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Journal of Experimental and Theoretical Physics

On the Adiabatic Approximation for the Density Matrix of an Isolated Spin system

Alexander Alexandrovitch Samokhin

Published Mar 1, 1967• 2 citations
OpenAlex

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Adiabatic approximation for the density matrix

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An adiabatic approximation for the Liouville density-matrix equation which includes decay terms is developed. The adiabatic approximation employs the eigenvectors of the non-normal Liouville operator. The approximation is valid when there exists a complete set of eigenvectors of the non-normal Liouville operator (i.e., the eigenvectors span the density-matrix space), the time rate of change of the Liouville operator is small, and an auxiliary matrix is nonsingular. Numerical examples are presented involving efficient population transfer in a molecule by stimulated Raman scattering, with the intermediate level of the molecule decaying on a time scale that is fast compared with the pulse durations of the pump and Stokes fields. The adiabatic density-matrix approximation can be simply used to determine the density matrix for atomic or molecular systems interacting with cw electromagnetic fields when spontaneous emission or other decay mechanisms prevail.

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On the quantum statistical theory of relaxation in isolated spin systems

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We study the adiabatic limit in the density matrix approach for a quantum system coupled to a weakly dissipative medium. The energy spectrum of the quantum model is supposed to be non-degenerate. In the absence of dissipation, the geometric phases for periodic Hamiltonians obtained previously by M.V. Berry are recovered in the present approach. We determine the necessary condition satisfied by the coefficients of the linear expansion of the non-unitary part of the Liouvillian in order to the imaginary phases acquired by the elements of the density matrix, due to dissipative effects, be geometric. The results derived are model-independent. We apply them to spin 1/2 model coupled to reservoir at thermodynamic equilibrium.

•Jan 1, 2002•5 citations

Time dependent reduced density matrix functional theory at strong correlation: insights from a two-site Anderson impurity model

S. Di Sabatino, Claudio Verdozzi, Pina Romaniello

The one-body density matrix has recently attracted considerable attention as a promising key quantity for the description of systems out of equilibrium. Its time evolution is given in terms of the two-body density matrix, and thus the central challenge is to find approximations to the latter. An extra layer of difficulty is added when dealing with strong electron correlations. In this work, we explore precisely this regime by looking at the two-site Anderson impurity model as a case study. To address the system's dynamics, we use an adiabatic approximation based on the exact ground-state two-body density matrix. We find that this adiabatic extension does not reproduce the exact results even for a slow switch-on of the external perturbation, and we trace back this behavior to the lack of an accurate imaginary part of the adiabatic approximation to the two-body density matrix. The attempt to restore an approximate imaginary part through a Hilbert transform of the real part works well only for very short times, but quickly deteriorates for longer times, with the one-body density matrix being pushed out of its N-representability domain. Our results thus pose an important constraint on practical prescriptions to perform the time evolution of the one-body density matrix.

Physical Chemistry Chemical Physics•Jan 1, 2021•9 citations

The adiabatic approximation for quantum spin systems with a spectral gap

Claudio Albanese

Communications in Mathematical Physics•Jul 1, 1996•0 citations
An adiabatic approximation for the Liouville density-matrix equation which includes decay terms is developed. The adiabatic approximation employs the eigenvectors of the non-normal Liouville operator. The approximation is valid when there exists a complete set of eigenvectors of the non-normal Liouville operator (i.e., the eigenvectors span the density-matrix space), the time rate of change of the Liouville operator is small, and an auxiliary matrix is nonsingular. Numerical examples are presented involving efficient population transfer in a molecule by stimulated Raman scattering, with the intermediate level of the molecule decaying on a time scale that is fast compared with the pulse durations of the pump and Stokes fields. The adiabatic density-matrix approximation can be simply used to determine the density matrix for atomic or molecular systems interacting with cw electromagnetic fields when spontaneous emission or other decay mechanisms prevail.
We study the adiabatic limit in the density matrix approach for a quantum system coupled to a weakly dissipative medium. The energy spectrum of the quantum model is supposed to be non-degenerate. In the absence of dissipation, the geometric phases for periodic Hamiltonians obtained previously by M.V. Berry are recovered in the present approach. We determine the necessary condition satisfied by the coefficients of the linear expansion of the non-unitary part of the Liouvillian in order to the imaginary phases acquired by the elements of the density matrix, due to dissipative effects, be geometric. The results derived are model-independent. We apply them to spin 1/2 model coupled to reservoir at thermodynamic equilibrium.
The one-body density matrix has recently attracted considerable attention as a promising key quantity for the description of systems out of equilibrium. Its time evolution is given in terms of the two-body density matrix, and thus the central challenge is to find approximations to the latter. An extra layer of difficulty is added when dealing with strong electron correlations. In this work, we explore precisely this regime by looking at the two-site Anderson impurity model as a case study. To address the system's dynamics, we use an adiabatic approximation based on the exact ground-state two-body density matrix. We find that this adiabatic extension does not reproduce the exact results even for a slow switch-on of the external perturbation, and we trace back this behavior to the lack of an accurate imaginary part of the adiabatic approximation to the two-body density matrix. The attempt to restore an approximate imaginary part through a Hilbert transform of the real part works well only for very short times, but quickly deteriorates for longer times, with the one-body density matrix being pushed out of its N-representability domain. Our results thus pose an important constraint on practical prescriptions to perform the time evolution of the one-body density matrix.