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Advanced Quantum Technologies

Effects of Counter‐Rotating Couplings on the Dynamics of a Λ System: Long‐Lived Coherence Oscillations

Jun-Dong Xiang, Zhiguo Lü, Yiying Yan

Published Dec 24, 2023• 0 citations
Publisher

Abstract

Abstract The time evolution of the ground‐state population and coherence of a Λ‐type three‐level emitter strongly coupled to a radiation reservoir is studied by using a transformed rotating‐wave approximation (TRWA) and time‐dependent variational approach with the multiple Davydov ansatz. The study aims at illustrating the role of counter‐rotating couplings in the long‐time evolution of the Λ system. The TRWA and variational approach are found to be consistent with each other and predict that there exist long‐lived oscillatory behaviors of ground‐state populations and coherence, which are robust in the presence of strong dissipation and arise from the counter‐rotating‐coupling‐induced virtual photon processes. The oscillation amplitude and frequency depend on the system‐reservoir coupling strength and the frequency difference between the two ground states, which can be qualitatively and quantitatively captured by the TRWA analytical results. Present results offer insights into the impact of the counter‐rotating couplings on the long‐time evolution of the strongly dissipative multilevel system as well as the protocol of generation of coherent oscillations by relaxation in a strong‐coupling regime.

DOI
10.1002/qute.202300263

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Abstract The time evolution of the ground‐state population and coherence of a Λ‐type three‐level emitter strongly coupled to a radiation reservoir is studied by using a transformed rotating‐wave approximation (TRWA) and time‐dependent variational approach with the multiple Davydov ansatz. The study aims at illustrating the role of counter‐rotating couplings in the long‐time evolution of the Λ system. The TRWA and variational approach are found to be consistent with each other and predict that there exist long‐lived oscillatory behaviors of ground‐state populations and coherence, which are robust in the presence of strong dissipation and arise from the counter‐rotating‐coupling‐induced virtual photon processes. The oscillation amplitude and frequency depend on the system‐reservoir coupling strength and the frequency difference between the two ground states, which can be qualitatively and quantitatively captured by the TRWA analytical results. Present results offer insights into the impact of the counter‐rotating couplings on the long‐time evolution of the strongly dissipative multilevel system as well as the protocol of generation of coherent oscillations by relaxation in a strong‐coupling regime.
Abstract We investigate the dynamics of the Λ system driven by two resonant laser fields in presence of dissipation for coupling strengths where the rotating‐wave approximation starts to break down. This regime is characterised by Rabi frequencies being approximately equal or smaller than the field frequencies. A systematic procedure to obtain an expansion for the solution of the Bloch evolution equations of the system is presented. The lowest contribution results to be the well‐known rotating‐wave approximation. The method is based on a semi‐classical treatment of the problem, and its predictions are interpreted fully quantum mechanically. The theory is illustrated by a detailed study of the disappearance of coherent population trapping as the intensities of the fields increase.
cavity mode and a radiation reservoir. Three methods are employed: a numerically exact variational approach with a multiple Davydov ansatz, the rotating-wave approximation (RWA), and the transformed RWA. Agreement between the variational results and the transformed RWA results is found in the regime of validity of the latter, where the RWA breaks down completely. We illustrate that the Lamb shift plays an essential role in modifying the vacuum Rabi splitting in the ultrastrong coupling regime, leading to off-resonant qubit-cavity coupling even though the cavity frequency equals the bare transition frequency of the qubit. Specifically, the emission spectrum exhibits one broad low-frequency peak and one narrow high-frequency peak in the presence of relatively weak cavity-qubit coupling. As the cavity-qubit coupling increases, the low-frequency peak narrows while the high-frequency peak broadens until they have similar widths.
Abstract We study the resonance and dynamics of a three-level V-type system strongly driven by a quasiperiodic bichromatic field by making use of a unitary-transformation (UT) method. The UT is used to construct a periodic effective Hamiltonian which incorporates the effects of counter-rotating couplings and takes a physically transparent form, and the Floquet theory is used to calculate transition probabilities. By comparing the UT results with those from the rotating-wave approximation (RWA) and generalized Floquet theory, we illustrate that the UT method captures not only accurate resonance structures but also dynamics of the driven V system in a strong-driving regime where the RWA breaks down. We show that there are two dynamical features, namely, peak-dip structures and beat behaviors, which are ascribed to a time-dependent coupling between the two upper levels due to the counter-rotating terms and cannot be described by a RWA Hamiltonian. Moreover, we show that the effective Hamiltonian can be used to calculate higher-order Bloch–Siegert shifts, which is another feature caused by the counter-rotating terms. Present work offers insights into the role of counter-rotating couplings in a bichromatically driven three-level system.
Abstract We apply a simple analytical method based on a unitary transformation to calculate the ground state, its excitation spectrum and quantum dynamic evolution of physical quantities for the double-photon quantum Rabi Hamiltonian over the wide coupling-strength range. The concise analytical method possesses the same mathematical simplicity as the approach of the rotating wave approximation (RWA). By quantitative comparison with the numerically exact result obtained by matrix diagonalization, we confirm that our calculated results obtained by transformed rotating-wave method are not only accurate in the weak coupling regime but also correct in intermediate strong-coupling case. In the intermediate ultrastrong-coupling regime, the calculated values of the ground state and lower lying excited states are nearly the same as the exact ones. It turns out that our calculation for the energy spectrum is beyond the ordinary-RWA. Meanwhile, we demonstrate the signatures resulting from the counter-rotating wave terms by monitoring the population, the coherence, the squeezing of the photon under the ultra-strong conditions. In particular, we find that when the frequency of the photon is much larger than the transition frequency of the system, the lineshape of the time evolution becomes complicated with the increase of the coupling strength, which may be verified experimentally.
The effects of counter-rotating terms (CRTs) on Rabi splitting and the dynamic evolution of atomic population in the Jaynes—Cummings model are studied with a coherent-state approach. When the coupling strength increases, the Rabi splitting becomes of multi-Rabi frequencies for the initial state of an excited atom in a vacuum field, and the collapses and revivals gradually disappear, and then reappear with quite good periodicity. Without the rotating-wave approximation (RWA), the initial excited state contains many eigenstates rather than two eigenstates under the RWA, which results in the multi-peak emission spectrum. An analytical approximate solution for the strong coupling regime is obtained, which gives a new oscillation frequency and explains the recovery of collapses and revivals due to the equal energy spacing.
Dark state as a consequence of interference between different quantum states has great importance in the fields of chip-scale atomic clock and quantum information. For the Λ-type three-level system, this dark state is generally regarded as being dissipation-free because it is a superposition of two lowest states without dipole transition between them. However, previous studies are based on the rotating-wave approximation (RWA) by neglecting the counter-rotating terms in the system-environment interaction. In this work, we study non-Markovian quantum dynamics of the dark state in a Λ-type three-level system coupled to two bosonic baths and reveal the effect of counter-rotating terms on the dark state. In contrast to the dark state within the RWA, leakage of the dark state occurs even at zero temperature, as a result of these counter-rotating terms. Also, we present a method to restore the quantum coherence of the dark state by applying a leakage elimination operator to the system.