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arXiv:cond-mat/0207418 · Physical review. B, Condensed matter

Effects of nonlinear sweep in the Landau-Zener-Stueckelberg effect

Dmitry A. Garanin, R. Schilling

Published Nov 27, 2002• 60 citations
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Abstract

We study the Landau-Zener-Stueckelberg (LZS) effect for a two-level system with a time-dependent nonlinear bias field (the sweep function) $W(t).$ Our main concern is to investigate the influence of the nonlinearity of $W(t)$ on the probability P to remain in the initial state. The dimensionless quantity $\ensuremath{\varepsilon}=\ensuremath{\pi}{\ensuremath{\Delta}}^{2}/(2\ensuremath{\Elzxh}v)$ depends on the coupling $\ensuremath{\Delta}$ of both levels and on the sweep rate $v.$ For fast sweep rates, i.e., \ensuremath{\varepsilon}\ensuremath{\ll}1, and monotonic, analytic sweep functions linearizable in the vicinity of the resonance we find the transition probability $1\ensuremath{-}P\ensuremath{\cong}\ensuremath{\varepsilon}(1+a),$ where $a>0$ is the correction to the LSZ result due to the nonlinearity of the sweep. Further increase of the sweep rate with nonlinearity fixed brings the system into the nonlinear-sweep regime characterized by $1\ensuremath{-}P\ensuremath{\cong}{\ensuremath{\varepsilon}}^{\ensuremath{\gamma}}$ with \ensuremath{\gamma}\ensuremath{\ne}1, depending on the type of sweep function. In the case of slow sweep rates, i.e., \ensuremath{\varepsilon}\ensuremath{\gg}1, an interesting interference phenomenon occurs. For analytic $W(t)$ the probability ${P=P}_{0}{e}^{\ensuremath{-}\ensuremath{\eta}}$ is determined by the singularities of $\sqrt{{\ensuremath{\Delta}}^{2}{+W}^{2}(t)}$ in the upper complex plane of t. If $W(t)$ is close to linear, there is only one singularity, which leads to the LZS result ${P=e}^{\ensuremath{-}\ensuremath{\varepsilon}}$ with important corrections to the exponent due to nonlinearity. However, for, e.g., $W(t)\ensuremath{\propto}{t}^{3}$ there is a pair of singularities in the upper complex plane. Interference of their contributions leads to oscillations of the prefactor ${P}_{0}$ that depends on the sweep rate through \ensuremath{\varepsilon} and turns to zero at some \ensuremath{\varepsilon}. Measurements of the oscillation period and of the exponential factor would allow one to determine \ensuremath{\Delta}, independently.

DOI
10.1103/physrevb.66.174438

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We study the Landau-Zener-Stueckelberg (LZS) effect for a two-level system with a time-dependent nonlinear bias field (the sweep function) W(t).W(t).W(t). Our main concern is to investigate the influence of the nonlinearity of W(t)W(t)W(t) on the probability P to remain in the initial state. The dimensionless quantity \ensuremathε=\ensuremathπ\ensuremathΔ2/(2\ensuremath\Elzxhv)\ensuremath{\varepsilon}=\ensuremath{\pi}{\ensuremath{\Delta}}^{2}/(2\ensuremath{\Elzxh}v)\ensuremathε=\ensuremathπ\ensuremathΔ2/(2\ensuremath\Elzxhv) depends on the coupling \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ of both levels and on the sweep rate v.v.v. For fast sweep rates, i.e., \ensuremath{\varepsilon}\ensuremath{\ll}1, and monotonic, analytic sweep functions linearizable in the vicinity of the resonance we find the transition probability 1\ensuremath−P\ensuremath≅\ensuremathε(1+a),1\ensuremath{-}P\ensuremath{\cong}\ensuremath{\varepsilon}(1+a),1\ensuremath−P\ensuremath≅\ensuremathε(1+a), where a>0a>0a>0 is the correction to the LSZ result due to the nonlinearity of the sweep. Further increase of the sweep rate with nonlinearity fixed brings the system into the nonlinear-sweep regime characterized by 1\ensuremath−P\ensuremath≅\ensuremathε\ensuremathγ1\ensuremath{-}P\ensuremath{\cong}{\ensuremath{\varepsilon}}^{\ensuremath{\gamma}}1\ensuremath−P\ensuremath≅\ensuremathε\ensuremathγ with \ensuremath{\gamma}\ensuremath{\ne}1, depending on the type of sweep function. In the case of slow sweep rates, i.e., \ensuremath{\varepsilon}\ensuremath{\gg}1, an interesting interference phenomenon occurs. For analytic W(t)W(t)W(t) the probability P=P0e\ensuremath−\ensuremathη{P=P}_{0}{e}^{\ensuremath{-}\ensuremath{\eta}}P=P0​e\ensuremath−\ensuremathη is determined by the singularities of \ensuremathΔ2+W2(t)\sqrt{{\ensuremath{\Delta}}^{2}{+W}^{2}(t)}\ensuremathΔ2+W2(t)​ in the upper complex plane of t. If W(t)W(t)W(t) is close to linear, there is only one singularity, which leads to the LZS result P=e\ensuremath−\ensuremathε{P=e}^{\ensuremath{-}\ensuremath{\varepsilon}}P=e\ensuremath−\ensuremathε with important corrections to the exponent due to nonlinearity. However, for, e.g., W(t)\ensuremath∝t3W(t)\ensuremath{\propto}{t}^{3}W(t)\ensuremath∝t3 there is a pair of singularities in the upper complex plane. Interference of their contributions leads to oscillations of the prefactor P0{P}_{0}P0​ that depends on the sweep rate through \ensuremath{\varepsilon} and turns to zero at some \ensuremath{\varepsilon}. Measurements of the oscillation period and of the exponential factor would allow one to determine \ensuremath{\Delta}, independently.
The Landau-Zener-Stueckelberg (LZS) effect in a model system of interacting tunneling particles is studied numerically and analytically. Each of N tunneling particles interacts with each of the others with the same coupling J. This problem maps onto that of the LZS effect for a large spin S=N/2.S=N/2.S=N/2. The mean-field limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ corresponds to the classical limit $\stackrel{\ensuremath{\rightarrow}}{S}\ensuremath{\infty}$ for the effective spin. It is shown that the ferromagnetic coupling J>0J>0J>0 tends to suppress the LZS transitions. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ there is a critical value of J above which the staying probability P does not go to zero in the slow sweep limit, unlike the standard LZS effect. For J<0J<0J<0 LZS transitions are boosted and P=0P=0P=0 for a set of finite values of the sweep rate for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ Various limiting cases such as strong and weak interaction, slow and fast sweep are considered analytically. It is shown that the mean-field approach works well for arbitrary N if the interaction J is weak.
We consider the inverse Landau-Zener problem which consists in finding the energy sweep functions W ( t ) ≡ ε 1 ( t ) − ε 2 ( t ) resulting in the required time dependences of the level populations for a two-level system crossing the resonance one or more times during the sweep. We find sweep functions of particular forms that let manipulate the system in a required way, including complete switching from the state 1 to the state 2 and preparing the system at the exact ground and excited states at resonance.
In the standard Landau-Zener-St\"uckelberg-Majorana (LZSM) problem, the bias sweep rate and gap are both time independent and fully characterize the LZSM problem. We consider the nonlinear LZSM problem, in which at least one of the two characteristic parameters varies as the system traverses the avoided-crossing region. This situation results in what could be thought of as a more accurate description of any realistic situation as compared with the idealized linear LZSM problem. We consider both the case of perturbative nonlinearities, where the nonlinearity adds small corrections to the linear problem, and the case of essential nonlinearities, where the sweep and/or minimum-gap functions are qualitatively different from those of the linear LZSM problem. In the case of perturbative nonlinearities, we derive analytic expressions for the LZSM transition probability based on the Dykhne-Davis-Pechukas (DDP) formula, taking into account the leading corrections to the standard LZSM formula. We compare the derived approximate expressions with numerical simulation results and comment on the validity of the approximations. In particular, if the nonlinear term is small in comparison to the linear term throughout the finite duration of the avoided crossing traversal, the perturbative approximation is valid. Our results also provide information about the validity of the DDP formula. In addition to reviewing cases of essential nonlinearity treated previously in the literature, we analyze the case of an essentially nonlinear sweep function that describes an almost square pulse.
We present a comprehensive analysis of the Landau-Zener tunneling of a nonlinear three-level system in a linearly sweeping external field. We find the presence of nonzero tunneling probability in the adiabatic limit (i.e., very slowly sweeping field) even for the situation that the nonlinear term is very small and the energy levels keep the same topological structure as that of the linear case. In particular, the tunneling is irregular with showing an unresolved sensitivity on the sweeping rate. For the case of fast-sweeping fields, we derive an analytic expression for the tunneling probability with stationary phase approximation and show that the nonlinearity can dramatically influence the tunneling probability when the nonlinear ``internal field'' resonate with the external field. We also discuss the asymmetry of the tunneling probability induced by the nonlinearity. Physics behind the above phenomena is revealed and possible application of our model to triple-well trapped Bose-Einstein condensate is discussed.
Two aspects of the classic two-level Landau-Zener (LZ) problem are considered. First, we address the LZ problem when one or both levels decay, i.e., ${\ensuremath{\varepsilon}}_{j}(t)\ensuremath{\rightarrow}{\ensuremath{\varepsilon}}_{j}(t)\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$. We find that if the system evolves from an initial time $\ensuremath{-}T$ to a final time +T+T+T such that $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|$ is not too large, the LZ survival probability of a state $|j\ensuremath{\rangle}$ can increase with increasing decay rate of the other state $|i\ensuremath{\ne}j\ensuremath{\rangle}$. This surprising result occurs because the decay results in crossing of the two eigenvalues of the instantaneous non-Hermitian Hamiltonian. On the other hand, if $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|\ensuremath{\rightarrow}\ensuremath{\infty}$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$, the probability is independent of the decay rate. These results are based on analytic solutions of the time-dependent Schr\"odinger equations for two cases: (a) the energy levels depend linearly on time, and (b) the energy levels are bounded and of the form ${\ensuremath{\varepsilon}}_{1,2}(t)=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}tanh(t/\mathcal{T})$. Second, we study LZ transitions affected by dephasing by formulating the Landau-Zener problem with noise in terms of a Schr\"odinger-Langevin stochastic coupled set of differential equations. The LZ survival probability then becomes a random variable whose probability distribution is shown to behave very differently for long and short dephasing times. We also discuss the combined effects of decay and dephasing on the LZ probability.
Landau–Zener–Stückelberg (LZS) interference has drawn renewed attention to quantum information processing research because it is not only an effective tool for characterizing two-level quantum systems but also a powerful approach to manipulate quantum states. Superconducting quantum circuits, due to their versatile tunability and degrees of control, are ideal platforms for studying LZS interference phenomena. We use a superconducting Xmon qubit to study LZS interference by parametrically modulating the qubit transition frequency nonlinearly. For dc flux biasing of the qubit slightly far away from the optimal flux point, the qubit excited state population shows an interference pattern that is very similar to the standard LZS interference in linear regime, except that all bands shift towards lower frequencies when increasing the rf modulation amplitude. For dc flux biasing close to the optimal flux point, the negative sidebands and the positive sidebands behave differently, resulting in an asymmetric interference pattern. The experimental results are also in good agreement with our analytical and numerical simulations.