D (cid:105) ↔ | ϕ k, (cid:105) due to the counter-rotatingterms in the Hamiltonian (1). ulation of E with the previous amplitude ε E, = 5 × − Ω , weights w (1) E, = 10 / , w (2) E, = 7 /
17 and phases φ (1) E, = 0, φ (2) E, = π (the weights were adjusted to equalizethe two transition rates). We see that the total numberof excitation exhibits the same qualitative behavior asfor the qubit, but the transition rate undergoes a 30-fold enhancement. The behavior of n ph and n at differsdrastically from the one observed for the 2-level atompartly due to the oscillations between the bare states | , k (cid:105) ↔ | , k − (cid:105) for k ≥
2, and partly due to the os-cillations between the dressed states | ϕ k,D (cid:105) ↔ | ϕ k, (cid:105) , aswill be discussed shortly. In figure 3c we consider thesimultaneous two-tone modulation of E and E withparameters ε E, = 5 × − Ω , ε E, = 9 × − Ω , w (1) E, = w (1) E, = 10 / w (2) E, = w (2) E, = 7 /
17 and phases φ (1) E, = φ (2) E, = 0, φ (2) E, = φ (1) E, = π . We see that theADCE rate suffers an additional 50% enhancement com-pared to the sole modulation of E , while the averagenumber of total annihilated excitations is roughly thesame as in the previous cases.Finally, in Fig. 4 we plot the probabilities offinding the system in the dressed states P ( m, S ) =Tr[ˆ ρ ( t ) | ϕ m, S (cid:105)(cid:104) ϕ m, S | ] as function of time for the 2-tonedouble-modulation discussed in Fig. 3c. As predictedby Eq. (9) there is a simultaneous periodic transferof populations from the states | ϕ ,D (cid:105) and | ϕ , (cid:105) to thestate | ϕ , − D (cid:105) , which correspond to the coherent annihi-lation of two system excitations. Other states | ϕ k (cid:54) =2 , − D (cid:105) are not affected by the modulation, as illustrated forthe state | ϕ , − D (cid:105) which undergoes just minor fluctua-tions due to off-resonant couplings neglected under RWA.Moreover, one also observes periodic oscillations betweenthe dressed states | ϕ k,D (cid:105) ↔ | ϕ k, (cid:105) for k ≥
2. This occursbecause for large | ∆ | we have ˜ λ k, ≈ ˜ λ k,D , as seen fromEq. (17), hence the first term on the RHS of Eq. (9)becomes nearly resonant and couples these states withthe strength ∼ | ς k,D, | [this behavior is due solely to thecounter-rotating terms in Eq. (1) and is independent ofmodulation]. VI. CONCLUSIONS
In conclusion, we showed that the resonant externalmodulation of a three-level artificial atom is highly ad-vantageous for the implementation of the antidynami-cal Casimir effect (ADCE) in comparison to a two-levelatom, since the transition rate can suffer almost 50-fold increase while the total amount of annihilated ex-citations is roughly the same. The strongest enhance-ment takes place in the double-resonant regime (when ∆ = − ∆ , so that Ω = 2 ω ) and for large detun-ing | ∆ | , though weaker enhancement may occur also inother regimes. Beside speeding up the ADCE, the use ofqutrits also loosens the requirements for accurate tuningof the modulation frequency, and reproduces the charac-teristic ADCE behavior of a qubit when all the atomictransitions are largely detuned from the cavity field (andΩ (cid:54) = 2 ω ). However, for the optimum annihilationof excitations from a thermal state the usage of qutritsalso brings some inconveniences, such as two-tone driv-ing and the necessity of controlling the phase differencebetween different components of the modulation. Never-theless, our results indicate that the substantial gain inthe transition rate compensates for the additional com-plexity in the external control, favoring the experimentalimplementation of ADCE. Acknowledgments
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