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arXiv:quant-ph/0702032 · Physical Review A

Two-level systems driven by large-amplitude fields

Sahel Ashhab, J. R. Johansson, Alexandre M. Zagoskin, Franco Nori

Published Jun 19, 2007• 245 citations
arXivHTMLPDF

Abstract

We analyze the dynamics of a two-level system subject to driving by large-amplitude external fields, focusing on the resonance properties in the case of driving around the region of avoided level crossing. In particular, we consider three main questions that characterize resonance dynamics: (1) the resonance condition, (2) the frequency of the resulting oscillations on resonance, and (3) the width of the resonance. We identify the regions of validity of different approximations. In a large region of the parameter space, we use a geometric picture in order to obtain both a simple understanding of the dynamics and quantitative results. The geometric approach is obtained by dividing the evolution into discrete time steps, with each time step described by either a phase shift on the basis states or a coherent mixing process corresponding to a Landau-Zener crossing. We compare the results of the geometric picture with those of a rotating wave approximation. We also comment briefly on the prospects of employing strong driving as a useful tool to manipulate two-level systems.

DOI
10.1103/physreva.75.063414

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We analyze the dynamics of a two-level system subject to driving by large-amplitude external fields, focusing on the resonance properties in the case of driving around the region of avoided level crossing. In particular, we consider three main questions that characterize resonance dynamics: (1) the resonance condition, (2) the frequency of the resulting oscillations on resonance, and (3) the width of the resonance. We identify the regions of validity of different approximations. In a large region of the parameter space, we use a geometric picture in order to obtain both a simple understanding of the dynamics and quantitative results. The geometric approach is obtained by dividing the evolution into discrete time steps, with each time step described by either a phase shift on the basis states or a coherent mixing process corresponding to a Landau-Zener crossing. We compare the results of the geometric picture with those of a rotating wave approximation. We also comment briefly on the prospects of employing strong driving as a useful tool to manipulate two-level systems.
Abstract We analyze the dynamics of a two-level quantum system (TLS) under the influence of a strong sinusoidal driving signal whose origin is the interaction of the two-level system with a quantum field. In this approach the driving field is replaced by a harmonic oscillator that is either strongly coupled to the TLS or populated with a large number of photons. Starting from the Rabi model, we derive expressions for the TLS’s oscillation frequencies and compare the results with those obtained from the model where the driving signal is treated classically. We show that in the limits of weak coupling and large photon number, the well-known expression for the Rabi frequency in the strong driving regime is recovered. In the opposite limit of strong coupling and small photon number, we find differences between the predictions of the semiclassical and quantum models. The results of the quantum picture can therefore be understood as Landau–Zener–Stueckelberg interferometry in the fully quantum regime.
The device for the Josephson flux qubit can be considered as a solid-state artificial atom with multiple energy levels. When a large-amplitude harmonic excitation is applied to the system, transitions at the energy levels avoided crossings produce visible changes in the qubit population over many driven periods that are accompanied by a rich pattern of interference phenomena. We present a Floquet treatment of the periodically time-dependent Schr\"odinger equation of the strongly driven qubit beyond the standard two-level approach. For low amplitudes, the average probability of a given sign of the persistent current qubit exhibits, as a function of the static flux detuning and the driving amplitude, Landau-Zener-St\"uckelberg (LZS) interference patterns that evolve into complex diamondlike patterns for large amplitudes. In the case of highly coherent flux qubits we show that the higher-order diamonds can not be simply described relying on LZS transitions in each avoided crossing considered separately. In addition we propose a spectroscopic method based on starting the system in the first excited state instead of in the ground state, which can give further information on the energy-level spectrum and dynamics in the case of highly coherent flux qubits. We compare our numerical results with recent experiments that perform amplitude spectroscopy to probe the energy spectrum of the artificial atom.
Quantum systems driven by strong oscillating fields are the source of many interesting physical phenomena. In this work, we experimentally study the dynamics of a two-level system of a single spin driven in the strong-driving regime where the rotating-wave approximation is not valid. This two-level system is a subsystem of a single nitrogen-vacancy (NV) center in diamond coupled to a first-shell 13C^{13}\mathrm{C}13C nuclear spin at a level anticrossing point. This near degeneracy occurs in the ${m}_{s}=\ifmmode\pm\else\textpm\fi{}1$ manifold of the electron spin when the energy level splitting between the ${m}_{s}=\ensuremath{-}1$ and +1+1+1 states due to the static magnetic field is $\ensuremath{\approx}127$ MHz and thus equal to the splitting due to the 13C^{13}\mathrm{C}13C hyperfine interaction. The transition frequency of this electron spin two-level system in a static magnetic field of 28.9 G is 1.7 MHz and it can be driven only by the component of the radio-frequency (RF) field along the NV symmetry axis. Electron spin Rabi frequencies in this system can reach tens of MHz even for moderate RF powers. The simple sinusoidal Rabi oscillations that occur when the amplitude of the driving field is small compared to the transition frequency evolve into complex patterns when the driving field amplitude is comparable to or greater than the energy level splitting. We observe that the system oscillates faster than the amplitude of the driving field and the response of the system shows multiple frequencies.
In this work, we study the time-averaged populations obtained for a fluxonium circuit under a large amplitude nonresonant periodic drive. We present numerical simulations of the time evolution which consider the multi-level structure of the driven quantum circuit, looking for a realistic modeling closer to experimental implementations. The Landau-Zener-Stückelberg spectra show resonances that can be understood as originated from constructive interference favoring transitions to higher levels. For a truncated two-level system (TLS) the resonance patterns can be interpreted using a simplified description of the avoided crossing that takes into account the dynamic phase accumulated at each operation point. For the multilevel case, we derive an effective two-level Hamiltonian using a Schrieffer-Wolff transformation starting from the Floquet Hamiltonian in the Sambe space. Our study provides predictive insight into experimental outcomes, offering an intuitive interpretation that could also support the implementation of fast-non-adiabatic single-qubit gates and entangling protocols.
Abstract Three analytic solutions to the Schrödinger equation for the time-dependent Landau–Zener (LZ) Hamiltonian are presented. They correspond to specific finite-time driving paths in a bounded parameter space of a two-level system. Two of these paths go through the avoided crossing of levels, either with a constant speed or with variable speed that decreases in the region of reduced energy gap, the third path bypasses the crossing such that the energy gap remains constant. The solutions yield exact time dependencies of the excitation probability for the system evolving from the ground state of the initial Hamiltonian. The LZ formula emerges as an approximation valid within a certain interval of driving times for the constant-speed driving through the avoided crossing. For long driving times, all solutions converge to the prediction of the adiabatic perturbation theory. The excitation probability vanishes at some discrete time instants.
We investigate Floquet dynamics of a particle held in a three-well system driven by a two-frequency field and identify integer and fractional photon resonances due to the dual-frequency driving. It is found that pairs of photon-assisted tunneling near the resonance originate from avoided level crossings in the Floquet spectra which, in essence, are quantum features of the hybridization between different quantum states. In particular, we establish a close connection between fractional-order resonances and Floquet mode properties under two-frequency driving conditions and illustrate their dependence on driving parameters. These results provide us a possibility to realize coherent control of quantum states with the assistance of classical external driving fields.