Liouville space

The state of a mechanical system of / degrees of freedom is specified by the vector r = (qi,. . ., qf, pi,. .., pf), whose components are the / generalized coordinates (qi,. . ., qf) and /conjugate momenta pi. p/). Geometrically, the state is represented by the point / in phase-space, which is a 2/ dimensional Cartesian space whose coordinate axes are labeled by (qi. p/) respectively. According to mechanics the state / changes with time according to the canonical equations of motion [Eq. (2.3.1)]. Equation (2.3.1) can be written in the compact vector form 

The Liouvillian is obviously a linear partial differential operator. As we shall soon see, 

This chapter is mathematically involved and should be skipped, at least on the first reading, by those not conversant with the mathematics of quantum mechanics. 

L has many of the. mathematical properties of the Hamiltonian operator in quantum mechanics. 

Here To and Ft are, respectively, the states of the system at times 0 and t, and the operator eiLt generates the state /) from the initial state. The operator eiLt is called the propagator. It defines a mapping in phase space. 

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Figure Al.6.18. Liouville space lattice representation in one-to-one correspondence with the diagrams in figure A1.6.17. Interactions of the density matrix with the field from the left (right) is signified by a vertical (liorizontal) step. The advantage to the Liouville lattice representation is that populations are clearly identified as diagonal lattice points, while coherences are off-diagonal points. This allows innnediate identification of the processes subject to population decay processes (adapted from [37]).

Yan Y J, Gillilan R E, Whitnell R M, Wilson K R and Mukamel S 1993 Optimal control of molecular dynamics -Liouville space theory J. Chem. Phys. 97 2320... 

Szymanski S, Gryff-Keller A M and Binsch G A 1986 Liouville space formulation of Wangsness-Bloch-Redfield theory of nuclear spin irelaxation suitable for machine computation. I. Fundamental aspects J. Magn. Reson. 68 399-432... 

It is more convenient to re-express this equation in Liouville space [8, 9 and 10], in which the density matrix becomes a vector, and the commutator with the Hamiltonian becomes the Liouville superoperator. In tliis fomuilation, the lines in the spectrum are some of the elements of the density matrix vector, and what happens to them is described by the superoperator matrix, equation (B2.4.25) becomes (B2.4.26). 

This Liouville-space equation of motion is exactly the time-domain Bloch equations approach used in equation (B2.4.13). The magnetizations are arrayed in a vector, and anything that happens to them is represented by a matrix. In frequency units (1i/2ti = 1), the fomial solution to equation (B2.4.26) is given by equation (B2.4.27) (compare equation (B2.4.14H. 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

Relaxation or chemical exchange can be easily added in Liouville space, by including a Redfield matrix, R, for relaxation, or a kinetic matrix, K, to describe exchange. The equation of motion for a general spin system becomes equation (B2.4.28). 

In Liouville space, both the density matrix and the operator are vectors. The dot product of these Liouville space... 

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. 

In (7.90) a slightly modified notation is introduced for convenience for the bra and ket vectors in the Liouville space for the resolvent superoperator... 

S. Tanaka, V. Chernyak, and S. Mukamel, Time-resolved X-ray spectroscopies nonlinear response functions and Liouville space pathways. Phys. Rev. A 63(6), 063405 (2001). 

Figure 14. Liouville space coupling schemes and their respective double-sided Feynman diagrams for three of the six pathways in Liouville space which contribute to p 2. The complex conjugates are not shown. All pathways proceed only via coherences, created by the interactions with the two fields shown as incoming arrows. Solid curves pertain to e( 11 and dashed curves to r/2T (Reproduced with permission from Ref. 47, Copyright 2005 American Institute of Physics.)...

S. Ramakrishna and T. Seideman, Coherence spectroscopy in dissipative media a Liouville space approach, J. Chem. Phys. 122, 084502 (2005). 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

In Liouville space, both the density matrix and the 4 operator become vectors. The scalar product of these Liouville space vectors is the trace of their product as operators. Therefore, the NMR signal, as a function of a single time variable, t, is given by (10), in which the parentheses denote a Liouville space scalar product ... 

Fig. 1. Two showing the fifth-order Raman pulse sequence. (A) Definition of fields, (B) Energy level diagram showing one possible Liouville space pathway.

Equation (5.38) can be interpreted as the scalar product of a forward-moving density and a backward-moving time-dependent operator. The optimal field at time t is determined by a time-dependent objective function propagated from the target time T backward to time t. A first-order perturbation approach to obtain a similar equation for optimal chemical control in Liouville space has been derived in a different method by Yan et al. [28]. 

Phase-space wavepackets for nuclear motions have been applied to the interpretation of nonlinear optical measurements using the Liouville space... 

This formula, unlike Eq. (2.7), maintains a complete bookkeeping of the time ordering of the various interactions with the radiation field [17] and has six terms, the Liouville space paths corresponding to three of them are shown in Fig. 2, and the paths for the complex-conjugate terms are obtained by interchanging the left and the right portions of each path. 

Figure 1- Liouville space diagram corresponding to the only term that contributes to the spontaneous light emission from a two-level system within the rotating-wave approximation [Eq. (2.7)]. Here ]g) and e) denote the ground and the excited states, respectively.

Figure 2. Liouville space paths for the three terms of Eq. (2.10).

Figure 3. Diagrams showing how to divide the triple integral in (2.7) to get the six terms of (2.10). Domains (a), (b), and (c) correspond to the three Liouville space paths given in Fig. 2 and domains (d), (e), and (/) to the complex-conjugate paths.

Figure 4. TW° ways to describe the time-evolution of the dipole operators single (a) and double (b) Liouville space diagrams.

Unlike the quantum response (2), which contains an interference of two Liouville space paths, the classical expression (3) contains no inter-... 

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