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International Journal of Quantum Chemistry

Effective description of the dissipative interaction between simple model-systems and their environment

Dieter Schuch

Published Jan 1, 1999• 29 citations
Publisher

Abstract

Chemical reactions are generally connected with energy transfer between reactands and their environment. There are different ways of taking this environment into account. The traditional system-plus-reservoir (S+R) approach couples the relevant system to a large number of environmental degrees of freedom, represented, e.g., by harmonic oscillators. However, this has the disadvantage—from a computational point of view—that a system of many coupled differential equations has to be solved due to the large number of degrees of freedom of the reservoir. The number of degrees of freedom can be drastically reduced if an effective description of the interaction between system and reservoir is applied. On the quantum mechanical level, the effect of the environment can be included, e.g., by the use of explicitly time-dependent Hamiltonians or nonlinear additions to the Schrödinger equation (SE). An advantage of the former method is the preservation of the canonical formalism and the linearity of the theory as well as the fact that its operators can be derived directly from the S+R approach. Therefore, it seems to be even more of a paradox that this method apparently violates the uncertainty principle. This violation does not occur in the nonlinear approaches but the nonlinearity seems, at first sight, to be problematic. However, it can be shown that a particular logarithmic nonlinear SE is equivalent to the approach using the time-dependent Hamiltonian and can be transformed into this. The fact that this transformation is nonunitary provides the key for the solution of the paradox and also allows one to show the equivalence of all three of the above-mentioned approaches. In investigating the classical counterpart, a noncanonical transformation is used between the dissipative physical level and the canonical formal level, which makes it even possible to transform the physical interacting system into a formal isolated system that can be treated in the conventional way. As a result, the effect of the interaction with the environment can be taken into account classically by noncanonical transformations and quantum mechanically by nonunitary transformations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 537–547, 1999

DOI
10.1002/(sici)1097-461x(1999)72:6<537::aid-qua1>3.0.co;2-q

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Chemical reactions are generally connected with energy transfer between reactands and their environment. There are different ways of taking this environment into account. The traditional system-plus-reservoir (S+R) approach couples the relevant system to a large number of environmental degrees of freedom, represented, e.g., by harmonic oscillators. However, this has the disadvantage—from a computational point of view—that a system of many coupled differential equations has to be solved due to the large number of degrees of freedom of the reservoir. The number of degrees of freedom can be drastically reduced if an effective description of the interaction between system and reservoir is applied. On the quantum mechanical level, the effect of the environment can be included, e.g., by the use of explicitly time-dependent Hamiltonians or nonlinear additions to the Schrödinger equation (SE). An advantage of the former method is the preservation of the canonical formalism and the linearity of the theory as well as the fact that its operators can be derived directly from the S+R approach. Therefore, it seems to be even more of a paradox that this method apparently violates the uncertainty principle. This violation does not occur in the nonlinear approaches but the nonlinearity seems, at first sight, to be problematic. However, it can be shown that a particular logarithmic nonlinear SE is equivalent to the approach using the time-dependent Hamiltonian and can be transformed into this. The fact that this transformation is nonunitary provides the key for the solution of the paradox and also allows one to show the equivalence of all three of the above-mentioned approaches. In investigating the classical counterpart, a noncanonical transformation is used between the dissipative physical level and the canonical formal level, which makes it even possible to transform the physical interacting system into a formal isolated system that can be treated in the conventional way. As a result, the effect of the interaction with the environment can be taken into account classically by noncanonical transformations and quantum mechanically by nonunitary transformations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 537–547, 1999
Caldirola [Nuovo Cimento 18, 393 (1941)] and Kanai's [Prog. Theor. Phys. 3, 440 (1948)] time-dependent approach for dissipative systems can be traced back to the conventional system-plus-reservoir approach; however, it apparently violates the uncertainty principle. This discrepancy can be avoided by considering a consistent transition between the Caldirola-Kanai formalism and a logarithmic nonlinear Schr\"odinger equation that requires not only a transformation of the operators, but also a nonunitary transformation of the wave function. This procedure also shows the equivalence of three at first sight incompatible approaches to describe dissipative quantum systems.
Couplings of a system to other degrees of freedom (that is, environmental degrees of freedom) lead to energy dissipation when the number of environmental degrees of freedom is large enough. Here we discuss quantal treatments for such energy dissipation. To this end, we discuss two different time-dependent methods. One is to introduce an effective time-dependent Hamiltonian, which leads to a classical equation of motion as a relationship among expectation values of quantum operators. We apply this method to a heavy-ion fusion reaction and discuss the role of dissipation on the penetrability of the Coulomb barrier. The other method is to start with a Hamiltonian with environmental degrees of freedom and derive an equation which the system degree of freedom obeys. For this, we present a new efficient method to solve coupled-channels equations, which can be easily applied even when the dimension of the coupled-channels equations is huge.
Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex wave function are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics and has a counterpart in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of nonlinear real Ermakov equations or equivalent complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. In this context, different approaches for describing open quantum systems, particularly effective ones, are discussed and compared. Certain kinds of nonlinear modifications of the Schrödinger equation are discussed as well as their interrelations and their relations to linear approaches via non-unitary transformations. The modifications of the aforementioned Ermakov and Riccati equations when environmental effects are included can be determined in the time-dependent case. From formal similarities conclusions can be drawn how the equations of time-independent quantum mechanics can be modified to also incluce the enviromental aspects.
A previously developed novel theory for the formal canonical quantization of classically dissipating systems will be the starting point for a detailed discussion of the quantum statistical aspects of the simple linearly damped harmonic oscillator. The formalism essentially involves complex classical canonical coordinates and momenta, and thus quite naturally leads to the possibility of creation and annihilation operators. Furthermore, the occurrence of quantal noise operators appears to be of principal importance for the conservation of the fundamental commutator in the course of time, as will be expressed in a simple fluctuation-dissipation relation. Making a canonical transformation back to the real, Cartesian Hermitian position and momentum an "effective" Hermitian Hamiltonian will be derived, with which a transformation is made from the Heisenberg frame to the Schr\"odinger frame where the density operator equation will be computed. This will make it obvious that no proper Schr\"odinger equation exists for the dissipative subsystem on its own, thus reflecting an incomplete knowledge. The master equation will then be translated into its Wigner representation. The intimate connection between the diffusion coefficients in the resulting Fokker-Planck equation and the uncertainty relation will be demonstrated in a clear fashion.
In this thesis, the influence of an environment on molecules and, in particular, on the quantum control of such systems is investigated. Different approaches to describe system-bath dynamics are implemented and applied. The inclusion of a dissipation term in the system Hamiltonian leads to energy loss and relaxation to the ground state. As a first application, the isomerisation reaction in an aromatic complex is treated. It is shown that this simple model is able to reproduce results of time-resolved spectroscopic measurements. Next, the influence of noise is investigated. The incorporation of fluctuations reveals that energy is not conserved and coherences are destroyed. As an example, the quantum control of a population transfer in Na2 is examined. The efficiency of control processes is studied in dependence on the strength of the noise and different system-bath couplings. Starting with the unperturbed system, Local Control Theory is applied to construct a field which selectively transfers population into a single excited electronic state. The coupling to the bath is then switched on to monitor the dependence of the coupling strength on the transfer efficiency. The perturbation of the bath effects the Na2 molecule in such a way that potential energy curves and transition dipole moments are distorted. An important result is that already elastic collisions lead to a substantial loss of control efficiency. The most promising approach used in this thesis is the stochastic Schrödinger equation. It is equivalent to the commonly employed descriptions of system-bath dynamics within the reduced density matrix formalism. It includes decoherences and dissipation caused by elastic and inelastic collisions. Our contribution is the incorporation of laser excitation into the kinetic Monte-Carlo scheme. Thus we are able to apply this stochastic approach to the quantum control of population transfer in the sodium dimer. Because within our description it is possible to separate pure dephasing, inelastic transitions, and coherent time-evolution, we can identify the relative influence of these processes on the control efficiency. This leads to a far more physical picture of the basic processes underlying the perturbations of an environment then what a reduced density matrix description can provide. In utilising the stochastic wave function approach instead of the density matrix formalism, the computations are quite efficient. The stochastic Schrödinger equation is realised by N independent runs, where, in our case, an ensemble size of N = 1000 gives converged results. The efficiency of the laser control process is studied as a function of temperature and collision rates. A rise in temperature (or collision rate) reeffects a stronger fluctuation and thus results in a less efficient transfer by the control field. Though the Gaussian fluctuations used here do not strictly represent 'white'- noise, since a deterministic machine is not able to produce uncorrelated random numbers, an acceptable distribution is achieved by simple procedures. An improvement of the here applied algorithms would, for instance, include a more sophisticated sampling of the dephasing rates. Only one example of a control process is studied here and an application of the developed approach to other problems of quantum control is to be performed. This thesis established a systematic approach to understand quantum control in the presence of an environment.
We suggest a new approach for describing quantum dissipation in a small systems for which the system-plus-reservoir approach is not relevant. We first analyze the fact that equilibrium thermodynamics may reveal the existence of an underlying dynamics. This is true in the algebraic approach of quantum mechanics via the Tomita-Takesaki theorem. A similar result is obtained if we start from the path integral expression of the partition function, an equation of motion is introduced. In both cases a parameter has to be identified with the physical time. Several arguments are presented showing that it is so. By investigating the dynamics for short times we introduce an equilibrium condition from which a natural unit of time appears. The equation of motion is extended to non equilibrium situations for which a H-theorem can be derived. The equation of motion can be transformed into a quantum Smoluchovski equation, its solution is a probability having a clear physical meaning. The relaxation of this probability towards its equilibrium value requires a time during which a thermodynamic description is possible or not. A standard bistable model is investigated and the chemical rate k(t)k(t)k(t) is calculated as a function of time, it appears to be a non monotonic function of time. In some conditions the stationary value of k(t)k(t)k(t) can be identified with the Kramers result. Finally we compare our approach which is naturally irreversible with an analysis based on the Schrödinger equation which is reversible. From all these results it appears that our equation of motion or its Smoluchovski version appear as a realistic starting point for describing quantum dissipation in small systems.