Efficient Generation of Many-Body Entangled States by Multilevel Oscillations

We generate high-fidelity massively entangled states in an antiferromagnetic spin-1 Bose-Einstein condensate (BEC) by utilizing multilevel oscillations. Combining the multilevel oscillations with additional adiabatic drives, we greatly shorten the necessary evolution time and relax the requirement on the control accuracy of quadratic Zeeman splitting, from microgauss to milligauss, for a ${}^{23}\mathrm{Na}$ spinor BEC. The achieved high fidelities over 96% show that two kinds of massively entangled states, the many-body singlet state and the twin-Fock state, are almost perfectly generated. The generalized spin squeezing parameter drops to a value far below the standard quantum limit even with the presence of atom number fluctuations and stray magnetic fields, illustrating the robustness of our protocol under real experimental conditions. The generated many-body entangled states can be employed to achieve the Heisenberg-limit quantum precision measurement and to attack nonclassical problems in quantum information science.

Figure 1

Schematic of multilevel oscillation for a harmonic oscillator. An initial state (red circle) oscillates from $-x_0$ to $x_0$ in the harmonic potential (black line) and reaches the target state (blue circle). In the eigenenergy basis of the final harmonic potential (blue line), the initial wide probability distribution in many levels shrinks to a Kronecker $\delta$ distribution in a single (ground) level after the multilevel oscillation. The grey dashed lines denote the corresponding energy levels.

Figure 2

(a) Phase diagram of an antiferromagnetic spin-1 BEC. Three critical quantum ground states (i.e., polar, singlet, and twin-Fock states) are illustrated by their atom distribution in three spin components. The blue solid line (red dashed line) denotes the energy gap $\mathrm{\Delta }E$ for a total atom number $N=1000$ ($N=100$). (b) AMO and AMOA processes in the instantaneous eigenenergy basis with A representing adiabatic and MO the multilevel oscillation. For the AMO process, starting from an initial polar state ($P$), the condensate evolves adiabatically at first, and after three multilevel oscillations, the system reaches the singlet state ($S$). For the AMOA process, three more multilevel oscillations are followed by another adiabatic evolution in order to generate the twin-Fock state. The blue-and-cyan ribbons show the fidelities of the state on the instantaneous low-energy eigenstates for $N=1000$. The color scales show the fidelity. The red dot-dashed line describes $q$ (right axes) as a piecewise function of time. When $t<4.25\mathrm{s}$ ($t>4.57\mathrm{s}$), $q>0$ ($q<0$); $q=0$ if $t\in [4.25,4.57]\mathrm{s}$.

Figure 3

(a) Dynamics of the occupied levels during the three multilevel oscillations for $N=1000$, in the eigenenergy basis for $q=0$. The number of occupied levels decreases as time goes by ($q$ decreases piecewise). The radii of circles scale the fidelity. (b) Evolution of the conversion efficiency $p_c=(N-N_0)/N$ during the generation of the twin-Fock state for $N=1000$. The process is divided into three multilevel oscillations and an additional adiabatic process, separated by three vertical grey dotted lines where the conversion efficiencies reach a local maximum. In both (a) and (b), the red dot-dashed lines represent the quadratic Zeeman splitting $q$ (right axis).

Figure 4

Dynamics of the generalized spin-squeezing parameter ${\xi }^2$ and fidelities of the state on the singlet state during the multilevel oscillation processes: (a) with fluctuations in the bias field and in the initial atom number for an average $\overline{N}=1000$, where the red (blue) solid line represents ${\xi }^2$ for even (odd) atom numbers and the black solid line is for the average, and the red dot-dashed line represents the fidelity for even $N$; (b) with relaxation and dephasing noises, where the red solid (dot-dashed) line is for ${\xi }^2$ (fidelity), and with atom loss and dephasing noises, where the blue solid (dot-dashed) line is for ${\xi }^2$ (fidelity) for $N=100$. The grey dashed lines represent the perfect multilevel oscillations without any noise.