Nonadiabatic diffraction of matter waves

Diffraction phenomena usually can be formulated in terms of a potential that induces the redistribution of a wave's momentum. Using an atomic Bose-Einstein condensate coupled to the orbitals of a state-selective optical lattice, we investigate a hitherto unexplored nonadiabatic regime of diffraction in which no diffracting potential can be defined and in which the adiabatic dressed states are strongly mixed. We show how, in the adiabatic limit, the observed coupling between internal and external dynamics gives way to standard Kapitza-Dirac diffraction of atomic matter waves. We demonstrate the utility of our scheme for atom interferometry and discuss prospects for studies of dissipative superfluid phenomena.

Figure 1

(a) Experimental scheme, shown for resonant coupling to the lowest orbital $|n\rangle \left(n=0\right)$ (see text). (b) Fraction of atoms in the lattice after applying a pulse (Rabi frequency $\mathrm{\Omega }/2\pi =8.1$ kHz, duration $\tau =60 \mu \mathrm{s}$) with variable detuning $\delta$. The red solid line is the incoherent sum of predicted line shapes for resonances centered at detunings given by the $q=0$ level structure of the lattice; the pulse areas are $\pi {\gamma }_n$, where ${\gamma }_n=0.72,0,0.61,0,0.32$ for $n=0,1,2,....$ The light gray points and line refer to a reference measurement without lattice (pulse area $\pi$). The inset shows time-of-flight absorption images after band mapping. (c) Population dynamics in orbital $|0\rangle$ for resonant driving with $\mathrm{\Omega }/2\pi =8.8$ kHz and $\delta /2\pi =19.2$ kHz. The curve is a resonant Rabi oscillation with frequency ${\gamma }_0\mathrm{\Omega }$. All error bars represent statistical standard deviations of at least three experimental iterations.

Figure 2

(a) Resonant transfer to $|0\rangle$ and diffraction for $\mathrm{\Omega }/2\pi =16.9$ kHz. Shown are the transferred fraction (red solid points) and the diffracted fraction (blue open circles) in the momentum state $|\pm 2\hslash k\rangle$. The red- and blue-shaded regions are the results of a simulation including experimental uncertainties in $\delta$ and $\mathrm{\Omega }$. The population in higher orders (not shown) is less than $2\%$ of the total number. (b) Observed diffraction patterns for $|b\rangle$ atoms. Each line is a single time-of-flight image summed over the direction perpendicular to the lattice. (c) Total diffraction signal as a function of $\mathrm{\Omega }$ for ${\gamma }_0\mathrm{\Omega }\tau =\pi$. The shaded blue (gray) area shows the band-structure prediction for atoms in the first diffracted order $N_{b,\pm 2}/N_{r+b}$ (undiffracted fraction $N_{b,0}/N_{r+b}$) allowing for effective detunings up to $1.1$ kHz (see text).

Figure 3

Calculated dynamics from the nonadiabatic to the adiabatic regime. (a) (i–iii) Adiabatic potentials ${\tilde{V}}_i$ for coupling strengths $\hslash \mathrm{\Omega }=\{0,1,15\}{E}_r^{\left(2\right)}$ and $s=30$. The position-dependent internal state composition is encoded in color. Panel (i) shows the bare potentials, while panel (iii) shows potentials in the adiabatic limit characterized by a near-constant dressed-state mixing angle. For weak coupling (ii), gaps are small and the mixing angle varies rapidly. (b) Nonadiabatic $q=0$ band energies for the four lowest states as a function of $\mathrm{\Omega }$ for resonance with $|n=0\rangle$. The dashed line is a state of opposite parity that is decoupled from the other three states and the red (gray)-shaded lines show the two-level ac-Stark shifts described in the text. (c) Spatial structure of the ground and first excited states in the bare-state basis, for $\hslash \mathrm{\Omega }=1.25E_r^{\left(2\right)}$. The red- and blue-shaded areas denote the square of the wave function, and the solid lines denote the wave function itself. (d) Diffractive dynamics in three regimes characterized by $\mathrm{\Omega }/V_0=0.01,0.05,15$, with $V_0=100E_r$. (e) Internal state dynamics in the case $\mathrm{\Omega }/V_0=15$. The red and blue lines trace the total population in $|r\rangle$ and $|b\rangle$ while the dashed lines represent the expected envelopes for the $m=0,1,2,3$ (from top to bottom) diffraction orders for standard Kapitza-Dirac diffraction.

Figure 4

In situ probing of matter-wave interference. (a) A pump pulse (${\gamma }_0\mathrm{\Omega }\tau =2\pi /3,\tau =40 \mu \mathrm{s}$) creates excitations in the condensate by removing atoms into a second trapped state $|p\rangle \equiv |2,-1\rangle$ and is followed by free evolution of the condensate density profile for time $T_{\mathrm{delay}}$ before a probe pulse (${\gamma }_0\mathrm{\Omega }\tau =2\pi /5,\tau =14 \mu \mathrm{s}$) transfers atoms into the trapped state $|r\rangle$. (b) Probe pulse yield as a function of delay time, displaying oscillations at the recoil frequency. The solid line is a sinusoidal fit with frequency $\omega /2\pi =15.0$ kHz and exponential damping time $\tilde{\tau }=0.66$ ms, corresponding to the beat note and characteristic time for the wave packets separating after the probe pulse. The insets illustrate the evolving wave-packet interference after application of the pump pulse.