Dynamic scheme for generating number squeezing in Bose-Einstein condensates through nonlinear interactions

We develop a scheme to generate number squeezing in a Bose-Einstein condensate by utilizing interference between two hyperfine levels and nonlinear atomic interactions. We describe the scheme using a multimode quantum field model and find agreement with a simple analytic model in certain regimes. We demonstrate that the scheme gives strong squeezing for realistic choices of parameters and atomic species. The number squeezing can result in noise well below the quantum limit, even if the initial noise on the system is classical and much greater than that of a Poissonian (shot-noise limit) distribution.

Figure 1

Timing for the coupling pulses in the proposed experiment. The coupling field is turned on at $t_0$ and then off again at $t_1$. After a duration $t_{\text{hold}}$, the coupling pulse is turned back on again at $t_2$ and finally turned off at $t_3$. After this, the population of state $|2⟩$ atoms is measured.

Figure 2

(Color online) ${\text{log}}_{10}\left[v\left({\widehat{N}}_2\right)\right]$ as a function of $t_{\text{hold}}$ and $\varphi$. For some values of $t_{\text{hold}}$ and $\varphi$, $v\left({\widehat{N}}_2\right)$ dips below 0.01, indicating significant squeezing. Parameters: ${\chi }_{11}={\chi }_{12}=0.018{\text{s}}^{-1}$, ${\chi }_{22}=0.019{\text{s}}^{-1}$. The strength of the coupling pulses was ${\theta }_1=0.3\text{rad}$ and ${\theta }_2=0.025\text{rad}$ for the first and second coupling pulses, respectively. The initial occupation of the BEC was chosen to be a Poisson distribution with $⟨\widehat{N}⟩={10}^7$. These parameters correspond to the scattering properties of sodium in a 500 Hz spherical harmonic trap.

Figure 3

(Color online) Normalized density profile $⟨{\widehat{\psi }}_2^{†}\left(x\right){\widehat{\psi }}_2\left(x\right)⟩/⟨{\widehat{N}}_2⟩$ at $t_2$ (blue solid line) compared to the normalized density profile at $t_1$ (black dashed line), as calculated by the TW model for two different parameter regimes. In case II, more atoms are transferred in the first coupling pulse, which creates significant multimode dynamics. In case I, the dynamics are much less pronounced and the density profile at $t_2$ differs only slightly from that at $t_1$. Parameters: case I: $\Delta t_1=1\mu \text{s}$, $\Delta t_2=0.5\mu \text{s}$, and $t_{\text{hold}}=16\text{ms}$. Case II: $\Delta t_1=6\mu \text{s}$, $\Delta t_2=0.5\mu \text{s}$, and $t_{\text{hold}}=3.8\text{ms}$. In both cases, we assumed ${\Omega }_0=50\text{rad}{\text{s}}^{-1}$, and a 500 Hz spherical harmonic trap. We assumed an initial number distribution, which was Poissonian, with a mean number of atoms $N_0={10}^7$.

Figure 4

(Color online) Results from the multimode TW model (red dots) compared to the analytic two-mode model (black trace), for two different parameter regimes [case I (left column) and case II (right column), respectively]. (a) and (b) show number variance $v\left({\widehat{N}}_2\right)$, while (c) and (d) show $⟨{\widehat{N}}_2⟩$, at $t_3$. In both cases, we assumed an initial number distribution, which was Poissonian, with a mean number of atoms $N_0={10}^7$. In case I [(a) and (c)], there is an excellent agreement between the multimode TW model and the two-mode model. In case II [(b) and (d)], there is a significant disagreement between the two models because a larger fraction of atoms is transferred during the first coupling pulse, which creates significant multimode dynamics, as can be seen in Fig. 3a. The blue dashed trace indicated results from the two-mode model, when a super-Poissonian distribution was used at the initial state, with 5% number uncertainty (approximately 150 times noisier than a Poisson distribution). (a) shows that even with large amounts of classical noise, it is possible to number squeeze below the quantum limit. Parameters: case I: $\Delta t_1=1\mu \text{s}$, $\Delta t_2=0.5\mu \text{s}$, and $t_{\text{hold}}=16\text{ms}$. Case II: $\Delta t_1=6\mu \text{s}$, $\Delta t_2=0.5\mu \text{s}$, and $t_{\text{hold}}=3.8\text{ms}$. In both cases, we assumed ${\Omega }_0=50\text{rad}{\text{s}}^{-1}$, and a 500 Hz spherical harmonic trap.

Figure 5

(Color online) $v\left({\widehat{N}}_2\right)$ as $a_{22}$ is varied. All other parameters (${\Omega }_0$, $\Delta t_1$, $\Delta t_2$, and $t_{\text{hold}}$) are the same as in Fig. 4 (case I), and the phase of the second coupling pulse was fixed at $\varphi =1.4\pi$, which was the optimum phase for Fig. 4 (case I). The squeezing is rapidly degraded as $a_{22}$ moves away from $a_0$. Black trace: results of the analytic two-mode model. Red dots: results from TW simulation.

Figure 6

$v\left({\widehat{N}}_2\right)$ as $a_{22}$ is varied, as calculated by the two-mode model. All other parameters (${\Omega }_0$, $\Delta t_1$, $\Delta t_2$, and $t_{\text{hold}}$) are the same as in Fig. 4 (case I). The phase of the second coupling pulse $\varphi$ has been optimized for maximum number squeezing for each value of $a_{22}$. Significant number squeezing can still be obtained as $a_{22}$ is altered by 30% in either direction. The exception is as $a_{22}$ approaches $a_{11}$ and $a_{12}$ the squeezing vanishes.