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arXiv:2007.03108 · Quantum Science and Technology

Breakdown signatures of the phenomenological Lindblad master equation in the strong optomechanical coupling regime

Ralf Betzholz, Bruno G Taketani, Juan Mauricio Torres

Published Oct 21, 2020• 9 citations
arXivHTMLPDF

Abstract

Abstract The Lindblad form of the master equation has proven to be one of the most convenient ways to describe the impact of an environment interacting with a quantum system of interest. For single systems the jump operators characterizing these interactions usually take simple forms with a clear interpretation. However, for coupled systems these operators take significantly different forms and the full dynamics cannot be described by jump operators acting on the individual subsystems only. In this work, we investigate the differences between a common phenomenological model for the master equation and the more rigorous dressed-state master equation for optomechanical systems. We provide an analytical method to obtain the absorption spectrum of the system for both models and show the breakdown of the phenomenological model in both the bad cavity and the ultra-strong coupling limit. We present a careful discussion of the indirect dephasing of the optical cavity in both models and its role in the differences of their predicted absorption spectra. Our work provides a simple experimental test to determine whether the simpler phenomenological model can be used to describe the system and is a step forward toward a better understanding of the role of the coupling between subsystems for open-quantum-system dynamics.

DOI
10.1088/2058-9565/abc39d

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Abstract The Lindblad form of the master equation has proven to be one of the most convenient ways to describe the impact of an environment interacting with a quantum system of interest. For single systems the jump operators characterizing these interactions usually take simple forms with a clear interpretation. However, for coupled systems these operators take significantly different forms and the full dynamics cannot be described by jump operators acting on the individual subsystems only. In this work, we investigate the differences between a common phenomenological model for the master equation and the more rigorous dressed-state master equation for optomechanical systems. We provide an analytical method to obtain the absorption spectrum of the system for both models and show the breakdown of the phenomenological model in both the bad cavity and the ultra-strong coupling limit. We present a careful discussion of the indirect dephasing of the optical cavity in both models and its role in the differences of their predicted absorption spectra. Our work provides a simple experimental test to determine whether the simpler phenomenological model can be used to describe the system and is a step forward toward a better understanding of the role of the coupling between subsystems for open-quantum-system dynamics.
Open-system dynamics play a key role in the experimental and theoretical study of cavity optomechanical systems. In many cases, the quantum Langevin equations have enabled excellent models for optical decoherence, yet a master-equation approach to the fully nonlinear optomechanical Hamiltonian has thus far proven more elusive. To address this outstanding question and broaden the mathematical tool set available, we derive a solution to the Lindblad master equation that models optical decoherence for a system evolving with the nonlinear optomechanical Hamiltonian. The method combines a Lie-algebra solution to the unitary dynamics with a vectorization of the Lindblad equation, and we demonstrate its applicability by considering the preparation of optical cat states via the optomechanical nonlinearity in the presence of optical loss. Our results provide a direct way of analytically assessing the impact of optical decoherence on the optomechanical intracavity state.
Abstract Recent advancements in strong single-photon optomechanical coupling also demand a deeper understanding of environmental interactions in this regime. In this regime the standard Lindblad master equation, which is derived in the eigenbasis of the bare optical and mechanical modes, misassigns the dephasing rate. We therefore use the dressed-state master equation, which is formulated in the eigenbasis of the strongly coupled photon–phonon Hamiltonian, the dressed states of the system. This work investigates the impact of squeezed vacuum and thermal reservoirs on the decoherence of cavity photon Fock states in the strong coupling regime. We demonstrate that decoherence can be effectively controlled by tuning reservoir parameters, with the control mediated through a cavity dephasing term that becomes significant at high temperatures. The findings presented provide critical insights into reservoir engineering for precise control of quantum decoherence, advancing the understanding of strongly coupled optomechanical systems in engineered environments.
Abstract In this chapter we study with the tools developed in Chapter 3 the basic models that are the foundations of light–matter interaction. We start with Rabi dynamics, then consider the optical Bloch equations that add phenomenologically the lifetime of the populations. As decay and pumping are often important, we cover the Lindblad form, a correct, simple and powerful way to describe various dissipation mechanisms. Then we go to a full quantum picture, quantizing also the optical field. We first investigate the simpler coupling of bosons and then culminate with the Jaynes–Cummings model and its solution to the quantum interaction of a two-level system with a cavity mode. Finally, we investigate a broader family of models where the material excitation operators differ from the ideal limits of a Bose and a Fermi field.
Ultrastrong light-matter interaction in an optomechanical system can result in nonlinear optical effects such as photon blockade. The system-bath couplings in such systems play an essential role in observing these effects. Here we study the quantum coherence of an optomechanical system with a dressed-state master equation approach. Our master equation includes photon-number-dependent terms that induce dephasing in this system. Cavity dephasing, second-order photon correlation, and two-cavity entanglement are studied with the dressed-state master equation.
With the recent advancements made in the coupling of quantum systems, reaching the (ultra)strong coupling regime, where the coupling constant exceeds the dissipation rate and approaches the frequencies of the coupled systems, is getting closer. In this regime, the much used semiclassical model is no longer valid, and the solving of the full quantum master equation can be computationally too demanding. Thus, we must develop a new approach to be able to simulate the system. One solution is to use the semiquantum approximation. In this thesis, we present a systematic way of employing the semiquantum approximation and compare it to the full master equation. Our aim is to determine the range for the coupling strength where the semiquantum model is applicable. We chose to do the comparison of the semiquantum approximation and the master equation within the Rabi model, which can be used to describe e.g. the interaction of light and matter. The Rabi model describes a system with a coupled two-level system and a harmonic oscillator. It was chosen because its behaviour is reasonably well known. We compared the semiquantum approximation to the quantum master equation in two cases. First we studied the steady state results from the two models, and then moved on to the spectral properties. The results show that the steady state and the spectrum obtained from the semiquantum model agree with the ones obtained by using the master equation, until the coupling reaches a considerable fraction of the resonant frequency of the oscillator, while still exceeding the experimentally reasonable dissipation rate by an order of magnitude. In the future, one could use the semiquantum approximation in the eld of cavity optomechanics. There a mechanical oscillator is coupled to optical radiation con ned in a cavity, e.g. a Fabry-Perot cavity. Especially in optomechanics, the solving of the master equation can turn out to be a formidable task, and by using the semiquantum approximation one could reduce the computation time considerably.
We derive a quantum master equation in Lindblad form for the open quantum Rabi model in the perturbative deep strong coupling regime, where the spectrum is close to degenerate and harmonic. The Lindblad form is obtained by combining secular and singular-coupling approximations, working in a Schrieffer-Wolff transformed frame. We benchmark our equation against a Bloch-Redfield one, analyzing the time-evolution and steady state properties of several observables and find excellent agreement. Analytical results are obtained in limiting cases.