Generating macroscopic superpositions with interacting Bose-Einstein condensates: Multimode speedups and speed limits

We theoretically investigate the effect of multimode dynamics on the creation of macroscopic superposition states (spin-cat states) in Bose-Einstein condensates via one-axis twisting. A two-component Bose-Einstein condensate naturally realizes an effective one-axis twisting interaction, under which an initially separable state will evolve toward a spin-cat state. However, the large evolution times necessary to realize these states is beyond the scope of current experiments. This evolution time is proportional to the degree of asymmetry in the relative scattering lengths of the system, which results in the following trade-off: faster evolution times are associated with an increase in multimode dynamics, and we find that generally multimode dynamics reduce the degree of entanglement present in the final state. However, we find that highly entangled catlike states are still possible in the presence of significant multimode dynamics and that these dynamics impose a speed limit on the evolution of such states.

Figure 1

(a)–(e) Single-mode $Q$ functions of $N=20$ atoms $[Q/Q_{\max }$ with $Q(\theta ,\varphi )=|\langle \alpha (\theta ,\varphi \left)\right|\mathrm{\Psi }\left(t\right)\rangle \left|\right]$, showing evolution under one-axis twisting for the state $|\mathrm{\Psi }\left(t\right)\rangle =exp(-i{\widehat{J}}_z^2\chi t)|\alpha (\pi /2,\pi /2)\rangle$. (f)–(j) The corresponding probability distributions in the ${\widehat{J}}_y$ eigenbasis. The system is prepared entirely in a single component $\left[\right|\mathrm{\Psi }\rangle =\left|\alpha \right(0,0)\rangle \equiv |N/2\rangle ]$ (a),(f) before a $\pi /2$ pulse places the state on the equator of the Bloch sphere $\left[\right|\mathrm{\Psi }\rangle =e^{-i{\widehat{J}}_x\pi ,2}|N/2\rangle =|\alpha (\pi /2,\pi /2)\rangle ]$ (b),(g). Using this as the initial state, the one-axis twisting interaction [Eq. (10)] creates a nonclassical state (c),(h) and quickly reaches the oversqueezed regime (d),(i). Eventually, after $t_{\mathrm{cat}}=\pi /2\chi$ the state becomes a spin-cat state in the ${\widehat{J}}_y$ basis (e),(j).

Figure 2

QFI vs time for the state $|\psi \left(t\right)\rangle =e^{-i{\widehat{J}}_z^2\chi t}|\alpha (\pi /2,0)\rangle$, calculated analytically (Ref. [14], solid black), and with the widely used truncated Wigner method (TW, dashed red). At $\chi t=\pi /2$ the QFI exhibits a “cat peak” in the QFI, associated with the creation of a spin-cat state, which is absent in the TW calculation.

Figure 3

Dynamics of $|{\varphi }_{a,m}{|}^2/{\left|{\varphi }_{\max }\right|}^2$ (a)–(c) and $|{\varphi }_{b,m}{|}^2/{\left|{\varphi }_{\max }\right|}^2$ (d), labeled by $m=(n_a-n_b)/2$ with, $N=100$, $\kappa =0$, and $\lambda =1$. (a) When the chemical potential is close to the single-particle ground-state energy the system is well described by a single-mode treatment, i.e., the dynamics are almost identical for populations $a,b$ and for all number components $m$. (b) A larger chemical potential gives rise to breathing dynamics shown here for the $m=0$ number component, which is different for each $m$, shown in (c). Multimode dynamics also differ between the components, for instance (d) shows the density $|{\varphi }_{b,m}{|}^2/{\left|{\varphi }_{\max }\right|}^2$. For $m=N/4$, $|{\varphi }_{b,m}{|}^2$ differs significantly from $|{\varphi }_{a,m}{|}^2$. The reason is that $N_a=(N/2+m)=3/4N$, while $N_b=(N/2-m)=N/4$, such that the initial condition for ${\varphi }_{b,m}$ is significantly further from a stationary state of Eq. (22).

Figure 4

Quantum Fisher information of a maximally nonlinear ($\kappa =0$), symmetric ($\lambda =1$) condensate with $N=100$ as a function of evolution time for several $\mu$. Times ${\tau }_{\mathrm{cat}}$ were taken directly from numerics by finding the peak, except for $\mu =200.0$, which was estimated from Eq. (33) and Eq. (13). The times are ${\tau }_{\mathrm{cat}}=1570$, 5.735, 0.9817 for $\mu =0.6$, 32.08, 200.0, respectively. Chemical potential $\mu$ is in units of $\hslash \omega$.

Figure 5

1D twisting rate $\chi$ under breathing dynamics with $\lambda =1$ and $\kappa =0$, far from the single-mode regime $\mu =32.08$ (solid cyan) and close to the single-mode regime $\mu =0.6$ (dashed green), which is close to zero. We also include twisting rate $\chi$ of the TF ground state [Eq. (33)] (dashed cyan), which agrees excellently with the numeric result at $\tau =0$. Inset: magnification of $\mu =0.6$ data which shows small, but nonzero dynamics. In the TF regime the period of the breathing motion agrees well with the $n=2$ excitation period, Eq. (35).

Figure 6

$F_0$, $F_1$, $F_2$ and $F_0+F_1+F_2=4\mathrm{var}\left({\widehat{J}}_y\right)$ for $N=100$ as a function of evolution time for single-mode (SM) dynamics (top) and (middle) a multimode simulation performed using the method of Sinatra and Castin. $F_2$ is the term responsible for the cat peak. (Bottom) The magnitude of the overlap ${\left[{\gamma }_j^{bb}(m=0,\tau )\right]}^{N/2}$ for $j=1,2,3$, and $\mu =32.08$. Due to the binomial coefficients in Eq. (27), the $m=0$ term contributes the most to $F_2$ [cf. Eq. (41c)].

Figure 7

For $N=100$ and $\kappa =0$ we compare the QFI of a condensate with symmetric interactions ($\lambda =1$) and an asymmetric condensate ($\lambda =0.5$) for different $\mu$ (a),(b). Asymmetric condensates are far more susceptible to the deleterious effects of multimode dynamics. The cat times for the symmetric (asymmetric) condensates are ${\tau }_{\mathrm{cat}}=1570\left(1814\right)$, $18.06\left(20.33\right)$ for $\mu =0.6$, 10.29, respectively. We also compare the magnitude of the overlap ${\gamma }_0^{ab}$ (c) for symmetric and asymmetric condensates.

Figure 8

(Top row) For a symmetric condensate $\lambda =1$, the maximum QFI is displayed for a range of parameters. The QFI is compared to $F_Q=N^2/2$ (dashed magenta), which is quickly (relative to ${\tau }_{\mathrm{cat}}$) reached under OAT. (Bottom row) For $N=100$, we display the time at which the maximum QFI occurs (not necessarily $t_{\mathrm{cat}}$). This is compared to the TF result [Eq. (34), solid black], and deviations from this curve indicate that the corresponding maximum QFI is not associated with a cat peak. The dashed cyan lines are integer multiples of the period of $|{\gamma }_2^{aa}(m=0)|$.