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arXiv:1005.1079

A probability-conserving dissipative Schrödinger equation

Michel van Veenendaal, Jun Chang, A. J. Fedro

Published May 6, 2010• 1 citation
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Abstract

Dissipative effects on a microscopic level are included in the Schrödinger equation. When the decay between different local levels as a result of the coupling to a bath, the Schrödinger equation no longer conserves energy, but the probability of the states is conserved. The procedure is illustrated with several examples that include direct electronic decay and damping of local phonons (vibrational levels). This method significantly reduces the calculational effort compared to conventional density matrix techniques.

DOI
10.48550/arxiv.1005.1079

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Abstract This chapter treats dissipation arising from coupling to the environment. Problems treated include photoionization, spontaneous emission, and cavity damping. A short-time perturbative analysis leads to decay of a single state coupled to a continuum at a rate determined by Fermi's golden rule. For longer times a non-perturbative solution of the amplitude equations leads to Weisskopf–Wigner decay. The final value theorem provides a simple method for obtaining the final state spectrum without first solving for the dynamics. A Heisenberg picture analysis leads to operator Langevin equations in which environment operators introduce quantum noise. A parallel analysis is possible working in the Schroedinger picture and leads to a master equation for the density operator of the damped system. This is solved to provide a complete description of the influence of dissipation on the system of interest. It is sometimes simpler to solve directly for the evolution of the field statistics.

•Nov 14, 2002•1 citation
Dissipative effects on a microscopic level are included in the Schrödinger equation. When the decay between different local levels as a result of the coupling to a bath, the Schrödinger equation no longer conserves energy, but the probability of the states is conserved. The procedure is illustrated with several examples that include direct electronic decay and damping of local phonons (vibrational levels). This method significantly reduces the calculational effort compared to conventional density matrix techniques.
Dissipative effects on a microscopic level are included in the Schr\odinger equation. When the decay between different local levels as a result of the coupling to a bath, the Schr\odinger equation no longer conserves energy, but the probability of the states is conserved. The procedure is illustrated with several examples that include direct electronic decay and damping of local phonons (vibrational levels). This method significantly reduces the calculational effort compared to conventional density matrix techniques.
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 investigate the effect of inter-mode coupling on the vibrational relaxation dynamics of molecules in weak dissipative environments. The simulations are performed within the reduced density matrix formalism in the Markovian regime, assuming a Lindblad form for the system-bath interaction. The prototypical two-dimensional model system representing two CO molecules approaching a Cu(100) surface is adapted from an ab initio potential, while the diatom-diatom vibrational coupling strength is systematically varied. In the weak system-bath coupling limit and at low temperatures, only first order non-adiabatic uni-modal coupling terms contribute to surface-mediated vibrational relaxation. Since dissipative dynamics is non-unitary, the choice of representation will affect the evolution of the reduced density matrix. Two alternative representations for computing the relaxation rates and the associated operators are thus compared: the fully coupled spectral basis, and a factorizable ansatz. The former is well-established and serves as a benchmark for the solution of Liouville-von Neumann equation. In the latter, a contracted grid basis of potential-optimized discrete variable representation is tailored to incorporate most of the inter-mode coupling, while the Lindblad operators are represented as tensor products of one-dimensional operators, for consistency. This procedure results in a marked reduction of the grid size and in a much more advantageous scaling of the computational cost with respect to the increase of the dimensionality of the system. The factorizable method is found to provide an accurate description of the dissipative quantum dynamics of the model system, specifically of the time evolution of the state populations and of the probability density distribution of the molecular wave packet. The influence of intra-molecular vibrational energy redistribution appears to be properly taken into account by the new model on the whole range of coupling strengths. It demontrates that most of the mode mixing during relaxation is due to the potential part of the Hamiltonian and not to the coupling among relaxation operators.
In a quantum system damping and noise enter through the coupling with an external large bath. The usual treatments computes the density matrix of the system by expanding in the coupling strength to the bath and averaging over the bath. The result, however, is known to be unacceptable as it leads to negative probabilities. It is here proposed to study correlation functions rather than the density matrix itself. Equations for them are derived and compared with the result obtained by the usual method.
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. By embedding elements of this matrix in a higher-dimensional Liouville-Bloch equation, the methods of unitary integration are adapted to solve for the density matrix as a function of time, including the nonunitary effects of dissipation and decoherence. The input requires only solutions of classical, initial value time-dependent equations. Results are illustrated for a damped, driven two-level system.
Abstract This chapter treats dissipation arising from coupling to the environment. Problems treated include photoionization, spontaneous emission, and cavity damping. A short-time perturbative analysis leads to decay of a single state coupled to a continuum at a rate determined by Fermi's golden rule. For longer times a non-perturbative solution of the amplitude equations leads to Weisskopf–Wigner decay. The final value theorem provides a simple method for obtaining the final state spectrum without first solving for the dynamics. A Heisenberg picture analysis leads to operator Langevin equations in which environment operators introduce quantum noise. A parallel analysis is possible working in the Schroedinger picture and leads to a master equation for the density operator of the damped system. This is solved to provide a complete description of the influence of dissipation on the system of interest. It is sometimes simpler to solve directly for the evolution of the field statistics.