Sculpting the Vortex State of a Spinor BEC

We use Raman-detuned laser pulses to achieve spatially varying control of the amplitude and phase of the spinor order parameter of a Bose-Einstein condensate. We present experimental results confirming precise radial and azimuthal control of amplitude and phase during the creation of vortex-antivortex superposition states.

Figure 1

Experimental configuration. A ${}^{87}\mathrm{Rb}$ BEC in the $|F=1,m_F=-1⟩$ state is acted on by simultaneous, copropagating pulses of ${\sigma }^{+}$ (light gray) and ${\sigma }^{-}$ (dark gray) polarized laser light, with intensities $I_{+}$, $I_{-}$. The fields are near resonant with the $5^2{P}_{1/2}$ excited states, and couple the states indicated in the energy level diagram (right).

Figure 2

Vortex coupling and interference using beams of different orbital angular momentum. Left: the laser pulse sequence and beam modes used to interact with the BEC, and the state of the BEC as the sequence progresses (a)–(c). An initial laser pulse pair using Gaussian (G) and Laguerre-Gaussian (${\mathrm{LG}}_0^1$) beam modes causes the initally spin-polarized BEC (a) to evolve into a coreless vortex state (b). Application of another pulse pair using G, ${\mathrm{LG}}_0^{-1}$ beam modes, transforms this into a coreless vortex-antivortex superposition state (c). Right: Experimental images of the BEC (a′)–(c′) corresponding to schematics (a)–(c). The spin components are separated by a magnetic field gradient prior to absorption imaging.

Figure 3

(a) Radial intensity profiles for the G, LG beam modes used in the experiment, with the BEC density shown for comparison. The curves are Gaussian and LG mode functions fit to data points from azimuthally averaged lineouts of images. (b) Experimental images of vortex-antivortex superpositions created in $|1,1⟩$ by the procedure in Fig. 2, using the beam modes of (a). The spiral distortion is caused by inhomogeneous light shifts, and increases with beam intensity.

Figure 4

(a) Calculated relative phase shift induced by a ${\sigma }^{-}$ polarized laser field as a function of detuning, for a pulse duration of $5\mu \mathrm{s}$ and fixed intensity of $44\mathrm{mW}/{\mathrm{cm}}^2$. (b) Modified pulse sequence: A pulse of the ${\sigma }^{-}$ polarized field (LG mode) is inserted between the “transfer” pulses, causing an additional spatially inhomogeneous phase shift, with a detuning-dependent magnitude and sign. The detunings of the beams used in the experiment are indicated by the labels ${\Delta }^{\prime }$ and ${\Delta }^{\prime \prime }$

Figure 5

Experimental images of vortex-antivortex superpositions in the $|1,1⟩$ component of the BEC, created with the pulse sequence of Fig. 4b. (a) A control pulse with the same detuning as the transfer pulses tightens the spiral compared to the nominal result when no control pulse is applied (b). A control pulse detuned above the $F^{\prime }=2$ manifold causes the light shifts in the sequence to cancel out, resulting in an undistorted interference pattern in the BEC (c). Increasing the intensity of the control pulse in the “cancelling” configuration overcompensates the radial phase gradients of the transfer pulses, creating spirals of opposite handedness (d), (e).

Figure 6

Comparison of experimental vortex interference patterns and corresponding numerical predictions from a model based on the Hamiltonian of Eq. (1).