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Influence of damping on quantum interference: An exactly soluble model. A. O. Caldeira. Institute for Theoretical Physics,University
Table of contents

Here, the relevant action functional is determined by the real Lagrangian. We expand somehow on this issue in the following since it does not seem to be widely appreciated, despite its deep significance and its potential practical utility. To this end, let us first briefly introduce the concept of Hamiltonian flows and symplectic manifolds [ 46 ]. With this premise in mind, we smoothly introduce a set of real variational parameters x , forming a coordinate system in the sample space.

In fact, the variational equations of motion take the form. Under such circumstances, the equations of motion can also be written with the help of the Poisson brackets. The importance of the symplectic structure thus described is hardly overemphasized. Apart from its possible consequences on fundamental issues such as the emergence of classicality, or the coexistence of quantum and classical worlds, it offers in practice the possibility of introducing robust propagation schemes in solving the variational equations of motion.

These symplectic propagators would not only conserve energy and norm but also the whole symplectic structure, a property that might be of great help when numerically investigating the emergence of irreversibility in Hamiltonian systems like the ones described by Eq. In this section, we give a brief account of wave packet quantum dynamical methods that have been applied in the past to system—bath problems of the kind discussed in this Chapter. As a result, several different methods exist which stem from the same multi-configurational ansatz. Among the various possibilities, the conceptually simplest method is the multi-configuration time-dependent Hartree MTDH , developed decades ago by Meyer et al.

The ansatz is a straightforward expansion of the wave function in terms of orthonormal single-particle states,. The orthogonality condition proves to be a key strength of the method, since it guarantees that the configurations entering Eq.

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Their derivation is straightforward, though lengthy, and the result can be summarized as follows. The orbital equations, on the other hand, are mean-field like [1] -. Details on the method and its numerical implementation can be found in the literature [ 16 , 32 , 33 ]. Some general considerations are in order. As such, it is extremely flexible and allows a systematic search of the convergence of the solution with respect to the length of the expansion of Eq.

In fact, provided large enough computational resources are available how large depends of course on the problema at hand , the MCTDH solution can be made numerically exact. Apart from this, there exists no limitation in the form of the system Hamiltonian and indeed, MCTDH has been applied with success to a very large number of different problems.

The application to system—bath problems to be described below represents just one possible problem where the method applies; further applications can be found in the original research papers and in the extensive review literature [ 16 ].

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  4. Here, we just mention that a user-friendly, highly efficient, general MCDTH code which takes arbitrary Hamiltonians as input is freely available upon request to the author [ 47 ]. A second class of multi-configurational methods is represented by the Gaussian - multi-configuration time-dependent Hartree G-MTDH developed a while ago by Burghardt et al. It is still a general-purpose method that can handle different kinds of quantum dynamical problems, and it is obtained from Eq.

    As a result, the equations of motion for the Gaussians become classical-like with considerable saving of memory and computer time in fact, one propagates the few parameters needed to define the Gaussians , at the expense of introducing overlap matrix elements between them. Though the method has several variants depending on the number of Gaussians introduced, it was originally formulated for system—bath-like problems, where one easily identifies primary modes to be described at the high, fully flexible level and secondary, less important modes that can be managed with moving Gaussians.

    Along this line of thought, LCSA was specifically designed as a local coherent-state approximation [ 34 ] to the dynamics of system—bath Hamiltonians of the general form. In this case, the shaping of the wave function relies on the fact that i the coupling to the bath is loca l in system coordinates, and ii the bath is approximately harmonic. Upon introducing a set of system discrete variable representation DVR states [1] - one expands the wave function as.

    The latter are then written as products of HO coherent states CSs , that is,. When using conventional phase factors for the CSs, they take the following form.

    Unitary Approaches to Dissipative Quantum Dynamics

    The bath equations are pseudo-classical. The latter is essential for a quantum, though approximate, description of the bath dynamics. For a detailed derivation of the equations see Ref. This concludes the description of the original LCSA method. Several variants are possible e. In fact, among the features of LCSA, one key strength of the method is that it reduces the bath dynamics to classical-like evolution, with a number of trajectories that scales linearly with the bath dimensions. This means that the method itself has a power-low scaling with such dimensions, the exponent of this scaling depending eventually on the interaction between bath modes.

    For bath modes coupled to the system only [as in Eq. This good scaling is in common with mixed quantum-classical methods, which, however, fail to correctly represent the system—bath correlations. Coupled trajectories also arise in a number of closely related approaches, namely the coupled coherent-state method of Shalashilin and Child [ 50 , 51 ] and the G-MCTDH method mentioned above. The main difference between the two is that in LCSA all the configurations are orthogonal to each other, as a consequence of the presence of a different DVR state in each of them.

    This leads to considerable simplifications in the resulting equations, at the price of a reduced accuracy. Finally, one interesting property about the pseudo-classical description of the bath degrees of freedom is that it suits well to induce dissipative dynamics into the total system.

    This possibility has been exploited, especially in conjunction with the need of removing numerical instabilities of the method without altering the system dynamics. We consider here a model Hamiltonian describing an anharmonic Morse oscillator coupled to a heat bath. A typical problem considered in this context is the small amplitude, damped motion of the oscillator. This type of simulations was used in Ref. A Markovian exponential decay of the energy was found for all but the strongest coupling case, where some energy oscillations are clearly evident.

    The graph illustrates the main problems of standard LCSA, an inherent numerical instability related to saturation of the bath.

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    Similarly, in Ref. Here, MCTDH was used and different degrees of correlation were introduced along the chain, namely a small number of oscillators were described by a full, many particle expansion, whereas the rest of the chain was treated with one spf per mode.

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    6. In this way, we exploited the strengths of the linear-chain representation of the bath to enlarge the physical time window of the simulation i. Small amplitude motion for the two non-Ohmic models of Ref. Also shown for comparison the benchmark obtained with the bath in normal form, as in Eq. The agreement with the benchmark is rather satisfactory and, as expected, the minor discrepancies were removed by increasing the correlation level. This is true both for the average system energy and for more detailed quantities like the position correlation functions.

      The Morse oscillator was also used to model a dissipative scattering event, one in which the system is initially asymptotically free and moves toward the potential well where energy exchange with the bath occurs. Typically, in the interaction region, the wave-packet splits into two parts: One gets trapped in the well and fully relaxes on the long run, while the other returns to the asymptotic region.

      Sticking probability as a function of the incident energy, for the Morse scattering problem described in the main text.

      A Josephson relation for e/3 and e/5 fractionally charged anyons

      Right panel: The structured SD of Ref. Black circles are benchmark results obtained with the bath in normal form. Lines serve as a guide to the eye. Simulations with standard LCSA showed that the numerical instabilities were too severe to extract meaningful sticking coefficients, even if the energy dissipation was described quite accurately [ 35 ]. On the contrary, the energy-local variant eLCSA gave stable results but only in semi-quantitative agreement with the benchmark.

      A detailed analysis showed that this is due to an inadequate system—bath correlation in the adopted ansatz , which is crucial for the energy transfer and hence the sticking process. Excellent results, on the other hand, were obtained by applying MCTDH with a partially correlated linear chain of oscillators [ 27 ].

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      Importantly, the results were shown to steadily and quickly converge toward the benchmark when increasing the level of correlation. Despite the apparent simplicity of the system, the presence of both dissipative and quantum features makes it a challenging dynamical problem.

      Single molecules in a quantum interference movie

      Recently, we have devised a rather elaborate system—bath model to describe hydrogen chemisorption of graphene and used it in a fully quantum study of the sticking dynamics with the MCDTH method. The model consists of an accurate description of the hydrogen atom and its bonding carbon atom a 4D system , which were then coupled to the graphene sheet described by a phonon bath. It rests on the following, reasonable assumptions: i The energy exchange that occurs between the system and the lattice for near equilibrium configurations is representative of energy dissipation; ii relaxation proceeds through sequential energy transfer from the hydrogen atom to the carbon atom; iii a mapping holds, at least approximately, which relates the classical Hamiltonian dynamics of the interesting C and H atoms to a GLE description.

      On this basis, the following form was adopted for the Hamiltonian. Details on how it was extracted and thoroughly checked can be found in the original research paper [ 40 ]. The SD has been obtained from the inversion procedure described in the text and used for the high-dimensional quantum dynamics calculations of Refs.

      Decoherence of trapped-atom motional state superpositions

      Once the coupling of the C atom with its environment was introduced, the Hamiltonian model of Eq. We start here by considering the relaxation problem of the C—H bond. The panel on the left gives the vibrational energy spectrum of the system.