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Physical Review A

Quasiclassical approximation for the spin-boson Hamiltonian with counterrotating terms

G. A. Finney, Julio R. Gea-Banacloche

Published Sep 1, 1994• 25 citations
Publisher

Abstract

We derive quasiclassical (approximately factorizable) trajectories for the spin-boson Hamiltonian without the rotating-wave approximation (RWA) and show how they may be used to calculate the evolution of the fully quantized system when the initial state of the oscillator (boson field) is a coherent state with sufficiently large amplitude. We give a general prescription to obtain such trajectories for other systems. We study how well-known features of the model in the RWA are modified by the counterrotating terms and explore some of the new phenomenology introduced by these terms.

DOI
10.1103/physreva.50.2040

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We derive quasiclassical (approximately factorizable) trajectories for the spin-boson Hamiltonian without the rotating-wave approximation (RWA) and show how they may be used to calculate the evolution of the fully quantized system when the initial state of the oscillator (boson field) is a coherent state with sufficiently large amplitude. We give a general prescription to obtain such trajectories for other systems. We study how well-known features of the model in the RWA are modified by the counterrotating terms and explore some of the new phenomenology introduced by these terms.
We study the interaction between a two-level system and a harmonic oscillator without the rotating-wave approximation (RWA). In quantum optics, this model (sometimes called the spin-boson model) is traditionally thought of as a two-Ievel atom coupled to a single-mode radiation field, but it could also describe (1) a two-level ion in a trap coupled to the motion of its own center of mass or (2) a two-level particle in a solid coupled to a phonon mode. For sufficiently strong coupling, this model is known to exhibit chaos in a variety of semi-classical approximations.1
The single-mode spin-boson model exhibits behavior not included in the rotating-wave approximation (RWA) in the ultra and deep-strong coupling regimes, where counter-rotating contributions become important. We introduce a symmetric rotating-wave approximation that treats rotating and counter-rotating terms equally, preserves the invariances of the Hamiltonian with respect to its parameters, and reproduces several qualitative features of the spin-boson spectrum not present in the original rotating-wave approximation both off-resonance and at deep-strong coupling. The symmetric rotating-wave approximation allows for the treatment of certain ultra- and deep-strong coupling regimes with similar accuracy and mathematical simplicity as does the RWA in the weak-coupling regime. Additionally, we symmetrize the generalized form of the rotating-wave approximation to obtain the same qualitative correspondence with the addition of improved quantitative agreement with the exact numerical results. The method is readily extended to higher accuracy if needed. Finally, we introduce the two-photon parity operator for the two-photon Rabi Hamiltonian and obtain its generalized symmetric rotating-wave approximation. The existence of this operator reveals a parity symmetry similar to that in the Rabi Hamiltonian as well as another symmetry that is unique to the two-photon case, providing insight into the mathematical structure of the two-photon spectrum, significantly simplifying the numerics, and revealing some interesting dynamical properties.
We study the spin-boson model without the counterrotating terms by a numerically exact method based on variational matrix product states. Surprisingly, the second-order quantum phase transition (QPT) is observed for the sub-Ohmic bath in the rotating-wave approximations. Moreover, first-order QPTs can also appear before the critical points. With the decrease of the bath exponents, these first-order QPTs disappear successively, while the second-order QPT remains robust. The second-order QPT is further confirmed by multi-coherent-state variational studies, while the first-order QPT is corroborated with the exact diagonalization in the truncated Hilbert space. Extension to the Ohmic bath is also performed, and many first-order QPTs appear successively in a wide coupling regime, in contrast to previous findings. The previous pictures for many physical phenomena for the spin-boson model in the rotating-wave approximation have to be modified at least at strong coupling.
We derive state-dependent bounds on the difference between two quantum evolutions generated by unbounded Hamiltonians sharing a common form domain. The main technical tool is a second integration by parts, performed at the level of sesquilinear forms rather than at the operator level, which removes the need for a common invariant operator domain. The resulting estimate involves the norm of the time-integrated difference of the two generators, rather than the integral of its norm, and is therefore sensitive to the averaging effects produced by fast-oscillating terms. As an application we prove a quantitative bound on the rotating-wave approximation for spin-boson models with a structured boson field, described by an arbitrary massive dispersion relation on a general measure space and by a suitable class of form factors. The proof involves a careful analysis of the high-frequency scaling. The bound holds on a dense subspace of states, is fully explicit, and all the constants entering it depend only on the parameters of the model and not on the frequency scale, so that the approximation becomes exact in the limit of large frequency.
By a model of a two-level particle coupled with boson field, we made it clear that an evolution problem can be solved beyond the rotating wave approximation. We have applied the coherent approximation method, which had been proved to be effective in dealing with stationary state problems of polaron, to the evolution problem of the system mentioned above. The results obtained showed that the coherent approximation method is effective to treat the evolution problem and, in general cases, the non-rotating wave terms in Hamiltonian should not be ignored. Our results may provide a deep physical insight for further experiments to test the effects of non-rotating wave terms.
Statistical and dynamical properties of a quasiparticle coupled to polarization vibrations in a dimer model are investigated using a full quantum-mechanical approach. The propagation of the system is described in terms of spin-boson coherent states and compared to various quasiclassical descriptions of the system. The spectrum of the energy eigenvalues and the level spacing distributions of the system are calculated and turn out to be almost regular, especially in the strong coupling limit. It is demonstrated that the usual quasiclassical approach to this problem is not quite satisfactory. This is because the quantum fluctuations in the spin variable give rise to entangled states and correlations which hence should be included in a quasiclassical description. We show that a limited but satisfactory quasiclassical approximation may be given using a suitable integrable reference Hamiltonian.