Nonadiabatic transition probabilities in a time-dependent Gaussian pulse or plateau pulse: Toward experimental tests...

For a quantum system subject to a time-dependent perturbing field, Dirac's analysis gives the probability of transition to an excited state |k⟩ in terms of the norm square of the entire excited-state coefficient c_k(t) in the wave function. By integrating by parts in Dirac's equation for c_k(t) at first order, Landau and Lifshitz separated c_k ⁽¹⁾(t) into an adiabatic term a_k ⁽¹⁾(t) that characterizes the gradual adjustment of the ground state to the perturbation without transitions and a nonadiabatic term b_k ⁽¹⁾(t) that depends explicitly on the time derivative of the perturbation at times t' ≤ t. Landau and Lifshitz stated that the probability of transition in a pulsed perturbation is given by |b_k(t)|², rather than by |c_k(t)|². We use the term "transition probability" to refer to the probability that a true excited-state component is present in the time-evolved wave function, as opposed to a smooth modification of the initial state. In recent work, we have examined the differences between |b_k(t)|² and |c_k(t)|² when a system is perturbed by a harmonic wave in a Gaussian envelope. We showed that significant differences exist when the frequency of the harmonic wave is off-resonance with the transition frequency. In this paper, we consider Gaussian perturbations and pulses that rise via a half Gaussian shoulder to a level plateau and later return to zero via a down-going half Gaussian. While the perturbation is constant, the transition probability |b_k(t)|² does not change. By contrast, |c_k(t)|² continues to oscillate while the perturbation is constant, and its time averaged value differs from |b_k(t)|². We suggest a general type of experiment to prove that the transition probability is given by |b_k(t)|², not |c_k(t)|². We propose a ratio test that does not require accurate knowledge of transition matrix elements or absolute field intensities.