Optimal control of number squeezing in trapped Bose-Einstein condensates

We theoretically analyze atom interferometry based on trapped ultracold atoms and employ optimal control theory in order to optimize number squeezing and condensate trapping. In our simulations, we consider a setup where the confinement potential is transformed from a single to a double well, which allows us to split the condensate. To avoid in the ensuing phase-accumulation stage of the interferometer dephasing due to the nonlinear atom-atom interactions, the atom-number fluctuations between the two wells should be sufficiently low. We show that low number fluctuations (high number squeezing) can be obtained by optimized splitting protocols. Two types of solutions are found: in the Josephson regime we find an oscillatory tunnel control and a parametric amplification of number squeezing, while in the Fock regime squeezing is obtained solely due to the nonlinear coupling, which is transformed to number squeezing by peaked tunnel pulses. We study splitting and squeezing within the frameworks of a generic two-mode model, which allows us to study the basic physical mechanisms and the multiconfigurational time-dependent Hartree for bosons method, which allows for a microscopic modeling of the splitting dynamics in realistic experiments. Both models give similar results, thus highlighting the general nature of these two solution schemes. We finally analyze our results in the context of atom interferometry.

Figure 1

(Color online) Schematics of the splitting process. When splitting a condensate, it breaks up into two parts (red solid lines) with equal mean atom-number and number fluctuations $\Delta n$. The distribution of atoms between the wells is controlled by the tunnel coupling $\Omega$ and the nonlinear intrawell energy $\kappa$.

Figure 2

(Color online) Bloch-sphere representation of $\mathbf{C}$. The $z$ axis corresponds to the number difference ($\Delta n$ and $\Delta \varphi$ are number and phase fluctuations, respectively). The action of the pseudospin operators ${\widehat{J}}_x$ and ${\widehat{J}}_z$ on $\mathbf{C}$ corresponds to rotations about the $x$ and $z$ axes, respectively. In (a) a binomial state is shown and in (b) a number-squeezed one.

Figure 3

(Color online) Number squeezing for quasiadiabatic exponential (gray, many lines) two-parameter (blue) and Fock optimization (symbols) for different atom numbers. For the optimized solutions we only report the final value of the number fluctuations. Both optimization strategies allow for number squeezing at least one order of magnitude faster than exponential splitting. The green lines report the limitations of the two-parameter optimization for $N=10000$ (limitations are very similar for smaller $N$). Solutions for realistic splitting within MCTDHB are only obtained for sufficiently long exponential decay constants $t_c$ reported in the panel. Smaller values of $t_c$ result in strongly oscillating condensates and reduced number squeezing. We note that in all our figures time is measured in units of 1.37 ms and position in $\mu \text{m}$ (rubidium units). For the generic model we use units such that $2\kappa N=1$.

Figure 4

(Color online) Typical results from Josephson OCT for the generic two-mode model for different splitting times and (a) $N=100$ and (b) $N=1000$. The initial (optimized) $\Omega$ is shown by the dashed (solid) line in the lower panels of the figures. The upper panels of the figures show the number fluctuations. The bright lines show estimates from a parametric oscillator model (see Sec. 3F). (a) The Bloch spheres demonstrate the state for different times.

Figure 5

(Color online) Typical results from Fock OCT for the generic two-mode model for (a) $N=100$ and (b) $N=1000$. The initial (optimized) $\Omega$ is shown by the dashed (solid) line in the lower panels of the figures. The upper panels of the figures show the number fluctuations. (a) The Bloch spheres demonstrate the state for different times.

Figure 6

(Color online) Inverse adiabatic squeezing time scale $U_{0}N$ versus 1D-interaction strength $U_0$ for several $N$, where each point along a straight line corresponds to a different aspect ratio (connected by black lines to points with same aspect ratio) as given in the figure. The straight lines are guides for the eyes.

Figure 7

(Color online) (a) Mean population difference $\alpha =\left({\rho }_{gg}-{\rho }_{ee}\right)/N$ of the orbitals normalized by $N$ ($\alpha$ is the coherence factor) and number fluctuations, (b) nonlinear coefficients of the orbitals equations from Eq. (17), (c) the tunnel coupling, and (d) the two-body matrix elements of Eq. (21). Parameters are $U_{0}N=1$ and $N=100$ (in the following figures we always use these values unless stated otherwise).

Figure 8

(Color online) (a) Trapping cost function for a two-parameter control for several $U_{0}N$ and $t_c$. (b) Example for the density and the tunnel coupling for a specific example ($U_{0}N=1$, $t_c=0.23$, marked by the red point), which both strongly oscillate. Here we have $N=100$, but results are similar for different $N$.

Figure 9

(Color online) In the left panel, a typical solution for Josephson optimization for $N=100$. Changing interactions (red line) or atom number (green line) by 10% do not appreciably change results. However, the control scheme is sensitive to the (transverse) trap parameters. Here, the deviation of the initial harmonic trapping frequency should not exceed 0.1% (dashed blue line) and is crucial for 1% (solid blue line). In the right panels, the corresponding orbitals are shown.

Figure 10

(Color online) Tunnel coupling within MCTDHB for the Josephson (dark blue line) and the Fock (bright red line) solution from Figs. 9, 11a, respectively, where the second one is scaled by a factor of 10.

Figure 11

(Color online) In the left panels typical OCT solution for Fock optimization for $N=100$ with (a) weak interactions $U_{0}N=1$ and (b) strong interaction $U_{0}N=5$. Exponential (OCT) control and squeezing is displayed by the dashed (solid) lines. In the right panels, the orbitals are shown.

Figure 12

(Color online) Comparison of squeezing (symbols) within the generic model and MCTDHB for $U_{0}N=1$ compared to quasiadiabatic exponential splitting (gray lines). The latter is calculated within the generic model since MCTDHB results are very similar. Best results are obtained for Fock optimization and large $N$.

Figure 13

(Color online) In the upper panels, the density for (a) an exponential and (b) an optimized splitting is shown. In the last sequence the trap is switched off and the condensates start to expand and overlap (TOF). In the lower panels, the interference in momentum space, obtained from the Wigner function $W\left(x=0,p\right)$ [53], is plotted. When the condensates are released, the interference in momentum space is transformed to an interference in position space. The fringe contrast is reduced in case of exponential splitting due to phase diffusion and squeezed states are much more robust. For a perfect fringe contrast, the limitations to the phase sensitivity are given by the quantum noise.

Figure 14

(Color online) ${\xi }_R$ for optimized (symbols) and exponential splitting (gray lines). The quasiadiabatic exponential splitting is shown for several time scales for $N=100$. After it reaches a minimum, it rises again due to loss of phase coherence. With optimized splitting, lower ${\xi }_R$ is obtained at shorter time scales. The symbols are the same as in Fig. 12. The results are compared to the Heisenberg limit ${\xi }_R=\sqrt{\frac{2}{N}}$ (horizontal lines). For larger atom-number $N=500$ (MCTDHB) and $N=1000$ (generic model) optimized splitting yields lower ${\xi }_R$, whereas for adiabatic splitting the curves are similar.

Figure 15

(Color online) Coherence factor $\alpha$ for typical Josephson OCT (dark red) and Fock OCT (bright green) solutions, where it decays slower than for the quasiadiabatic exponential splitting (broken line). The Josephson solution shows up oscillations in the phase coherence during splitting because the driving of the parametric oscillator affects both conjugate observables of number and phase.