Colloquium: Coherently controlled adiabatic passage

The merging of coherent control (CC) and adiabatic passage (AP) and the type of problems that can be solved using the resulting coherently controlled adiabatic passage (CCAP) method are discussed. The discussion starts with the essence of CC as the guiding of a quantum system to arrive at a given final state via a number of different quantum pathways. The guiding is done by “tailor-made” external laser fields. Selectivity in a host of physical and chemical processes is shown to be achieved by controlling the interference between such quantum pathways. The AP process is then discussed, in which a system is navigated adiabatically along a single quantum pathway, resulting in a complete population transfer between two energy eigenstates. The merging of the two techniques (CCAP) is shown to achieve both selectivity and completeness. Application of CCAP to the solution of the nondegenerate quantum control problem is first discussed and shown that it is possible to completely transfer population from an initial wave packet of arbitrary shape, composed of a set of nondegenerate energy eigenstates, to a final arbitrary wave packet, also composed of nondegenerate states. The treatment is then extended to systems with degenerate states and shown how to induce isomerization between the broken-symmetry local minima of a Jahn-Teller ${\mathrm{Al}}_3\mathrm{O}$ molecule. These approaches can be further generalized to situations with many initial, intermediate, and final states and applied to quantum coding and decoding problems. CCAP is then applied to cyclic population transfer (CPT), induced by coupling three states of a chiral molecule in a cyclic fashion, $\mid 1⟩\leftrightarrow \mid 2⟩\leftrightarrow \mid 3⟩\leftrightarrow \mid 1⟩$. Interference between two adiabatic pathways in CPT allows for a complete population transfer, coupled with multichannel selectivity, by virtue of its phase sensitivity. CPT can be used to show the purification of mixtures of right-handed and left-handed chiral molecules. Finally, quantum-field coherent control is introduced, where CCAP is extended to the use of nonclassical light. This emerging field may be used to generate new types of entangled radiation-matter states.

Figure 1

A schematic illustration of an experiment in which the directionality of current formed in the conduction band of a semiconducting quantum well system was controlled by irradiating the sample by a combination of a fundamental frequency (that of the $\mathrm{C}{\mathrm{O}}_2$ laser) and its second harmonics. From 34.

Figure 2

Experimental control of current directionality. Shown is the dc current generated by the experimental setup of Fig. 1 as a function of the phase difference between the fundamental source (wavelength of $10.6\mu \mathrm{m}$) and its second harmonic. From 34.

Figure 3

The model and dynamics in the adiabatic passage. (Left) The three-level $\Lambda$ configuration adiabatic passage scheme. (Right) The pulses [panel (a), where $P$ stands for the pump and $S$ for the Stokes pulse], eigenvalues [panel (b)], and populations [panel(c)] as a function of time in this adiabatic passage with counterintuitive pulse ordering.

Figure 4

Experiments on STIRAP in the Ne atom. Shown is the percentage of population transfer to level ∣2⟩ $(={}^{3}P_2)$ as a function of the delay between the pump and Stokes pulses. As the pulses begin to overlap, with the Stokes pulse preceding the pump pulse, the population transfer to the ∣2⟩ state approaches 100%. From 10.

Figure 5

An illustration of a transfer process in which arbitrary wave packets made of nondegenerate states can be populated over and over, $\mid \Psi ⟩\to \mid 0⟩\to \mid {\Psi }^{\prime }⟩$.

Figure 6

The time evolution of the $p_k={\mid c_k\mid }^2$ populations of $\mid k⟩$ $(k=1-5)$ energy eigenstates realized in the chain of transfer processes described in the text.

Figure 7

(Color online) The DQC scheme. (Top) The first step: population in an initial wave packet $\mid {\Psi }^i⟩={\mid 1⟩}_{\mathrm{loc}}$, composed of $n$ nearly degenerate eigenstates, is transferred to a single state ∣0⟩ by a two-photon adiabatic passage via $n$ nondegenerate intermediate states. (Bottom) The second step: population transfer by a time-reversed process with different Rabi frequencies from ∣0⟩ to the target wave packet $\mid {\Psi }^f⟩={\mid 2⟩}_{\mathrm{loc}}^{\prime }$, composed of another set of nearly degenerate eigenstates.

Figure 8

(Color online) The three Jahn-Teller-distorted stable T-shaped configurations of the ${\mathrm{Al}}_3\mathrm{O}$ molecule and the three equilateral transition-state saddle points which separate them. In the center, we show the potential along $s$—the pseudorotation—and $z$—the out-of-plane—motion of O relative to the ${\mathrm{Al}}_3$ plane.

Figure 9

(Color online) The DQC dynamics: (a) The three lowest {0,0} energy eigenstates of ${\mathrm{Al}}_3\mathrm{O}$ as a function of the $s$ and $z$ coordinates. (b) Superpositions of these eigenstates form localized wave packets about each T-shaped minimum. (c) The $\mid j⟩$ intermediate states. (d) The ∣0⟩ parking state. (e) The time dependence of the DQC dynamics, showing controlled transfer between two T-shaped minima.

Figure 10

(Color online) The adiabatic decoder. Quantum information that encoded in the phases of the population amplitudes of the $\mid 1_A⟩,\dots ,\mid N_A⟩$ states is used to transfer the population to predominantly one of the $\mid 1_C⟩,\dots ,\mid L_C⟩$ states.

Figure 11

(Color online) The results of the binary-to-hexadecimal decoder for differently encoded information on the $N=5$ initial states (front). In the back, we see the populations of the $2^{N-1}=16$ final levels for these cases, while the intermediate states, in the midregion, are unpopulated.

Figure 12

(Color online) The double-well potential energy curve for the torsional motion of the ${\mathrm{D}}_2{\mathrm{S}}_2$ molecule. The two enantiomers connected by this stereomutation torsional motion are shown in the upper panel.

Figure 13

(Color online) The enantiodiscriminator. (Top) A schematic plot of the CPT scheme, in which three levels of each enantiomer are resonantly coupled by three fields, as used in the discrimination step. (Middle) The time evolution of population of three levels: Both enantiomers start in the ∣1⟩ state. At the end of the process the $L$ enantiomer is transferred to the $\mid 3_L⟩$ state, while the $D$ enantiomer remains in the initial $\mid 1_D⟩$ state. (Bottom) The time dependence of the eigenvalues of the Hamiltonian of Eq. (27): The population initially follows the $\mid E_0⟩\equiv \mid {\lambda }_0⟩$ null eigenvalue state. At $t\approx \tau$ it crosses over diabatically to $\mid E_{-}⟩\equiv \mid {\lambda }_{-}⟩$ for one enantiomer and to $\mid E_{+}⟩\equiv \mid {\lambda }_{+}⟩$ for the other. ($\mid {\lambda }_{0,\pm }⟩$ are defined in the text.)

Figure 14

(Color online) The enantioconverter scheme. (Top) Population is transferred from the $\mid 3_L⟩$ state to the $\mid 4_D⟩$ state, while going through the superposition of $\mid 5_{S,A}⟩$ states. (Bottom) The populations $p_i$ of the broken-symmetry states, as a function of time, during the conversion step.

Figure 15

(Color online) The nonclassical enantioconverter. In the classical-field converter, an initial (left-handed) $\mid 1_L⟩$ state is transformed into the $\mid 3_D⟩$ state. The nonclassical converter uses the $\mid \alpha ⟩+\mid -\alpha ⟩$ cat state to produce the $\mid 3_L⟩\mid \alpha ⟩+\mid 3_D⟩\mid -\alpha ⟩$ entangled state.

Figure 16

(Color online) Photon distributions ${\rho }_{L,L}^{\beta ,\beta }$ (upper panels) and ${\rho }_{D,D}^{\beta ,\beta }$ (lower panels) in the quantum field enantioconverter for entangled chiral states. The distributions are presented in the $\Delta {\beta }_x=\pm 6.4$, $\Delta {\beta }_y=\pm 5$ range. (Left column) The driving field is given by the cat state $\mid \mathcal{C}⟩$ with ${\mid \alpha \mid }^2=7$. (Right column) The driving field is given by the $\mid n=7⟩$ Fock state.