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arXiv:1005.0652 · Physical Review B

Exact solution for quantum dynamics of a periodically driven two-level system

Anirban Gangopadhyay, Maxim Dzero, Victor M. Galitski

Published Jul 22, 2010• 35 citations
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Abstract

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.

DOI
10.1103/physrevb.82.024303

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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.
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 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.
Coherent Control of Two-level Systems with Periodic NNN-Steps Driving Fields
In this work, we derive the exact solutions of the dynamical equation, which can describe all two-level Hermitian systems driven by periodic NNN-steps driving fields. The solutions are a set of quasi-periodic functions, and the expressions of the amplitudes and phases are obtained in the frequency-domain. Particularly, we develop a general formalism for the effective Hamiltonian without any approximations, and give the validity range according to the transition probability of the micromotion operator. With the guidance of the exact solutions and the effective Hamiltonian, we find that the coherent destruction of tunneling phenomenon, the resonance phenomenon, and the beat-frequency phenomenon can emerge in this system. Finally, some applications are demonstrated in quantum information processing by periodic NNN-steps driving fields.
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Using the Hubbard representation for S U ( 2 ) , we write the time-evolution operator of a two-level system in the disentangled form. This allows us to map the corresponding dynamical law into a set of nonlinear coupled equations. In order to find exact solutions, we use an inverse approach and find families of time-dependent Hamiltonians whose off-diagonal elements are connected with the Ermakov equation. A physical model with the so-obtained Hamiltonians is discussed in the context of the nuclear magnetic resonance phenomenon.
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.