Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Experiment

We demonstrate a method of adiabatic population transfer from a single quantum state into a coherent superposition of states. The transfer is executed with femtosecond pulses, spectrally shaped in a simple and intuitive manner, which does not require iterative feedback-controlled loops. In contrast to nonadiabatic methods of excitation, our approach is not sensitive to the exact value of laser intensity. We show that the population transfer is complete, and analyze the possibility of controlling the relative phases and amplitudes of the excited eigenstates. We discuss the limitations of the proposed control methods due to the dynamic level shifts and suggest ways of reducing their influence.

Figure 1

(Color online) Illustration of the interaction of a multilevel system with polychromatic coherent radiation consisting of a number of discrete spectral components (long red arrows). Dashed lines represent the ground state “dressed” with a photon. Bare excited states of the system (solid lines) are dynamically shifted in energy due to the presence of nonresonant components of the laser field. This results in incomplete population transfer via nonadiabatic process (a). When the spectral components are simultaneously chirped in frequency (b), adiabatic transfer into a coherent superposition of excited states is executed. Different timing of the level crossings (short arrows) complicates the control over the resulting wave packet, but does not affect the completeness of excitation.

Figure 2

(Color online) Results of numerical simulations of piecewise adiabatic passage for potassium atom. Population transfer is shown in (a) as a function of time. Blue solid line represents the ground state population while red dashed and green dotted lines correspond to the populations of $4P_{1/2}$ and $4P_{3/2}$ excited states, respectively. Excitation field amplitude is plotted in (b) on the same time scale.

Figure 3

(Color online) (a) Relevant quantum states of potassium. Blue double-peak curve represents the shaped spectrum of pump pulses, whereas red arrows depict possible channels of ionizing the atoms with probe pulses. (b) Spectral shaping of pump pulses with local frequency chirping around two electronic resonances at 766.5 and 766.9 nm: intensity (solid black) and phase (dashed magenta). Dotted green line represents flat spectral phase used for a nonadiabatic excitation scheme. (c) Experimental setup. Two optical parametric amplifiers (OPA) are pumped by a Ti:sapphire regenerative amplifier (RGA) producing 130 fs 2 mJ pulses at 1 kHz repetition rate. One OPA is used to generate pump pulses of variable energy, controlled by an attenuator and measured by a fast photodiode. Pump pulses are shaped with a pulse shaper and are weakly focused on a cloud of potassium atoms inside a vacuum chamber. Probe pulses are delayed by a variable time delay and tightly focused on the atomic cloud. Potassium atoms, ionized by probe pulses, are accelerated toward and detected by a multichannel plate based ion detector.

Figure 4

(Color online) Adiabatic and nonadiabatic excitation of a single ($4P_{1/2}$ or $4P_{3/2}$) excited state. (a) Pump pulse spectra used to excite either $D_1$ or $D_2$ transitions separately (solid blue and dashed red lines, respectively). Spectral profile of the initial unshaped femtosecond pulse is shown as dashed black curve. [(b), (c)] Nonadiabatic (blue diamonds) and adiabatic (red circles) population transfer into $4P_{1/2}$ (b) and $4P_{3/2}$ (c) states. Ion signal, proportional to the target state population, is plotted as a function of the corresponding pulse area, $A_{1,2}$. Solid lines show the anticipated ${\text{sin}}^2\left(A_{1,2}\right)$ dependence fitted to the experimental data.

Figure 5

(Color online) Nonadiabatic population transfer into a superposition of $4P_{1/2}$ and $4P_{3/2}$ states. The two-dimensional plots [(a) experiment, (c) calculation] show the ion signal (color coded) as a function of the effective pulse area and pump-probe time delay. Oscillations along the vertical (time) axis reflect quantum beating due to the time evolution of the wave packet. The quantum beats are shown in (b) for pulse areas of $\pi$ (solid blue) and $2\pi$ (dashed red), corresponding to the dashed vertical white lines in (a). In plot (d), we present the calculated populations of the two excited states, $4P_{1/2}$ (dash-dotted blue) and $4P_{3/2}$ (dashed green), and the relative phase (dotted red) between the corresponding wave functions plotted on the same scale (in the units of $\pi$), as a function of the effective pulse area of the excitation field. The ground-state population is shown as a solid black line.

Figure 6

(Color online) Adiabatic population transfer into a superposition of $4P_{1/2}$ and $4P_{3/2}$ states. The two-dimensional plots [(a) experiment, (c) calculation] show the ion signal (color coded) as a function of the effective pulse area and pump-probe time delay. The oscillations along the vertical (time) axis reflect quantum beating due to the time evolution of the wave packet. The quantum beats are shown in (b) for pulse areas of $\pi$ (solid blue) and $2\pi$ (dashed red), corresponding to the dashed vertical white lines in (a). In plot (d), we present the calculated populations of the two excited states, $4P_{1/2}$ (dash-dotted blue) and $4P_{3/2}$ (dashed green), and the relative phase (red dots) between the corresponding wave functions, as a function of the effective pulse area of the excitation field. The ground state population is shown as a solid black line.

Figure 7

(Color online) Demonstration of the population transfer completeness. (a) Ion signal as a function of the pump-probe time delay for a sequence of frequency chirped $D_1$ and $D_2$ pulses separated by 8 ps (the time when the second pulse is present is between two dashed vertical lines). Filled black (red squares) dots correspond to the signal before (after) the arrival of the second pulse. (b) Fourier transform of the ion signal before (solid blue) and after (dashed red) the arrival of the second pulse (log scale). Strong peak is expected at the frequency of quantum beating (1.73 THz) if both states are populated.

Figure 8

(Color online) Experimental control over the quantum phase of an excited wave packet. (a) To control the relative phase of the eigenstates in the target superposition state, an extra constant phase shift of 0 (dotted red), $\pi /2$ (dashed-dotted green) and $\pi$ radian (solid blue) is added to the spectral phase of $D_1$ pulses. Black dashed line shows the double-peaked spectral amplitude of the pulse. The introduced phase shift results in a corresponding vertical shift of the detected quantum beats shown in (b). (c) Vertical cross-sections of the two-dimensional scans in (b) at pulse area of $\pi$. Blue diamonds, red squares, and green circles corresponds to 0, $\pi /2$, and $\pi$ phase shifts of $D_1$ pulse, respectively.

Figure 9

(Color online) Quality of population control with piecewise adiabatic passage. Numerical simulations demonstrate possible discrepancy between the achieved population ratio $\beta$ of two $4P$ states (vertical axis) and its target value (horizontal axis), for different control methods (see text for symbol description).