Quantum Dissipative Systems

The following sections are included:

• Preface
• Preface to the Fourth Edition
• Preface to the Third Edition
• Preface to the Second Edition
• Preface to the First Edition
• Contents

Quantum-statistical mechanics is a very rich and checkered field. It is the theory dealing with the dynamical behavior of spontaneous quantal fluctuations.

When probing dynamical processes in complex many-body systems, one usually employs an external force which drives the system slightly or far away from equilibrium, and then measures the time-dependent response to this force. The standard experimental methods are quasielastic and inelastic scattering of light, electrons, and neutrons off a sample, and the system’s dynamics is analyzed from the line shapes of the corresponding spectra. Other experimental tools are, e. g., spin relaxation experiments, study of the absorptive and dispersive acoustic behaviors, and investigation of transport properties. In such experiments, the system’s response gives information about the dynamical behavior of the spontaneous fluctuations. Theoretically, the response is rigorously described in terms of time correlation functions. Therefore, time correlation functions and their Fourier transforms are at the core of interest in theoretical studies of the relaxation dynamics of nonequilibrium systems…

Often in condensed phases, a rather complex physical situation can adequately be described by a global model system consisting of only one or few relevant dynamical variables in contact with a huge environment, of which the number of degrees of freedom is very large or even infinity. If we are interested in the physical properties of the small relevant system alone, we have to handle this system as an open system which exchanges energy with its surroundings in a random manner. In the last fifty years, a great variety of different theoretical methods for open quantum systems has been developed and employed. In this book, emphasis is put on the functional integral approach to open quantum systems. Over the years, this method has turned out to be very powerful and has found broad application. Nevertheless, I find it appropriate to begin with a survey of various other formalisms. Clearly, the discussion given subsequently can not do justice to all of them. However, I hope that the interested reader will be able to get a line along the given references for deeper studies. I find it appropriate to begin with a brief discussion of the classical regime.

The following sections are included:

• Classical Langevin equation and quantum mechanical generalization
  • Langevin equation for a classical damped system
  • Quantum mechanical and quasiclassical Langevin equation
• New schemes of quantization
• Traditional system-plus-reservoir methods
  • Quantum-mechanical master equations for weak coupling
  • Lindblad theory
  • Operator Langevin equations for weak coupling
  • Phenomenological methods
• Stochastic dynamics in Hilbert space

A complex quantum system in nature can often clearly be divided into a subsystem of real interest with only few relevant degrees of freedom and a mostly large residual system. Such composite system is usually referred to as a system-plus-reservoir entity, in which the system corresponds to the subsystem of interest and the reservoir to the residual system. In general, the system-reservoir coupling leads to exchange of energy between system and reservoir, which, in a phenomenological classical description, usually shows up as damping of the system. Therefore, it is important to understand, how the classical Langevin equation and its quasiclassical generalization discussed above in Section 2.1, emerge from a reduction of the system-plus-reservoir entity…

In a canonical ensemble – the entirety of all microstates with fixed particle number and volume – the system is kept at equilibrium by contact with a heat reservoir at temperature T, and thus only the energy fluctuates. The equilibrium characteristics of a canonical ensemble of quantum systems governed by the Hamiltonian $\widehat{H}$ is determined by properties of the canonical density operator ${\widehat{W}}_{\beta }=Z^{-1}{e}^{-\beta \widehat{H}}$, where $Z=\text{t}\text{r}{e}^{-\beta \widehat{H}}=\underset{k}{{{\displaystyle \sum }}^{\text{}}}{e}^{-\beta E_k}$ is the partition function. The source of the imaginary-time path integral approach to equilibrium thermodynamics is the fact that the canonical density operator $e^{-\beta \widehat{H}}$ is related to the time-evolution operator $e^{-i\widehat{H}t/ħ}$ by analytic continuation of time, t → z = –iħβ, known in field theory as Wick rotation from Minkowskian to Euclidean field theory. Feynman taught us that the coordinate matrix elements of $e^{-i\widehat{H}t/ħ}$ and $e^{-\beta \widehat{H}}$ can be written as a sum over histories of real- and imaginary-time paths in configuration space, respectively. From a contemporary point of view, path integrals represent not only an approach alternative to canonical quantization of classical mechanics, but are also basic to the foundation and interpretation of quantum mechanics. Besides that, they form a perfect basis for numerical simulations of quantum thermodynamics and dynamics by means of Monte-Carlo methods.

The following sections are included:

• Statement of the problem and general concepts
• Feynman-Vernon method for product initial state
  • Influence functional
  • Friction and quantum noise
  • Damping and decoherence
  • Quasiclassical Langevin equation
• Influence functional method for general initial states
  • General initial states and preparation function
  • Complex-time path integral for propagating function
  • Real-time path integral for propagating function
  • Closed time contour representation
• Stochastic unraveling of influence functionals
• Non-Markovian dissipative dynamics in the semiclassical limit
  • Van Vleck and Herman-Kluk propagator
  • Semiclassical dissipative dynamics
• Energy transfer between system, reservoir and interaction
• Brief summary and outlook

So far I have employed functional integral methods in order to establish exact formal expressions which describe the thermodynamics and dynamics of damped quantum systems. Now we move to new grounds and consider the physical properties of exactly solvable damped linear systems, i.e. the damped quantum oscillator and quantum Brownian particle. Subsequently, I discuss a variational method which makes it possible for us to treat nonlinear damped quantum systems in much the same way as linear systems. I conclude this part with a brief general discussion of quantum decoherence.

The interplay of friction and quantum-statistical fluctuations with impact on thermodynamics and dynamics can be studied quantitatively on a microscopic basis for linear systems. Of particular interest is a harmonic oscillator subjected to linear (state-independent) friction of arbitrary frequency-dependence…

Historically, the Langevin equation (2.3) was used to analyze the irregular motion (Zitterbewegung) of a heavy particle moving in a thermally equilibrated molecular medium. This phenomenon is known as Brownian motion, named after Robert Brown, who observed the random motion of pollen grains immersed in a fluid. In many cases, a systematic external force is absent. Then we are left with the problem of free Brownian motion. Often, the memory times of the fluctuating forces are negligibly short compared to the time scale over which one observes the Brownian particle. The assumption of memoryless friction is usually referred to as Markov limit…

It is appealing to study quantum statistical expectation values with concepts familiar from classical statistical mechanics. The phase space representation of the reduced density matrix in terms of Wigner’s function discussed in Subsection 4.4.2 is just one example. Another important concept which goes back to Feynman [272] is the effective classical potential…

There are two main lines of investigations on decoherence in quantum mechanics. On the one hand, one is interested in fundamental problems in the interpretation of quantum theory, in particular in questions connected with measurement and the quantum to classical transition (cf., e.g., Ref. [58]). The importance of decoherence in the appearance of a classical world in quantum theory is well known [287]. In cosmology, the decoherence which must have been occurred during the inflationary era of the Universe has been attributed to quantum fluctuations. On the other hand, environment-induced decoherence is omnipresent in the microscopic world. It is found, e.g., for an atom confined in a quantum optical trap, or for electron propagation in a mesoscopic device. Striking examples for coherence effects are the anomalous magnetoresistance in disordered systems due to weak localization [288], and Aharonov-Bohm type oscillations in the resistance of mesoscopic rings [289]…

In the second part, I considered the exactly solvable case of linear dissipative quantum systems, the thermodynamic variational approach useful for nonlinear quantum systems, and issues of quantum decoherence. Now I turn to a study of the frequent situation in which a metastable state is separated from the outside region by a free energy barrier. The decay of a metastable state plays a central role in many scientific areas including low-temperature physics, nuclear physics, chemical kinetics and transport in biomolecules. At high temperatures, the system escapes from the metastable well predominantly over the barrier by thermal activation. At zero temperature on the other hand, the system is in the localized ground state of the metastable well and can escape only by quantum mechanical tunneling through the barrier. I shall focus the discussion on the influence of friction on the decay process. The treatment will be based on an effective method which allows to study this problem in a unified manner for temperatures ranging from T = 0 up to the classical regime.

There are many processes in physics, chemistry, and biology in which a system makes transitions between different states by traversing a barrier. The theory of rate coefficients for barrier crossing has a long history extending back to the work by Arrhenius in 1889 [308]. H. A. Kramers’ article of 1940 [266] represents a cornerstone in the quantitative analysis of thermally activated rate processes. This work provided a thorough theoretical description in the classical regime both for very weak and for moderate-to-strong damping. It includes important limiting cases such as the transition state theory or the Smoluchowski model of diffusion controlled processes…

The following sections are included:

• Classical transition state theory
• Moderate-to-strong-damping regime
• Strong damping regime
• Weak-damping regime

With the advent of quantum mechanics, the first who introduced quantum tunneling was Friedrich Hund in 1927, when he described the intramolecular rearrangement in ammonia molecules [319]. Shortly later, the tunneling effect was popularized by Oppenheimer [320], who used it to explain the ionization of atoms in strong electric fields, and by Gamow [321], and Gurney and Condon [322], who explained the radioactive decay of nuclei. Perhaps the oldest guess of a quantum transition state theory was made by Wigner, when he proposed as a quantum mechanical generalization of the classical TST expression (11.1) the formula [323]

The following sections are included:

• The global metastable potential
• Periodic orbit and bounce
  • The case T = 0
  • The case 0 < T < T₀

Under the condition of weak metastability, Eq. (10.3), the functional integral (4.214) for the partition function is dominated by the stationary points of the action. Besides the periodic bounce q_B(τ), there are two trivial constant solutions of Eq. (4.215). On the one hand, we have $\overline{q}(\tau )=q_{\text{b}}$, in which the particle sits in the minimum of the upside-down potential − V (q), and accordingly on the barrier top of the original potential. On the other hand, there is the constant path $\overline{q}(\tau )=0$, in which the particle sits in the minimum of the original potential, and accordingly at the local maximum of the upside-down potential (see Figs. 10.1 and 10.2)…

The following sections are included:

• Rate formula above the crossover regime
• Quantum corrections in the pre-exponential factor
• The quantum Smoluchowski equation approach
• Multidimensional quantum transition state theory

We have seen in Section 15.2 that the quantum correction factor c_qm increases with decreasing temperature, and that it diverges as T → T₀. The divergence is, because the eigenvalue ${\text{Λ}}_1^{(\text{b})}={\text{Λ}}_{-1}^{(\text{b})}={\nu }_1^2-{\omega }_{\text{b}}^2+\left|{\nu }_1\right|\widehat{\gamma }\left(\left|{\nu }_1\right|\right)$ in Eq. (15.7) vanishes at T = T₀…

We are now skilled and ready to study the decay of a metastable state at temperature T well below the crossover regime at which the escape from the well goes off by dissipative quantum tunneling through the barrier.

One of the most intriguing aspects of quantum theory is the phenomenon of constructive and destructive interference. The phase coherence between different quantum mechanical states becomes apparent in oscillatory behaviors of observables. The simplest model involving quantum coherence is a two-state system. The problem of a quantum system whose state is effectively confined to a two-dimensional Hilbert space is often encountered in physics and chemistry. For instance, imagine a quantum mechanical particle tunneling clockwise forth and back between two different localized states. In reality, such system is strongly influenced by the surroundings. The coupling can lead to qualitative changes in the behaviors: the environment-induced fluctuations can destroy quantum coherence and can even lead to a “phase transition” to a state in which quantum tunneling is quenched. A manipulable two-state system is the quantum version of the classical binary bit, usually called qubit or quantum bit. In quantum computing, a qubit is the basic unit of quantum information and quantum computing devices. Since decoherence is the biggest enemy of qubit implementation, understanding, control and strategies for reduction of environmental influences are indispensable.

Anderson et al. [121] and, independently, Phillips [122] postulated in 1972 the existence of two-level systems in glasses to explain low temperature anomalies of the specific heat in these amorphous materials. Although the true microscopic nature of the tunneling entities in glasses is unclear, they can be visualized by a particle tunneling in a double well along an (unknown) reaction coordinate. It turned out in ultrasonic experiments as an important difference between dielectric and metallic glasses that the lifetime of the tunneling eigenstates is drastically shorter in the case of metals. Golding et al. [386] explained this unexpected behavior by a nonadiabatic coupling of the tunneling entity to the conduction electrons in metals. Shortly later, Black and Fulde [387] extended this idea to describe relaxation processes in a superconducting environment by following the lines sketched above in Subsections 3.3.3 and 4.2.8. The theoretical predictions were confirmed by G. Weiss et al. [388] in ultrasonic experiments on superconducting amorphous metals. They showed that the lifetime is reduced when the environment is switched from superconducting to normalconducting, thus demonstrating the significance of the electronic coupling. A survey of the early developments and the perturbative treatment of this coupling is given in Ref. [123]. For a nonperturbative treatment see Ref. [389]. The nonlinear acoustic response of amorphous metals has been studied by Stockburger et al. [390], and the results for the dynamical susceptibility were found in good agreement with experiments [391]. A comprehensive review of the physics of tunneling systems in amorphous and crystalline solids with emphasis on the thermodynamic, acoustic, dielectric and optical properties has been given in Ref. [392]…

In this chapter, we consider the environmental influences on thermodynamic properties of selected quantum systems. First, we investigate the equilibrium properties of the open two-state system. Then we discuss, based on exact formal expressions for the partition function, the relationship of the Ohmic spin-boson model with variants of the Kondo model and with the 1/r² Ising model. Finally, we calculate quantum-statistical tunneling rates with the method of analytic continuation of the free energy. Emphasis is put on electron transfer in a solvent and on interstitial tunneling in solids.

Before we turn in Chapter 21 to the real-time dynamics of the TSS, we first study quantum-statistical tunneling with the thermodynamic method discussed above in Part III. The results shall be confirmed in Chapter 21 by a dynamical approach.

So far, we have mainly discussed quantum statistical properties of the open two-state system. As far as dynamics is concerned, we have been limited to the study of nonadiabatic tunneling rates in the incoherent regime. We now turn towards real-time dynamics based on the methods given in Chapter 5. The approach will cover the full dynamics in a unified manner for different kinds of the initial preparation and for arbitrary linear dissipation, both in the incoherent and oscillatory regime. We shall provide explicit expressions in most regions of the parameter space. Emphasis is put on the regime in which the system shows quantum coherent oscillations.

The following sections are included:

• Symmetric TSS in the NIBA
  • Ohmic scaling limit
  • The super-Ohmic case
• White-noise regime
  • Power spectrum of the stochastic force
  • Symmetric Ohmic TSS at moderate-to-high temperature
  • Biased Ohmic TSS at moderate-to-high temperature
• Weak quantum noise in the biased TSS
  • The one-boson self-energy
  • Populations and coherences (super-Ohmic and Ohmic)
• Pure dephasing
• 1/f noise and decoherence
  • 1/f noise from fluctuating background charges
  • 1/f noise from coherent two-level systems
  • Decoherence from 1/f noise
• The Ohmic TSS at and close to the Toulouse point
  • Grand-canonical sums of collapsed blips and sojourns
  • The function ${〈{\sigma }_z〉}_t$ for $K=\frac{1}{2}$
  • The case $K=\frac{1}{2}-\kappa$; coherent-incoherent crossover
  • σ_z equilibrium correlation function and response function
  • σ_x equilibrium correlation function and response function
  • Correlation and response functions in the Toulouse model
• Long-time behavior at T = 0 in the regime 0 < K < 1: general
  • The populations
  • The population correlations and Shiba relation
  • The coherence correlation function
• From weak to strong tunneling: relaxation and decoherence
  • Incoherent tunneling beyond the nonadiabatic limit
  • Decoherence at zero temperature: analytic results
• Thermodynamics from dynamics

With the advance of laser technology and the possibility for experimental time resolution in the sub-picosecond regime, there has been growing interest in the study of the dynamics of quantum systems that are driven by strong time-dependent external fields. Quantum dynamics of explicitly time-dependent Hamiltonians shows a variety of novel effects, such as the phenomenon of coherent destruction of tunneling [492], stabilization of localized states which would otherwise decay [493], and the possibility of controlling the tunneling dynamics with pulsed monochromatic light [494, a] [495] and sinusoidal fields [494, b] [495]. For a review, see Refs. [496, 497]. Here we restrict the attention to a two-state system which is simultaneously exposed to a fluctuating force by the surroundings and to deterministic time-dependent forces.

A quantum Brownian particle in a multi-well potential coupled to a dissipative environment is archetypal for many problems in physics and chemistry. Examples include super-ionic conductors, atoms on surfaces, and interstitials in dielectrics and metals. Also the current-voltage characteristics of a Josephson junction, and charge transport in a quantum wire hindered by impurity scattering are described by this model. At high temperature, the particle moves forward or backward from well to well by incoherent tunneling or thermally activated transitions. As the temperature is lowered, coherent tunneling across many wells may become significant, and the competing different tunneling paths may interfere constructively or destructively. The model has a profound and powerful duality symmetry between the weak-binding and strong-binding representation. In view of the broad area of applications, the understanding of quantum transport in multi-well systems is a central issue.

The following sections are included:

• Introduction
• Weak- and tight-binding representation

The following sections are included:

• Quantum transport and quantum-statistical fluctuations
  • Product initial state
  • Characteristic functions of moments and cumulants
  • Thermal initial state and correlation functions
• Poissonian quantum transport
  • Incoherent nearest-neighbor transitions (weak tunneling)
  • The general case (strong tunneling)

In the tight-binding model (24.4), the quantum particle makes sudden transitions between neighboring states of the density matrix with a probability amplitude ±iΔ/2 per unit time. The sequence of states of the density matrix visited in succession can be visualized in terms of a paths of a walker on an infinite square lattice spanned by the sites of the discrete (q, q′) double path. The walker starts on some site on the principal diagonal, say at (0, 0), and then randomly makes horizontal and vertical steps along the q- and q′-axis, respectively. At each site of the square lattice there are four possible directions to proceed further. Starting out from the initial diagonal state (n, n) and ending at the diagonal state (m, m) requires an even number of steps with the minimum number 2|n − m|…

In the scaling limit, it is expedient to scale the mobility with that of a free Brownian partice, ${\mu }_0=1/\eta =q_0^2/2\pi ħK$ [cf. Eq. (7.29)]. The normalized mobility is defined as

Duality is a recognized concept by now. In electromagnetism, the simplest form of duality is the invariance of the source free Maxwell equations under interchange of electric and magnetic field, B → E, E → − B. In the presence of sources, the product of electric and magnetic charges obeys the Dirac quantization condition. In general, duality maps a theory with strong coupling to one with weak coupling. Thus, if a duality symmetry exists, one can study the strong-coupling regime via the perturbative analysis of the weak-coupling regime. A theory is self-dual when there is an exact map between the strong- and the weak-coupling sector of the same theory.

The following sections are included:

• Tunneling rates and cumulant generating function
• Leading low temperature contribution

Analytic evaluation of the grand-canonical sum of the real-time Coulomb gas (27.2) for any temperature, bias and Kondo parameter K is still unresolved. A route has been proposed in which the problem is solved crabwise [491, 544]. There it was shown that the grand-canonical sum in imaginary time can be solved in analytic form for twisted partition functions. From these the nonlinear mobility is found with a conjecture relying on analytical continuation of winding numbers to the real physical bias. Here we show how the method works. In addition, we confirm results of this approach obtained in particular regions of the parameter space. The regime K ≪ 1 for general T and ϵ, and the regime T = 0 for general ϵ and K have been studied in Ref. [545]. Here we present the method for the TB model (24.4). Translation to the WB model (24.1) is straightforward along the lines given above.

The physics of interacting particles in one dimension is drastically different from the physics of interacting particles in two or three dimensions. The theoretical methods and techniques relevant to quantum physics in one dimension have been reviewed in compendia by Gogolin, Nersesyan, and Tsvelik [547], and by Giamarchi [548]. Signatures of many-body correlations have attracted a great deal of interest in recent years. Many investigations have been focussed on one-dimensional (1D) electron systems, in which the usual Fermi liquid behavior is destroyed by the interaction. The generic features of many 1D interacting fermion systems are well described in terms of the Tomanaga-Luttinger liquid (TLL) model [549, 415]. In the TLL model, the effects of the electron-electron interaction are captured by a dimensionless parameter g. A sensitive experimental probe of a Luttinger liquid state is the tunneling conductance through a point contact in a 1D quantum wire, as observed in Ref. [514]. Of interest are also the dc nonequilibrium current noise [550] and higher cumulants. The generic model is a quantum impurity embedded in a Luttinger liquid environment (QI-TLL model). Tunneling of edge currents in the fractional quantum Hall (FQH) regime provides another realization of a Luttinger phase. As shown by Wen [515], the edge state excitations are described by a (chiral) Luttinger liquid with Luttinger parameter g = v, where v is the fractional filling parameter.

The following sections are included:

• Introduction
• Universal tradeoff between power, efficiency and relative uncertainty
  • Power and power fluctuations
  • Trade-off relation
  • Harmonic Brownian duet
• Linear response
  • Maximum output power
  • Maximal efficiency
• Nonlinear response

The following sections are included:

• Bibliography
• Index