Dissipation and thermal noise in hybrid quantum systems in the ultrastrong-coupling regime

Alessio Settineri, Vincenzo Macrí, Alessandro Ridolfo, Omar Di Stefano, Anton Frisk Kockum, Franco Nori, and Salvatore Savasta

Phys. Rev. A 98, 053834 – Published 20 November 2018

Abstract

The interaction among the components of a hybrid quantum system is often neglected when considering the coupling of these components to an environment. However, if the interaction strength is large, this approximation leads to unphysical predictions, as has been shown for cavity-QED and optomechanical systems in the ultrastrong-coupling regime. To deal with these cases, master equations with dissipators retaining the interaction between these components have been derived for the quantum Rabi model and for the standard optomechanical Hamiltonian. In this article, we go beyond these previous derivations and present a general master equation approach for arbitrary hybrid quantum systems interacting with thermal reservoirs. Specifically, our approach can be applied to describe the dynamics of open hybrid systems with harmonic, quasiharmonic, and anharmonic transitions. We apply our approach to study the influence of temperature on multiphoton vacuum Rabi oscillations in circuit QED. We also analyze the influence of temperature on the conversion of mechanical energy into photon pairs in an optomechanical system, which has been recently described at zero temperature. We compare our results with previous approaches, finding that these sometimes overestimate decoherence rates and underestimate excited-state populations.

Received 18 July 2018

DOI:https://doi.org/10.1103/PhysRevA.98.053834

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Open quantum systems Optomechanics Quantum optics

Quantum master equation

Atomic, Molecular & Optical

Authors & Affiliations

Alessio Settineri¹, Vincenzo Macrí², Alessandro Ridolfo², Omar Di Stefano², Anton Frisk Kockum^2,3, Franco Nori^2,4, and Salvatore Savasta^1,2

¹Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, I-98166 Messina, Italy
²Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
³Wallenberg Centre for Quantum Technology, Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96 Gothenburg, Sweden
⁴Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

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Vol. 98, Iss. 5 — November 2018

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Figure 1
(a) Master-equation approach valid in the weak- and strong-coupling regimes. The light-matter coupling is neglected while deriving the dissipators. (b) The master-equation approach considering the light-matter coupling. As the coupling strength between the two subsystems increases, it becomes necessary to treat dissipation effects including the coupling between the subsystems. This can be done by developing the system operators describing the coupling to the reservoirs in the eigenbasis of the coupled light-matter system.
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Figure 2
Dynamics of anomalous two-photon vacuum Rabi oscillations. Results obtained using the generalized-master-equation approach, varying the temperature of both subsystems. (a) Time evolution of the mean cavity photon number $〈 {\hat{A}}^{(-)} {\hat{A}}^{(+)} 〉$ after the arrival of a Gaussian $π$ pulse to the qubit. The system starts in the ground state. (b) Two-photon correlation function for the cavity, obtained with the same parameters and conditions. After the arrival of the pulse, independent of the temperature of the reservoirs, the system undergoes vacuum Rabi oscillations showing the reversible exchange of photon pairs between the qubit and the resonator. However, when raising the temperature, due to the increasing decoherence, the oscillations become more damped and the correlation function reaches higher stationary values due to larger incoherent, thermal contributions. Note that the second and the fifth dips are shallower because of some spurious effects generated by other transitions excited by the coherent pulse. All parameters for the simulations are given in the text. $Ω_{eff}$ on the $x$ axis indicates the effective resonant coupling.
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Figure 3
Comparison between the results obtained using the generalized-master-equation approach (red solid curves) and the standard dressed master equation (black dashed curves). (a) Time evolution of the mean cavity photon number $〈 {\hat{A}}^{(-)} {\hat{A}}^{(+)} 〉$ at temperature $T / ω_{c} = 0.75$ with all other parameters the same as in Fig. 2. (b) Two-photon correlation functions, obtained with the same parameters and conditions. After the arrival of the pulse, both approaches show the system undergoing two-photon Rabi oscillations and relaxing to thermal equilibrium. However, with the standard dressed master equation, the coherence losses are slightly overestimated (because of the post-trace RWA), so the oscillations are more damped and the stationary value is reached sooner.
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Figure 4
Results for the DCE at different temperatures, obtained using the generalized-master-equation approach. (a), (b) System dynamics for $ω_{c} ≃ ω_{m}$ , under coherent mechanical pumping, in perfect cooling conditions $T_{γ} = T_{κ} = 0$ , starting the dynamics from the ground state. (c), (d) The same, but with $T_{γ} / ω_{m} = T_{κ} / ω_{m} = 0.5$ and the initial state being the thermal state with $T / ω_{m} = 0.5$ . The blue dashed curves show the mean phonon number $〈 {\hat{B}}^{(-)} {\hat{B}}^{(+)} 〉$ in (a), (c) and the phonon-phonon correlation function $g_{B}^{(2)} (t, t)$ in (b), (d). The red solid curves describe the mean cavity photon number $〈 {\hat{A}}^{(-)} {\hat{A}}^{(+)} 〉$ in (a), (c) and the zero-delay normalized photon-photon correlation function $g_{A}^{(2)} (t, t)$ in (b), (d). All parameters for the simulations are given in the text.
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Figure 5
State populations obtained using the master-equation approach of Ref. [75]. The red (blue) solid curve shows the time evolution of the population of the one-photon state $| 1, 0 〉$ (the one-phonon state $| 0, 1 〉$ ) labeled $ρ_{22} (t) [ρ_{11} (t)]$ under perfect cooling conditions $T_{γ} = T_{κ} = 0$ and without any pumping. The initial state is the ground state $| 0, 0 〉$ . All other parameters are the same as in Fig. 4. In these conditions, without any external driving or thermal excitations, the system is expected to remain in the ground state. However, the plot clearly shows a nonzero population in both the one-photon and one-phonon states. This indicates that this approach is not able to correctly describe optomechanical systems when the ${\hat{V}}_{DCE}$ contribution no longer can be neglected.
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