Adiabatic generation of arbitrary coherent superpositions of two quantum states: Exact and approximate solutions

The common objective of the application of adiabatic techniques in the field of quantum control is to transfer a quantum system from one discrete energy state to another. These techniques feature both high efficiency and insensitivity to variations in the experimental parameters, e.g., variations in the driving field amplitude, duration, frequency, and shape, as well as fluctuations in the environment. Here we explore the potential of adiabatic techniques for creating arbitrary predefined coherent superpositions of two quantum states. We show that an equally weighted coherent superposition can be created by temporal variation of the ratio between the Rabi frequency $\mathrm{\Omega }\left(t\right)$ and the detuning $\mathrm{\Delta }\left(t\right)$ from 0 to $\infty$ (case 1) or vice versa (case 2), as it is readily deduced from the explicit adiabatic solution for the Bloch vector. We infer important differences between cases 1 and 2 in the composition of the created coherent superposition: The latter depends on the dynamical phase of the process in case 2, while it does not depend on this phase in case 1. Furthermore, an arbitrary coherent superposition of unequal weights can be created by using asymptotic ratios of $\mathrm{\Omega }\left(t\right)/\mathrm{\Delta }\left(t\right)$ different from 0 and $\infty$. We supplement the general adiabatic solution with analytic solutions for three exactly soluble models: two trigonometric models and the hyperbolic Demkov-Kunike model. They allow us not only to demonstrate the general predictions in specific cases but also to derive the nonadiabatic corrections to the adiabatic solutions.

Figure 1

Pulse shapes in the sin-cos (left) and cos-sin (right) models. (Top row) Full-cycle pulse shapes ($0\leqq t/T\leqq \pi$). Panel (a) shows a typical level crossing case in which the detuning $\mathrm{\Delta }\left(t\right)$ crosses resonance at the time when the Rabi frequency $\mathrm{\Omega }\left(t\right)$ is maximum. In panel (b) the shapes of the Rabi frequency $\mathrm{\Omega }\left(t\right)$ and the detuning $\mathrm{\Delta }\left(t\right)$ from frame (a) are exchanged. (Middle row) Half-cycle pulse shapes ($0\leqq t/T\leqq \pi /2$). (Bottom row) The truncated pulse shapes from panels (c) and (d) are extended until they vanish.

Figure 2

Bloch variables vs the pulse area $A=\mathrm{\Lambda }T$ for the sin-cos (top) and cos-sin (bottom) models.

Figure 3

Pulse shapes $\mathrm{\Omega }\left(t\right)$ and detuning $\mathrm{\Delta }\left(t\right)$ in the Demkov-Kunike (top), Allen-Eberly (middle), and Rosen-Zener (bottom) models. Left, full duration; right, half duration.

Figure 4

Bloch variables for the AE model vs the dimensionless Rabi frequency $\alpha ={\mathrm{\Omega }}_{0}T/2$ for chirp rate $B=2/T$ (top) and vs the dimensionless chirp rate $\beta =BT/2$ for Rabi frequency ${\mathrm{\Omega }}_0=6/T$ (bottom). The solid curves are the exact expressions (44), while the dashed curves display the approximate expressions (45). Wherever the dashed curves are invisible they coincide with the solid ones (except for the range $\alpha <1$ in the top frame, where the approximations are not applicable).

Figure 5

Bloch variables for the half-RZ model. (Top) Bloch variables vs the dimensionless Rabi frequency $\alpha ={\mathrm{\Omega }}_{0}T/2$ for detuning $\mathrm{\Delta }=3/T$. (Bottom) Bloch variables vs the dimensionless detuning $\delta =\mathrm{\Delta }T/2$ for Rabi frequency ${\mathrm{\Omega }}_0=3/T$. The solid curves are the exact expressions from Eqs. (48), while the dashed curves display the approximate expressions (50). Wherever the dashed curves are invisible they coincide with the solid ones.