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Quantum dynamics of a two-level system under extrenal field

A. M. Korostił, Ju. Korostil

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

We present exact analytic solutions for non-linear quantum dynamics of \n twolevel \nsystem (TLS) subject to \n periodic-in-time external field. >n constructing the \nexactly solvable models, we use \n approach where the form of external perturbation \nis chosen to preserve \nn integrability constraint, which yields \n single non-linear \ndifferential equation for the ac-field. solution to this equation is expressed in terms \nof Jacobi elliptic functions with three independent parameters that allows n to \nchoose the frequency, average value, and amplitude of the time-dependent field at \nwill. This form of the ac-drive is especially relevant to the problem of dynamics of \nTLS charge defects that cause dielectric losses ?n superconducting qubits.

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•Oct 2, 2009•0 citations
We present exact analytic solutions for non-linear quantum dynamics of \n twolevel \nsystem (TLS) subject to \n periodic-in-time external field. >n constructing the \nexactly solvable models, we use \n approach where the form of external perturbation \nis chosen to preserve \nn integrability constraint, which yields \n single non-linear \ndifferential equation for the ac-field. solution to this equation is expressed in terms \nof Jacobi elliptic functions with three independent parameters that allows n to \nchoose the frequency, average value, and amplitude of the time-dependent field at \nwill. This form of the ac-drive is especially relevant to the problem of dynamics of \nTLS charge defects that cause dielectric losses ?n superconducting qubits.
We present a family of exact analytic solutions for nonlinear quantum dynamics of a two-level system (TLS) subject to a periodic-in-time external field. In constructing the exactly solvable models, we use a ``reverse engineering'' approach where the form of external perturbation is chosen to preserve an integrability constraint, which yields a single nonlinear differential equation for the ac field. A solution to this equation is expressed in terms of Jacobi elliptic functions with three independent parameters that allows one to choose the frequency, average value, and amplitude of the time-dependent field at will. This form of the ac drive is especially relevant to the problem of dynamics of TLS charge defects that cause dielectric losses in superconducting qubits. We apply our exact results to analyze nonlinear dielectric response of such TLSs and show that the position of the resonance peak in the spectrum of the relevant correlation function is determined by the quantum-mechanical phase accumulated by the TLS wave function over a time evolution cycle. It is shown that in the nonlinear regime, this resonance frequency may be shifted strongly from the value predicted by the canonical TLS model. We also analyze the ``spin'' survival probability in the regime of strong external drive and recover a coherent destruction of tunneling phenomenon within our family of exact solutions, which manifests itself as a strong suppression of ``spin-flip'' processes and suggests that such nonlinear dynamics in LC resonators may lead to lower losses.
Analytical solutions to the time-dependent Schrödinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presenting several new exact solutions to the time-dependent Schrödinger equation. Our general method and many of the new solutions we present are particularly relevant to qubit control in quantum computing applications.
The time-dependent average population in a two-level system interacting with an Ohmic boson bath and driven by a dc-ac field is obtained. The study is performed within the noninteracting-blip approximation and for the high-frequency driving case. The integro-differential master equation and the exact formal solution for the average transition probability are obtained. The dissipative quantum decay and the transition temperature from the coherent to the incoherent motion are investigated. The effect of the external electric field on the the dissipative two-level system is found to destroy its coherence.
In this work, we derive exact solutions of a dynamical equation, which can represent all two-level Hermitian systems driven by periodic NNN-step driving fields. For different physical parameters, this dynamical equation displays various phenomena for periodic NNN-step driven systems. The time-dependent transition probability can be expressed by a general formula that consists of cosine functions with discrete frequencies, and, remarkably, this formula is suitable for arbitrary parameter regimes. Moreover, only a few cosine functions (i.e., one to three main frequencies) are sufficient to describe the actual dynamics of the periodic NNN-step driven system. Furthermore, we find that a beating in the transition probability emerges when two (or three) main frequencies are similar. Some applications are also demonstrated in quantum state manipulations by periodic NNN-step driving fields.
We present a new field-theoretic approach to analyze nonequilibrium dynamics of two-level systems (TLS), which is based on a correspondence between a driven TLS and a Majorana fermion field theory coupled to bosonic fields. This approach allows us to calculate analytically properties of nonlinear TLS dynamics with an arbitrary accuracy. We apply our method to analyze specific TLS dynamics under a monochromatic periodic drive that is relevant to the problem of decoherence in Josephson junction qubits. It is demonstrated that the method gives the precise positions of the resonance peaks in the nonlinear dielectric response function that are in agreement with numerical simulations.
We study the dissipative dynamics of a two-level system (TLS) exposed to strong ac driving. By combing Floquet theory with Van Vleck perturbation theory in the TLS tunneling matrix element, we diagonalize the time-dependent Hamiltonian and provide corrections to the renormalized Rabi frequency of the TLS, which are valid for both a biased and unbiased TLS and go beyond the known high-frequency and rotating-wave results. In order to mimic environmental influences on the TLS, we couple the system weakly to a thermal bath and solve analytically the corresponding Floquet-Bloch-Redfield master equation. We give a closed expression for the relaxation and dephasing rates of the TLS and discuss their behavior under variation of the driving amplitude. Further, we examine the robustness of coherent destruction of tunneling (CDT) and driving-induced tunneling oscillations (DITO). We show that also for a moderate driving frequency an almost complete suppression of tunneling can be achieved for short times and demonstrate the sensitiveness of DITO to a change of the external parameters.