Time-dependent quantum systems and PT symmetry

In this work we intend to study a class of time-dependent quantum systems with nonHermitian Hamiltonians, particularly those whose Hermitian counterpart are important for the comprehension of posed problems in quantum optics and quantum chemistry, which consists of an oscillator with time-dependent mass and frequency under the action of a time-dependent imaginary potential. We obtain the propagator for a general timedependence of the parameters and the wave-functions are obtained explicitly for constant frequency and mass and a linear time-dependence in the potential. The wave-functions are used to obtain the expectation value of the Hamiltonian. Although not PT symmetric the case of study exhibits real values of energy. ∗E-mail: [email protected] 1 In the last few years, the so called PT symmetric systems introduced in the seminal paper of Bender and Boettcher [1] has attracted much attention. They consist of non-Hermitian Hamiltonians with real eigenvalues, which however exhibit parity and time-reversal symmetries or , in other words, when one make the space-time inversion (P : x → −x; T : t → −t, i → − i), the Hamiltonian remains invariant. These symmetries are believed to be responsible for the reality of the energy eigenvalues, although a general proof seems to be lacking. Most of the papers dedicated to the subject refers to time-independent system and stationary states.. Despite an enormous list of works devoted to the developing and the understanding of this new kind of situation [2-6], as far as we know, no one has discussed the case of time-dependent systems, which are very important for the comprehension of many problems in quantum optics, quantum chemistry and others areas of physics [7][8]. In particular we can cite the case of the electromagnetic field intensities in a Fabry-Pérot cavity [9]. In fact this kind of problem constitutes a line of investigation which still attracts the interest of the physicists [11],[12],[15]. Here we intend to solve a class of time-dependent quantum system with non-Hermitian Hamiltonians, particularly those which consist of an oscillator with time-dependent mass and frequency, plus a linear non-Hermitian term with time-dependent coupling. After that, we will discuss the effect of that time-dependence on the reality of the spectrum, versus the PT symmetry by treating a particular case. The system to be studied is represented by the onedimensional non-Hermitian Hamiltonian, namely H = p 2m (t) + m (t) ω (t) 2 x + i λ (t) x. (1) which in principle can respect the PT symmetry, depending on the time-dependence of their parameters. In fact it is easy to realize that the PT symmetry imposes that λ (t) must be an even function of time. The corresponding Schroedinger equation is written as − 1 2m (t) ∂ψ (x, t) ∂x2 + [ 1 2 m (t) ω (t) x + i λ (t) x ] ψ (x, t) = i ∂ψ (x, t) ∂t , (2) where m (t), ω (t) and λ (t) are time-dependent functions (We set h̄ = 1 for a while). Now, performing the following transformation x = s (τ) y + i η (t) , (3) where τ is a single-valued function related to the original time t by τ (t) = ∫ t μ (ξ) dξ, dτ (t) dt = μ (t) . (4) After the transformation of variables we are led to the following equation i μ ∂ψ (y, τ) ∂τ − i μ (