Mesoscopic quantum superpositions in bimodal Bose-Einstein condensates: Decoherence and strategies to counteract it

We study theoretically the interaction-induced generation of mesoscopic coherent spin state superpositions (small-particle-number cat states) from an initial coherent spin state in bimodal Bose-Einstein condensates and the subsequent phase revival, including decoherence due to particle losses and fluctuations of the total particle number. In a full multimode description, we propose a preparation procedure of the initial coherent spin state and we study the effect of preexisting thermal fluctuations on the phase revival, and on the spin and orbito-spinorial cat-state fidelities.

Figure 1

Fidelity and absolute value of contrast vs time for a split Bose-Einstein condensate of $N=300 |F=1,m_F=-1\rangle {}^{87}\mathrm{Rb}$ atoms in two identical and spatially separated harmonic potentials in the presence of three-body losses. Scattering length $a=100.4a_0$, trapping frequency $\omega /2\pi =500$ Hz, with a three-body loss constant rate $K_3=6\times {10}^{-42}{\mathrm{m}}^6$/s [24]. This gives $\chi =12{\mathrm{s}}^{-1}$ and ${\gamma }^{\left(3\right)}=2.6\times {10}^{-7}{\mathrm{s}}^{-1}$.

Figure 2

Top: Trapping configuration for rubidium 87 atoms: two cigar-shaped harmonic traps displaced by $\mathrm{\Delta }z$ along the “long” trap axis. Bottom: representation of the initial state on the Bloch sphere: $\left|b\right.〉=\left|F=2,m_F=1\right.〉$ is the majority component and $\left|a\right.〉=\left|F=1,m_F=-1\right.〉$ is the minority component spin state. The initial state is close to the south pole ($\theta$ close to $\pi$).

Figure 3

Optimal cat state that we predict for realistic experimental conditions with the two rubidium 87 states $|a\rangle =|F=1,m_F=-1\rangle$ and $|b\rangle =|F=2,m_F=1\rangle$. Top: Fisher information (24) as a function of time, calculated with the Gross-Pitaevskii Hamiltonian (22) (green dash-dotted line), and with the general two-mode model of Sec. 4a (red solid line); for comparison, the blue dotted curve, Eq. (26), gives the maximal Fisher information that one could obtain for the time-dependent mean atom numbers. Bottom: Wigner function at $t_{\mathrm{cat}}=112$ ms calculated with the Gross-Pitaevskii Hamiltonian (22). The Wigner function, in the south hemisphere of the Bloch sphere, is projected onto the $x-y$ plane. Parameters: The total atom number of average $\overline{N}=150$ has Poissonian fluctuations, ${\overline{N}}_a=5.71,{\overline{N}}_b=144.29,{\omega }_{\perp }=2\pi \times 1000$ Hz, ${\omega }_{za}=2\pi \times 850$ Hz, ${\omega }_{zb}=2\pi \times 50$ Hz, $\mathrm{\Delta }z=1.620a_{\perp }$ (distance between the trap centers). Scattering lengths $a_{aa}=100.4a_0,a_{bb}=95.44a_0,a_{ab}=98.13a_0$ [24]. One-body, two-body, and three-body loss rate constants $K_a^{\left(1\right)}=K_b^{\left(1\right)}=0.2{\mathrm{s}}^{-1},K_b^{\left(2\right)}=8.1\times {10}^{-20}{\mathrm{m}}^3/\mathrm{s},K_{ab}=1.51\times {10}^{-20}{\mathrm{m}}^3/\mathrm{s},K_a^{\left(3\right)}=6\times {10}^{-42}{\mathrm{m}}^6/\mathrm{s}$ [24]. In the general two-mode model of Sec. 4a, these parameters lead to $\chi =12.895{\mathrm{s}}^{-1},\tilde{\chi }=12.888{\mathrm{s}}^{-1},{\gamma }_b^{\left(2\right)}=6.436\times {10}^{-3}{\mathrm{s}}^{-1},{\gamma }_{ab}=1.032\times {10}^{-3}{\mathrm{s}}^{-1},{\gamma }_a^{\left(3\right)}=5.15\times {10}^{-6}{\mathrm{s}}^{-1}$.

Figure 4

Optimal cat state that we predict for realistic experimental conditions with the two sodium 23 states $|a\rangle =|F=1,m_F=0\rangle$ and $|b\rangle =|F=2,m_F=-2\rangle$. Top: Fisher information (24) as a function of time, calculated with the Gross-Pitaevskii Hamiltonian (22) (green dash-dotted line), and with the general two-mode model of Sec. 4a (red solid line); for comparison, the blue dotted curve, Eq. (26), gives the maximal Fisher information that one could obtain for the time-dependent mean atom numbers. Bottom: Wigner function at $t_{\mathrm{cat}}=178$ ms calculated with the Gross-Pitaevskii Hamiltonian (22). The Wigner function, in the south hemisphere of the Bloch sphere, is projected onto the $x-y$ plane. Parameters: The total atom number of average $\overline{N}=150$ has Poissonian fluctuations, ${\overline{N}}_a=22,N_b=128,{\omega }_{\perp a}=2\pi \times 1415$ Hz, ${\omega }_{\perp b}=2\pi \times 115$ Hz, ${\omega }_{za}=2\pi \times 612$ Hz, ${\omega }_{zb}=2\pi \times 772$ Hz, $\mathrm{\Delta }z=0.62\times {10}^{-6}$ m (distance between the trap centers). Scattering lengths $a_{aa}=52.91a_0,a_{bb}=64.25a_0,a_{ab}=64.25a_0$ [28]. One-body loss rate constants $K_a^{\left(1\right)}=K_b^{\left(1\right)}=0.01{\mathrm{s}}^{-1}$. In the general two-mode model of Sec. 4a, these parameters lead to $\chi =8.763{\mathrm{s}}^{-1},\tilde{\chi }=8.729{\mathrm{s}}^{-1}$. We expect no relevant $a-b$ or $b-b$ two-body losses here [28], and we have checked that for the considered parameters the contribution of three-body losses is negligible.

Figure 5

Density cuts along the $z$ axis of the majority (blue solid line) and the minority (red dashed line) component for parameters of Fig. 3 for rubidium 87 (left), and of Fig. 4 for sodium 23 (right).

Figure 6

Fidelities of the multimode cat-state preparation as a function of the temperature $T$ of the initial ideal gas in the canonical ensemble, in a cubic box of size $L$ with periodic boundary conditions. Black (lower) solid line: fidelity $\mathcal{F}$ of the orbitospinorial cat state as given by Eq. (62). Red: peak fidelity ${\mathcal{F}}_{\mathrm{spin}}$ of the spin cat state as defined by Eq. (84), obtained from a Monte Carlo thermal average of Eq. (89) over 4000 realizations and temporal maximization around the cat-state time (circles with error bars) or peak fidelity ${\mathcal{F}}_{\mathrm{spin}}^{\mathrm{Bog}}$ from the Bogoliubov approximation (95) at the cat-state time such that $A\left(t\right)=\pi /2$ (upper solid line). Dashed red line: lower bound ${\mathcal{F}}_{\mathrm{spin}}^{\mathrm{Bog},\mathrm{minor}}$ on ${\mathcal{F}}_{\mathrm{spin}}^{\mathrm{Bog}}$, as given by Eq. (100). Contrarily to $\mathcal{F}$ and to ${\mathcal{F}}_{\mathrm{spin}}^{\mathrm{Bog},\mathrm{minor}}$, which are universal functions of $k_{B}T/\mathrm{\Delta }, {\mathcal{F}}_{\mathrm{spin}}$ and ${\mathcal{F}}_{\mathrm{spin}}^{\mathrm{Bog}}$ depend on the particle number $N$ and on the interaction strength, adiabatically ramped up and down between 0 and the $a-a$ and $b-b$ scattering length $a_{\mathrm{f}}$; see Sec. 5e. Here the parameters are the ones of Fig. 7: $N=300,4\pi a_{\mathrm{f}}/L=0.0667$, and $t_{\mathrm{ramp}}=20t_{\mathrm{ramp}}^{\mathrm{adiab}}$. The temperature is expressed in units of $\mathrm{\Delta }/k_B$, where $\mathrm{\Delta }=\frac{{\hslash }^2{\left(2\pi \right)}^2}{2m{L}^2}$ is the minimal excitation energy.

Figure 7

First-order coherence function $g^{\left(1\right)}\left(t\right)$ of the condensate, as given by Eq. (68) in the multimode Bogoliubov approximation (82). $N=300$ lossless ${}^{87}\mathrm{Rb}$ atoms are initially prepared in internal state $|a\rangle =|F=1,m_F=-1\rangle$ in a box ${[0,L]}^3$ with periodic boundary conditions, $L=1\mu \mathrm{m}$, at temperature $T$ in the absence of interactions. At $t=0^{+}$ they are subjected to a $\pi /2$ pulse towards the internal state $|b\rangle =|F=1,m_F=1\rangle$ and to an adiabatic Hann ramping (52) of the interaction strength with a duration $t_{\mathrm{ramp}}=20t_{\mathrm{ramp}}^{\mathrm{adiab}}=0.21$ ms, where $t_{\mathrm{ramp}}^{\mathrm{adiab}}$ is defined in Eq. (C10), up to the final value $g_{\mathrm{f}}=4\pi {\hslash }^2{a}_{\mathrm{f}}/m$, with the $a-a$ and $b-b$ scattering lengths $a_{\mathrm{f}}=100.4a_0$. In (a): $g^{\left(1\right)}$ as a function of time for temperatures $k_{B}T=0$ (black solid line), $k_{B}T=\mathrm{\Delta }/4$ (red solid line), $k_{B}T=\mathrm{\Delta }/2$ (blue solid line), from bottom to top at the first and third revival times and from top to bottom at the second revival time. (b) Its absolute value as a function of temperature at the first revival time (at which $|g^{\left(1\right)}\left(t\right)|$ has a maximum). We recall that $\mathrm{\Delta }=\frac{{\hslash }^2{\left(2\pi \right)}^2}{2m{L}^2}$ is the energy of the first single-particle excited state in the box.

Figure 8

Relative deviations between $g^{\left(1\right)}\left(t_{\mathrm{rev}}\right)$ and ${\mathcal{F}}^2\left(t_{\mathrm{cat}}\right)$ in presence of one-body losses. The approximate formula (A9) is represented as a full line, while the dashed lines are exact solutions for $N=100$ and $N=300$. The values of the abscissa $2N{({\gamma }^{\left(1\right)}/\chi )}^2$ corresponding to the trapping angular frequency $\omega =2\pi \times 500$ Hz and the scattering length $a=100.4$ Bohr radii and one-body loss rate equal to $K_1=0.01\mathrm{Hz}$ are marked as dotted vertical lines for $N=100$ (left line) and $N=300$ (right line). Note that as in Sec. 2, we restrict here to the case in which the two components are symmetric and spatially separated.

Figure 9

Relative deviations between $g^{\left(1\right)}\left(t_{\mathrm{rev}}\right)$ and ${\mathcal{F}}^2\left(t_{\mathrm{cat}}\right)$ in presence of three-body losses. The approximate formula (A21) is represented as a full line, while the symbols linked by dashed lines are exact solutions for $N=100$ and $N=300$. The values of the abscissa $N^5{({\gamma }^{\left(3\right)}/\chi )}^2/8$ corresponding to the trap and loss parameters of Fig. 1 are marked as dotted vertical lines for $N=100$ (left line) and $N=300$ (right line). Note that as in Sec. 2, we restrict here to the case in which the two components are symmetric and spatially separated.

Figure 10

The algorithm searches for good configurations in the parameter space by maximizing the Fisher information and verifies the presence of a small-amplitude cat state signaled by high contrast fringes in the Wigner function.

Figure 11

Initial (red peak on the right) and the final (blue peak on the left) probability distribution of the total number of atoms. Around 20% of the atoms are lost, but practically only in the majority component $b$. Table: Budget of losses at time $t_{\mathrm{cat}}$. Parameters are the same as in Fig. 3.

Figure 12

Example of output of the optimization program described in Appendix pp2 for the Fisher information at the cat-state time. $N_a$ on the $x$ axis is in the initial mean number of atoms in the minority component. For a given $N_a$ the different points correspond to successive configurations explored by the algorithm in its convergence process. Restricting to configurations with trap aspect ratios smaller than 20 we select out the darker points (in blue). Scattering lengths and loss rates are as in Fig. 3 for ${}^{87}\mathrm{Rb}$. Figure 3 corresponds to one of the blue points with maximal Fisher information around 1500.