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Classical dynamics of interacting qubits driven by a quantum field

B. M. Rodríguez-Lara, Ray‐Kuang Lee

Published Feb 18, 2011• 0 citations
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Abstract

B. M. Rodŕiguez-Lara 2 and Ray-Kuang Lee 3 Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore 117542 Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan Abstract We present a stability analysis of a qubit ensemble driven by a quantum field in the semiclassical limit. Transitions from Rabi to Josephson dynamics are identified depending on both the intra-qubit interactions to field-ensemble coupling ratio and the ratio between the total excitation number and the ensemble size. The quantized field is found to produce asymmetric dynamics for symmetric initial conditions for both Rabi and Josephson oscillations.

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B. M. Rodŕiguez-Lara 2 and Ray-Kuang Lee 3 Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore 117542 Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan Abstract We present a stability analysis of a qubit ensemble driven by a quantum field in the semiclassical limit. Transitions from Rabi to Josephson dynamics are identified depending on both the intra-qubit interactions to field-ensemble coupling ratio and the ratio between the total excitation number and the ensemble size. The quantized field is found to produce asymmetric dynamics for symmetric initial conditions for both Rabi and Josephson oscillations.
Rabi oscillations are coherent transitions in a quantum two-level system under the influence of a resonant drive, with a much lower frequency dependent on the perturbation amplitude. These serve as one of the signatures of quantum coherent evolution in mesoscopic systems. It was shown recently (Grønbech-Jensen N and Cirillo M 2005 Phys. Rev. Lett. 95 067001) that in phase qubits (current-biased Josephson junctions) this effect can be mimicked by classical oscillations arising due to the anharmonicity of the effective potential. Nevertheless, we find qualitative differences between the classical and quantum effects. Firstly, while the quantum Rabi oscillations can be produced by the subharmonics of the resonant frequency ω 10 (multiphoton processes), the classical effect also exists when the system is excited at the overtones, n ω 10 . Secondly, the shape of the resonance is, in the classical case, characteristically asymmetric, whereas quantum resonances are described by symmetric Lorentzians. Thirdly, the anharmonicity of the potential results in the negative shift of the resonant frequency in the classical case, in contrast to the positive Bloch–Siegert shift in the quantum case. We show that in the relevant range of parameters these features allow us to distinguish confidently the bona fide Rabi oscillations from their classical Doppelgänger.
We investigate the emergence of semiclassical dynamics in the quantum Rabi model using a recently developed limiting procedure that formally establishes correspondence with the semiclassical Rabi Hamiltonian [E. K. Twyeffort Irish and A. D. Armour, Phys. Rev. Lett. 129, 183603 (2022)]. While the limit itself is defined at the Hamiltonian level, how it is reached depends on the choice of quantum states. Defining a set of quantitative measures that capture the differences between quantum and semiclassical dynamics, we examine convergence to the semiclassical limit when the field is prepared in a displaced number state. These states, which interpolate to Fock states for zero displacement, are more general than the set of coherent states usually employed when considering the emergence of semiclassical behavior. Numerical computations of these measures consistently demonstrate the progressive emergence of semiclassical behavior as the joint limit of vanishing coupling and infinite displacement is approached. Complementing the numerical results, analytical approximations are developed that reproduce the behavior in the vicinity of the semiclassical limit with a high degree of fidelity and allow scaling relations to be derived. Although any initial displaced number state will eventually converge to the corresponding semiclassical dynamics as the limit is taken, the rate of convergence depends on the Fock number nnn of the state. States with larger values of nnn, which behave less classically than coherent states, converge more slowly to the limit.