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Theoretical and Mathematical Physics

Quantum dissipative systems. I. Canonical quantization and quantum Liouville equation

Vasily Evgenevich Tarasov

Published Sep 1, 1994• 22 citations
Publisher
DOI
10.1007/bf01018575

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The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system. But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.

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From Liouville equation to universal quantum control: A study of generating ultra highly squeezed states

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We find that the seemingly disparate control approaches for classical and quantum continuous-variable systems can be unified via differential manifolds of the ancillary representations. For classical systems, the ancillary representation is defined by the time-dependent ancillary canonical variables resulting from symplectic transformation over the original canonical variables. Under the Hamilton-Jacobi theory, the ancillary canonical variables act as dynamical invariants to guide the system nonadiabatically through the entire phase space. The second quantization of the Liouville equation for the dynamical invariants leads to the Heisenberg equation for the relevant ancillary operators, which is found to be a sufficient condition to activate nonadiabatic passages towards arbitrary target states in both Hermitian and non-Hermitian systems and yield constrained exact solutions of the time-dependent Schroedinger equation. Using the non-Hermitian Hamiltonian rigorously derived from the Lindblad master equation, our theory is exemplified by the generation of single-mode squeezed states with a squeezing level about 29.3 dB and double-mode squeezed states with 20.6 dB, respectively.

quant-pharXiv:2604.02635•Physical Review A•Apr 6, 2026•0 citations
The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system. But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.
We find that the seemingly disparate control approaches for classical and quantum continuous-variable systems can be unified via differential manifolds of the ancillary representations. For classical systems, the ancillary representation is defined by the time-dependent ancillary canonical variables resulting from symplectic transformation over the original canonical variables. Under the Hamilton-Jacobi theory, the ancillary canonical variables act as dynamical invariants to guide the system nonadiabatically through the entire phase space. The second quantization of the Liouville equation for the dynamical invariants leads to the Heisenberg equation for the relevant ancillary operators, which is found to be a sufficient condition to activate nonadiabatic passages towards arbitrary target states in both Hermitian and non-Hermitian systems and yield constrained exact solutions of the time-dependent Schroedinger equation. Using the non-Hermitian Hamiltonian rigorously derived from the Lindblad master equation, our theory is exemplified by the generation of single-mode squeezed states with a squeezing level about 29.3 dB and double-mode squeezed states with 20.6 dB, respectively.