Rabi oscillations of a two-level atomic system with damping effects are described by the use of a pseudo-Hermitian Hamiltonian. The Rabi frequency and the amplitudes for being in the upper and lower level depend in the present system on both the Hermitian resonant interaction between the em field and the atomic dipole and on damping effects. The time development of the two-level system is studied by using the metric of the pseudo-Hermitian Hamiltonian and its bi-orthonormal basis of states. The special characteristics of the present system are related to PT (parity-time reversal) and C (generalized conjugation operator) invariant properties of the pseudo-Hermitian Hamiltonian.
It is generally believed that Parity-Time (PT)-symmetry breaking occurs when eigenvalues or both eigenvalues and eigenvectors coincide. However, we show that this well-accepted picture of PT-symmetry breaking is incorrect. Instead, we demonstrate that the physical mechanism of PT-symmetry breaking is the resonance between positive- and negative-action modes. It is proved that PT-symmetry breaking occurs when and only when this resonance condition is satisfied, and this mechanism applies to all known PT-symmetry breakings observed in different branches of physics. The result is achieved by proving a remarkable fact that in finite dimensions, a PT-symmetric Hamiltonian is necessarily pseudo-Hermitian, regardless whether it is diagonalizable or not.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (𝒫), time-reversal (𝒯), and charge-conjugation (𝒞) operators and establish the 𝒫𝒯- and 𝒞𝒫𝒯-invariance of H.
In the presence of Lindblad decoherence, i.e., dissipative effects in an open quantum system due to interaction with an environment, we examine the transition probabilities between the eigenstates in the two-level quantum system described by non-Hermitian Hamiltonians with the Lindblad equation, for which the parity-time-reversal (PT) symmetry is conserved. First, the density matrix formalism for
PT-symmetric non-Hermitian Hamiltonian systems is developed. It is shown that the Lindblad operators
Lj are pseudo-Hermitian, namely, $\ensuremath{\eta}{L}_{j}^{}{\ensuremath{\eta}}^{\ensuremath{-}1}={L}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}$ with $\ensuremath{\eta}$ being a linear and positive-definite metric, and respect the
PT symmetry as well. We demonstrate that the generalized density matrix ${\ensuremath{\rho}}_{\mathrm{G}}^{}(t)\ensuremath{\equiv}\ensuremath{\rho}(t)\ensuremath{\eta}$, instead of the normalized density matrix ${\ensuremath{\rho}}_{\mathrm{N}}^{}(t)\ensuremath{\equiv}\ensuremath{\rho}(t)/\mathrm{tr}\left[\ensuremath{\rho}(t)\right]$, should be implemented for the calculation of the transition probabilities in accordance with the linearity requirement. Second, the density matrix formalism is used to derive the transition probabilities in general cases of
PT-symmetric non-Hermitian Hamiltonians. In some concrete examples, we calculate compact analytical formulas for the transition probabilities and explore their main features with numerical illustrations. We also make a comparison between our present results and our previous ones using state vectors in the absence of Lindblad decoherence.
The PT-symmetric cubic oscillator with Hamiltonian H_3 = - (1/2) d_x^2 + (1/2) x^2 + i G x^3 is a paradigmatic example of a pseudo-Hermitian, or PT-symmetric Hamiltonian with a purely real spectrum when endowed with L^2(R) boundary conditions. Eigenfunctions of the stationary Schr\{o}dinger equation H_3 psi_n(x) = E_n psi_n(x) are manifestly complex, while the energy eigenvalues E_n are real and positive. Although H_3 does not commute with the parity operator, we find that for a natural choice of the global complex phase of the eigenstate wave function, the real and imaginary parts of the eigenfunctions [i.e., Re psi_n(x) and Im psi_n(x)] are eigenstates of parity. Both the real and imaginary parts of the eigenfunctions are found to have an infinite number of zeros, even for the ground state and even for infinitesimal G, but the real and imaginary parts of psi_n(x) never vanish simultaneously when G>0. Furthermore, we find that the eigenfunctions are concentrated in an allowed region where the energy E_n is larger than the complex modulus of the complex potential. PT-symmetric Hamiltonians constitute natural generalizations of Hermitian Hamiltonians as time-evolution operators. Our results suggest that PT-symmetric (pseudo-Hermitian) time evolution can naturally be interpreted as time evolution in a situation where manifestly complex gain and loss terms mutually compensate and lead to the real energy eigenvalues.