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Journal of Physics A Mathematical and General

Two-level system coupled to a boson mode: the large n limit

M Schmutz

Published Dec 1, 1986• 11 citations
Publisher

Abstract

The classical limit n>>0 is investigated for a two-level system coupled to a boson mode. The usual formulations are found to neglect a shift lambda 2 of the spectrum where lambda is the strength of the coupling. A scaling law for the highly excited energy levels is derived and tested numerically. An insight is obtained into the characteristic properties of the energy level distribution which is of interest in connection with possible 'quantum chaotic' behaviour. The interpretation of the asymptotically effective Hamiltonian as a one-dimensional tight-binding model for the two-band Stark ladder is discussed.

DOI
10.1088/0305-4470/19/17/021

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The classical limit n>>0 is investigated for a two-level system coupled to a boson mode. The usual formulations are found to neglect a shift lambda 2 of the spectrum where lambda is the strength of the coupling. A scaling law for the highly excited energy levels is derived and tested numerically. An insight is obtained into the characteristic properties of the energy level distribution which is of interest in connection with possible 'quantum chaotic' behaviour. The interpretation of the asymptotically effective Hamiltonian as a one-dimensional tight-binding model for the two-band Stark ladder is discussed.
We consider the quantum Rabi model, a fundamental model of quantum optics describing a two-level system coupled to a bosonic mode, including counter-rotating terms. In the limit of large separation between the two-level transition frequency and the bosonic mode frequency, the model exhibits a quantum phase transition at a critical coupling strength. Although the model is exactly solvable, simple analytic expressions for important physical quantities are generally unavailable. We derive rigorous upper and lower bounds for the ground-state energy and boson occupation number across the quantum phase transition at finite frequency ratios. Our results characterize the subcritical, critical, and supercritical regimes and establish the scaling behavior of the boson occupation near the critical point. Exact numerical diagonalization demonstrates the tightness of the derived bounds, while the resulting finite-frequency inequalities provide experimentally testable predictions for trapped-ion quantum simulators.
The Hamiltonian is derived for the interaction of a Λ-type three-level atom with a two-mode quantum cavity field from the general interaction Hamiltonian between a multi-level atom and a multi-mode radiation field, and it is reduced to an effective two-mode Raman-coupled model under a large detuning condition. We propose a modified effective Hamiltonian for the two-mode Raman-coupled model, find the time-dependent state vectors, and present the validity conditions for the involved interaction Hamiltonians. It is shown that in the study of the two-mode Raman-coupled model it is not enough to retain only the usually used effective Hamiltonian, one must also take into account the ac Stark shift of the atomic levels (at least one of the levels). Finally, we study the atomic dynamics in the interaction of a Λ-type three-level atom with a two-mode quantum cavity field and in the two-mode Raman-coupled model. We find that the number of collapse-revivals, the collapse time and the revival time show new characteristics.
The time-dependent behavior of a two-level system interacting with a quantum oscillator system is analyzed in the case of a coupling larger than both the energy separation between the two levels and the energy of the quantum oscillator ($\ensuremath{\Omega}<\ensuremath{\omega}<\ensuremath{\lambda}$, where $\ensuremath{\Omega}$ is the frequency of the transition between the two levels, $\ensuremath{\omega}$ is the frequency of the oscillator, and $\ensuremath{\lambda}$ is the coupling between the two-level system and the oscillator). Our calculations show that the amplitude of the expectation value of the oscillator coordinate decreases as the two-level system undergoes the transition from one level to the other, while the transfer probability between the levels is staircaselike. This behavior is explained by the interplay between the adiabatic and nonadiabatic regimes encountered during the dynamics with the system acting as a quantum counterpart of the Landau-Zener model. The transition between the two levels occurs as long as the expectation value of the oscillator coordinate is driven close to zero. On the contrary, if the initial conditions are set such that the expectation values of the oscillator coordinate are far from zero, the system will remain locked on one level.
We study the spectral properties of a spin-boson Hamiltonian that depends on two continuous parameters 0 < or = Lambda < infinity (interaction strength) and 0 < or = alpha < or = pi/2 (integrability switch). In the classical limit, this system has two distinct integrable regimes, alpha=0 and alpha=pi/2 . For each integrable regime we can express the quantum Hamiltonian as a function of two action operators. Their eigenvalues (multiples of variant Planck's over 2pi ) are the natural quantum numbers for the complete level spectrum. This functional dependence cannot be extended into the nonintegrable regime (0<alpha<pi/2) . Here level crossings are prohibited and the level spectrum is naturally described by a single (energy sorting) quantum number. In consequence, the tracking of individual eigenstates along closed paths through both regimes leads to conflicting assignments of quantum numbers. This effect is a useful and reliable indicator of quantum chaos-a diagnostic tool that is independent of any level-statistical analysis.
Quantum light-matter systems at strong coupling are notoriously challenging to analyze due to the need to include states with many excitations in every coupled mode. We propose a nonperturbative approach to analyze light-matter correlations at all interaction strengths. The key element of our approach is a unitary transformation that achieves asymptotic decoupling of light and matter degrees of freedom in the limit where light-matter interaction becomes the dominant energy scale. In the transformed frame, truncation of the matter or photon Hilbert space is increasingly well justified at larger coupling, enabling one to systematically derive low-energy effective models, such as tight-binding Hamiltonians. We demonstrate the versatility of our approach by applying it to concrete models relevant to electrons in crystal potential and electric dipoles interacting with a cavity mode. A generalization to the case of spatially varying electromagnetic modes is also discussed.