Numerics.LvNE

Anonymous Burkhard Schmidt
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Solving the Liouville-von Neumann equation

Quantum dynamics of open systems is usually formulated in terms of reduced density operator (diagonal matrix elements referred to as "populations", off-diagonal ones as "coherences"). Its evolution is governed by a quantum master equation i. e. a differential equation for the entire density matrix, both the "densities" (diagonal elements) and the "coherences" (off-diagonal elements), the latter ones being intrinsically quantum mechanical. Note that, due to a formal similarity to the Liouville equation in classical mechanics, the quantum master equation is also known as (quantum) Liouville-von Neumann equation or simply LvNE

klzzwxh:0002rac{klzzwxh:0003artial}{klzzwxh:0004artial t}klzzwxh:0005at{klzzwxh:0006ho}={klzzwxh:0007at{klzzwxh:0008at{klzzwxh:0009athcal L}}}_Hklzzwxh:0010atklzzwxh:0011ho+{klzzwxh:0012at{klzzwxh:0013at{klzzwxh:0014athcal L}}}_Dklzzwxh:0015atklzzwxh:0016ho

The first term of the r.h.s. represents the closed system dynamics in terms of a Liouvillian superoperator for the Hermitian (aka Hamiltonian) part (using atomic units throughout)

{klzzwxh:0019at{klzzwxh:0020at{klzzwxh:0021athcal L}}}_Hklzzwxh:0022atklzzwxh:0023ho=-iklzzwxh:0024eftklzzwxh:0025klzzwxh:0026at H,klzzwxh:0027atklzzwxh:0028ho klzzwxh:0029ightklzzwxh:0030_-

with the Hamiltonian introduced on our Wiki page about Schrödinger's equations

klzzwxh:0034at{H}=klzzwxh:0035at{T}+klzzwxh:0036at{V}-F(t)klzzwxh:0037dotklzzwxh:0038at{klzzwxh:0039u}

for a quantum system interacting (within the semiclassical dipole approximation) with electrical field F(t) and where μ is a Hermitian matrix with dipole moments. In our Matlab codes we make use of an energy representation where the Hamiltonian of the unperturbed system, H0=T+V, is diagonal.

In order to retrieve the expectation value of a quantum-mechanical operator/observable O from the solution of the LvNE, the following trace formula is used

klzzwxh:0044angleklzzwxh:0045at{O}klzzwxh:0046angle=klzzwxh:0047athrm{Tr}(klzzwxh:0048at{O}klzzwxh:0049at{klzzwxh:0050ho})

where the time-dependence can arise from time-dependent O or time-dependent ρ or both.

The second term on the r.h.s. of the first equation represents the coupling to the environment accounting for time-directed irreversibility, i. e., dissipation and/or dephasing. The most widely used formalisms in use are the following ones:

In conclusion, the above Ansatz of the quantum-master (Liouville-von Neumann) equation in Born-Markov approximation (weak coupling, no memory) provides a description useful in many cases ranging from NMR over quantum optics to molecular/chemical applications.

Tetradic notation

In order to bring the above evolution equations into a shape compatible with the standard form of linear/bilinear control systems and also with standard ODE solvers, the density operators (or, rather their matrix representations) have to be mapped onto vectors and, accordingly, the superoperators have to be mapped onto matrices.

Frequently used choices of mapping density matrices onto vectors are the following:

  • Column-wise: this mimics the memory organization in Fortran and MATLAB, in particular, the (:) operator of the latter. This is equivalent to the mathematical concept of vectorization which converts a matrix into a column vector by stacking the columns of the matrix A on top of one another.

  • Row-wise: this mimics the memory organization in C and in C++.

  • Diagonal first: First the populations, then the coherences (where the order of the latter ones is arbitrary but fixed), which leads to block structures of the Lindblad Liouvillian superoperators, see e. g. Appendix A of [1] and/or our Wiki page on linear/bilinear control systems.

klzzwxh:0054ho=klzzwxh:0055eft(klzzwxh:0056egin{array}{ll}klzzwxh:0057ho_{00} & klzzwxh:0058ho_{01}klzzwxh:0059klzzwxh:0060ho_{10} & klzzwxh:0061ho_{11}klzzwxh:0062 klzzwxh:0063nd{array} klzzwxh:0064ight) klzzwxh:0065apsto klzzwxh:0066eft( klzzwxh:0067egin{array}{l} klzzwxh:0068ho_{00} klzzwxh:0069 klzzwxh:0070ho_{11} klzzwxh:0071 klzzwxh:0072ho_{01} klzzwxh:0073 klzzwxh:0074ho_{10} 
klzzwxh:0075nd{array}klzzwxh:0076ight)

References

  1. B. Schäfer-Bung, C. Hartmann, B. Schmidt, and Ch. Schütte: J. Chem. Phys. 135, 014112 (2011)

Related

Wiki: Numerics.Control.Lindblad
Wiki: Numerics.Control
Wiki: Numerics.LvNE.Lindblad
Wiki: Numerics.LvNE.Redfield
Wiki: Numerics.Main

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