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arXiv:nlin/0306017 · Waves in Random Media

Irreversible quantum graphs

Uzy Smilansky

Published Jan 1, 2004• 31 citations
arXivHTMLPDF

Abstract

Irreversibility is introduced to quantum graphs by coupling the graphs to a bath of harmonic oscillators. The interaction which is linear in the harmonic oscillator amplitudes is localized at the vertices. It is shown that for sufficiently strong coupling, the spectrum of the system admits a new continuum mode which exists even if the graph is compact, and a {\it single} harmonic oscillator is coupled to it. This mechanism is shown to imply that the quantum dynamics is irreversible. Moreover, it demonstrates the surprising result that irreversibility can be introduced by a "bath" which consists of a {\it single} harmonic oscillator.

DOI
10.1088/0959-7174/14/1/016

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Irreversibility is introduced to quantum graphs by coupling the graphs to a bath of harmonic oscillators. The interaction which is linear in the harmonic oscillator amplitudes is localized at the vertices. It is shown that for sufficiently strong coupling, the spectrum of the system admits a new continuum mode which exists even if the graph is compact, and a {\it single} harmonic oscillator is coupled to it. This mechanism is shown to imply that the quantum dynamics is irreversible. Moreover, it demonstrates the surprising result that irreversibility can be introduced by a "bath" which consists of a {\it single} harmonic oscillator.
A system of a particle and a harmonic oscillator, which have pure point spectra if uncoupled, is known to acquire an absolutely continuous spectrum when they are coupled by a sufficiently strong point interaction. Here, the dynamical mechanism underlying this spectral phenomenon is exposed. The energy of the oscillator is proven to exponentially diverge in time, while the spatial probability distribution of the particle collapses into a δ-function at the interaction point. On account of this result, a generalized model with many oscillators which interact with the particle at different points is argued to provide a formal model for the approximate measurement of position and collapse of wave packets.
A pure quantum state of large number N of oscillators, interacting via harmonic coupling, evolves such that any small subsystem n<<N of the global state approaches equilibrium. This provides a different example where stability emerges as natural phenomena under quantum dynamics alone, with no necessity to bring in any additional statistical postulates. Mixing of equilibrated subsystems consisting of 1,2,...,n<<N clearly indicates that small subsystems are entangled with the rest of the state, i.e., the bath. Every single mode oscillator is found to relax in a mixed density matrix of the Boltzmann canonical form. In two oscillator stationary subsystems, intraentanglement within the "system" oscillators is found to exist when the magnitude of the squeezing parameter of the bath is comparable in magnitude with that of the coupling strength.
We analyze the strong-coupling dynamics of a driven harmonic oscillator whose energy is modulated by a continuum of other bosonic modes. This type of system-bath interaction appears, for example, in optomechanical or equivalent circuit QED setups, where the frequency of a confined photonic mode depends linearly on a fluctuating boundary. Compared to the canonical spin-boson model, where coupling to bath modes only leads to decoherence, the role of the environment in such systems is more complex, since it also provides the only source of nonlinearity. We show that even for an unstructured bath, these environment-induced nonlinearities can dominate over decoherence processes resulting in Rabi oscillations and the formation of highly non-classical states. These findings provide important insights into the non-Markovian dynamics of higher-dimensional open quantum systems and for realizing few-photon optical nonlinearities through strong interactions with a bath.
Ullersma's model consistently describes the quantum-mechanical damped harmonic oscillator. Proper limiting processes in the initially reversible solutions lead to irreversible behavior. It is shown that the irreversible solutions can directly be derived from a field version of the discrete Ullersma Hamiltonian H (H\ensuremath{\rightarrow}HF{H}_{F}HF​). In the ground state of the system the phonon occupation number of the oscillator does not vanish. Therefore, these ``bare'' phonons cannot be interpreted as ``physical'' quanta or particles. By suitable canonical transformations, the bare-field description is transformed into a physical description (HF{H}_{F}HF​\ensuremath{\rightarrow}Hph{H}_{\mathrm{ph}}Hph​), which allows the definition of physical particles and which is characterized by a renormalized frequency, a screened interaction between oscillator and bath, and a symmetry of position and momentum operators. The physical description is closely related to the rotating-wave approximation of the original model. The relaxation behavior and the equilibrium properties of the different descriptions are discussed and compared.