Proposal for the Creation and Optical Detection of Spin Cat States in Bose-Einstein Condensates

We propose a method to create “spin cat states,” i.e., macroscopic superpositions of coherent spin states, in Bose-Einstein condensates using the Kerr nonlinearity due to atomic collisions. Based on a detailed study of atom loss, we conclude that cat sizes of hundreds of atoms should be realistic. The existence of the spin cat states can be demonstrated by optical readout. Our analysis also includes the effects of higher-order nonlinearities, atom number fluctuations, and limited readout efficiency.

Figure 1

Spin cat state creation (a)–(d) and detection (e)–(g). In (a)–(f) the radially symmetric photons and spherically symmetric BECs are represented by spatial density distributions. (a) A coherent light pulse is sent into the BEC. (b) The light state is absorbed in the BEC (see inset), creating a CSS. The shape of the input pulse is chosen such that the two-component BEC is in its ground state after the absorption. (c) The trapping frequency ${\omega }_b$ for the small component is increased adiabatically. The density of the small component now exceeds that of the large component at the center. (d) The collision-induced Kerr nonlinearity drives the system into a spin cat state (CAT). (e) The trapping frequency is adiabatically reduced to its initial value. (f) The spin state is reconverted into light, whose Husimi $Q$ function [24] is determined via homodyne detection [25]. (g) Expected shape of the $Q(\beta )$ function in phase space. The coherent state at $t=0$ gives a single peak, while the cat state at $t={\tau }_c$ yields two peaks. Further evolution for another interval ${\tau }_c$ returns the output light to a coherent state, yielding a single peak at $t=2{\tau }_c$. This would not be possible if the two peaks at $t={\tau }_c$ corresponded to an incoherent mixture, thus proving the existence of a coherent superposition of CSSs in the BEC at ${\tau }_c$ [26].

Figure 2

(a) The time to create a spin cat state ${\tau }_c=\pi /|2{\eta }_2|$ (thick red curve) versus the time to lose one atom ${\tau }_{\ell }$ in component $B$ (thick black curve) as a function of the trapping frequency for the small component ${\omega }_b$. The size of the cat $\overline{n}=100$ in this example. One sees that ${\tau }_c<{\tau }_{\ell }$ is possible for sufficiently large ${\omega }_b$. The plot also shows the main individual loss channels contributing to the calculation of ${\tau }_{\ell }$, where ${\tau }_{m,c}$ is the individual time of losing one atom through $m$-body collision with particle combinations $c$. It furthermore shows analytic approximations for ${\tau }_c\sim {\omega }_b^{-3/2}$ (red dotted curve) and ${\tau }_{3,bbb}\sim {\omega }_b^{-3}$ (blue dashed curve), see text. (b) Achievable cat size $\overline{n}$ as a function of ${\omega }_b$. The shaded region corresponds to ${\tau }_c<{\tau }_{\ell }$ for a condensate size $N={10}^5$ as in (a). The cat size can be increased somewhat by reducing $N$ (dashed line). We also show that there is a region where ${\tau }_c<0.1{\tau }_{\ell }$ so that loss should really be negligible. The green circles correspond to ${\tau }_c=10,1,0.1\mathrm{s}$ (from left to right). The star corresponds to the values used in Fig. 3, and the corresponding density distributions are shown in Fig. 1. Both plots are for ${}^{23}\mathrm{Na}$ with spin states $|A⟩=|F=1,m=0⟩$, $|B⟩=|F=2,m=-2⟩$, scattering lengths $a_{aa}=2.8\mathrm{nm}$, $a_{bb}=a_{ab}=3.4\mathrm{nm}$ [20], loss coefficients $L_1=0.01/\mathrm{s}$, $L_2=0$, $L_3=2\times {10}^{-42}{\mathrm{m}}^6/\mathrm{s}$ [41], and a trapping frequency ${\omega }_a=2\pi \times 20\mathrm{Hz}$ for the large component.

Figure 3

Optical demonstration of the spin cat state in the presence of various imperfections for the parameter values corresponding to the star in Fig. 2 ($\overline{n}=100$ and ${\omega }_b=2\pi \times 500\mathrm{Hz}$). The spin state is reconverted into light and the Husimi phase space distribution function $Q(\beta )$ is determined via homodyne tomography. (a) Includes the effects of the higher-order nonlinearities ${\eta }_3$ and ${\eta }_4$. Two far separated peaks corresponding to the cat state are clearly visible at ${\tau }_c^{*}=0.68\mathrm{s}$, and one peak corresponding to the revived coherent state at $2{\tau }_c^{*}$. The shift of the cat creation time due to the higher-order terms is ${\tau }_c^{*}/{\tau }_c=1.06$. (b) Furthermore includes 90% photon retrieval loss, which moves the peaks towards the origin, and 5% uncertainty in the total atom number, which spreads the peaks in the angular direction.