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arXiv:quant-ph/9804003 · Physical Review A

Fluctuations, time-correlation functions, and geometric phase

Arun Kumar Pati

Published Jul 1, 1999• 9 citations
arXivHTMLPDF

Abstract

The adiabatic approximation usually applies to a collection of lighter particles (the ``fast'' system) and heavier particles (the ``slow'' system), where the geometric phase naturally appears. With the help of the recently introduced gauge potential in the context of adiabatic Berry phase for open paths, we establish a fluctuation-correlation theorem by relating the quantum fluctuations in the generator of the parameter change (of the ``slow'' system) to the time integral of the quantum correlation function between the projection operator and force operator acting on the ``fast'' system. By taking a cue from linear response theory we relate the quantum fluctuation in the generator to the generalized susceptibility. The relation between the open-path geometric phase, diagonal elements of the quantum metric tensor, and the force-force correlation function are provided and the classical limit of the fluctuation-correlation theorem is discussed.

DOI
10.1103/physreva.60.121

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The adiabatic approximation usually applies to a collection of lighter particles (the fast'' system) and heavier particles (the slow'' system), where the geometric phase naturally appears. With the help of the recently introduced gauge potential in the context of adiabatic Berry phase for open paths, we establish a fluctuation-correlation theorem by relating the quantum fluctuations in the generator of the parameter change (of the slow'' system) to the time integral of the quantum correlation function between the projection operator and force operator acting on the fast'' system. By taking a cue from linear response theory we relate the quantum fluctuation in the generator to the generalized susceptibility. The relation between the open-path geometric phase, diagonal elements of the quantum metric tensor, and the force-force correlation function are provided and the classical limit of the fluctuation-correlation theorem is discussed.
Geometrical concepts have played an important role in quantum mechanics. In the Schrodinger representation, the Berry–Simon connection and the quantum geometric tensor appear as gauge potentials in the effective Hamiltonian for a heavy system under the Born–Oppenheimer approximation. On the other hand, the Berry phase should arise as the amplitude of each path in the pathintegral formalism. In their pioneering work, Kuratsuji and Iida have developed the treatment of the Born– Oppenheimer approximation in the path integral quantization, where the Berry phase of the light system naturally appears in the effective action for the heavy system. However, they did not derive the gauge potentials themselves in the effective Hamiltonian. In this short note, we show that the gauge potentials which relate to the geometry of the Hilbert space of the light system are also derived in the adiabatic scheme of path integrals. Our result presents the path-integral formalism to consider the effective dynamics of the heavy system under the adiabatic approximation. Let us consider the following Hamiltonian:
The autocorrelation function of the force acting on a slow classical system, resulting from interaction with a fast quantum system is calculated following Berry-Robbins and Jarzynski within the leading order correction to the adiabatic approximation. The time integral of the autocorrelation function is proportional to the rate of dissipation. The fast quantum system is assumed to be chaotic in the classical limit for each configuration of the slow system. An analytic formula is obtained for the finite time integral of the correlation function, in the framework of random matrix theory (RMT), for a specific dependence on the adiabatically varying parameter. Extension to a wider class of RMT models is discussed. For the Gaussian unitary and symplectic ensembles for long times the time integral of the correlation function vanishes or falls off as a Gaussian with a characteristic time that is proportional to the Heisenberg time, depending on the details of the model. The fall off is inversely proportional to time for the Gaussian orthogonal ensemble. The correlation function is found to be dominated by the nearest neighbor level spacings. It was calculated for a variety of nearest neighbor level spacing distributions, including ones that do not originate from RMT ensembles. The various approximate formulas obtained are tested numerically in RMT. The results shed light on the quantum to classical crossover for chaotic systems. The implications on the possibility to experimentally observe deterministic friction are discussed. PACS: 05.45.Mt, 05.45.Ac, 05.45.-a 1
An isolated classical chaotic system, when driven by the slow change in several parameters, responds with two reaction forces: geometric friction and geometric magnetism. By using the theory of quantum fluctuation relations, we show that this holds true also for open quantum systems and provide explicit expressions for those forces in this case. This extends the concept of Berry curvature to the realm of open quantum systems. We illustrate our findings by calculating the geometric magnetism of a damped charged quantum harmonic oscillator transported along a path in physical space in the presence of a magnetic field and a thermal environment. We find that, in this case, the geometric magnetism is unaffected by the presence of the heat bath.
The autocorrelation function of the force acting on a slow classical system, resulting from interaction with a fast quantum system is calculated following Berry-Robbins, Wilkinson and Jarzynski within the leading order correction to the adiabatic approximation. The time integral of the autocorrelation function is often proportional to the rate of energy transfer between the systems. The fast quantum system is assumed to be chaotic in the classical limit for each configuration of the slow system. An analytic formula is obtained for the finite-time integral of the correlation function, in the framework of random matrix theory (RMT), for a specific dependence on the adiabatically varying parameter. Extension to a wider class of RMT models is discussed. For the Gaussian unitary and symplectic ensembles for long times the time integral of the correlation function vanishes or falls off as a Gaussian with a characteristic time that is proportional to the Heisenberg time, depending on the details of the model. The fall-off is inversely proportional to time for the Gaussian orthogonal ensemble. The correlation function is found to be dominated by the nearest-neighbour level spacings. It was calculated for a variety of nearest-neighbour level spacing distributions, including ones that do not originate from RMT ensembles. The various approximate formulae obtained are tested numerically in RMT. The results shed light on the quantum to classical crossover for chaotic systems. The implications on the possibility to experimentally observe deterministic friction are discussed.
A precise definition of an adiabaticity parameter \ensuremath{\nu} of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(\ensuremath{\tau})=${\ensuremath{\sum}}_{\mathrm{\ensuremath{\ell}}}$${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) with ${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) being at least of the order ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\ell}}}$. In particular, U(0){\mathrm{U}}^{(0)}U(0)(\ensuremath{\tau}) corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the 1/\ensuremath{\tau} expansion of U(\ensuremath{\tau}). It is also shown that the nonadiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. This suggests the introduction of an adiabatic product expansion for U(\ensuremath{\tau}) which turns out to yield exact expressions for U(\ensuremath{\tau}) for a large number of quantum systems. In particular, a simple application of the adiabatic product expansion is used to show that for the Hamiltonian describing the dynamics of a magnetic dipole in an arbitrarily changing magnetic field, there exists another Hamiltonian with the same eigenvectors for which the Schr\"odinger equation is exactly solvable. Some related issues concerning geometric phases and their physical significance are also discussed.
Adiabatic theorem and non-adiabatic corrections have been widely applied in modern quantum technology. Recently, non-adiabatic linear response theory has been developed to probe the many-body correlations in closed systems. In this work, we generalize the non-adiabatic linear response theory to open quantum many-body systems. We show that, similar to the closed case, the first-order deviation from the instantaneous steady state is memoryless—it depends only on the final parameters and not on the initial state or ramping path. When ramping the Hamiltonian, the linear response of observables is governed by the derivative of the retarded Green’s function, as in closed systems. In contrast, ramping the dissipation gives rise to a different response, characterized by a high-order correlation function defined in the steady state. Our results offer a compact and physically transparent formulation of non-adiabatic response in open systems, and demonstrate that ramping dynamics can serve as a versatile tool for probing many-body correlations beyond equilibrium.