Antidynamical Casimir effect as a resource for work extraction

A. V. Dodonov, D. Valente, and T. Werlang

Phys. Rev. A 96, 012501 – Published 5 July 2017

Abstract

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 as a consequence of the dynamical Casimir effect. Excitations can also be systematically subtracted from the atom-field system by suitably adjusting the modulation frequency, in the so-called antidynamical Casimir effect (ADCE). We evaluate the quantum thermodynamical work and show that a realistic out-of-equilibrium finite-time protocol harnessing ADCE allows for work extraction from the system, whose amount can be much bigger than the modulation amplitude, $| W_{ADCE} | ≫ ℏ ε_{Ω}$ , in contrast to the case of very slow adiabatic modulations. We provide means to control work extraction in state-of-the-art experimental scenarios, where precise frequency adjustments or complete system isolation may be difficult to attain.

Received 16 April 2017

DOI:https://doi.org/10.1103/PhysRevA.96.012501

Physics Subject Headings (PhySH)

Casimir effect & related phenomena Cavity quantum electrodynamics Quantum optics with artificial atoms Superconducting qubits

Quantum cavities

Jaynes-Cummings model Rabi model Rotating wave approximation

Atomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

A. V. Dodonov^1,*, D. Valente^2,3, and T. Werlang²

¹Institute of Physics and International Centre for Condensed Matter Physics, University of Brasilia, 70910-900, Brasilia, Federal District, Brazil
²Instituto de Física, Universidade Federal de Mato Grosso, Cuiabá MT 78060-900, Brazil
³Laboratoire Pierre Aigrain (LPA), École Normale Supérieure (ENS) – Paris, Centre National de la Recherche Scientifique (CNRS), 24 rue Lhomond, 75005 Paris, France

^*adodonov@fis.unb.br

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Vol. 96, Iss. 1 — July 2017

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Images

Figure 1
Quantum work and average total number 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 $τ^{- 1} = g ε_{Ω} / 2 Δ_{+}$ with $Δ_{\pm} = ω \pm Ω_{0}$ . For the DCE and AJC regimes the initial state is $| g, 0 〉$ ; for the JC and ADCE regimes it is $| g, 3 〉$ . Parameters $g / ω = 5 \times 10^{- 2}$ , $Δ_{-} = 8 g$ , $ε_{Ω} = 0.05 Ω_{0}$ , and $ϕ = 0$ . The modulation frequencies are $η_{DCE} = 2.0089 ω$ , $η_{AJC} = 0.9943 Δ_{+}$ , $η_{JC} = 1.07 Δ_{-}$ , and $η_{ADCE} = 1.0076 (3 ω - Ω_{0})$ . Panels (e1) and (e2) show the dynamics of $〈 σ_{z} 〉$ over short time scales, exhibiting fast oscillations synchronized with $\dot{Ω}$ . In the DCE regime [panel (e1)] the oscillations are nonsinusoidal, but the characteristic period is still $\approx 2 π / η$ .
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Figure 2
Panel (a): quantum work as a function of dimensionless time $t / τ$ for single-tone modulations with ${η^{(1)} = 1.0076 (3 ω - Ω_{0}), ε_{Ω}^{(1)} = 0.05 Ω_{0}}$ (black curve), ${η^{(2)} = 1.0113 (3 ω - Ω_{0}), ε_{Ω}^{(2)} = 0.025 Ω_{0}}$ (red curve), and for the two-tone modulation with the above frequencies and modulation amplitudes (blue curve). All curves correspond to the ADCE regime and $ϕ^{(1, 2)} = 0$ . We used the initial state $ρ (0) = | g 〉 〈 g | \otimes ρ_{T}^{f}$ , where $ρ_{T}^{f} = \sum_{n} p_{n} | n 〉 〈 n |$ is a field thermal state with $p_{n} = {\bar{n}}^{n} / {(\bar{n} + 1)}^{n + 1}$ and $\bar{n} = 1.5$ (other parameters are as in Fig. 1). $τ_{m}$ denotes the instant of time when the extracted work is maximum. Panel (b): dynamics of quantum work around the instant of time $τ_{m}$ for a single-tone modulation with frequency $η^{(1)}$ . Panel (c): level diagram related to the modulation-induced transitions between the approximate eigenstates 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 and atom damping rates are $κ = γ = 2 \times 10^{- 5} g$ and the reservoirs' temperature is $k_{B} T_{r} / ω = 0.33$ . Panel (e): behavior of $N$ under unitary and dissipative dynamics.
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Figure 3
Panels (a),(b): quantum work and $N$ under unitary (black curves) and dissipative (green curves) evolution for aperiodic modulation with frequency $η (t) = η^{(1)} - 10 λ + λ^{2} t$ , where $η^{(1)} = 1.0076 (3 ω - Ω_{0})$ , $ϕ = 0$ , and $λ = 1.67 \times 10^{- 5} ω$ . The initial state and other parameters are as in Fig. 2. The dynamics resembles the typical Landau-Zener behavior for an effective TLA undergoing adiabatic frequency sweep. Panel (c): quantum work under dissipative dynamics for two different choices of the modulation frequency: $η (t) = η^{(1)} - 10 λ + λ^{2} t$ (red curve) and $η (t) = η^{(1)} - 6 λ + λ^{2} t$ (blue curve). One can see that for the blue curve the work extraction is larger and faster; hence there is room for further optimization of our protocol.
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