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arXiv:1605.04798 · Physical Review A

Dissipative two-level systems under ultrastrong off-resonant driving

A. P. Saǐko, Siarhei A. Markevich, Ryhor Fedaruk

Published Jun 20, 2016• 14 citations
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Abstract

We study the dissipative dynamics of a two-level system under ultrastrong driving when the frequency and strength of the exciting field exceed significantly the transition frequency. We find three qualitatively different regimes of such dynamics: (1) the collapse and revival of oscillations in the population difference, (2) the simple exponential decay of the oscillations resulting in their steady state with the finite amplitude, and (3) the steady-state stabilization of the equally populated levels. The nonmonotonic Bessel-function-like dependence on the driving strength is also predicted for the decay rate of these oscillations. The features of this dependence are determined by the relative rates of energetic relaxation and pure dephasing.

DOI
10.1103/physreva.93.063834

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We study the dissipative dynamics of a two-level system under ultrastrong driving when the frequency and strength of the exciting field exceed significantly the transition frequency. We find three qualitatively different regimes of such dynamics: (1) the collapse and revival of oscillations in the population difference, (2) the simple exponential decay of the oscillations resulting in their steady state with the finite amplitude, and (3) the steady-state stabilization of the equally populated levels. The nonmonotonic Bessel-function-like dependence on the driving strength is also predicted for the decay rate of these oscillations. The features of this dependence are determined by the relative rates of energetic relaxation and pure dephasing.
We study the dissipative Rabi model under deep strong far-off-resonant driving when the driving frequency and strength exceed significantly the qubit transition frequency. We find analytical expressions for the density matrix, the steady-state first-order field correlation function and the multiphoton emission spectrum. This spectrum consists of a series of narrow delta-like lines, caused by coherent processes and of Lorenzian peaks resulted from the incoherent scattering of photons. It is shown that the width of emission lines as well as the decay rate of the time oscillations in the population difference have oscillating dependence on the driving strength. We show how the features of this dependence are controlled by the relative rates of energetic relaxation and pure dephasing.
Dynamics of a two-level system with Ohmic dissipation driven by a time-dependent cw field in the noninteracting-blip approximation are studied. It is found that at low temperatures when \ensuremath{\alpha}, a friction parameter, lies between 1/2 and 1, the localization predicted by Chakravarty [Phys. Rev. Lett. 49, 681 (1982)] and Bray and Moore [Phys. Rev. Lett. 49, 1546 (1982)] disappears, and the rate constant for tunneling of the particle increases with the intensity and frequency of the field. For 01/2 quantum coherence is observed, and the transition temperature from the coherent to the decay regime depends on the field intensity. When the intensity parameter approaches the zeros of the zeroth Bessel function coherence disappears. This transition from coherent to incoherent motion oscillates with the intensity of the field, giving rise to a cascade of transitions at different temperatures.
The model of a two-level system (TLS) coupled to a harmonic oscillator has found a widespread application in physics and chemistry. In this thesis we focus on the field of circuit quantum electrodynamics where the TLS stands for a quantum bit (qubit), which is the basic building unit of a potential quantum computer, and the oscillator represents a transmission-line resonator, which can be used, e.g., to store information contained on the qubit or for communication between several qubits. Furthermore, the oscillator can model a superconducting quantum interference device (SQUID) determining the state of the qubit. In first experimental realizations the coupling between the qubit and the oscillator was small compared to characteristic frequencies of the two devices. In such a situation the Jaynes-Cummings model provides a realistic and completely analytical description of the energy spectrum and the dynamics of the setup. However, it relies on a rotating-wave approximation (RWA) which neglects counter-rotating terms in the full qubit-oscillator Hamiltonian. These counter-rotating terms become important under increase of the coupling strength, as it has been achieved in recent experiments. We present two analytical approaches to take these additional contributions into account: The first one treats the qubit-oscillator Hamiltonian perturbatively in the coupling strength and predicts a frequency shift in its energy spectrum, the Bloch-Siegert shift. The second one considers the qubit's tunneling splitting as small parameter and thus treats the problem to all orders in the coupling allowing to enter the ultrastrong coupling regime, where the coupling strength becomes comparable to the qubit's and/or oscillator's frequency. For both cases we give a thorough analysis of the qubit's dynamics. In order to perform operations the qubit is usually probed by an external driving force. We model this situation by coupling the TLS to a classical oscillator and examine the resulting time dependent Hamiltonian using a combination of Floquet theory and Van Vleck perturbation theory. Thus, we provide an analysis of the qubit's energy spectrum and dynamics which is nonperturbative in the driving amplitude. We investigate effects like coherent destruction of tunneling and driving-inducing tunneling oscillations. By coupling the driven TLS to a quantized oscillator we give for the first time an analytical treatment of the qubit-oscillator system being simultaneously exposed to both ultrastrong coupling and extreme driving. We observe interesting phenomena in the dynamics like the suppression of specific frequencies under a variation of the coupling strength. To mimic environmental influences on the qubit we apply the Caldeira-Leggett master equation approach to the driven TLS and to the qubit-oscillator system and provide for both cases analytical expressions for the relaxation and dephasing rates.
We study the dynamics of an initially localized symmetric two-level system coupled to high-temperature dissipative environments and driven by a strong time-periodic force which corresponds to high-frequency monochromatic light. Qualitative arguments based on the quantized representation of the radiation field predict a wealth of intriguing behaviors which are confirmed and quantified via accurate numerical path integral calculations. With the exception of very strong friction we find that high-frequency driving always helps stabilize localized states. At intermediate friction the delocalization rate approaches a “universal’’ limiting value which is largely independent of the parameters of the environment and of the specifics of the driving force, depending only on its overall strength. This robust behavior implies that localized states can be stabilized in these systems without much finetuning of external conditions. In the weak friction regime the interplay between phase interference and dissipation results in nonmonotonic variation of the decay rate with friction and driving frequency. The path integral results are compared to those obtained earlier via analytical treatments.
A quantum dynamical problem has been analytically solved for a two-level system where localized states L 0 and R 0 are strongly coupled with reservoirs of local oscillations {L n } and {R n }. It is additionally assumed that the spectra of reservoirs are equidistant and the coupling constants are the same. It has been shown that the evolution of states L 0 and R 0 in recurrence cycles depends on three independent factors, which characterize exchange with the two-level system, exchange of L 0 with {L n } (R 0 with {R n }) and the phonon-induced decay of {L n } and {R n }. In addition to coherent oscillations with the frequency of the two-level system, Δ, and dissipative tunneling with a rate Δ2/πC 2 (where C is the matrix element of the coupling of L 0 and R 0 with L n and R n ), a new regime appears where L-R transitions are induced by the partial recovery of the populations of L 0 and R 0 in each recurrence cycle due to synchronous transitions from reservoirs. These transitions induce repeating changes in the populations of the states of the two-level system (Loschmidt echo). The number and width of the echo components increase with the cycle number. Evolution becomes irregular because of the mixing of the contributions from pulses of the neighboring cycles, when the cycle number k exceeds the critical value k c = π2 C 2. Unlike the populations, their cycle-average values remain regular at k ≫ k c. When Δ ≪ πC 2, the cycle-average populations oscillate with a frequency of ΔΩ/πC 2 irrespective of mixing. The frequency of oscillations of the populations of the states {L n } and {R n } is approximately nΩ(Δ/2πC 2)2, where Ω is the spacing between the neighboring levels of the reservoir and nΩ is the difference between the energies of the states L 0 and L n . The appearance of the mentioned low-frequency oscillations is due to the formation of collective states of the two-level system that are “dressed” by the reservoir. The predicted oscillations can be detected by femtosecond spectroscopy methods.
Recent experiments on a range of engineered quantum systems have highlighted the important role of interacting two-level systems (TLSs) in modifying device properties and generating fluctuations. Focusing on the case of an oscillator coupled to a single near-resonant TLS, we explore how interactions between the TLS and lower-frequency thermally activated two-level fluctuators (TLFs) degrade the oscillator's coherence. Depending on the strength of the couplings, a single TLF can give rise to coherence oscillations that appear alongside, or supplant, Rabi oscillations of the oscillator-TLS system. Bath-driven transitions in the TLF cause irreversible coherence decay at a rate that is highly sensitive to both the couplings and the transition rate. For an ensemble of TLFs, we identify and characterise the different regimes of non-exponential phase-averaging-driven coherence decay that the oscillator can display. Using numerical calculations, we examine the extent to which systems with just a few TLFs differ from the limit of a large (continuum) TLF ensemble. Our work provides a theoretical framework for understanding the interplay of coherent TLS interactions and TLF-induced dephasing in quantum devices such as superconducting and phononic resonators.