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Physical Review A

Dispersive readout with non-Markovian environments

H. Z. Shen, Q. Wang, X. X. Yi

Published Feb 10, 2022• 20 citations
Publisher

Abstract

We study the non-Markovian dispersive readout of a single-mode bosonic (SMB) quantum system coupled to the open cavity with two non-Markovian environments. Assuming that the SMB quantum system is initially prepared in a thermal equilibrium state, the susceptibility of the measured SMB quantum system to the cavity can be obtained by the nonequilibrium linear response theory, which provides the dispersive frequency shift of the cavity. We analytically derive the transmission and reflection of the cavity in the non-Markovian regime and discuss the non-Markovian dispersive readout of the SMB quantum system, which is in good agreement with that obtained by the Markovian approximation. We show that the effect of non-Markovian dynamics on the system's behavior leads to the decrease of the transmission and shifts of the frequency corresponding to the peak of the transmission in the dispersive regime compared with the Markovian case and the increase of the sum of the transmission and reflection due to the excitation backflowing induced by the couplings of the cavity and two non-Markovian environments, which could be applied to quantum secure direct communication with quantum memory. Finally, we generalize the readout theory to the periodically driven SMB quantum system based on the Floquet theory and quantum network, which contain all influences from the non-Markovian environments. The formalism presented in this paper opens an alternative field of possible applications in quantum information and quantum communication with non-Markovianity by manipulating the spectral densities of the environments.

DOI
10.1103/physreva.105.023707

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We study the non-Markovian dispersive readout of a single-mode bosonic (SMB) quantum system coupled to the open cavity with two non-Markovian environments. Assuming that the SMB quantum system is initially prepared in a thermal equilibrium state, the susceptibility of the measured SMB quantum system to the cavity can be obtained by the nonequilibrium linear response theory, which provides the dispersive frequency shift of the cavity. We analytically derive the transmission and reflection of the cavity in the non-Markovian regime and discuss the non-Markovian dispersive readout of the SMB quantum system, which is in good agreement with that obtained by the Markovian approximation. We show that the effect of non-Markovian dynamics on the system's behavior leads to the decrease of the transmission and shifts of the frequency corresponding to the peak of the transmission in the dispersive regime compared with the Markovian case and the increase of the sum of the transmission and reflection due to the excitation backflowing induced by the couplings of the cavity and two non-Markovian environments, which could be applied to quantum secure direct communication with quantum memory. Finally, we generalize the readout theory to the periodically driven SMB quantum system based on the Floquet theory and quantum network, which contain all influences from the non-Markovian environments. The formalism presented in this paper opens an alternative field of possible applications in quantum information and quantum communication with non-Markovianity by manipulating the spectral densities of the environments.
In this paper we study the dispersive readout of a driven three-level atom weakly interacting with a passive cavity that couples to an active cavity. The system excluding the three-level atom exhibits a parity-time (PT\mathcal{P}\mathcal{T}PT) symmetric phase transition at the exceptional point (EP), which originates from the balance between dissipation of the passive cavity and gain of the active cavity. The perturbation to the passive cavity induced by the three-level atom can be amplified near the EP, where the eigenvalues of the effective Hamiltonian have significant differences when the three-level atom is in the excited, metastable, and ground states, respectively. Applying the modified Laplace transformation and input-output theory near the EP, we can realize the dispersive readout of the three-level atom weakly interacting with coupled cavities, which can be characterized through the different distributions of the transmission spectrum of the passive cavity and the ratio of the active cavity excitation number to the input photon number corresponding to the different levels of the three-level atom. Moreover, we generalize the dispersive readout method to the non-Markovian regime, which is in good agreement with that obtained by the Markovian approximation. We find that the non-Markovian effects of the environment backacting on the system dynamics can lead to the enhancement of the dispersive readout for the three-level atom, which is revealed by the excitation backflow generated in the interaction between the passive cavity and environment. The dispersive readout proposed is conducive to understanding the influences of PT\mathcal{P}\mathcal{T}PT symmetry and the non-Markovian effect on dispersive readout, which offers the possibility of an alternative field of possible applications for quantum information and quantum communication.
We present a unified picture of dispersive readout of quantum systems in and out of equilibrium. A cornerstone of the approach is the backaction of the measured system to the cavity obtained with nonequilibrium linear-response theory. It provides the dispersive shift of the cavity frequency in terms of a system susceptibility. It turns out that already effortless computations of the susceptibility allow one to generalize former results beyond a rotating-wave approximation. Examples are the readout of detuned qubits and thermally excited multilevel systems. For ac-driven quantum systems, we identify the relevant Fourier component of the susceptibility and introduce a computational scheme based on Floquet theory. The usefulness is demonstrated for two-tone spectroscopy and interference effects in driven two-level systems. This also reveals that dispersive readout does not necessarily measure excitation probabilities.
We propose a protocol for the measurement of adiabatic phases of periodically driven quantum systems coupled to an open cavity that enables dispersive readout. It turns out that the cavity transmission exhibits peaks at frequencies determined by a resonance condition that involves the dynamical and the geometric phase. Since these phases scale differently with the driving frequency, one can determine them by fitting the peak positions to the theoretically expected behavior. For the derivation of the resonance condition and for a numerical study, we develop a Floquet theory for the dispersive readout of ac driven quantum systems. The feasibility is demonstrated for two test cases that generalize Landau-Zener-Stückelberg-Majorana interference to two-parameter driving.
For a bosonic (fermionic) open system in a bath with many bosonic (fermionic) modes, we derive the exact non-Markovian master equation in which the memory effect of the bath is reflected in the time-dependent decay rates. In this approach, the reduced density operator is constructed from the formal solution of the corresponding Heisenberg equations. As an application of the exact master equation, we study the active probing of the non-Markovianity of the quantum dissipation of a single bosonic mode of an electromagnetic field in a cavity-QED system. The non-Markovianity of the bath of the cavity is explicitly reflected by the atomic decoherence factor.
In the linear response theory, it is well known that the response of a quantum system to an external perturbation described by the susceptibility is formulated in the Schrödinger picture. The theory might apply to open quantum systems (or Floquet systems); however, it has ignored the non-Markovian effect in almost all works so far. In this Letter, we propose a new method to address those issues by introducing Heisenberg operators to derive an exact susceptibility for the non-Markovian Floquet periodic driving system. The susceptibility includes all the influences of the environment on the Floquet system. We will show that the susceptibility connects closely to the structure of the Floquet energy spectrum of the whole system (system plus environment). Moreover, we can read out Floquet bound states in the first Brillouin zone of the whole system from the susceptibility. The presented results may find applications in quantum engineering with open systems following modulated periodic evolution in quantum optics.
Non-Markovian dynamics go beyond the Markovian approximation by capturing memory effects and information backflow in open quantum systems, which are crucial for describing realistic physical processes. In this work, we study the exact non-Markovian dynamics of a driven cavity coupled to an anisotropic three-dimensional photonic-crystal environment via counterrotating-wave interactions. We derive an exact analytical expression for the cavity amplitude satisfying the integro-differential equation, which includes the contributions of the bound states outside the continuum and the dissipative parts with the continuum spectrum. Based on the characteristic function method, we derive the exact non-Markovian master equation for the cavity, which contributes to the gain of the cavity. We give the physical origin of non-Markovian gain in the presence of bound states in the system consisting of cavity and environment, which has no Markovian counterparts due to the nonexponential gain in the non-Markovian structured environment. We find that three different types of bound states can be formed in the system, containing one bound state with no inversion of photon number, two bound states with the periodic equal-amplitude oscillation, and the gain with two complex roots without the bound states formation. We derive a current equation including the source from the driving field, the transient current induced by the change in the number of photons, and the two-photon current caused by the counterrotating-wave term. The results are compared with those given by the rotating-wave interactions and extended to a more general quantum network involving an arbitrary number of coupled cavities. Our findings may pave the way for a deeper understanding of non-Markovian dynamics with gain in quantum networks involving counterrotating-wave effects.