Arbitrary coherent superpositions of quantized vortices in Bose-Einstein condensates via orbital angular momentum of light

We recently proposed a scheme for the creation of coherent superpositions of vortex states in Bose-Einstein condensates (BECs) using orbital angular momentum (OAM) states of light [Phys. Rev. Lett. 95, 173601 (2005)]. Here we discuss further technical details of the proposal, provide an alternative, time-reversal-symmetric scheme for transfer of a superposition of OAM states of light to the BEC via a procedure analogous to the traditional stimulated Raman adiabatic passage technique, and discuss an alternative trap configuration conducive for sustaining large charge vortices. Superpositions of OAM states of light, that can be created using the existing experimental techniques, can be transferred to an initially nonrotating BEC via a specially devised Raman coupling scheme. The techniques proposed here open up avenues to study coherent interaction of OAM states of light with matter. The study could also be employed for performing various quantum-information-processing tasks with OAM states of light—including a memory for a quantum state of the initial superposition.

Figure 1

(Color online). Dove prism as a sign shifter for the OAM states of light. The dove prism changes the handedness of the incident beam carrying the OAM and causes a sign change in its winding number. The red (thick) ray changes place with the diametrically oppositely placed blue (thin) ray after passage through the dove prism.

Figure 2

Scheme for creation of superposition of the OAM states $\left(|+\ell ⟩+|-\ell ⟩\right)/\sqrt{2}$.

Figure 3

(Color online). Internal level scheme of the trapped BEC atoms and the applied fields. The fields are specified through the corresponding Rabi frequencies and the polarization state. The fields are highly detuned from the excited states to avoid populating the excited states and ensure the stability of the condensate. The two-photon detunings $\delta$ are needed for one method of transfer of population from the nonrotating state to the vortex states but need to be zero for the STIRAP-like transfer mechanism. The internal quantum numbers of the states $|0⟩$,$|\pm ⟩$ are exactly the same. This makes specific polarization modes necessary for the coupling fields as shown. The intermediate states may have different internal quantum numbers, but they are not populated.

Figure 4

(Color online). Generation of the vortex state superposition via the linear frequency chirp technique. A 60:40 superposition of the component vortex states is generated here starting with a nonrotating state of the BEC. The population of the BEC cloud is normalized to 1. The quantity on the $x$ axis is essentially dimensionless time. The linear chirp used for complete population transfer is given by the two-photon detuning $\delta \left(t\right)=C\left(1-{\Omega }_{0}t\right)$, where $C=2{\Omega }_0$ and ${\Omega }_0=3\text{kHz}$. Other parameters are ${\omega }_{\perp }=132\text{Hz}$, $a=5\text{nm}$, $L_{\perp }=2.35\mu \text{m}$, $L_z=1.4\mu \text{m}$, and $\kappa =1.7\text{kHz}$.

Figure 5

(Color online). Generation of the vortex state superposition: Results of the numerical solutions of the equations for the STIRAP scheme show the superposition 60:40 of the $|+⟩$ and $|-⟩$ vortex states. The pulse profiles $f\left(t\right)$ and $g\left(t\right)$ are of the form in Eq. (23) with ${\sigma }_1={\sigma }_2=0.25$ and $t_1=1.0$ and $t_2=0.5$ in units of $1/{\Omega }_0$, with $f_0/g_0=0.5$ to obtain complete transfer of the nonrotating BEC into the vortex superposition. [See Eq. (24).] Other parameters are ${\Omega }_0=2\times {10}^5\text{Hz}$, $\Delta =10{\Omega }_0$, and $\delta =0$. The quantity on the $x$ axis is the scaled time ${\Omega }_{0}t$.

Figure 6

(Color online). Plot of the transfer function Eq. (21) versus the distance between the pulse centers, $t_1-t_2$. Note that $t_1-t_2>0$ (or $t_1>t_2$) suggests the counterintuitive pulse sequence needed for the STIRAP-based population transfer process. The other parameters are identical to those noted in the caption of Fig. 5.

Figure 7

(Color online). Spatial profile of the sombrero potential plotted in arbitrary units in two dimensions with $\sigma =2.0$ and $\lambda =0.005$.

Figure 8

Thomas-Fermi spatial wave function of the BEC in a sombrero potential plotted in arbitrary units versus one spatial dimension. The doughnutlike particle density distribution is quite clear.

Figure 9

(Color online). Generation of the vortex state superposition in a sombrero trap: Results of the numerical solutions of the equations for the STIRAP scheme show the superposition 60:40 of the $|+⟩$ and $|-⟩$ vortex states. The Rabi frequency ${\Omega }_0=1\text{kHz}$ and $\Delta =100{\Omega }_0$.

Figure 10

Detection of the vortex superposition through characteristic interference of a general normalized state $\alpha |+\ell ⟩+\beta {\text{e}}^{\text{i}\theta }|-\ell ⟩$. $\ell =3$ and $\theta =0$ unless specified otherwise. The plots show the particle density distribution of the superposition states in two spatial dimensions. Dark regions correspond to the absence of particles and lighter regions correspond to higher particle density. (a) ${\alpha }^2:{\beta }^2=1:1,V=1$, giving $m=2\ell$ lobes in the interference pattern. (b) ${\alpha }^2:{\beta }^2=0.1:0.9,V=0.6$. (c) ${\alpha }^2:{\beta }^2=0.9:0.1,V=0.6$. Patterns (b) and (c) are identical. (d) Phase determination: ${\alpha }^2:{\beta }^2=1:1,\theta =\pi ,V=1$. Note the rotation of the pattern with respect to that of (a) by an angle $\theta /m$. After one more unit of OAM is added in the superposition to arrive at $\alpha |+\ell +1⟩+\beta e^{i\theta }|-\ell +1⟩$, distinct patterns (e) and (f) are obtained instead of (b) and (c).

Figure 11

(Color online). Schematic representation of an ideal beam splitter with input ports 1 and 2 and output ports 3 and 4. The amplitudes for reflection and transmission from port 1 are, respectively, $r$ and $t$. The corresponding amplitudes for a photon entering port 2 are $r^{\prime }$ and $t^{\prime }$.