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AIP conference proceedings

Theory of Quantum Transitions Based on Non-adiabatic Dressed States

Ivan Georgiev Koprinkov, Michail D. Todorov, Christo I. Christov

Published Jan 1, 2011• 0 citations
Publisher

Abstract

Nonadiabatic dressed states of a quantum system interacting with an external electromagnetic field and the environment are presented. The electromagnetic field is described in terms of carrier‐envelope concept. The amplitude and phase of the field are arbitrary functions of time, subject to a generalized adiabatic condition. The interaction with the environment is described my means of a dumping term in the Hamiltonian. The field time derivatives and the dumping play role of nonadiabatic factors. The main matrix elements within the nonadiabatic dressed states are found. A theory of quantum transitions based on the nonadiabatic dressed states is proposed for the first time.

DOI
10.1063/1.3659938

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The Journal of Chemical Physics•Oct 26, 2012•12 citations
Nonadiabatic dressed states of a quantum system interacting with an external electromagnetic field and the environment are presented. The electromagnetic field is described in terms of carrier‐envelope concept. The amplitude and phase of the field are arbitrary functions of time, subject to a generalized adiabatic condition. The interaction with the environment is described my means of a dumping term in the Hamiltonian. The field time derivatives and the dumping play role of nonadiabatic factors. The main matrix elements within the nonadiabatic dressed states are found. A theory of quantum transitions based on the nonadiabatic dressed states is proposed for the first time.
A change in the geometry of the Hilbert space of a quantum system interacting with a nonadiabatic electromagnetic field is found. To identify its physical origin, the evolution of orthogonality to non-orthogonality of the states of the quantum system has been investigated based on the matrix elements of the quantum states in three main physical cases: a quantum system in zero external field (closed quantum system), a quantum system in slowly changing (adiabatic) field, and a quantum system in rapidly changing (nonadiabatic) field. The quantum states are orthogonal in the first two cases and change to non-orthogonal in a nonadiabatic field, following continuously the parameters of the field. A time-dependent Fubini-Study metric of the Hilbert space can be determined based on the matrix elements of the quantum states in the nonadiabatic case. Such a dynamical metric of the Hilbert space in quantum mechanics resembles the dynamical metric of spacetime in general relativity.
We contrast Dirac's theory of transition probabilities and the theory of nonadiabatic transition probabilities, applied to a perturbed system that is coupled to a bath. In Dirac's analysis, the presence of an excited state |k0⟩ in the time-dependent wave function constitutes a transition. In the nonadiabatic theory, a transition occurs when the wave function develops a term that is not adiabatically connected to the initial state. Landau and Lifshitz separated Dirac's excited-state coefficients into a term that follows the adiabatic theorem of Born and Fock and a nonadiabatic term that represents excitation across an energy gap. If the system remains coherent, the two approaches are equivalent. However, differences between the two approaches arise when coupling to a bath causes dephasing, a situation that was not treated by Dirac. For two-level model systems in static electric fields, we add relaxation terms to the Liouville equation for the time derivative of the density matrix. We contrast the results obtained from the two theories. In the analysis based on Dirac's transition probabilities, the steady state of the system is not an equilibrium state; also, the steady-state population ρkk,s increases with increasing strength of the perturbation and its value depends on the dephasing time T2. In the nonadiabatic theory, the system evolves to the thermal equilibrium with the bath. The difference is not simply due to the choice of basis because the difference remains when the results are transformed to a common basis.
The quantum coherence as a phase property of the most fundamental quantum mechanical entity, the wave function, is explained for the first time. Phase-sensitive nonadiabatic dressed states, arising from the interaction of a quantum system and a coherent external electromagnetic field, are used in these considerations. Two types of phase correlations in the multilevel phase-sensitive nonadiabatic dressed states are found: a rapidly changing phase correlation between the real and the virtual components and a stationary phase correlation between different virtual components of these states.
When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state ∣0> of an unperturbed Hamiltonian H(0), the wave function acquires excited-state components ∣k> with coefficients c(k)(t) exp(-iE(k)t/ℏ), where E(k) denotes the energy of the unperturbed state ∣k>. It is well known that each coefficient c(k)(t) separates into an adiabatic term a(k)(t) that reflects the adjustment of the ground state to the perturbation--without actual transitions--and a nonadiabatic term b(k)(t) that yields the probability amplitude for a transition to the excited state. In this work, we prove that the energy at any time t also separates completely into adiabatic and nonadiabatic components, after accounting for the secular and normalization terms that appear in the solution of the time-dependent Schrödinger equation via Dirac's method of variation of constants. This result is derived explicitly through third order in the perturbation. We prove that the cross-terms between the adiabatic and nonadiabatic parts of c(k)(t) vanish, when the energy at time t is determined as an expectation value. The adiabatic term in the energy is identical to the total energy obtained from static perturbation theory, for a system exposed to the instantaneous perturbation λH'(t). The nonadiabatic term is a sum over excited states ∣k> of the transition probability multiplied by the transition energy. By evaluating the probabilities of transition to the excited eigenstates ∣k'(t)> of the instantaneous Hamiltonian H(t), we provide a physically transparent explanation of the result for E(t). To lowest order in the perturbation parameter λ, the probability of finding the system in state ∣k'(t)> is given by λ(2) ∣b(k)(t)∣(2). At third order, the transition probability depends on a second-order transition coefficient, derived in this work. We indicate expected differences between the results for transition probabilities obtained from this work and from Fermi's golden rule.