Efficient vibrational state coupling in an optical tilted-washboard potential via multiple spatial translations and application to pulse echoes

We measure the application of simple and compound pulses consisting of time-dependent spatial translations to coupling vibrational states of ultracold ${}^{85}\text{R}\text{b}$ atoms in a far-detuned 1D optical lattice. The lattice wells are so shallow as to support only two bound states, and we prepare the atoms in the ground state. The lattice is oriented vertically, leading to a tilted-washboard potential analogous to those encountered in condensed-matter systems. Experimentally, we find that a square pulse consisting of lattice displacements and a delay is more efficient than single-step pulses or Gaussian pulses. This is described as an example of coherent control. It is striking that contrary to the intuition that soft pulses minimize loss, the Gaussian pulse is outperformed by the square pulse. Numerical calculations are in strong agreement with our experimental results and show the superiority of the square pulse to the single-step pulse for all lattice depths and to the Gaussian pulse for lattice depths greater than seven lattice recoil energies. We also compare the effectiveness of these pulses for reviving oscillations of atoms in vibrational superposition states using the pulse-echo technique. We find that the square and Gaussian pulses result in higher echo amplitudes than the single-step pulse. These improved echo pulses allow us to probe coherence at longer times than in the past, measuring a plateau which has yet to be explained. In addition, we show numerically that the vibrational state coupling due to such lattice manipulations is more efficient in shallow lattices than in deep lattices. The coupling probability for an optimized single-step pulse approaches $1/e$ as the depth goes to infinity (harmonic-oscillator limit), while in shallow lattices with large anharmonicity, the coupling probability reaches a maximum value of 0.51 for a lattice depth of five recoil energies. For square and Gaussian pulses the coupling in the lattice is even stronger, reaching maxima of 0.64 at 6 recoil energies and 0.67 at 5 recoil energies, respectively.

Figure 1

(a) Lattice potential $U\left(x\right)=U_o{\text{sin}}^2\left(k_{L}x\right)$ for the depth $U_o=18E_R$ and the first three bands as a function of dimensionless parameter $qx=x\pi /a$, where $a$ is the lattice spacing. (b) Schematic of the tilted lattice showing $U\left(x\right)+mgx$ and the quasibound states of the Wannier-Stark ladder. The energy levels are the averages of band energies and they differ from site to site by $2.86E_R$, where $2.86E_R$ is the tilt due to gravity per lattice site.

Figure 2

(Color online) (a) Experimental $P_{11}$ as a function of displacement after application of single-step, square and Gaussian pulses for a lattice depth of $U_o=18E_R$. The temporal widths of the square and Gaussian pulses are the experimental optimized temporal widths. (b) Calculated $P_{11}$ for the three pulses, with calculated optimized temporal widths for the square and Gaussian pulses. For experimental and calculated optimized pulse parameters see Table .

Figure 3

(Color online) (a) Experimental $P_{12}$ as a function of displacement after application of single-step, square, and Gaussian pulses for a lattice depth of $U_o=18E_R$. (b) Calculated $P_{12}$ for the three pulses. Experimental and calculated temporal pulse widths are the same as in Fig. 2.

Figure 4

(Color online) Calculated couplings $P_{12}$, $P_{11}$ and loss vs scaled delay for the optimized square pulse at fixed $\Delta x=0.16a$ in a lattice depth of $U_o=18E_R$. The ground- and excited-state populations coherently oscillate as a function of delay.

Figure 5

(Color online) Calculations: Loss vs coupling $P_{12}$ after the square, Gaussian, and single-step pulses in a lattice depth of $U_o=18E_R$. For the same coupling $P_{12}$, the square pulse results in less loss than the single-step and Gaussian pulses.

Figure 6

(Color online) Experimental data: Evolution of $P_f\left(t\right)$ as a function of time. The curve is a fit to data (sinusoid with Gaussian decay). $T_{\text{ave}}=\left(187\pm 2\right)\mu \text{s}$ and r.m.s. width of Gaussian is $\left(258\pm 6\right)\mu \text{s}$

Figure 7

(Color online) (a) Full echo curves after applying each pulse centered at time $t_0=1040\mu \text{s}$. All echo signals are centered at $2t_0=2080\mu \text{s}$. The initial oscillations are due to the echo pulse itself. (b) Subtracted echo curves where the initial oscillations due to pulses measured after 4 ms have been subtracted from the curves in (a). All curves have been displaced vertically for clarity.

Figure 8

(Color online) Bar plot of calculated ratio $P_{12}/P_{11}$ for the pulse parameters that result in maximum $P_{12}$ (optimized coupling pulse, checkered pattern bars) and the pulse parameters that result in maximum echo amplitude (optimized echo pulse, solid black bars). The blue bars show the experimental results.

Figure 9

(Color online) Echo amplitude as a function of time delay. The pulse used for the echo is the optimized experimental square pulse for the lattice depth of $U_o=20E_R$, and is centered at time $t_o$; the echo is observed at $2t_o$.

Figure 10

(Color online) (a) Band structure of the periodic potential with $U_o=18E_R$, showing the first four bands. The dashed line indicates the depth of lattice.(b) ${\psi }_{n,q=0}\left(x\pi /a\right)$, the Bloch functions for quasimomentum $q=0$ in the first four bands as a function of position. Lattice spacing is $\pi$ with the central well extending from $-\pi /2$ to $\pi /2$. The solid black curve is the lattice potential; the vertical dotted lines specify the center of wells and the horizontal dotted lines are drown for the illustration of wave functions. The Bloch functions have been rescaled and displaced vertically for clarity.

Figure 11

(Color online) (a) ${\left|c_{12}\right|}_q^2={\left|⟨{\Psi }_{n=2,q}\left(x\right)|{\Psi }_{1,q}\left(x-\Delta x\right)⟩\right|}^2$ as a function of quasimomentum $q$ for a single-step pulse (single lattice displacement) with the optimum displacement of $\Delta x=0.25a$ for the lattice depth of $U_o=18E_R$. (b) $P_{1n}=\left(1/2\pi a\right){\int }_{-\pi /a}^{\pi /a}dq{\left|c_{1n}\right|}_q^2$ as a function of displacement for bands $n=1$ and $n=2$ for the same lattice depth. For comparison, the measured values are shown on the same plot.

Figure 12

(Color online) ${\left|c_{12}\right|}_q^2$, the coupling probability due to (a) single-step pulse and (b) square pulse $\left(\tau =0.35\right)$ for different quasimomentum components as a function of displacement.

Figure 13

(Color online) (a) Optimum $P_{12}$ after a single-step pulse as a function of displacement for different lattice depths $U_o=s{E}_R$. (b) The same overlap for harmonic potentials with ${\omega }_{\text{ho}}={\overline{\omega }}_{12}$, where ${\overline{\omega }}_{12}$ is the average frequency difference of the two lowest bands in a lattice depth of $U_o=s{E}_R$. Displacement in the case of harmonic potential is expressed in units of lattice spacing $a=\pi s^{1/4}\sigma$, with $\sigma =\sqrt{\hslash /m{\omega }_{\text{ho}}}$.

Figure 14

(Color online) (a) $P_{12}$ after a square pulse as a function of displacement for different delays in a lattice of depth $U_o=18E_R$. $\tau =\text{delay}/{\overline{T}}_{12}$ is indicated in the legend for each curve. The optimum scaled delay between the two displacements for this lattice depth is $\tau =0.35$ (for $U_o=18E_R$, ${\overline{T}}_{12}=200\mu \text{s}$, and optimum $\text{delay}=70\mu \text{s}$). (b) $P_{12}$ for the harmonic oscillator potential with ${\omega }_{\text{ho}}={\overline{\omega }}_{12}$ of the lattice. The delays in (b) are the same as in (a). Displacement in the case of harmonic potential is expressed in terms of lattice spacing $a=\pi s^{1/4}\sigma$, with $\sigma =\sqrt{\hslash /m{\omega }_{\text{ho}}}$.

Figure 15

(Color online) The optimum smallest scaled delay of the square pulse $\tau =\text{delay}/{\overline{T}}_{12}$ as a function of lattice depth. The optimum displacement is different for different lattice depths.

Figure 16

(Color online) (a) $P_{12}$ after a square pulse as a function of scaled delay $\tau$ for a harmonic potential with ${\omega }_{\text{ho}}={\overline{\omega }}_{12}$ in a lattice depth of $U_o=18E_R$. Depending on the magnitude of displacement, the optimum delay changes but the maximum overlap $P_{12}$ stays the same. (b) $P_{12}$ after a square pulse as a function of scaled delay $\tau$ for a single quasimomentum component $q=0$ of the same lattice depth. $P_{12}$, the maximum overlap is different for different displacements. Displacements in (b) are the same as in (a).

Figure 17

(Color online) (a) Phase-space schematics (not to scale) for the harmonic oscillator illustrating the square pulse sequence. The optimum delay $t$ $\left(\theta ={\omega }_{\text{ho}}t\right)$ depends on the magnitude of displacement. (b) $\text{cos}\left(\theta \right)=1-1/\left[2{\left(\Delta x/r_1\right)}^2\right]$ as a function of $\left(\Delta x/r_1\right)$; $r_1$ is the phase-space radius of the first excited state of the harmonic oscillator. For the displacement of $\Delta x=0.5r_1$ the optimum delay is $t=T/2$ for $\text{cos}\left(\theta \right)=-1$. As the displacement is increased the optimum delay becomes smaller; at $\Delta x=r_1/\sqrt{2}$ (the blue dashed line), the delay of $t=T/4$ for $\theta =\pi /2$ results in maximum coupling.

Figure 18

(Color online) Calculated $P_{12}$ as a function of lattice depth for optimized single-step, square, and Gaussian pulses. The coupling probability increases as the lattice depth is decreased until the second band is no longer supported, at which part it drops off sharply. The inset shows $P_{12}$ for lattice depths less than $15E_R$.