Efficient generation of many-body entangled states by multilevel oscillations

TL;DR

This paper presents a method to efficiently generate high-fidelity many-body entangled states in a spinor Bose-Einstein condensate using multilevel oscillations combined with adiabatic drives, achieving robustness and high fidelity.

Contribution

The authors introduce a novel protocol combining multilevel oscillations and adiabatic drives to produce entangled states with high fidelity and robustness in BECs, reducing control precision requirements.

Findings

Achieved over 96% fidelity in generating entangled states.

Successfully produced many-body singlet and twin-Fock states.

Demonstrated robustness against atom number fluctuations and magnetic field stray.

Abstract

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 micro-Gauss to milli-Gauss, for a $^{23}$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.

Equations

$$H_o$$

$$H_e=c_2^{\prime }\frac{{\mathbf{L}}^2}{N}-q{a}_0^{†}{a}_0.$$

$${\xi }^2=\frac{1}{SN}\sum _{\alpha =x,y,z}(\Delta L_{\alpha }{)}^2,$$

$$H_e=c_2^{\prime }\frac{{\mathbf{L}}^2}{N}-q{a}_0^{†}{a}_0-p{L}_z-h{L}_x,$$

$$\dot{\rho }$$

Full text

Massive entanglement is of great importance for applications in quantum computing (e.g., logical qubit design utilizing decoherence-free subspace) Haffner et al. (2008); Horodecki et al. (2009); Lidar et al. (1998); West et al. (2010), quantum information processing Sorensen et al. (2001); Cabello (2002); Prevedel et al. (2007); Pezzé and Smerzi (2009), and quantum metrology beyond the standard quantum limit Wineland et al. (1992, 1994); Gross et al. (2010); Riedel et al. (2010); Wu and You (2016); Feldmann et al. (2018); Pezze et al. (2017); Pezzè et al. (2018). For these applications, it is desirable to involve as many particles as possible into entangled states. Two well-known massively entangled states are many-body singlet state and twin-Fock state. For the many-body singlet state, in which a large number of nonzero spins consist of a “giant” zero total spin, it has attracted a great amount of attention to enhance the sensitivity of a gradient magnetometer Urizar-Lanz et al. (2013) and to realize robust logical qubits in decoherence-free subspace Lidar et al. (1998); West et al. (2010); Prevedel et al. (2007). For the twin-Fock state, with half of particles each in two orthogonal modes, it is often employed to improve the precision of a quantum magnetometer to the Heisenberg limit Zhang and Duan (2013); Luo et al. (2017); Lücke et al. (2011); Kruse et al. (2016).

However, these entangled states are typically very fragile. To generate these states in current experiments, the main challenge comes from the extremely fine control of the experimental conditions and deep suppression of the environmental noises Luo et al. (2017); Sun et al. (2017); Koashi and Ueda (2000); Ho and Yip (2000). For a 23Na antiferromagnetic spinor condensate, both the bias field and the stray magnetic fields in a laboratory must be below micro-Gauss in order to observe its ground state for $N\sim 1,000$ Koashi and Ueda (2000); Ho and Yip (2000); Mueller et al. (2006); Jiang et al. (2014). As mentioned in previous papers, the antiferromagnetic spin-1 BEC exhibits two quantum phases Mueller et al. (2006); Liu et al. (2009); Jiang et al. (2014); Jacob et al. (2012); Dag et al. (2018); Bookjans et al. (2011); Vinit and Raman (2017); Frapolli et al. (2017). Ideally, by adiabatically tuning the quadratic Zeeman splitting from positive infinity through zero to negative infinity Zhao et al. (2014); Gerbier et al. (2006); Leslie et al. (2009), one can respectively generate the many-body singlet state and the twin-Fock state by passing through critical point of quantum phase transition. The adiabaticity usually requires a finite and moderate energy gap between the ground and the first excited states. However, such a requirement is impossible to meet in the antiferromagnetic spin-1 BEC, because the gap reduces inversely proportional to the number of atoms $N$, $\Delta E\sim 1/N$ Bookjans et al. (2011); Jacob et al. (2012); Dag et al. (2018); Sarlo et al. (2013); Sala et al. (2016); Zhao et al. (2018), which drops faster than that in a ferromagnetic spin-1 BEC with $\Delta E\sim 1/N^{1/3}$ Zhang and Duan (2013); Hoang et al. (2016); Luo et al. (2017); Zou et al. (2018); Xue et al. (2018). Indeed, given a 23Na BEC with $N=1,000$ atoms and a typical density of $10^{14}$ cm*-3* ($c^{\prime}_{2}\approx$ 25 Hz), the adiabatic evolution time to reach the ground state must be much larger than $N^{3}/(108c^{\prime}_{2})\sim 10^{5}$ seconds by a crude estimation, which is many orders of magnitude larger than the condensate lifetime of $\sim 100$ seconds Koashi and Ueda (2000); Sala et al. (2016). For almost two decades since the prediction of the many-body singlet state by Law et al in 1998 Law et al. (1998), a practical and experimentally feasible method has been longed to generate this highly entangled state in an antiferromagnetic spinor BEC Koashi and Ueda (2000); Ho and Yip (2000); Kawaguchi and Ueda (2012); Stamper-Kurn and Ueda (2013); Sala et al. (2016); Sun et al. (2017); Zhao et al. (2018).

In this Letter, we theoretically achieve the generation of massively entangled states, the singlet and twin-Fock states, in an antiferromagnetic 23Na spin-1 condensate by employing a rapid, efficient and robust method. This method accelerates the dynamics and relaxes the requirement on the control accuracy of quadratic Zeeman splitting by partially replacing the adiabatic evolution near the quantum critical point of the phase transition with multilevel oscillations Pu et al. (1999); Zhang et al. (2005); Chang et al. (2007); Li et al. (2015); Shore (2011); Claudon et al. (2004, 2008). We term the method as adiabatic and multilevel-oscillation (AMO) process for the generation of the singlet states, and as AMO and adiabatic (AMOA) process for the generation of the twin-Fock states.

The main advantage of the multilevel oscillation over an adiabatic process can be in principle illustrated by a harmonic oscillator as shown in Fig. 1. Consider an oscillator with a mass $M$ in an extra linear potential Scully and Zubairy (1997)

[TABLE]

where $P$ is the momentum, $x$ the position, $\omega$ the trapping angular frequency, and $V(x)=Fx$ with $F$ an additional force applied on the oscillator. To reach the desired target state, one may employ an adiabatic process by slowly tilting the linear potential from $F_{0}x$ to $-F_{0}x$, or by a multilevel oscillation process by setting $F=0$ for a half period and then setting $F=-F_{0}$, as illustrated in Fig. 1.

It is easy to calculate the required adiabatic evolution time $T_{A}\gg 10\sqrt{2}/\omega$ if we set $x_{0}=10\sqrt{\hbar/(M\omega)}$ and the multilevel oscillation time $T_{MO}=\pi/\omega$. Clearly, the multilevel oscillation time is much shorter than the adiabatic one when the oscillator is transferred from $x=-x_{0}$ to $x_{0}$, thus the process is greatly accelerated. In fact, the multilevel oscillation process is a generalized “Rabi” oscillation for a half period in a multilevel system not (a).

For an antiferromagnetic 23Na spin-1 condensate, the effective Hamiltonian under the single spatial mode approximation, which is valid up to $10,000$ atoms, is ($\hbar=1$) Stamper-Kurn et al. (1998); Stenger et al. (1998); Ho (1998); Law et al. (1998); Ohmi and Machida (1998); Pu et al. (1999); Yi et al. (2002); Zhang and Duan (2013); Luo et al. (2017); Zou et al. (2018); not (a)

[TABLE]

The first term describes the spin-exchange collision, where we set $c^{\prime}_{2}=25$ Hz for a typical condensate density, and $\bm{L}\equiv\sum_{mn}{a}_{m}^{{\dagger}}\bm{f}_{mn}{a}_{n}$ with $\bm{f}_{mn}$ the spin-1 matrices and $a_{m}(a_{m}^{\dagger})$ the annihilation (creation) operator in spin component $m$. The second term represents the magnetic energy with $q$ the quadratic Zeeman splitting of a single atom. Depending on $q/c^{\prime}_{2}$, the system exhibits two phases, resulting from the competition between the quadratic Zeeman term and the spin-exchange collision. Near the critical point where $q$ is small, the energy gap can be calculated perturbatively $\Delta E/c^{\prime}_{2}\simeq 6/N-0.1907\times N\times q+0.0253\times N^{3}\times q^{2}$ Sarlo et al. (2013). The minimal gap occurs at the critical point $q_{c}/c^{\prime}_{2}=3.7688/N^{2}$, very close to zero if $N$ is large, as shown in Fig. 2(a).

The spin-1 BEC and the harmonic oscillator share the same chain-form Schrödinger equation except for different coefficients, as derived in the Supplemental Materail not (a); Shore (2011); Sarlo et al. (2013). However, the effective potential for the spin-1 BEC is anharmonic so that a single large-amplitude oscillation may take an infinitely long time Zhang et al. (2005); Chang et al. (2007); Li et al. (2015); not (a). Instead, the total evolution time may be shorter if we stepwise change the quadratic Zeeman splitting $q$ so that the system evolves through many local harmonic oscillations.

Following the above strategy, we successfully generate with a high fidelity the many-body singlet state at $q=0$ and the twin-Fock state as $q\rightarrow-\infty$ by employing the AMO and AMOA processes respectively. The initial state is a polar state of a 23Na condensate, where all atoms are in the $m=0$ spin component of the $S=1$ ground hyperfine manifold. This polar state is easily accessible in experiment by setting a large bias magnetic field and optically pumping away the atoms in $m=\pm 1$ spin components Jiang et al. (2016); Luo et al. (2017). For a large but finite $q_{0}=277$ Hz, the initially prepared polar state overlaps with the ground state with a high fidelity about $1-(c^{\prime}_{2}/q_{0})^{2}/2$, which is over 99%.

The adiabatic process of the AMO is carried out numerically by slowly reducing $q$ according to $q(t)=q_{0}\times(1-{t}/{T_{0}})^{2},$ where $T_{0}=0.955$ s, and $t$ ends up at $0.9$ s with a final $q_{f}=0.788$ Hz. For convenience in experimental implementation, we linearly sweep the magnetic bias field thus a parabolic function for $q(t)$. In this adiabatic process as shown in Fig. 2(b), we calculate the adiabatic parameter $\beta=|\partial\langle e|qa^{\dagger}_{0}a_{0}|g\rangle/\partial t|/\Delta E^{2}$, with the instantaneous ground state $|g\rangle$ and the first excited state $|e\rangle$ of the Hamiltonian in Eq. (2). We find $\beta\leq 0.054$ during the whole adiabatic process, thus the adiabatic condition is satisfactorily fulfilled since $\beta\ll 1$.

For the three multilevel oscillations as shown in Fig. 2(b) to generate the singlet state, we observe significant excitations in the instantaneous eigenenergy basis, indicating these multilevel oscillations are diabatic. To better understand this process, we redraw the probability distribution in the eigenenergy basis $|l\rangle$ for $q=0$ in Fig. 3(a). The optimized values of $q(t)$ and the corresponding evolution times are also listed not (a). Briefly, any state $|\psi\rangle$ is expanded as $\sum_{l=0}^{N}\beta_{l}|l\rangle$, and we define an eigenenergy level as occupied if $|\beta_{l}|^{2}>0.1\%$. For a state at a given time, we calculate $|\beta_{l}|^{2}>0.1\%$ and count the number of occupied levels $K$. The goal of stepwise multilevel oscillations is to reduce $K$ to 1, i.e., to the singlet state. In each multilevel oscillation, for a given constant $q$, we evolve the system and monitor $K$ till $K$ reaches its first local minimum $K(q)$; then we sweep $q$ to further minimize $K(q)$ in order to find the optimal range of $q$, as detailed in the SM not (a).

As shown in Fig. 3(a), $K$ shrinks from 15 to 4 during the first multilevel oscillation. The number further shrinks to 2 and 1, respectively, during the second and the third multilevel oscillations. Eventually, the fidelity of the final state (with respect to the singlet state) is over 99%. We note that the required smallest magnetic field is about 0.8 mG, corresponding to $q=1.875\times 10^{-4}$ Hz. This field strength is easily accessible in experiments and about three orders of magnitude stronger than previous estimations of microGauss Koashi and Ueda (2000); Ho and Yip (2000). Remarkably, the total evolution time is only 4.25 s, at least five orders of magnitude shorter than a full adiabatic process Sala et al. (2016). Here we show only one set of $q(t)$, while there are many other sets resulting in fast generating the singlet state with similar or even higher fidelity not (a).

After generating the singlet state, we employ a reversed procedure but with negative $q$ to produce the twin-Fock state, as shown in Figs. 2(b) and 3(b). We notice in Fig. 2(a) that the energy gap is almost symmetric about $q=0$, reminding us that the twin-Fock state may be reached by simply reverse the AMO process with only sign change of $q(t)$. This whole process is the AMOA process. Indeed, the evolved final state overlaps with the twin-Fock state with a fidelity higher than 96%, indicating the success of the AMOA method. A so high efficiency contrasts sharply to a direct Landau-Zener transition by linearly sweeping $q$ from $q_{0}$ to $-q_{0}$ in the same time period 8.63 s, where the fidelity of the twin-Fock state is almost zero Zener (1998); Wittig (2005).

As elegant as the above AMO and AMOA processes to efficiently generate the many-body singlet state and the twin-Fock state, a practical final state is never a pure one in a real experiment. To evaluate the robustness of the AMO process under realistic experimental conditions, we need to include the effects of the stray magnetic fields (both dephasing and relaxation effects), the atom number shot noise, and the atom loss during the evolution. Although the control errors in $q$ and timing are non-negligible noise source, it is easy to prove that they are equivalent to the dephasing effect. Furthermore, in a real experiment, it is almost impossible to measure the quantum state fidelity, thus we choose the generalized spin-squeezing parameter to monitor the AMO process. In addition, this parameter can also estimate the entanglement degree of the evolved quantum state. The parameter is defined as

[TABLE]

where $(\Delta L_{\alpha})^{2}=\langle L_{\alpha}^{2}\rangle-\langle L_{\alpha}\rangle^{2}$. A spin state is squeezed if $\xi^{2}<1$, compared to a coherent spin state with $\xi^{2}=1$ which sets the standard quantum limit. For the singlet state, $\xi^{2}=0$.

First, we consider dephasing effect of stray magnetic fields and atom shot noise effect on the AMO process. The dephasing strength is set as uniformly distributed random numbers $\delta B_{z}\in[-0.1,0.1]$ mG and $q$ changes according to $q=q_{{}_{MW}}+(B_{z}+\delta B_{z})^{2}\times 277$ Hz/G2 with $q_{{}_{MW}}$ denoting the level shift induced by a driving microwave field Zhao et al. (2014); Gerbier et al. (2006); Leslie et al. (2009). We assume that the initially prepared atom number fluctuation of the condensate is uniformly distributed in the range $[N-\sqrt{N},N+\sqrt{N}]$. The numerical simulation results are presented by the black solid line in Fig. 4(a). Clearly, the generalized squeezing parameter monotonically decreases to a lowest value close to $2/N$, after the three multilevel oscillations. The deviation of the minimal $\xi^{2}$ from the singlet state’s value of zero is due to the odd atom numbers in the condensate, whose lowest value $\xi^{2}=2/N$ ($N$ is an odd integer) for its ground state $|l=1,m=0\rangle$. Allowed to distinguish the odd and even number of atoms by postselection not (b), we find the even number condensate continues decreasing to a value much smaller than $1/N$ and very close to the ideal case (grey dashed line), indicating the formation of the many-body singlet state with a very high fidelity above 99% and the robustness of the AMO process.

Second, we consider the relaxation and the dephasing effects of stray magnetic fields on the AMO process. Without loss of generality, we consider the external transversal stray magnetic field is just along the $x$-axis. The effective Hamiltonian becomes

[TABLE]

where $p=-\gamma(B_{z}+\delta B_{z})$ with a moderate bias $B_{z}=0.85$ G and the gyromagnetic ratio $\gamma=-0.7$ MHz/G is the linear Zeeman splitting, and $h=-\gamma\delta B_{x}$ is for the transversal magnetic field. We assume that $\delta B_{x}$ is also uniformly distributed in $[-0.1,0.1]$ mG. We are limited by the computational power to $N=100$, due to the explosion of the Hilbert space introduced by the $L_{x}$. The numerical results are shown in Fig. 4(b). We find that the dynamics of the parameter $\xi^{2}$ (with negligible error bars) overlaps with the ideal one, demonstrating the stay magnetic fields within $0.1$ mG rarely affect the multilevel oscillations. In fact, the final fidelity to the singlet state is still higher than $99\%$.

Finally, we take the atom loss and dephasing effects into consideration. The dynamics must be depicted by the following master equation,

[TABLE]

where we take $\Gamma=0.005$ s*-1* and $H_{e}$ is the Hamiltonian in Eq. (2). In the giant Hilbert space spanned by $|N,l,m\rangle$, we carry out numerical simulations for an “initial” atom number $N(t=0.9\;\text{s})=100$, focusing on the multilevel oscillation process. As shown in Fig. 4(b), the generalized spin squeezing parameter $\xi^{2}$ (with negligible error bars) also reaches $1/N$ but higher than the ideal case. Here we note that the final fidelity to the singlet state drops down to $70\%$ due to the atom loss, but it can be easily remedied by a postselection procedure and the fidelity is improved to a value higher than 99% not (c).

In conclusion, we almost perfectly generate the long-sought massively entangled states, both for the many-body singlet and twin-Fock states in an antiferromagnetic 23Na spin-1 condensate with the AMO and AMOA processes. The numerical simulations show that the generation efficiencies of both states are over $96\%$ in a few seconds and in moderate magnetic fields (from milliGauss to Gauss). Under realistic experimental conditions, the AMO process is robust against the dephasing and relaxation noises of stray magnetic fields, the atom shot noise, and the atom loss. It is worthy to explore in the future the potential of the multilevel oscillations, to replace the adiabatic evolution, near a quantum critical point in many physical systems, e.g., ferromagnetic 87Rb spin-1 condensates, two-level quantum systems, and adiabatic spin quantum computers Luo et al. (2017); Peng et al. (2008). The generated many-body spin singlet state provides a stepping stone to reach the Heisenberg limit gradient magnetometer Urizar-Lanz et al. (2013) and the twin-Fock state can be directly utilized to measure the external magnetic field beyond the standard quantum limit Zhang and Duan (2013); Luo et al. (2017).