Proposal of a Computational Approach for Simulating Thermal Bosonic Fields in Phase Space

Abstract

When a quantum field is in contact with a thermal bath, the vacuum state of the field may be generalized to a thermal vacuum state, which takes into account the thermal noise. In thermo field dynamics, this is realized by doubling the dimensionality of the Fock space of the system. Interestingly, the representation of thermal noise by means of an augmented space is also found in a distinctly different approach based on the Wigner transform of both the field operators and density matrix, which we pursue here. Specifically, the thermal noise is introduced by augmenting the classical-like Wigner phase space by means of Nosé-Hoover chain thermostats, which can be readily simulated on a computer. In this paper, we illustrate how this may be achieved and discuss how non-equilibrium quantum thermal distributions of the field modes can be numerically simulated.Published in Physics 2019, 1(3), 402-411; https://doi.org/10.3390/physics1030029 I. INTRODUCTIONAccording to quantum field theory, the vacuum is filled by fluctuating quantum fields.Such fields are present both in condensed matter systems [1-6] and, more generally, in empty space-time, where their existence is believed to be linked to dark matter and dark energy [7][8][9][10][11][12][13][14][15][16]. Universal phenomena such as the emergence of order, symmetry breaking, and phase transitions are related to the thermal degrees of freedom the first time in order to stress that we are not using the expression in a literal meaning. of the fields [17,18]. Hence, the study of the thermal excitation of the quantum vacuum is of interest to many research areas [19][20][21][22][23]. In thermo field dynamics [24][25][26], the vacuum state is generalized to a thermal vacuum state by doubling the dimension of the Fock space of the original vacuum. The additional dimension of the Fock space represent the degrees of freedom of the thermal bath, which are involved in the excitation and de-excitation processes of the thermal system.In this paper, we illustrate an approach to simulate the thermal distributions of bosonic * Electronic address: asergi@unime.it † Electronic address: roberto.grimaudo01@unipa.it ‡ Electronic address: gabriel.hanna@ualberta.ca § Electronic address: antonino.messina@unipa.it H (Q,P ) = J ω JP 2 J 2λ 2 J + ω J λ 2 JQ 2 J the Wigner transform of the field Hamiltonian in Equation (1) reads: H (Q,P ) = J P 2 J 2µ J + µ J ω 2 J Q 2 J 2 = J

Spin‐Chain‐Star Systems: Entangling Multiple Chains of Spin Qubits

“…In this case, likely, a numerical approach to solve the dynamics is necessary. Several approaches could be used to deal with this scenario, from the standard Lindblad theory [69,70] to the partial Wigner transform [71][72][73][74][75][76][77] and the non-Hermitian formalism. [78][79][80][81][82][83][84][85][86][87]…”

Spin-chain-star systems: entangling multiple chains of spin qubits

“…In this case, likely, a numerical approach to solve the dynamics is necessary. Several approaches could be used to deal with this scenario, from the standard Lindblad theory [67,68] to the partial Wigner transform [69][70][71][72][73][74][75] and the non-Hermitian formalism [76][77][78][79][80][81][82][83][84][85]. Let us consider the following N-spin model…”

Two-Qubit Entanglement Generation through Non-Hermitian Hamiltonians Induced by Repeated Measurements on an Ancilla

“…Moreover, one may wish to concentrate on the application of the theoretical method of [ 41 ] in more complex cases, like two-qubit systems immersed in a quantum oscillator environment. In this case, a fruitful comparison with other approaches [ 79 , 80 , 81 , 82 , 83 ] developed to face this kind of problem is possible.…”