Quantum control of spin-nematic squeezing in a dipolar spin-1 condensate

Versatile controllability of interactions and magnetic field in ultracold atomic gases ha now reached an era where spin mixing dynamics and spin-nematic squeezing can be studied. Recent experiments have realized spin-nematic squeezed vacuum and dynamic stabilization following a quench through a quantum phase transition. Here we propose a scheme for storage of maximal spin-nematic squeezing, with its squeezing angle maintained in a fixed direction, in a dipolar spin-1 condensate by applying a microwave pulse at a time that maximal squeezing occurs. The dynamic stabilization of the system is achieved by manipulating the external periodic microwave pulses. The stability diagram for the range of pulse periods and phase shifts that stabilize the dynamics is numerical simulated and agrees with a stability analysis. Moreover, the stability range coincides well with the spin-nematic vacuum squeezed region which indicates that the spin-nematic squeezed vacuum will never disappear as long as the spin dynamics are stabilized.

In this paper, we propose a scheme for storage of the maximal spin-nematic squeezing in a dipolar spinor condensate. We consider a system of dipolar spin-1 BEC with an initial state of all atoms in the state of m f = 0. The free dynamical process gives rise to quantum spin mixing and spin-nematic squeezing. By manipulating an external microwave pulse at a time that maximal spin-nematic squeezing occurs, the squeezing is stored for a long time with its squeezing angle maintained in a fixed direction. The dynamic stabilization of the system is demonstrated by applying periodic microwave pulses. The range of pulse periods and phase shifts with which the condensate can be stabilized is numerical calculated and compares well with a linear stability analysis in the mean field approximation. We also show that the existence range of the spin-nematic squeezed vacuum coincides well with the stabilization range, which indicates that the spin-nematic squeezed vacuum will always exist as long as the system is stabilized.

Results
Model. We consider a spin F = 1 condensate with N atoms trapped in an axially symmetric potential. For simplicity, we choose the symmetry axis to be the quantization axis z. The second quantized Hamiltonian of the system with short-range collisions and long-range magnetic dipolar interaction reads 31  where ψ α r ( ) is the atomic field annihilation operator associated with atom in spin state α α | = = 〉 = ± f m 1, ( 0, 1) f , F is the angular momentum operator and e = (r − r′)/|r − r′| is the unit vector. The mass of the atom is given by M and the trapping potential V ext (r) is assumed to be spin independent. Collisional interaction parameters for the spin-independent and spin-exchange are 24,25 , where a f (f = 0, 2) is the s-wave scattering length for spin-1 atoms in the combined symmetric channel of total spin f. The strength of the magnetic dipole-dipole interaction is given by with μ 0 being the vacuum magnetic permeability, μ B the Bohr magneton, and g F the Landé g-factor. For both the 87 Rb and 23 F is the total angular momentum operator, Ŝ z is its z-component, and =ˆ † n a a 0 0 0 . ′ c 2 and ′ c d are the rescaled collisional and dipolar interaction strengths, respectively, which are given by with θ e being the polar angle of (r − r ′ ). The sign of ′ c 2 is determined by the type of atoms: 87 Rb ( ′ < c 0 2 ) and 23 Na ( ′ > c 0 2 ), the sign and the magnitude of dipolar interaction strength ′ c d can be tuned via modifying the trapping geometry (see Methods).
Spin-nematic squeezing. Before discussing the dynamic properties of the system, we want to point out that commutes with all the other terms in the Hamiltonian. If we start with an initial state that is an eigenstate of Ŝ z , the dipolar term ′ĉ S 3 d z 2 has no effect and thus can be neglected. In the following, we consider an initially spin-polarized condensate where all atoms are prepared in the spin-0 state, i.e., ψ 0 denotes the usual Fock states. During the spin-mixing dynamical processing, the spin mixing Hamiltonian (2) conserves both the total particle number N and magnetization, the evolution state of the system in vector form is where N k , is so-called pairs basis with N the total particle number and k the number of pairs of atom in the m f = ±1 states. Thus the expected values of 〈S x,y,z (t)〉 equals to zero and then the mean spin vanishes and the spin squeezing parameter is divergent.
Fortunately, spin-1 has other higher order spin moments which could exhibit squeezing. Based on the commutation relationship of the operators Q i,j , we can define {S x , Q yz , Q + } and {S y , Q xz , Q − } as two subspaces of SU(3), where Q + and Q − are defined as Q + = Q zz − Q yy and Q − = Q xx − Q zz , respectively (see Methods x y where θ is the quadrature angle. Consider the evolution state are the ratio between the variance of the quadrature operator to the standard quantum limit of N which reduce to 41 x y x y yz xz ( ) indicates spin-nematic squeezed vacuum. In Fig. 1, we display the dynamics of the spin component m f = 0 (ρ 0 = N 0 /N) and the corresponding spin-nematic squeezing parameter ( ξ 10 log x 10 2 ) for different dipolar interactions. The spinor interaction strength is chosen as a realistic experimental parameter with As the dipolar interaction |c| increases, the speed of spin mixing slows down and the corresponding time of maximal squeezing t m becomes larger. It is due to the fact that the enhancement of dipolar interaction suppresses the spinor interaction. When the inter-spin interaction reduces to 0, there will be no spin mixing and squeezing.
In the recent experiment, the spin-nematic squeezing is measured by using an SU(3) rotation in spin-nematic phase space around the −Q zz axis 41 . The wave function after the rotation is given by which corresponds to an additional phase on different states N k , . The rotation (phase shifts) can be experimentally implemented by using 2π Rabi pluses on the where Δ is the detuning normalized to the on-resonance Rabi rate 41 .
The microwave pulse can also be used to control the dynamics of spin-nematic squeezing. As shown in Fig. 2, a pulse is added at the maximal-squeezing time t m with the phase shift Δθ = −0.98π, we can find that the maximal squeezing can be stored for a long time. In addition, with the help of the pulse, the direction of the squeezing can also be maintained along a fixed axis. Experimentally, it is possible that the parameter c may deviate from the value of c = −0.1. We varied the dipolar interaction parameter c near the value of c = −0.1, and found that the spin-nematic squeezing (ξ ≈ −22 pulse. Here, we emphasize that the maintained squeezing is not a squeezed vacuum; as shown in the inset of Fig. 2(b), the population in spin components m f = ±1 are macroscopically populated.
Dynamic stabilization and spin-nematic squeezed vacuum. Next we consider a spin-nematic squeezed vacuum which is associated with negligible occupation of the squeezed modes, and is analogous to optical two-mode vacuum squeezing [42][43][44][45] . To generate the spin-nematic squeezed vacuum, we shall control the stability of the dynamics which ensure that there is essentially no population transfer ( <1%) from the m f = 0 state. In our scheme, the dynamic stabilization is achieved by preventing the buildup of the correlations using the periodic phase shifts which is similar to that used in spin-1 condensate with quadratic Zeeman energies 23 . The numerical simulation result demonstration dynamic stabilization of the system are shown in Fig. 3. The spin population ρ 0 as a function of t is shown for two different microwave pulse parameters with δθ = −0.5π and −0.2π, which respectively corresponds to a stabilized condition and a unstable condition. For comparison, the unstabilized dynamics showing free evolution spin mixing with δθ = 0 is also displayed. The difference between the three different cases is the size of quadrature phase shift applied per pulse. It means that for a proper size of quadrature phase shift, the dynamic of the system can be stabilized and then measurement of the spin population  displays a map of the stability region versus pulse period and quadrature phase shift. For the shorter pulse periods, the system is stabilized with a wider range of quadrature phase shifts. For long pulse periods, the range of quadrature phase shifts capable of stabilizing the dynamics shrinks. Here we also note that the direction of the shrink only along quadrature phase shifts from 0 to −π.
The nature of the stability can be well understood in the classical spin-nematic phase space. In the mean field framework, the evolution dynamics of S ⊥ and Q ⊥ are given by (see Methods) With the period quadrature phase shift ∆θ e iQ zz , the stabilization condition of the dynamics is given by the inequality 2 , = − | ′ a c c 3 2 and Γ = ab . Such an inequality can be used to mark the boundaries of the analytic stability region. In Fig. 4, the analytical results of the range of the stabilization are plotted as black lines with dots in the plane of quadrature phase shifts and pulse periods. It is clearly seen that the numerical results coincide well with the analytical ones obtained with mean field approximation. Here we emphasize that the result of Eq. (7) is similar with that obtained in spin-1 condensate with external magnetic field 23 .
When the condensate is stabilized with ρ 0 = 1, the squeezing parameter ξ < 10 log 0 x 10 2 dB indicates the condensate exhibits spin-nematic squeezed vacuum. In Fig. 5(a), the evolution of the spin-nematic squeezed vacuum parameter are plotted for two different applied phase shifters. In the unstabilize case with Δθ = −0.3π, the squeezing phenomenon disappears after a certain time. While the stabilized pulse (Δθ = −0.8π) shows the expected periodic evolution of the spin-nematic squeezing and also show a dramatic reduction of the squeezing compared with the unstabilized one after a long time evolution. It can be noted that the system always exhibits spin-nematic vacuum squeezing with the stabilized pulse.
We also explore the range of pulse periods and quadrature phase shifts that provide the exhibition of spin-nematic squeezed vacuum for any time. The numerical results of ξ x 2 after 160 ms are shown in Fig. 5(b) which displays a map of the squeezed vacuum region versus pulse period and quadrature phase shift. For clearly shown in Figure, we set ξ = 2 x 2 dB when ξ > 0 x 2 dB, which denotes no squeezing. The numerical results are in good overall agreement with the stabilization condition, which indicates that the system can always exhibits spin-nematic squeezed vacuum as the spin dynamics is stabilized. We shall point out that the the squeezing region includes unstable pulse with Δθ = 0 and π, since the squeezing parameter ξ x 2 has not enough time to increase larger than 0 in 160ms for the marginally unstable case.

Discussion
In this article,we have investigated the coherent control spin-nematic squeezing and dynamic stabilization in a spin-1 condensate with dipolar interaction by periodically manipulating the phases of the states. By applying a microwave pulses at the time when the maximal spin-nematic squeezing occurs, the maximal squeezing can be stored with its squeezing angle maintained in a fixed axis. The dynamic stabilization of system is also demonstrated by the pulse. The stability diagram for the range of pulse period and phase shifts that stabilize the spin dynamics are numerical simulated and coincide well with a stability analysis in a mean field approximation. We further study the spin-nematic squeezed vacuum of the system and map the squeezing parameter region on the plane of pulse period and quadrature phase shift. The system always exhibits spin-nematic squeezed vacuum as the spin dynamics is stabilized.
Our scheme presented above demonstrate for the storage of spin-nematic squeezing and dynamical stabilization of the spin dynamics are quite robust for a wide range of parameters. Although the stabilization is demonstrated with a condensate in SMA for which the spatial dynamics are factored out, our scheme should be applicable to the control of the coupled spin or spatial dynamics that lead to domain formation in larger condensates. We hope our scheme will be realized in future experiment and also can be used to explore the quantum control of spin dynamics in other spin systems.

Methods
Dipole-dipole interaction. To calculate the parameters ′ c d and ′ c 0,2 , we consider φ(r) to be the single-particle ground state of the harmonic potential, i.e., φ κ π =  . Figure 6 shows the κ dependence of function χ(κ). It is seen that the value of the parameter χ(κ) can be tuned from −1 to 2 by changing the trapping geometry [31][32][33] . For κ < 1, the dipolar interaction is attractive, and it is repulsive for κ > 1. When κ = 1, we can obtain χ(κ) = 0, which indicates that the dipolar interaction disappears.  Spin and nematic operators. According to the definition of the operator Q i,j which are given by  , and x = 2ρ 0 − 1, we obtain We note that S ⊥ , Q ⊥ , and x have spin Poisson brackets and thus define as a spin representation which can be shown as a sphere. Stabilization Condition. The free spin mixing dynamics in a dipolar interaction is described by To discuss the problem of the dynamical stability, we shall adopt the linear stability analysis that has wide applications in various nonlinear systems. First, the infinitesimal variables δS ⊥ and δQ ⊥ are introduced by Since the expectation value of S ⊥ = Q ⊥ = 0 for the dynamical process, we will drop the notation δ of the expansion, and then Eqs (10) and (11) where Γ = ab.