The Dynamical Casimir Effect in a Dissipative Optomechanical Cavity Interacting with Photonic Crystal

Abstract

We theoretically study the dynamical Casimir effect (DCE), i.e., parametric amplification of a quantum vacuum, in an optomechanical cavity interacting with a photonic crystal, which is considered to be an ideal system to study the microscopic dissipation effect on the DCE. Starting from a total Hamiltonian including the photonic band system as well as the optomechanical cavity, we have derived an effective Floquet-Liouvillian by applying the Floquet method and Brillouin-Wigner-Feshbach projection method. The microscopic dissipation effect is rigorously taken into account in terms of the energy-dependent self-energy. The obtained effective Floquet-Liouvillian exhibits the two competing instabilities, i.e., parametric and resonance instabilities, which determine the stationary mode as a result of the balance between them in the dissipative DCE. Solving the complex eigenvalue problem of the Floquet-Liouvillian, we have determined the stationary mode with vanishing values of the imaginary parts of the eigenvalues. We find a new non-local multimode DCE represented by a multimode Bogoliubov transformation of the cavity mode and the photon band. We show the practical advantage for the observation of DCE in that we can largely reduce the pump frequency when the cavity system is embedded in a narrow band photonic crystal with a bandgap.

Fifty Years of the Dynamical Casimir Effect

“…The experiment based on the suggestion [462] was realized recently [465]. A simulation of the DCE in photonic lattices or photonic crystals was suggested in papers [466,467].…”

PT -Symmetric Qubit-System States in the Probability Representation of Quantum Mechanics

“…Since the vector components ρ Ej are expressed in terms of dichotomic probabilities p Ej 1, 2, 3 ; j, k = 1, 2 , 3, linear Equations ( 25) and (26) have the form of a system of equations for the set of probability distributions p Ej 1, 2, 3 determining the qutrit Hamiltonian eigenstates. The explicit probabilistic form of Equation ( 25) reads 12) p * ( 13) p (12) p ( 22) 3 p * ( 23) p (13) p (22) p 12) p * ( 13) p (12) p ( 22) 3 p * ( 23) p (13) p (22) p…”

“…and explicit probabilistic form of Equation ( 26) is 12) p * ( 13) p (12) p ( 22) 3 p * ( 23) p (13) p (22) p 12) p * ( 13) p (12) p ( 22) 3 p * ( 23) p (13) p (22) p…”

“…3 . Thus, for the three-level atom, we can obtain the energy spectrum for an arbitrary non-Hermitian Hamiltonian with 3×3-density matrix H by solving Equations ( 27) and ( 28) written for dichotomic probabilities p (11) 3 and p (22) 3 and complex numbers p (13) , p (13) , and p (23) expressed as linear combinations of dichotomic probabilities p (jk) 1, 2 , namely,…”

“…Recently, the problem of non-Hermitian Hamiltonian operators with real eigenvalues was discussed in [4][5][6][7][8] in connection with P T -symmetry (parity-time) of the physical problems, where the eigenvectors and eigenvalues of non-Hermitian Hamiltonians with such symmetries are associated with specific properties of the physical systems (see, for example, [9,10]), including the dynamical Casimir effect reviewed in [11,12] where the non-Hermiticity of the Hamiltonian is related to accounting for the dissipation processes [13]. The problems of non-Hermitian quantum mechanics were studied in [14][15][16][17][18].…”