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Physical Review A

Coherent control of a quantum transition by a phase jump

Boyan T. Torosov, Nikolay V. Vitanov

Published Nov 6, 2007• 29 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.76.053404

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We present an analytically exactly soluble two-state model, in which a hyperbolic-secant-shaped pulsed interaction has a phase jump of \ensuremathϕ\ensuremath{\phi}\ensuremathϕ at the time of its maximum. The detuning has a constant part and a hyperbolic-tangent chirp term. For \ensuremathϕ=0\ensuremath{\phi}=0\ensuremathϕ=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 \ensuremathϕ\ensuremath{\phi}\ensuremathϕ 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 \ensuremathϕ\ensuremath{\phi}\ensuremathϕ. The jump phase \ensuremathϕ\ensuremath{\phi}\ensuremathϕ 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.
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.
Abstract The Demkov-Kunike (DK) model, characterized by a hyperbolic tangent detuning and a hyperbolic secant coupling, serves as a universal framework for describing driven quantum systems. However, existing research has primarily focused on two-level systems, some gaps still persist in extending the model to three-level configurations to achieve coherent population transfer (CPT). This study constructs exact solutions for both two- and three-level DK models via a canonical transformation, providing a basis for the quantitative analysis of population transfer, and achieves effective regulation of CPT via a finite-time adiabatic driving mechanism under specific parameter constraints. Furthermore, we analyze the impact of pulse truncation—a common experimental constraint—as well as environmental dissipation on quantum state evolution. The results show that the proposed scheme achieves rapid, high-fidelity coherent population transfer not only under ideal conditions, but also in the presence of realistic pulse truncation and even when decoherence induced by system-environment coupling is included. This work achieves CPT in the three-level DK model, offering a promising approach for advancing quantum state control in practical applications.
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.
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.
We propose and analyze a robust population-transfer protocol in a driven two-level system based on a sudden sign change of the detuning at the maximum of a smooth coupling pulse. Away from the jump the dynamics is adiabatic, while the sign flip produces a single nonadiabatic kick in the adiabatic basis. Within a simple stepwise adiabatic-sudden approximation we obtain a compact analytic expression for the final transition probability, identify the parameter regimes that yield high-fidelity inversion, and show that the result depends only on the change of the mixing angle across the detuning jump, i.e., solely on the ratio of the peak Rabi frequency to the detuning. Numerical simulations of the full time-dependent Schrödinger equation confirm the validity and robustness of this description over a broad parameter range. We then use the Majorana decomposition to extend the scheme to an SU(2)-symmetric three-state chain driven by the same coupling and detuning functions. In this setting the three-state propagator is expressed in closed form through the two-level Cayley-Klein parameters, which allows us to derive explicit transition probabilities for all three initial states. In particular, we show that for strong coupling the protocol yields almost complete population transfer between the two outer states, with only small transient population of the middle state, while retaining the same intrinsic robustness as in the underlying two-level model.