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Physical review. B, Condensed matter

Variational study of dissipative two-state systems

Yunlong Shi, Hong Chen, Xiang Xia Wu

Published Jan 15, 1994• 4 citations
Publisher

Abstract

Two variational states, the displaced state and the displaced-squeezed state, are compared for a two-state system with super-Ohmic dissipation, and their dependence on the form of the coupling strength ${\mathit{g}}_{\mathit{l}}$\ensuremath{\sim}${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}^{\ensuremath{\lambda}}$ (${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}$ is the phonon frequency) is analyzed. We find that, in the strong-coupling region, the displaced-squeezed state is more stable and the ground-state properties depend not only on the spectral density but also on the index \ensuremath{\lambda}. Moreover, we find that the stable area of the displaced-squeezed state increases as \ensuremath{\lambda} decreases.

DOI
10.1103/physrevb.49.2931

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Two variational states, the displaced state and the displaced-squeezed state, are compared for a two-state system with super-Ohmic dissipation, and their dependence on the form of the coupling strength gl{\mathit{g}}_{\mathit{l}}gl​\ensuremath{\sim}\ensuremathωl\ensuremathλ{\mathrm{\ensuremath{\omega}}}_{\mathit{l}}^{\ensuremath{\lambda}}\ensuremathωl\ensuremathλ​ (\ensuremathωl{\mathrm{\ensuremath{\omega}}}_{\mathit{l}}\ensuremathωl​ is the phonon frequency) is analyzed. We find that, in the strong-coupling region, the displaced-squeezed state is more stable and the ground-state properties depend not only on the spectral density but also on the index \ensuremath{\lambda}. Moreover, we find that the stable area of the displaced-squeezed state increases as \ensuremath{\lambda} decreases.
摘要 采用变分方法，比较了超欧姆耗散两能态系统中位移变分态和位移－压缩变分态的特性。通过研究发现，在强耦合区域，位移－压缩态较稳定，其基态性质不仅取决于谱密度，而且还依赖于耦合强度ｇ1～ω1λ的指数λ；位移－压缩态稳定的区域将随λ减小而增加。对于欧姆耗散两能态系统，在位移－压缩态退局域相区域亦有同样的特性，且该区域亦随λ减小而增加。 Abstract The properties of two variational states, the displaced state and the displaced-squeezed state, are compared for a two-state system with superohmic dissipation by a variational approach. We find that, in strong coupling region, the displaced-squeezed state is the more stable and the ground-state properties depend not only on the spectral density but also on the index λ of the coupling strength ｇ1～ω1λ(ω1 is phonon frequenceis). Moreover, we obtain that the stable area of the displaced-squeezed state increases as λ reduces. Similar properties have been obtained for ohmic dissipation and it is shown that the delocalization area of the displaced-squeezed state increases as λ reduces. 作者及机构信息 石云龙2, 聂一行2, 陈鸿1, 吴翔1 (1)同济大学物理系; (2)雁北师范学院物理系 基金项目: 国家自然科学基金和山西省自然科学基金资助的课题. Authors and contacts SHI YUN-LONG2, NIE YI-HANG2, CHEN HONG1, WU XIANG1 (1)同济大学物理系; (2)雁北师范学院物理系 参考文献 施引文献
The static properties of a dissipative two-state system are studied by perturbation, mapping to the quantum sine-Gordon model, and a variational approach. We find that its properties at zero temperature depend not only on the spectral density J(\ensuremath{\omega})\ensuremath{\sim}tsuml{\mathit{tsum}}_{\mathit{l}}tsuml​gl2{\mathit{g}}_{\mathit{l}}^{2}gl2​\ensuremath{\delta}(\ensuremath{\omega}-${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}$), but also on the explicit form of the coupling strength gl{\mathit{g}}_{\mathit{l}}gl​\ensuremath{\sim}${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}^{\ensuremath{\lambda}}$ (${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}$ is the phonon frequency). For weak coupling (\ensuremath{\lambda}\ensuremath{\ge}1), the effect of displacement of the phonon state is dominant; while for strong couplings (\ensuremath{\lambda}1), one must take account of both displacement and deformation. Displaced squeezed states are proposed as ground states of the bath under strong coupling,and we show that the localization transition of the Ohmic dissipation occurs at ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$=3-2\ensuremath{\lambda} instead of ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$=1 as for the weak-coupling case.
The ground state of the model of a two-state system coupled with phonons is studied by a variational method. Two trial states, the displaced state, and the displaced-squeezed state, for the phonon subsystem are compared, and their dependence on tunneling parameter \ensuremath{\Delta}, phonon frequency \ensuremath{\omega}, and coupling strength g is analyzed. We find that for \ensuremath{\Delta}/\ensuremath{\omega}\ensuremath{\ll}1 the displaced state is more stable for g/\ensuremath{\omega}<1, while the displaced-squeezed state is preferable for g/\ensuremath{\omega}>1.
The ground states of phonons in a dissipative quantum tunneling system and their influences on the localiza ion-delocalization transtion are investigated by a variational approach. We find that: 1. There are two kind of influences, displacement and deformation, on phonon states due to coupling with a tunneling particle, and displaced -squeezed states are proposed to describe both these influences. 2. The localization-delocahzation transition depends on the choice of phonon ground state. The transition will occur at strong coupling as the energy of the ground state decreases.