Phase retrieval of vortices in Bose-Einstein condensates

We propose and demonstrate numerically a measurement scheme for complete reconstruction of the 2D quantum wave function of a Bose-Einstein condensate, amplitude and phase, from a time-of-flight measurement. We identify a fundamental ambiguity present in the measurement of phase structures of high-symmetry excitations (e.g., vortices) and show how to overcome it by allowing for different expansion durations in different directions. We demonstrate this approach with the reconstruction of matter-wave vortices and arrays of vortices.

Figure 1

(a) Amplitude distribution of single $m=\pm 1$ vortex state in arbitrary unit. (b) Phase structure of vortex state $m=1$. (c) Phase structure of vortex state $m=-1$. (d)–(i) Top row: TOF measurement of state (b) with linear (d), "weak”(e), and “strong” nonlinear (f) propogation. Bottom row: TOF measurement for state (c) with linear (g), weak (h), and strong nonlinear (i) propogation. See Appendix pp2 for simulation and propagation regimes details.

Figure 2

Top row: Augmented TOF measurement for vortex states of $m=1$ with linear (a), weak (b), and strong nonlinear (c) propagation. Bottom row: Augmented TOF measurement for vortex states of $m=-1$ with the same propagation parameters for “linear” (d), weak (e), and strong nonlinear (f) propagation.

Figure 3

Graphical depiction of the iterative phase-retrieval algorithm for the augmented TOF measurements of BEC.

Figure 4

Norm difference between augmented TOF measurements of vortex states $m=1$ and $m=-1$ as a function of propagation time $T_1$. Time axis is scaled by the energy difference between the second and first modes of the trap.

Figure 5

(a) Amplitude of wave function for reconstruction and (b) its phase; (c) is the augmented TOF and (d) is the original TOF. In (e), we present the phase reconstructed with the augmented TOF scheme and its corresponding phase-retrieval algorithm. The far-field result is shown in (f). In (g), an erroneous phase reconstruction is shown using the original TOF scheme, producing the “measured” far-field measurement (h) but with a wrong object phase.

Figure 6

(a), (b) Amplitude and phase of the reconstructed wave function, respectively. (c, d) Calculated atom density corresponding to augmented TOF measurement of (a), (b), without appreciable noise and with added Gaussian white noise, respectively. The noise in (d) is very strong, corresponding to SNR 0 dB, yet the measurement scheme and the algorithm facilitate correct reconstruction, amplitude, and phase, as shown in (e) and (f), respectively.

Figure 7

(a), (b) Amplitude and random phase pattern of the wave function, respectively. (c) Reconstructed phase based on (d)––which is the calculated augmented TOF measurement of (a), (b). (e) Augmented TOF of reconstructed wave function. Note the asymmetric aspect ratio in the augmented TOF images.