New nonclassical solutions for the harmonic oscillator with generalized time-dependent frequency have been found. Simple expression on energy level, creation operator a†(t) and annihilation operator a(t) have been obtained. Using new solutions we want to show how to study squeezing.
Harmonic oscillators with time-dependent frequencies describe physics of the Paul trap of ions, radiation fields in media of time-varying dielectric constants, and so on. Their ground states are generically squeezed states. Here the effects of small anharmonicity on such nonclassical states are discussed. A perturbative calculation is performed, in which the unperturbed states are prepared in terms of the Hermitian invariant of the time-dependent harmonic oscillator. A quartic potential perturbation is investigated in detail. It is found that the correction term as well as the unperturbed term in the ground-state persistence probability can be expressed by the moments of a number operator alone. The oscillatory behavior is found in that correction term, which shows a competition of excitation and deexcitation due to the time dependence of the frequency and the nonlinear interaction. This may be observed experimentally as the temporal fluctuation of the quasienergy levels. Some formulas are also presented for the variances and correlations of the quadratures in the perturbed states.
New generalized squeezed states for the time-dependent harmonic oscillator are found through the theory of invariants. Our method gives a comparatively clearer picture than methods using evolution operators because we can establish a direct connection between the classical and quantum solutions. The additional significance of our method is that it is possible to find new generalized quantum squeezed states from just one particular solution to the classical time-dependent oscillator. Accordingly, more general results for the variance of x (squared uncertainty) are found for the cases of linear sweep of the restoring force and compared to recent results encountered in the literature.
The problem of quantum harmonic oscillator with time-dependent frequency has been solved analytically only for special co(t) functions. In this paper (Sec. 1.) we give a recursive method to determine the creation and annihilation operators for any step-like 0)(0 (t) functions (i. e. functions being constant between jumps). In Sec. 2. we apply this method to jumps between two values of w. In Sec 3. we approximate more general functions with a series of small jumps, we examine the Fourier series of the function defined in Sec. 2, and show that even the first Fourier component leads to increasing squeezing.
With time and space transformation, we first solve the Schrödinger equation of the time-dependent harmonic oscillator (TDHO) system. The properties of the squeezing in the case of , i.e., the Paul trap and , namely a stable frequency interfered by a single-pulsing-type disturbance, are investigated by using function series expansion. Results in the former case are compared with the numerical calculations carried out before. It is found that the condition related to the most strong squeezing is the direct result of our method, not any approximation is needed. Solutions of the latter one are valuable for the study of the ion trap, where different cases are studied in detail. Finally we made a discussion for the measurement and observation of a trapped ion by using squeezing properties.