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

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.

Figure 1

Rabi frequency $\Omega \left(t\right)={\Omega }_{0}tanht/\tau$ as a function of time. The duration of the pulse is $T$.

Figure 2

Transition probability vs the pulse area $A$ for various detunings $\Delta$ and coupling rise times $\tau$. The values of $\Delta$ and $\tau$ are given above each frame. Solid curves show the tanh model [Eq. (20)]; dashed curves show the Rabi model [Eq. (29)].

Figure 3

Transition probability in logarithmic scale vs the pulse area $A$ for detuning $\Delta =2/T$ and two values of the coupling rise time $\tau$, given above each frame. Solid curves show the tanh model [Eq. (20)]; dashed curves show the Rabi model [Eq. (29)].

Figure 4

Transition probability vs the detuning for various pulse areas $A$ and coupling rise times $\tau$. The values of $A$ and $\tau$ are given above each frame. Solid curves show the tanh model [Eq. (20)]; dashed curves show the Rabi model [Eq. (29)].