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Physical Review A

Nonlinearity and temperature dependence of drive-induced shifts in a thermal environment

Arpan Chatterjee, Rangeet Bhattacharyya

Published Oct 21, 2020• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.102.043111

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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.
We report an alternate formulation of the quantum master equation (QME) to describe the dynamics of a quantum system weakly coupled to a heat bath, in the presence of weak external driving. A key feature of this approach is the introduction of an explicit Hamiltonian to model the thermal fluctuations in the heat bath. We show that the resulting time coarse-grained dynamical equation for the quantum system has dissipators with a natural regulator, which emerges from an ensemble average over the fluctuations. Importantly, such regularized dissipators arise from the second-order contributions of both the external drive as well as the system-environment coupling. We show that the second-order drive terms, regularized to time-scales set by the fluctuations, result in dynamic drive-induced frequency shifts as well as drive-dependent relaxation phenomena. Considering the specific case of an ensemble of two-level systems, subjected to a linearly polarized external drive, we derive the modified Bloch Equations with such drive-dependent shift and damping terms. The resulting drive-induced frequency shifts converge to the known forms of dynamic frequency shifts, such as the Bloch-Siegert shift or the dynamic Stark shift, in appropriate limits. The Kramers-Kronig pairs of these frequency shifts - manifest as drive-dependent damping terms in the modified Bloch Equations and help explain the Redfield limit of free-induction-decay (FID) rates as well as the non-Bloch decay of Rabi oscillations in isotropic medium. Our method predicts correct orders of magnitudes of non-Bloch decay rates in isotropic medium as well as their observed temperature dependence. The QME reported here, correctly describes all known aspects of the driven-dissipative dynamics up to second-order of an open quantum system with appropriate thermal signatures and as such is expected to provide deeper insights into the study of quantum information processing on real systems.
We report that the Bloch-Siegert shift which appears in Nuclear Magnetic Resonance (NMR) spectroscopy can also be shown to originate as a part of a complex drive-induced second-order susceptibility term. The shift terms thus obtained are shown to have an absorptive Kramers-Kronig pair. The theoretical treatment involves a finite time-propagation of a nuclear spin-1/21/21/2 system and the spin-bearing molecule under the action of thermal fluctuations acting on the latter. The finite propagator is constructed to account for many instances of thermal fluctuations occurring in a time-scale during which the spin density matrix changes infinitesimally. Following an ensemble average, the resulting quantum master equation directly yields a finite time-nonlocal complex susceptibility term from the external drive, which is extremely small but measurable in solution-state NMR spectroscopy. The dispersive part of this susceptibility term originating from the non-resonant component of the external drive results in the Bloch-Siegert shift. We have verified experimentally the existence of the absorptive Kramers-Kronig pair of the second-order shift term, by using a novel refocussed nutation experiment. Our method provides a single approach to explain both relaxation phenomena as well as Bloch-Siegert effect, which have been treated using non-concurrent techniques in the past.
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.
We apply a simple analytical method based on a unitary transformation to calculate the Bloch-Siegert (BS) shift over the entire driving-strength range. In quantitative comparison with the numerically exact BS shift obtained by Floquet formalism as well as the previous BS results, we confirm that our calculated results are not only accurate in the weak-driving regime but also correct in the strong-driving limit. In the intermediate strong-driving regime, the calculated values of the BS shift are nearly the same as the exact ones. It turns out that our calculation for the BS shift is beyond perturbation. Meanwhile, we demonstrate the signatures caused by the BS shift by monitoring the excited-state population and the probe-pump spectrum under the experiment accessible conditions. In particular, we find that when the driving frequency is fixed at the transition frequency of the system, the line shape of the probe-pump spectrum becomes asymmetric with the increase of the driving strength, which may be verified experimentally.
A superconducting qubit was driven in an ultrastrong fashion by an oscillatory microwave field, which was created by coupling via the nonlinear Josephson energy. The observed Stark shifts of the "atomic" levels are so pronounced that corrections even beyond the lowest-order Bloch-Siegert shift are needed to properly explain the measurements. The quasienergies of the dressed two-level system were probed by resonant absorption via a cavity, and the results are in agreement with a calculation based on the Floquet approach.
We suggest that Bose-Einstein condensates in optical lattices subjected to ac forcing with a smooth envelope may provide detailed experimental access to multiphotonlike transitions between ac-Stark-shifted Bloch bands. Such transitions correspond to resonances described theoretically by avoided quasienergy crossings. We show that the width of such anticrossings can be inferred from measurements involving asymmetric pulses. We also introduce a pulse tracking strategy for locating the particular driving amplitudes for which resonances occur. Our numerical calculations refer to a currently existing experimental setup [E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichs\"ollner, and H.-C. N\"agerl, Phys. Rev. Lett. 104, 200403 (2010)].