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arXiv:1409.5237 · Physical Review A

Qubit interference at avoided crossings: The role of driving shape and bath coupling

Ralf Blattmann, Peter Hänggi A, Sigmund Kohler

Published Apr 9, 2015• 27 citations
arXivHTMLPDF

Abstract

We derive the structure of Landau-Zener-St\"uckelberg-Majorana (LZSM) interference patterns for a qubit that experiences quantum dissipation and is additionally subjected to time-periodic but otherwise general driving. A spin-boson Hamiltonian serves as the model, which we treat with a Bloch-Redfield master equation in the Floquet basis. It predicts resonance peaks whose form depends significantly on the operator through which the qubit couples to the bath. The Fourier transforms of the LZSM patterns exhibit arc structures which reflect the shape of the driving. These features are captured by an effective time-independent Bloch equation which provides an analytical solution. Moreover, we determine the decay of these arcs as a function of dissipation strength and temperature.

DOI
10.1103/physreva.91.042109

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We derive the structure of Landau-Zener-St"uckelberg-Majorana (LZSM) interference patterns for a qubit that experiences quantum dissipation and is additionally subjected to time-periodic but otherwise general driving. A spin-boson Hamiltonian serves as the model, which we treat with a Bloch-Redfield master equation in the Floquet basis. It predicts resonance peaks whose form depends significantly on the operator through which the qubit couples to the bath. The Fourier transforms of the LZSM patterns exhibit arc structures which reflect the shape of the driving. These features are captured by an effective time-independent Bloch equation which provides an analytical solution. Moreover, we determine the decay of these arcs as a function of dissipation strength and temperature.
Abstract Landau–Zener–Stückelberg–Majorana (LZSM) interference occurs when qubit parameters are periodically modulated across avoided level crossings. We explore this phenomenon in nonlinear multilevel bosonic systems, where interference is influenced by multiple energy levels. We fabricate two superconducting resonators with flux-tunable Josephson junction arrays. The first device, exhibiting weak nonlinearity, behaves like a linear resonator under weak driving but shows LZSM interference akin to two-level systems. With stronger driving, nonlinear effects alter the interference pattern. We theoretically demonstrate that merging LZSM peaks can lead to dissipative quantum chaos. In the second device, where nonlinearity exceeds photon-loss rates, we observe additional LZSM peaks from Kerr multiphoton resonances. Under Floquet theory, these resonances represent synthetic modes of coupled nonlinear cavities, revealing effective coupling as modulation parameters vary. Our findings advance the understanding of LZSM physics and emphasize the control of nonlinear Floquet states and the emergence of chaos in engineered systems, with significant implications for novel applications in quantum dynamics and quantum control.
We study Landau-Zener-St\"uckelberg (LZS) interferometry in a cQED architecture under the effects of dissipation. To be specific, we consider a superconducting qubit driven by a dc+ac\mathrm{dc}+\mathrm{ac}dc+ac signal and coupled to a transmission line resonator, but our results are valid for general qubit-resonators devices. To take the environment into account, we assume that the resonator is coupled to an Ohmic quantum bath. The Floquet-Born-Markov master equation is numerically solved to obtain the dynamics of the system for an arbitrary amplitude of the drive and different timescales. We unveil important differences in the resonant patterns between the strong coupling and ultrastrong coupling regimes in the qubit-resonator interaction, which are mainly due to the magnitude of photonic gaps in the energy spectrum of the system. We identify in the LZS patterns the contribution of the qubit gap and the photonic gaps, showing that for large driving amplitudes the patterns present a weaving structure due to the combined intercrossing of the different gaps contributions.
Landau–Zener–Stückelberg–Majorana (LZSM) interference describes the coherent superposition of nonadiabatic transitions when a quantum system is driven through an avoided crossing, but its classical optical analog remains largely unexplored. This work establishes an optical-resonator-based platform for exploring LZSM interference in atomic polarization dynamics using temporal coupled-mode theory. By exploiting the formal correspondence between the cavity mode and atomic polarization in a weakly driven two-level system (TLS), exact analytical solutions are derived for linear frequency sweeps and sinusoidal modulation. The results reveal that the Landau–Zener transition of atomic polarization occurs even when the input frequency transiently sweeps across resonance. The finite temporal memory inherent in the cavity response gives rise to Stückelberg interference, manifesting as oscillations in transmission and cavity energy. The dependence of the interference pattern on sweep rate, linewidth, and modulation is systematically analyzed, and a non-monotonic behavior of oscillation amplitude versus the cavity linewidth is identified. Furthermore, the conventional critical coupling concept is reexamined in the nonadiabatic regime, where counterintuitively the deepest transmission dip occurs under over-coupling rather than critical coupling. Finally, an inverse-design approach engineering both the amplitude and frequency of the input frequency is introduced to tailor the LZSM interference pattern. Our results provide a new platform to all-optical simulation of complex quantum interference.
The interference between a sequence of Landau-Zener (LZ) transitions can produce Rabi oscillations (LZROs). This phenomenon is a kind of time-domain Landau-Zener-St\"uckelberg-Majorana (LZSM) interference. However, experimental demonstrations of this LZSM interference induced Rabi oscillation are extremely hard due to the short coherence time of the driven quantum system. Here, we study theoretically the time-domain LZSM interference between the clock transition in a one-dimensional 87Sr^{87}\mathrm{Sr}87Sr optical lattice clock (OLC) system. With the help of both the adiabatic-impulse model and Floquet numerical simulation method, the LZROs with special steplike structure are clearly found both in the fast- and slow-passage limit in the real experiment parameter regions. In addition, the dephasing effect caused by the system temperature can be suppressed with destructive interference in the slow-passage limit. Finally, we discuss the possible Bloch-Siegert shift while the pulse time is away from the half-integer and integer periods. Our work provides a clear road map to observe the LZROs on the OLC platform.
Driven-dissipative qubit-resonator dynamics, which are the basis of most dispersive superconducting qubit measurement schemes, are often modeled with Lindblad master equations built from subsystem local jump operators, even when the qubit and resonator are appreciably hybridized. In this work we revisit this setting using a microscopic Bloch-Redfield approach, where dissipation is constructed in the eigenbasis of the coupled qubit-resonator Hamiltonian with a complete, frequency dependent, open system description of the transmission line environment. Here, we show that the Lindblad and Bloch-Redfield decay rates can be quantitatively different in the absence of driving, while in the driven case we demonstrate that the time-independent Redfield dissipator and its time-dependent generalization can show qualitatively different behaviors as a function of driving strength. Finally, we investigate the effects of driving in structured spectral densities, recovering the suppression of measurement-induced relaxation in the presence of a so called Purcell filter.
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.