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Physical Review A

Exactly solvable two-state quantum model for a pulse of hyperbolic-tangent shape

Lachezar S. Simeonov, Nikolay V. Vitanov

Published Apr 14, 2014• 24 citations
Publisher

Abstract

We present an analytically exactly solvable two-state quantum model, in which the coupling has a hyperbolic-tangent temporal shape and the frequency detuning is constant. The exact solution is expressed in terms of associated Legendre functions. An interesting feature of this model is that the excitation probability does not vanish, except for zero pulse area or zero detuning; this feature is attributed to the asymmetric pulse shape. Two limiting cases are considered. When the coupling rises very slowly, it is nearly linear and the tanh model reduces to the shark model introduced earlier. When the coupling rises very quickly, the tanh model reduces to the Rabi model, which assumes a rectangular pulse shape and hence a sudden switch on. Because of its practical significance, we have elaborated the asymptotics of the solution in the Rabi limit, and we have derived the next terms in the asymptotic expansion, which deliver the corrections to the amplitude and the phase of the Rabi oscillations due to the finite rise time of the coupling.

DOI
10.1103/physreva.89.043411

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We present an analytically exactly solvable two-state quantum model, in which the coupling has a hyperbolic-tangent temporal shape and the frequency detuning is constant. The exact solution is expressed in terms of associated Legendre functions. An interesting feature of this model is that the excitation probability does not vanish, except for zero pulse area or zero detuning; this feature is attributed to the asymmetric pulse shape. Two limiting cases are considered. When the coupling rises very slowly, it is nearly linear and the tanh model reduces to the shark model introduced earlier. When the coupling rises very quickly, the tanh model reduces to the Rabi model, which assumes a rectangular pulse shape and hence a sudden switch on. Because of its practical significance, we have elaborated the asymptotics of the solution in the Rabi limit, and we have derived the next terms in the asymptotic expansion, which deliver the corrections to the amplitude and the phase of the Rabi oscillations due to the finite rise time of the coupling.
Abstract We present an analytically exactly solvable N -pod quantum model, in which the coupling has a hyperbolic-tangent temporal shape and the frequency detuning is constant. The solution is derived by using the Morris–Shore transformation to reduce the N -pod system to a -like one and then to a two state system and a set of decoupled states in two steps. In this work, we use the propagator for a reduced two state system and the inverse transformation to achieve the reduced three state propagator. It is shown that any preselected coherent superposition of all N ground states can be obtained from an initial superposition of M ground states in the presence of detuning. We use a train of tanh coincident pulses and we show that the effect of spontaneous emission is efficiently suppressed by increasing the number of pulses. Besides, we explore that a coherent superposition of ground states from the initial excited state can be implemented using tanh pulses.
It is shown that any pulse with a time-asymmetric envelope and a constant non-zero detuning excites a two-level quantum system with a probability of nearly one-half provided the pulse is very asymmetric, adiabatic in the slowly rising part of its envelope, sudden in the steep part and the peak Rabi frequency is much greater than the detuning. Three particular cases are considered analytically: the asymmetric hyperbolic-secant, the Bambini-Berman and the exponential pulse, which illustrate the general conclusions. The results are applied to a special three-level 'vee' system whose two transitions are driven by two asymmetric laser pulses of the same Rabi frequencies and the same detunings. Analysis shows that if such a system is prepared initially in its ground state, effective population depletion of this state can be achieved, the results being quite insensitive to variations in the Rabi frequency and the detuning.
A class of exact analytic solutions of the time-dependent Schrödinger equation is presented for a two-state quantum system coherently driven by a nonresonant external field. The coupling is a linear function of time with a finite duration and the detuning is constant. Four special models are considered in detail, namely the shark, double-shark, tent and zigzag models. The exact solution is derived by rotation of the Landau–Zener propagator at an angle of π/4 and is expressed in terms of Weber's parabolic cylinder function. Approximations for the transition probabilities are derived for all four models by using the asymptotics of the Weber function; these approximations demonstrate various effects of physical interest for each model.
Exact analytical solutions to the dissipative time-dependent Schrödinger equation are obtained for a decaying two-state system with decay rates Γ1 and Γ2 for the levels with extremal spin projections. The system is coherently driven with a pulse whose detuning is made up of two parts: a time-dependent part (chirp) of hyperbolic-tangent shape and a static part with real and imaginary terms. This gives us a wide range of possibilities to arbitrarily select the interaction terms. We considered two versions which led to decaying Demkov-Kunike (DK) models; the version in which the Rabi frequency (interaction) is a time-dependent hyperbolic-secant function (called decaying DK1 model) and the case when it is constant in time and never turns off (decaying DK2 model). Our analytical solutions account for all possible initial moments instead of only t0 = 0 or t0 = − ∞ as for non-decaying models and may be useful for the experiments on level crossings. Two complementary limits of the pulse detuning are considered and explored: the limit of fast (i) and slow rise (ii). In the case (i), the coupling between level positions in the first DK model collapses, while the second DK model reduces to a Rabi model (constant Hamiltonian), in the case (ii), both DK models reduce to the LZ model. In both cases (i) and (ii), analytical approximated solutions which conveniently approach the exact solutions are derived.
We present an analytically exactly soluble two-state model, in which a hyperbolic-secant-shaped pulsed interaction has a phase jump of $\ensuremath{\phi}$ at the time of its maximum. The detuning has a constant part and a hyperbolic-tangent chirp term. For $\ensuremath{\phi}=0$, this model reduces to the Demkov-Kunike model, which in turn contains as particular cases three other well-known models: the Rosen-Zener, Allen-Eberly, and Bambini-Berman models. A nonzero $\ensuremath{\phi}$ induces dramatic changes in the transition probability, ranging from complete population inversion to complete population return. The analytic results are particularly instructive in the adiabatic limit and demonstrate that complete population inversion can always occur for a suitable choice of $\ensuremath{\phi}$. The jump phase $\ensuremath{\phi}$ can therefore be used as a control parameter for the two-state transition probability.
An exact analytic solution to the time-dependent Schrödinger equation is presented for a two-state quantum system coherently driven by a nonresonant pulsed external field. The pulse has a hyperbolic-secant shape, with a sign jump (i.e. a phase step of π) at its maximum, which nullifies the overall pulse area. It is shown that off resonance, for detunings larger than the pulse bandwidth, the transition probability approaches unity as the field intensity increases. The population inversion is robust against small-to-moderate variations in the detuning and the intensity, a feature reminiscent of adiabatic passage. The population inversion, however, is not of adiabatic nature; it is induced by δ-function-shaped nonadiabatic coupling with an area of nearly π. An estimate of the required experimental resources shows that implementation with shaped femtosecond pulses is feasible.