Two-mode Bose gas: Beyond classical squeezing

The dynamical evolution of squeezing correlations in an ultracold Bose-Einstein condensate distributed across two modes is investigated theoretically in the framework of the Bose-Hubbard model. It is shown that the eigenstates of the Hamiltonian do not exploit the full region allowed by Heisenberg’s uncertainty relation for number and phase fluctuations. The development of nonclassical correlations and relative number squeezing is studied in the transition from the Josephson to the Fock regime. Comparing the full quantum evolution with classical statistical simulations allows us to identify quantum aspects of the squeezing formation. In the quantum regime, the measurement of squeezing allows us to distinguish even and odd total particle numbers.

Figure 1

Phase diagram for $N=100$ particles in a double-well potential, with (on average) equal populations in the two wells, $\langle \mathrm{n̂}\rangle =\langle {\mathrm{n̂}}_1-{\mathrm{n̂}}_2\rangle /2=0$, and the absolute phase chosen such that $\langle {\mathrm{Ŝ}}_2\rangle =i\langle {\mathrm{â}}_2^{†}{\mathrm{â}}_1-{\mathrm{â}}_1^{†}{\mathrm{â}}_2\rangle =0$; see Eq. (A9). For any given coherence parameter $\alpha =2\langle {\mathrm{Ŝ}}_1\rangle /N,$ the variance $(\Delta n){}^2=N{\xi }_3/4$ of the number difference between the wells is bounded below and above by Heisenberg’s uncertainty relation and the constraint that the total particle number $N$ is fixed. The allowed area is defined by the inequality (11) shaded in colors. In the medium (violet) and dark (blue) regimes, the squeezing parameter is below the standard quantum limit, ${\xi }_3⩽1$; see Eq. (5). In the dark (blue) regime, the squeezing can be used to gain precision in metrology; see Eq. (14).

Figure 2

Positions of the energy eigenstates of the Hamiltonian (1) for a system with $N=100$ particles in the phase diagram introduced in Fig. 1. Each black dot corresponds to one such state. The dotted lines are drawn to guide the eye between successive eigenstates of a Hamiltonian with fixed ratio $U/J$. From top to bottom, the sets of states are obtained for $U/J=0,1,10$, and $100$, respectively. The (red) solid line connects all ground states for different $U/J$. We emphasize that the energy eigenstates do not extend over the full (shaded) region allowed by Heisenberg’s uncertainty.

Figure 3

Evolution of the coherence $\alpha$ and the number variance $(\Delta n){}^2$ for a gas of $N=100$ atoms [(blue) solid line] under a slow ramp-up of $U(t)/J(t)$ over a time $t_{\max }=0.16\hspace{0.5em}$s as given in Fig. 14. The system starts in a state with $(\Delta n(t_0)){}^2=37.5$ and $\alpha (t_0)=0.994,$ which is obtained for an initial temperature $T_0=20$ nK. A range of time points is indicated by black dots. They are spaced by $0.1t_{\max }$, and the first completely distinct point is for $t=0.3t_{\max }$. The evolution under an adiabatic change of the parameters $U$ and $J$, that is, for $v_0\to 0$, is shown by the (red) dotted line. An isotherm for $T=0.17$ nK is shown as a dashed line. See Fig. 1 and Sec. 2 for the definition of the differently shaded areas allowed by the constraint Heisenberg, squeezing, and metrology gain limits.

Figure 4

Diagonal elements of the density matrix, $\langle E_i|\mathrm{\rho ̂}(t)|E_i\rangle ,$ in the basis of the energy eigenstates (A7) of the Hamiltonian (1), for five different times during the ramp-up of $U(t)/J(t)$ shown in Fig. 14 (Appendix ). Note the semilogarithmic scale. The initial temperature is $T_0=20$ nK. The inset shows the diagonal elements of the density matrix for $t=t_{\max }$ as a function of the energy on a logarithmic scale, demonstrating the thermal character of the state.

Figure 5

Semiclassical evolution [(purple) dotted line] of the coherence $\alpha$ and the number variance $(\Delta n){}^2=N{\xi }_3/4$ for a gas of $N=100$ atoms under a slow ramp-up ($t_{\max }=0.16\hspace{0.5em}$s) of $U(t)/J(t)$ as given in Fig. 14. The (blue) solid line shows the corresponding quantum evolution, as discussed in Sec. 3a and Fig. 3 for the same evolution time $t_{\max }$ and the same time-dependent parameters $U$ and $J$. The initial temperature is $20$ nK as in Fig. 3. The definition of the shadings was introduced in Fig. 1.

Figure 6

Difference between the ${\xi }_3$ obtained from the semiclassical and the quantum evolutions, normalized to the quantum result, for three different initial phase-space distributions: (Purple) solid line: Wigner function for the state (15). (Red) dotted line: Product of distributions of initial relative number and coherence derived from Eq. (15). (Blue) dashed line: Ellipsoidal Gaussian state with widths given by these relative number and coherence distributions. See the main text for a discussion.

Figure 7

Evolution of the phase-space probability distribution $P(n,\varphi ;t)$, Eq. (23), between $t=0$ (left panel) and $t=0.68t_{\max }$ (right panel) as obtained by classical simulations according to Eq. (23). White color indicates $P(n,\varphi ;t)\approx 0$ while other colors indicate a positive probability. The initial distribution is derived from a quantum gas of $N=100$ atoms at a temperature of $T_0=20$ nK. The tilt of the final phase-space distributions reflects classical correlations between the generalized position and momentum, which reduce the squeezing in $n$ and are due to the nonadiabatic settling to a thermal state; see Fig. 5.

Figure 8

Evolution of the coherence $\alpha$ and the squeezing parameter ${\xi }_3$ related to the number variance through $(\Delta n){}^2=N{\xi }_3/4$ for a gas of $N=100$ atoms under a very slow ramp-up of $U(t)/J(t)$ as given in Eq. (24). The quantum evolution is represented by the (blue) solid line, while the classical statistical evolution is the (purple) dotted line. The system starts in the ground state of the Bose-Hubbard Hamiltonian for $\mathit{NU}/J(t_0)\simeq 0.16$. This corresponds to $(\Delta n(t_0)){}^2=24$ and $\alpha (t_0)=0.993$. The total evolution time is $t_{\max }=10.18\hspace{0.5em}$s. The evolution under an adiabatic change of the parameters $U$ and $J$ (i.e., for $1/\tau \to 0)$ is shown by the (red) short-dashed line. See Fig. 1 and Sec. 2 for the definition of the areas allowed by the constraint Heisenberg and metrology gain limits, distinguished by different shading.

Figure 9

Evolution of the Wigner function for an inverse ramp speed $\tau =0.55\hspace{0.5em}$s, corresponding to the quantum evolution shown in Figs. 8 and 10. Colors encode the value of $W(n,\varphi )\times {10}^3$ [see Eq. (B7)], with $n=(|\alpha |{}^2-|\beta |{}^2)/2$ and $\varphi =2\hspace{0.5em}\arg (\alpha )$. $W$ is negative in the blue areas. The initial distribution shown in the upper left panel is derived from the ground state of the Hamiltonian with $J(t_0)/\mathit{NU}=6$ for $N=100$ atoms. The upper right panel shows the Wigner function for the number-squeezed state reached at $t/t_{\max }=0.5$, with ${\xi }_3(0.5t_{\max })\simeq 3\times {10}^{-2}$, $\alpha (0.5t_{\max })\simeq 0.86$. In the lower panels, we show the Wigner function at $t/t_{\max }=0.75$ (left) and $t/t_{\max }=0.9$ (right), that is, for maximally negative ${\xi }_3\simeq 2\times {10}^{-5}$ at $\alpha \simeq -0.02$ and maximally negative $\alpha \simeq -0.1$ at ${\xi }_3\simeq 2\times {10}^{-4}$, respectively (cf. Fig. 8). The contour lines help to show the position of the maxima in $\left|\varphi \right|=0$ and $\left|\varphi \right|=\pi$, respectively.

Figure 10

Evolution of the squeezing parameter ${\xi }_3$ related to the number variance through $(\Delta n){}^2=N{\xi }_3/4$ for a gas of $N=100$ atoms under a very slow ramp-up of $U(t)/J(t)$ as given in Eq. (24). The quantum evolution is represented by the (blue) solid line, while the classical statistical evolution is the (purple) short-dashed line. The initial state and all other parameters are chosen as in Fig. 8. The evolution under an adiabatic change of the parameters $U$ and $J$ is shown by the (red) long-dashed line.

Figure 11

Evolution of the squeezing parameter ${\xi }_3$ related to the number variance through $(\Delta n){}^2=N{\xi }_3/4$ for a gas of $N=101$ atoms under a very slow ramp-up of $U(t)/J(t)$ as given in Eq. (24). The quantum evolution is represented by the (blue) solid line, while the classical statistical evolution is the (purple) short-dashed line. All other parameters are chosen as in Fig. 10. The adiabatic evolution is again shown by the (red) long-dashed line.

Figure 12

Final value of ${\xi }_3(t_{\max })$ obtained after an evolution time $t_{\max }=18.4\tau$ vs ramp speed $\tau$. The solid line corresponds to the analytical result in Eq. (27), which is valid in the Rabi and Josephson regimes. The (blue) crosses and (purple) circles are the corresponding final values obtained from a full quantum evolution and semiclassical evolution of the initial state, respectively. In the Fock regime, which can be reached for ramp rates $1/\tau <2\pi \hspace{0.5em}$s${}^{-1}$, the squeezing becomes stronger than the classical limit ${\xi }_3\simeq 6\times {10}^{-3}$.

Figure 13

Trapping potential $V_{\mathrm{ext}}$ realised in the experiment [18] for two different barrier heights before and after an adiabatically conducted ramp-up.

Figure 14

Time evolution of the parameters $U(t)$ and $J(t)$ of the Bose-Hubbard Hamiltonian as realized in the experiment [18]. This evolution drives the system from the Rabi to the Fock regimes.

Figure 15

Spectrum of the Bose-Hubbard Hamiltonian in different regimes. The energies are given in units of $U/s$, with $s$ as a scale factor given in the legend. All energies are shifted such that $E_0=0$.

Figure 16

Distribution of the relative particle number $n=(n_1-n_2)/2$ in three different eigenstates of the Hamiltonian in the Josephson regime, $U/J=1$, for $N=100$ particles. For the states with odd $i$, the coefficients are odd under $n\to -n$, and vice versa for the even-$i$ states.