Wigner representation

Then, instead of performing the six-dimensional integral in Eq. (5.19) all at once, we perform successive three-dimensional integrals over s and R. The first step takes us to W R,P), the Wigner representation [130,131] of the density matrix, and the second step to the p-space density matrix, n(P — p/2 P + p/2). The reverse transformation of Eq. (5.20) can also be performed stepwise over P and p to obtain A( , p), the Moyal mixed representation [132], and then the r-space representation V R— s/2 R + s/2). These steps are shown schematically in Figure 5.2. 

The electron and momentum densities are just marginal probability functions of the density matrix in the Wigner representation even though the latter, by the Heisenberg uncertainty principle, cannot be and is not a true joint position-momentum probability density. However, it is possible to project the Wigner density matrix onto a set of physically realizable states that optimally fulfill the uncertainty condition. One such representation is the Husimi function [122,133-135]. This seductive line of thought takes us too far away from the focus of this... 

Figure 5.2. The relationships among the r-space density matrix F, the p-space density matrix n, the Wigner representation W, and the Moyal representation A. Two-headed arrows with a T beside them signify reversible, three-dimensional, Fourier transformations.

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

We have thus expressed the autocorrelation signals using the Wigner representation for both the external fields and the gate. The molecular properties are contained in the response function F(4). In the next section, we show how when the incoming external pulses and the detection gate are temporally well separated, we can use the Wigner representation for the material system as well. 

Substituting this into (4.1) and introducing the Wigner representation for the pump and the probe fields as was done for the fluorescence, we get... 

The classical Liouvillian operator Zc, which is the classical limit of the Landau-von Neumann superoperator in Wigner representation, can also be analyzed in terms of a spectral decomposition, such as to obtain its eigenvalues or resonances. Recent works have been devoted to this problem that show that the classical Liouvillian resonances can be obtained as the zeros of another kind of zeta function, which is of classical type. The resolvent of the classical Liouvillian can then be obtained as [60, 61]... 

Another useful form we get is the Wigner representation. Let us take into account additionally an external potential U(r, t). In this case it is convenient to introduce the correlation density matrix gx2 by... 

A systematic route to achieve a mixed quantum classical description of EET may start with the partial Wigner representation p(R,P t) of the total density operator referring to the CC solvent system. R and P represent the set of all involved nuclear coordinates and momenta, respectively. However, p(R,P t) remains an operator in the space of electronic CC states (here 4>o and the different first order of the -expansion one can change to electronic matrix elements. Focusing on singly excited state dynamics we have to consider pmn( It, / /,) = 4>m p R, P t) 4>n) which obeys the following equation... 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

If we expand Eq. (47) in the coordinate Q [-representation for the environmental degrees of freedom only we obtain the second line. Taking the Wigner representation for these degrees of freedom and finally, defining the general coordinate X = (R,P) gives Eq. (48). 

FIGURE 15 Wigner representation of the optimal electric field for l2 wave packet control. [Reproduced with permission from Krause, Whitnell, J. L., Wilson, R. M., K. R., and Yan, Y. (1993). J. Chem. Phys. 99, 6562. Copyright American Institute of Physics.]... 

In doing so it proves convenient to carry out the computations in the Wigner representation. That is, as in quantum mechanics based upon the wave function, it is necessary to deal with a representation of the density operator p. The (convenient) Wigner representation pw of p is defined, for an N degree of freedom system, by... 

This Wigner representation of the density pw q, p) proves particularly useful since it, satisfies a number of properties that are similar to the classical phase-space distribu tion pd(q, p). For example, if p = pure state, then fdppw = probability density in coor- dinate space. Similarly, integrating pw over q gives the probability density in ( momentum space. These features are shared by the classical density p p, q) in phase space. Note, however, that pw is not a probability density, as evidenced by if the fact that it can be negative, a reflection of quantum features of the dynamics, ) [165], 3... 

The Wigner function has the valuable property that the time evolution equation for the quantum dynamics in the Wigner representation resembles that for the classical Liouville dynamics. Specifically, the Schrodinger equation can be transformed to [70]... 

Equilibrium time correlation function expressions for transport properties can be derived using linear response theory [3]. Linear response theory can be carried out directly on the Wigner transformed equations of motion to obtain the transport properties as correlation functions involving Wigner transformed quantities. Alternatively, we may carry out the linear response analysis in terms of abstract operators and insert the Wigner representation of operators in the final form for the correlation function. We use the latter route here. 

As discussed in the Introduction, our interest is in quantum-classical systems where the environmental degrees of freedom can be treated classically. The above formulation of quantum dynamics and quantum statistical mechanics in the Wigner representation suggests that we consider another formulation of quantum mechanics based on a partial Wigner representation where only the degrees of freedom in the subsystem are Wigner transformed [4]. We now sketch how this program can be carried out. 

The time evolution of a quantum operator C = AB, which is the product of two operators, can be written in the partial Wigner representation as... 

Quantum mechanics in the partial Wigner representation is exact and the partially Wigner transformed quantum bracket satisfies the Jacobi identity,... 

Its time evolution is given by full quantum mechanics in the Wigner representation. In order to obtain a computationally tractable form, we consider a limit where the time evolution of W X, X2,t) is approximated by quantum-classical dynamics. 

Since Wigner transformed quantum mechanics is difficult to compute, we express the S subsystem degrees of freedom in an adiabatic basis rather than in a Wigner representation. To this end, we first observe that Aw Xi) can be written as... 

V. S. Filinov, Y. V. Medvedev, and V. L. Kamskyi (1995) Quantum dynam-ics and wigner representation of quantum-mechanics. Mol. Phys. 85, pp. 711-726 V. S. Filinov (1996) Wigner approach to quantum statistical mechanics and quantum generalization molecular dynamics method. Part 1. Mol. Phys. 88, pp. 1517-1528... 

The optimization of the pump pulse leads to a localization of the phase space density around the intermediate target. The intermediate target operator can be represented in the Wigner representation [Eq. (20)] by a minimum uncertainty wavepacket ... 

By comparing Eq. (95) with Eqs. (88) and (94), we can conclude that the differences between the generalized and truncated squeezed vacuums are smaller than those between them and the conventional squeezed vacuum. All these states coincide in the high-dimension limit. In Fig. lOb-f, we have presented the Wigner representation of the generalized squeezed vacuum for those values of the squeeze parameter , which correspond to maximum and minimum values of the vacuum-state probability given by Ps(0) = 2 (see Fig. 10a). We... 

The subscript W refers to this partial Wigner transform, N is the eoordinate space dimension of the bath and X = R, P). In this partial Wigner representation, the Hamiltonian of the system takes the form Hw R,P) = P /2M + y-/2m+ V q,R). If the subsystem DOF are represented using the states of an adiabatic basis, a P), which are the solutions of hw R) I R)=Ea R) I where hw K)=p /2m+ V q,R) is the Hamiltonian for the subsystem with fixed eoordinates R of the bath, the density matrix elements are p i -, 0 = ( I Pw( 01 )- From the solution of the quantum Liouville equation given some initial state of the entire quantum system, the reduced density matrix elements of the quantum subsystem of interest can be obtained by integrating over the bath variables, p f t) = dX p X,t), in order to find the populations and off-diagonal elements (coherences) of the density matrix. 

Another approach has been chosen by Bonacic-Koutecky and coworkers [136]. Together with Jortner, they applied the density matrix method in the Wigner representation (see Sect. 2.2.3) to investigate the vibrational dynamics of the trimer. Within this approach the simulation of the real-time photoion spectra involves three steps ... 

Fig. 5.9. Comparison of the real-time charge reversal process Ag —> Ags - Ag. (a) Simulation applying the density matrix method in the Wigner representation (Tanions = 300 K, 100fs pulse width), (b) Pump probe experiment (Tanions = 300 K, 100fs pulse width), (c) Simulation applying the density matrix method in the Wigner representation in case of a 5 times shorter probe pulse width (by courtesy of M. Hartmann taken from [136])...

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