Vortex Phase Qubit: Generating Arbitrary, Counterrotating, Coherent Superpositions in Bose-Einstein Condensates via Optical Angular Momentum Beams

We propose a scheme for the generation of arbitrary coherent superpositions of vortex states in Bose-Einstein condensates (BEC) using the orbital-angular-momentum states of light. We devise a scheme to generate coherent superpositions of two such counterrotating states of light using well-known experimental techniques. We show that a specially designed Raman scheme allows for transfer of the optical vortex-superposition state onto an initially nonrotating BEC. This creates an arbitrary and coherent superposition of a vortex and antivortex pair in the BEC. The ideas presented here could be extended to generate entangled vortex states, design memories for the orbital-angular-momentum states of light, and perform other quantum information tasks. Applications to inertial sensing are also discussed.

Figure 1

Generation of superposition of the OAM states. The first beam splitter (BS) at the input is a $|\tilde{r}{|}^2/|\tilde{t}{|}^2$ beam splitter as $\tilde{r}$ and $\mathrm{i}\tilde{t}$ are the reflection and transmission amplitudes. The second beam splitter is $50:50$, and the mirrors (M) are perfectly reflecting. The dove prism (DP) performs the operation $|\ell ⟩\to |-\ell ⟩$. Choosing $\varphi =\pi$, and discarding the output at port 2, state $(\tilde{t}|+\ell ⟩+\tilde{r}|-\ell ⟩)$ is obtained at port 1.

Figure 2

The level scheme for generation of vortex state superposition. A nonrotating state $|0⟩$ is coupled to the vortex states $|+⟩$ and $|-⟩$ through the optical vortex field components providing coupling strengths ${\Omega }_{+}$, ${\Omega }_{-}$ and a strong drive field with Rabi frequency ${\Omega }_c$. The optical vortex beam is ${\sigma }_{+}$ polarized, whereas the drive field is ${\sigma }_{-}$ polarized such that the hyperfine quantum number $m_F$ is the same for the components $|0⟩,|\pm ⟩$. The vorticity of the $|\pm ⟩$ states is $\pm \ell$, respectively.

Figure 3

(a) Transfer function $f(t)$ for various detunings. (A) $\delta =0$, (B) $\delta =900$, (C) $\delta =380$, (D) linearly varying detuning $\delta (t)=3000–{400}^{2}t$. All detunings are given in Hertz. The complete population transfer to the vortex state, corresponding to $f(t)=-1$ for sufficiently large $t$ (measured in seconds), is possible only with time-dependent detuning. (b)–(d) show various superpositions of the vortex states. The steady state ratios are give in the notation $|{\beta }_{+}{|}^2:|{\beta }_{-}{|}^2$.

Figure 4

Detection of the vortex superposition through characteristic interference of a general normalized state $\alpha |+\ell ⟩+\beta {\mathrm{e}}^{\mathrm{i}\varphi }|-\ell ⟩$. $\ell =3$ and $\varphi =0$ unless specified otherwise. (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: ${\alpha }^2:{\beta }^2=1:1$, $\varphi =\pi$; $V=1$, notice rotation of the pattern with respect to that of (a) by an angle $\varphi /m$. After adding one more unit of OAM in the superposition to arrive at $\alpha |+\ell +1⟩+\beta {\mathrm{e}}^{\mathrm{i}\varphi }|-\ell +1⟩$ distinct patterns (e) and (f) are obtained instead of (b) and (c).