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Fundamental Aspects of Quantum Brownian Motion CHAOS 15, 026105 ͑2005͒ Fundamental aspects of quantum Brownian motion Peter Hänggi and Gert-Ludwig Ingold Institut für Physik, Universität Augsburg, 86135 Augsburg, Germany ͑Received 1 December 2004; accepted 9 December 2004; published online 17 June 2005͒ With this work we elaborate on the physics of quantum noise in thermal equilibrium and in stationary nonequilibrium. Starting out from the celebrated quantum fluctuation-dissipation theorem we discuss some important consequences that must hold for open, dissipative quantum systems in thermal equilibrium. The issue of quantum dissipation is exemplified with the fundamental problem of a damped harmonic quantum oscillator. The role of quantum fluctuations is discussed in the context of both, the nonlinear generalized quantum Langevin equation and the path integral ap- proach. We discuss the consequences of the time-reversal symmetry for an open dissipative quan- tum dynamics and, furthermore, point to a series of subtleties and possible pitfalls. The path integral methodology is applied to the decay of metastable states assisted by quantum Brownian noise. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1853631͔ This work deals with the description of quantum Brown- the substitution of the energy kBT from the classical equipar- ian motion in linear and nonlinear quantum systems that tition law4 by the thermally averaged quantum energy ͑but exhibit frictional influences. The symmetries of thermal leaving out the zero point energy contribution͒ of the har- equilibrium impose severe constraints on the time evolu- monic oscillator. Nyquist’s remark thus constitutes a precur- tion properties of open quantum systems. These lead to a sor of the celebrated work by Callen and Welton5 who gen- quantum generalization of the classical Einstein relation eralized the relations by Einstein, Nyquist, and Johnson to that connects friction with the strength of thermal quan- include quantum effects: In their work they put forward a tum fluctuations. There exist a variety of theoretical generally valid connection between the response function roadways to model quantum dissipation. Here, we discuss and the associated quantum fluctuations in equilibrium, the the topic for the prominent case of a damped harmonic oscillator upon combining thermodynamics and linear re- quantum fluctuation-dissipation theorem. sponse theory. A dissipative nonlinear quantum dynamics Without doubt, quantum fluctuations constitute a promi- can be dealt with a generalized quantum Langevin equa- nent noise source in many nanoscale and biological systems. tion, a path integral formulation, or in terms of a gener- For example, the tunnelling and the transfer of electrons, alized quantum master equation for the corresponding quasiparticles, and alike, is assisted by noise for which the reduced dynamics. We illustrate the situation for the quantum nature cannot be neglected. The features of this problem of the dissipative decay out of a metastable state. noise change drastically as a function of temperature: At suf- Furthermore, we point out a series of subtleties, pitfalls ficiently high temperatures a crossover does occur to classi- and shortcomings that one must be aware of when con- cal Johnson–Nyquist noise. fronted with the world of quantum noise driven With this work we shall present various methods and phenomena. schemes of modelling quantum Brownian motion from first principles. In particular, the thermal noise must at all times I. INTRODUCTION obey the quantum version of the fluctuation-dissipation theo- ͑ ͒ Albert Einstein explained the phenomenon of Brownian rem à la Callen-Welton . This latter property is necessary in motion in his annus mirabilis of 1905 by use of statistical order to be consistent with the second law of thermodynam- ͑ ͒ methods which he ingeniously combined with the laws of ics and the principle of quantum detailed balance. We thermodynamics.1 In this pioneering work he as well pro- elaborate on several alternative but equivalent methods to vided a first link between the dissipative forces and the im- describe quantum noise and quantum Brownian motion per peding thermal fluctuations, known as the Einstein relation se: These are the functional integral method for dissipative 6,7 which relates the strength of diffusion to the friction. This quantum systems and time-dependent driven quantum 8 9 intimate connection between dissipation and related fluctua- systems, the quantum Langevin ͑operator͒ approach, sto- 10,11 tions was put on a firm basis much later when Nyquist2 and chastic schemes, or the concept of stochastic Schrödinger 12 Johnson3 considered the spectral density of voltage- and equations. In doing so, we call attention to distinct differ- current-fluctuations. ences to the classical situation and, as well, identify a series What role do quantum mechanics and the associated of delicate pitfalls which must be observed when making quantum fluctuations play in this context? After the birth of even innocent looking approximations. Such pitfalls involve, quantum mechanics in the early 1920’s we can encounter in among others, the rotating-wave approximation, the use of the very final paragraph of the 1928 paper by Nyquist for the quasiclassical Langevin forces, the quantum regression hy- first time the introduction of quantum mechanical noise via pothesis and/or the Markov approximation.6,8,13 1054-1500/2005/15͑2͒/026105/12/$22.5015, 026105-1 © 2005 American Institute of Physics Downloaded 21 Jun 2005 to 137.250.81.48. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp 026105-2 P. Hanggi and G.-L. Ingold Chaos 15, 026105 ͑2005͒ II. THE QUANTUM FLUCTUATION-DISSIPATION =Y͑␻͒␦V͑␻͒ where the admittance Y͑␻͒ is identical to the THEOREM AND ITS IMPLICATIONS ␹ ͑␻͒ ˙ susceptibility IQ . As a consequence of I=Q, the symme- As already mentioned, in 1951 Callen and Welton trized power spectrum of the current fluctuations is given by ͑␻͒ ␻ ͑␻͒ proved a pivotal relation between equilibrium fluctuations SII =i SIQ so that we obtain and dissipative transport coefficients. Note also that this quantum fluctuation-dissipation relation holds true indepen- ប␻ dent of particle statistics. The following cornerstone achieve- ͑␻͒ ប␻ ͩ ͪ ͑␻͒ 5 SII = coth Re Y ments can be found in this primary work: 2kBT ͑1͒ The generalization of the classical Nyquist’s formula to ប␻ ប␻ =2ͫ + ͬRe Y͑␻͒. ͑6͒ the quantum case; 2 e␤ប␻ −1 ͑2͒ the quantum mechanical proof that susceptibilities are ӷប␻ related to the spectral densities of symmetrized correla- In the high temperature limit kBT , we recover the ͑␻͒! ͑␻͒ tion functions. results of Nyquist and Johnson, i.e. SII 2kBT Re Y . For the Markovian limit of an Ohmic resistor, where Y͑␻͒ ͑␻͒ For a single degree of freedom, linear response theory =1/R, this result simplifies to read SII =2kBT/R. The yields for the change of the expectation value of an operator- quantum version was already anticipated by Nyquist in the valued observable B due to the action of a ͑classical͒ force last paragraph of his 1928 paper.2 However, he made use of F͑t͒ that couples to the conjugate dynamical operator A the original expression of Planck which yields only the sec- ond contribution present in the lower line of ͑6͒. Nyquist t ͗␦ ͑ ͒͘ ͵ ␹ ͑ ͒ ͑ ͒ ͑ ͒ thus missed the first term arising from the vacuum energy B t = ds BA t − s F s . 1 −ϱ which already appears in a paper by Planck published in 1911.14 ␦ ͑ ͒ ͑ ͒ ͗ ͘ Here, B t =B t − B 0 denotes the difference with respect On the other hand, in the extreme quantum limit kBT to the thermal equilibrium average ͗B͘ in the absence of the Ӷប␻ ͑␻͒!ប␻ ͑␻͒ 0 , we find that SII Re Y . In particular, this force. The reaction of the system is contained in the response implies that at zero frequency the spectral weight of the cur- ␹ ͑ ͒ function BA t with a so-called dissipative part rent fluctuations vanishes in the generic case where the ad- 1 mittance does not exhibit an infrared divergence. ␹d ͑t͒ = ͓␹ ͑t͒ − ␹ ͑− t͔͒. ͑2͒ We cannot emphasize enough that the quantum BA 2i BA AB fluctuation-dissipation relation ͑5͒ and corresponding impli- ␹d ͑ ͒ ␹d ͑␻͒ The Fourier transform of BA t will be denoted by ˜ BA .It cations hold true for any isolated, closed quantum system. is worth noting here that only when A=B does this part in Thus, upon contracting the dynamics in full phase space onto fact coincide with the imaginary part of the complex-valued a reduced description of an open quantum system exhibiting ␹Љ ͑␻͒ susceptibility ˜ BA . dissipation these relations hold true nevertheless. Therefore, The fluctuations are described by the equilibrium corre- care must be taken when invoking approximations in order to lation function avoid any violation of these rigorous relations. We next con- sider the role of quantum dissipation for an exactly solvable ͑ ͒ ͗␦ ͑ ͒␦ ͑ ͒͘ ͑ ͒ CBA t = B t A 0 ␤ 3 situation: the damped quantum harmonic oscillator dynam- ␤ ics. at inverse temperature =1/kBT. The correlation function is complex-valued because the operators B͑t͒ and A͑0͒ in gen- ͑ ͒ eral do not commute. While the antisymmetric part of CBA t is directly related to the response function by linear response theory, the power spectrum of the symmetrized correlation III. QUANTUM DISSIPATION: THE DAMPED function HARMONIC OSCILLATOR ͑ ͒ 1 ͗␦ ͑ ͒␦ ͑ ͒ ␦ ͑ ͒␦ ͑ ͒͘ ͑ ͒ A. Equilibrium correlation functions SBA t = 2 B t A 0 + A 0 B t 4 Let us next consider the most fundamental case of a depends on the Fourier transform of the dissipative part of simple open quantum system, namely, the damped harmonic the response function via oscillator.
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