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arXiv:cond-mat/0305467 · Physical review. B, Condensed matter

Theoretical analysis of continuously driven dissipative solid-state qubits

Marlies C Goorden, Frank K. Wilhelm

Published Jul 28, 2003• 29 citations
arXivHTMLPDF

Abstract

We study a realistic model for driven qubits using the numerical solution of the Bloch-Redfield equation as well as analytical approximations using a high-frequency scheme. Unlike in idealized rotating-wave models suitable for NMR or quantum optics, we study a driving term which neither is orthogonal to the static term nor leaves the adiabatic energy value constant. We investigate the underlying dynamics and analyze the spectroscopy peaks obtained in recent experiments. We show, that unlike in the rotating-wave case, this system exhibits nonlinear driving effects. We study the width of spectroscopy peaks and show, how a full analysis of the parameters of the system can be performed by comparing the first and second resonance. We outline the limitations of the NMR linewidth formula at low temperature and show, that spectroscopic peaks experience a strong shift which goes much beyond the Bloch-Siegert shift of the eigenfrequency.

DOI
10.1103/physrevb.68.012508

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We study a realistic model for driven qubits using the numerical solution of the Bloch-Redfield equation as well as analytical approximations using a high-frequency scheme. Unlike in idealized rotating-wave models suitable for NMR or quantum optics, we study a driving term which neither is orthogonal to the static term nor leaves the adiabatic energy value constant. We investigate the underlying dynamics and analyze the spectroscopy peaks obtained in recent experiments. We show, that unlike in the rotating-wave case, this system exhibits nonlinear driving effects. We study the width of spectroscopy peaks and show, how a full analysis of the parameters of the system can be performed by comparing the first and second resonance. We outline the limitations of the NMR linewidth formula at low temperature and show, that spectroscopic peaks experience a strong shift which goes much beyond the Bloch-Siegert shift of the eigenfrequency.
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We study the coherent dynamics of relaxing qubits driven by a bichromatic radiation in the dispersive regime, when detuning of the frequency ωrfω_{rf}ωrf​ of a longitudinal radiofrequency field from the Rabi frequency ω1ω_{1}ω1​ in a transverse microwave field is comparable in magnitude to ωrfω_{rf}ωrf​ and ω1ω_{1}ω1​. We analytically describe this regime beyond the rotating wave approximation and find that the dominant feature of dynamics of qubits is the shift of the Rabi frequency caused by the dynamical Zeeman and Bloch-Siegert-like effects. These fundamental effects can be experimentally separated because, unlike the Bloch-Siegert effect, the dynamical Zeeman effect depends on the detuning sign. Our theoretical results are in a good agreement with the experimental data obtained in pulse EPR for the E1′E'_{1}E1′​ centers in crystalline quartz.
We show that the counter-rotating terms of the dispersive qubit-cavity Rabi model can produce relatively large and nonmonotonic Bloch-Siegert shifts in the cavity frequency as the system is driven through a quantum-to-classical transition. Using a weak microwave probe tone, we demonstrate experimentally this effect by monitoring the resonance frequency of a microwave cavity coupled to a transmon and driven by a microwave field with varying power. In the weakly driven regime (quantum phase), the Bloch-Siegert shift appears as a small constant frequency shift, while for a strong drive (classical phase) it presents an oscillatory behavior as a function of the number of photons in the cavity. The experimental results are in agreement with numerical simulations based on the quasienergy spectrum.
The transient spectrum of scattered radiation from a qubit (represented by a spin-1/2 system or a two-level atom) driven by a resonant rectangular pulse is calculated analytically without adopting the rotating wave approximation (RWA). Operator solutions for the non-autonomous model of Heisenberg equations of motion are obtained within an iterative approach to O (λ) where λ is the system parameter representing the ratio of the Rabi frequency of the driving pulse to its circular frequency. The spectrum in the case of an initial ground-state atom and for increasing pulse area shows asymmetric profiles outside the RWA compared with a symmetric profile within the RWA. For the qubit this asymmetry is much pronounced with splitting of side peaks, which develops to ‘ringing’ for larger pulse area (within and outside the RWA). As shown analytically, this is attributed to the strong Rabi oscillations and interference between various amplitude spectral components initiated by the finite atomic coherent dispersion.
Drive-induced shifts, such as ac Stark shift and Bloch-Siegert shift, are routinely used in various spectroscopies. These shifts are experimentally known to show dispersive Lorentzian behavior as a function of its characteristic frequencies in optical pumping experiments. However, the drive-induced Stark shifts, as calculated using Floquet or dressed atom approaches, do not show the above nonlinear behavior. To address this, we theoretically investigated the drive-induced shifts using a previously reported fluctuation-regulated quantum master equation [A. Chakrabarti and R. Bhattacharyya, Phys. Rev. A 97, 063837 (2018)]. The shifts are obtained as closed-form expressions over the entire detuning range of the drive. The predicted shifts match satisfactorily with the known experimental data of the light shifts. We show that the calculated shifts are a Kramers-Kronig pair of the drive-induced dissipation in conformity with experimental findings. Moreover, we show that at low temperatures, i.e., for less thermal fluctuations, our results asymptotically match with the known theoretical form of the shifts. In the high-temperature regime, we predict that the shifts decrease in magnitude and are inversely proportional to the square of the temperature.