Phase diagram and spin mixing dynamics in spinor condensates with a microwave dressing field

Spinor condensates immersed in a microwave dressing field, which access both negative and positive values of the net quadratic Zeeman effect, have been realized in a recent experiment. In this report, we study the ground state properties of a spinor condensate with a microwave dressing field which enables us to access both negative and positive values of quadratic Zeeman energy. The ground state exhibits three different phases by varying the magnetization and the net quadratic Zeeman energy for both cases of ferromagnetic and antiferromagnetic interactions. We investigate the atomic-number fluctuations of the ground state and show that the hyperfine state displays super-Poissonian and sub-Poissonian distributions in the different phases. We also discuss the dynamical properties and show that the separatrix has a remarkable relation to the magnetization.

After successful experimental realizations of spinor condensates in 23 Na and 87 Rb atoms [1][2][3] , experimental and theoretical studies on spinor condensates have emerged as one of the most fast moving frontiers in degenerate quantum gases. An optical trap enables simultaneous and equal confinement of atoms in different hyperfine states. In comparison to scalar condensates, spinor condensates can exhibit richer quantum phenomena due to their internal spin degrees of freedom. In addition to Feshbach resonances and optical latices which tune the interatomic interactions, spinor condensates systems are offering an unprecedented degree of control over many other parameters, such as the spin, the temperature and the dimensionality of the system 4,5 .
In this report, we study the ground state properties of a spin-1 condensate with a microwave dressing field where both negative and positive values of δ net can be accessed. The phase diagrams of the ground state for both ferromagnetic 87 Rb and antiferromagnetic 23 Na interactions are demonstrated. Based on the fractional population ρ 0 of the hyperfine state m F = 0, we define three distinct phases with ρ 0 = 0, 0 < ρ 0 < 1, and ρ 0 = 1 representing the antiferromagnetic (AFM) phase, the broken axis symmetry phase (BA) and the longitudinal polar phase, respectively. By tuning the parameters of δ net and the magnetization m, quantum phase transitions will occur. We also investigate the atom number fluctuations of the ground state with different values of δ net and m. It is found that the hyperfine state of m F = 0 exhibits super-Poissonian distributions in the AFM phase for both ferromagnetic and antiferromagnetic interactions, while sub-Poissonian distributions are presented in the BA phase. At the boundary of BA and AFM phases, the atom number fluctuation attains its maximum value. In the polar phase, the Mandel Q parameter equals to − 1, which means no atom number fluctuation. For the dynamical properties, we find a remarkably different relationship between the total magnetization |m| and a separatrix in the phase space. Finally, we show that our results agree with the prediction that spin dynamics in spin-1 condensates substantially depend on the sign of the ratio between the net quadratic Zeeman effect and the spin-dependent interaction 31 .

Results
Model and its requantization. In the second quantized form, the system of an interaction atomic spin-1 Bose gas in the presence of an external field is given by 41,42 Here E HFS is the hyperfine splitting and B is the magnetic field. Since the total atom number and the total magnetization m are conserved in the dynamical process, the linear term does not affect the system dynamics. By using a microwave dressing field, the net quadratic Zeeman shifts δ net can be expressed as δ net = δ B + δ M with δ M induced by the microwave dressing field 12,40 . Microwave dressing field is an off-resonant microwave field. In the recent experiment, the microwave dressing field was realized by plus a polarized microwave field with πor σ ± -polarizations in a sodium condensate ( 23 Na or 87 Rb atoms) with a constant and homogeneous magnetic field B. Far from any hyperfine resonance, a level shift δ | E m F for the state = , F m 1 F can be found 37,38 where k = 0 or ± 1 correspond to the πor σ ± -polarized microwave pulses respectively, and Δ is the detuning of the microwave pulses from the transition between the states = , . The quadratic Zeeman shift induced by a microwave dressing field then reads as , which can be tuned from − ∞ to + ∞ 37,38 . For a given k, the allowed Rabi flopping is between the states = , F m For both 87 Rb and 23 Na atoms, the spin-dependent interaction is much weaker than the density-dependent interaction. As a consequence, one can adopt the single mode approximation (SMA) 23,43,44 . The effective energy of system can be written as where ρ 0 is the fractional population of the hyperfine state m F = 0, m = ρ 1 − ρ −1 is the magnetization, and c is the renormalized spin-dependent interaction. The cases c > 0 and c < 0 correspond to antiferromagnetic ( 23 Na) and ferromagnetic ( 87 Rb) spinor condensates, respectively. θ = θ 1 + θ −1 − 2θ 0 is the relative phase among the three different hyperfine states. The corresponding time evolutions of ρ 0 and θ are governed by the following differential equation 22 Scientific RepoRts | 5:14464 | DOi: 10.1038/srep14464 Since ρ 0 and θ can be interpreted as the effective conjugate variables, the above equations can also be derived from the canonical equation of Hamiltonian dynamics Ground state properties. By minimizing the energy, ρ 0 in the MF ground state of the system can be obtained. If where the sign ± correspond to the cases of ferromagnetic and antiferromagnetic interactions, respectively. As shown in Fig. 1, the fractional population ρ 0 in the MF ground state as a function of |m| and δ net is plotted for both antiferromagnetic ( 23 Na) and ferromagnetic ( 87 Rb) spinor condensates. It can be seen from Fig. 1(a) that, for the case of ferromagnetic interactions, the value of ρ 0 may start to become nonzero at δ net = − 2|c| and grow to its maximum as δ net is increased. Particularly, ρ 0 = 1 at δ net ≥ 2|c| when m = 0. For the case of antiferromagnetic interactions shown in Fig. 1(b), a sharp jump for the value of ρ 0 near δ net = 0 takes place. When δ net < 0, ρ 0 = 0 for all values of |m|. When δ net > 0, there exists a region of values of |m| in which ρ 0 > 0 and especially ρ 0 = 1 for m = 0. According to the behavior of ρ 0 in the MF ground state, the phase diagrams are plotted in Fig. 2 for both cases of ferromagnetic and antiferromagnetic interactions. There are three different phases in the MF ground state, i.e., the longitudinal polar phase (ρ 0 = 1), the AMF phase (ρ 0 = 0), and the BA phase (0 < ρ 0 < 1). When m = 0, the order parameter of the AMF phase is ρ = , ρ 0 = 0 and that of the polar state is ρ 0 = 1. In the Spherical-harmonic representation, the former state is obtained by rotating the latter about the x axis by π/2, and therefore, these two states are equivalent and degenerate 4,5 . In the BA phase, the ground state features spontaneous breaking of axisymmetry or spontaneous breaking of the SO(2) symmetry 45 . In the recent experiment, the phase diagram for the antiferromagnetic interaction ( Fig. 1(a)) has been realized via adiabatically tuning the microwave field 40 .
To study the ground state atom number fluctuations, the effective Hamiltonian will be requantized by treating ρ 0 and θ as operators which satisfy the following commutation relation 21 As a result, θ can be represented as ρ ( / )∂/∂î N 2 0 in the ρ 0 representation. As long as ρ 0 stays away from the two ends at ρ 0 = 0 and 1, θ is a Hermitian operator. By symmetrizing the nonlinear term containing both θ and ρ 0 with Wigner-representation technique in conventional phase space, we obtain an effective Hamiltonian 21,27 exhibits a Poissonian fluctuation when the number of excitation a † a is counted. The fluctuation level for such a state is also called shot noise.
In Fig. 3, the Mandel Q parameters are plotted as functions of |m| and δ net with ferromagnetic and antiferromagnetic interactions. For the ferromagnetic case, in the AFM phase, we find Q ≃ 1, which means the hyperfine state m F = 0 displays a super-Poissonian distribution. At the boundary between the AFM and BA phases, Q attains its maximum. In the BA phase, we can see Q decreases with increasing the parameter δ net for a fixed magnetization |m|. In this phase, we can see Q < 0 except near the boundary between the two different phases, which indicates the ground state exhibits a sub-Poissonian distribution. In the BA phase, we can also find the value of Q parameter shows a sudden transition from Q > − 1 to Q = − 1. It is due to the fact that when the quadratic Zeeman shift δ net is large enough, ρ 0 keeps constant as ρ 0 = 1 − |m|. In such a situation, there is no atomic-number fluctuations and Q = − 1 indicating the polar phase. For the antiferromagnetic interactions, the behavior of Q is similar to that in the ferromagnetic interactions.
Dynamical properties. Because of the spin-exchange interaction, spinor condensates present dynamical responses significantly which are different from that in scalar condensates. In addition to density excitations or waves, the coherent spin mixing between different spin components will arise in spinor condensates. As shown in Fig. 4(a,b), the time evolution of ρ 0 is plotted as a function of time t with different δ net for antiferromagnetic interactions. We can see ρ 0 exhibits a periodical oscillation. The equal energy contours of Eq. (3) are also plotted in Fig. 4(c,d) for m = 0 with the initial condition θ = 0 and ρ 0 = 0.5. At any δ net , we can define a separatrix, i.e., the energy contour E sep , a point on which is called a saddle point and satisfies the equation . This defines the boundary between the two different regions in phase space. When the initial energy of the system E 0 > E sep , the value of θ will be restricted in the dynamical evolution process, while for E 0 < E sep , there will be no bound. With this definition, the regions with an oscillation phase and a running phase can be well judged. In both regions, ρ 0 oscillates with the oscillation period defined as When E 0 = E sep , the oscillation becomes anharmonic and the period T diverges.
For the antiferromagnetic interactions case, E sep = δ net when δ net > 0. We plot the oscillation period T as a function of δ net and |m| in Fig. 5(a-d) for the ferromagnetic and antiferromagnetic interactions cases, respectively with two different initial conditions. For the initial condition with θ = 0 in Fig. 5(a,c), there are two peaks of the oscillation period located in the regions δ net > 0 and δ net < 0, respectively. By contrast, for the initial state with θ = π in Fig. 5(b,d), we can only find one peak of the oscillation period  in the regions δ net /c > 0 and δ net /c < 0 for the ferromagnetic and antiferromagnetic interactions cases, respectively. It can be concluded that the spin oscillation is always harmonic in the other half region of δ net /c without any peak. This observation agrees with the prediction that the spin dynamics in spin-1 spinor condensates substantially depends on the sign of δ net /c 31 .
From Fig. 5 we can also find a remarkably different relationship between the total magnetization m and the separatrix in phase space. With the initial condition of θ = 0, the position of the separatrix moves slightly when the total magnetization is varied in the positive δ net /c region. In the negative δ net /c region, the separatrix quickly disappears when m is away from zero. For the initial condition with θ = π, the position of the separatrix displays a strong dependence on the magnetization |m|. As magnetization |m| increases, the position of the separatrix moves to a larger value of δ net . In fact, the spin dynamics in the negative region for our antiferromagnetic interaction has been realized in a recent experiment 12 , which are similar to those reported with ferromagnetic spinor in magnetic fields where δ net > 0 4,13 . However, the relationship between the separatrix and m for the ferromagnetic system with both positive and negative values of δ net has not been experimentally explored yet. We hope that our results for the ferromagnetic interaction case can be realized in future experiments.

Discussion
In conclusion, we have studied the ground state properties of a spin-1 condensate in a microwave dressing field. Three distinct phases in the MF ground state are demonstrated based on the fractional population of the hyperfine state m F = 0. For the antiferromagnetic interactions case, there is a phase transition between the BA and AFM phases in the positive δ net region. When m = 0, the ground state stays in the polar phase for positive δ net . In the negative δ net region the system always stays in the AFM phase. By contrast, for the ferromagnetic interactions case, the phase transition occurs between the BA and AFM phases in the negative δ net region. The ground state stays in the BA phase in the positive δ net region except for the situation of m = 0 and δ net ≥ 2|c|. The results of the atom number fluctuations show that the m F = 0 state exhibits a super-Poissonian distribution in the AFM phase for both cases of ferromagnetic and antiferromagnetic interactions. In the BA phase, the m F = 0 state displays a sub-Poissonian distribution, while in the polar phase, there is no fluctuation.
Moreover, the dynamical properties for different initial conditions are also studied. With the initial condition of θ = 0, the position of the separatrix is nearly independence of the total magnetization in the positive δ net /c region. In the negative δ net /c region, the separatrix quickly disappears when m is away from zero. For the initial condition of θ = π, there is no separatrix in the negative δ net /c region. In the positive δ net /c region, the position of the separatrix exhibits a strong dependence on the magnetization |m|. Comparing the results for both cases of ferromagnetic and antiferromagnetic interactions, our results convince the prediction that spin mixing dynamics in spin-1 condensate strongly depends on the sign of δ net /c.

Methods
The static properties for the spin-1 condensate can be studied by numerically solving the eigenquation  φ ρ φ ρ ( ) = ( ) E n n n eff 0 0 , where n = 0, 1, 2,... labels the states with increasing eigenenergy. According to the eigenfunction φ n (ρ 0 ), we can obtain the atom number distributions for different stationary states. In the numerical approach, the term θ cos is implemented as ( + )/ θ θ −ê e 2 i i , i.e., as a superposition of the left-and right-shift operators in the ρ 0 representation. Then the effective Hamiltonian  eff can be expressed as a symmetric tridiagonal matrix and calculated by the numerical diagonalization. Since the spin-exchange collisions couple the states with definite parities, the space of even particle number for spin populations in the hyperfine state m F = 0 is decoupled from the odd one. Thus, the Hilbert space of the system can be divided into two subspaces. Without loss of generality, in our calculations, we have restricted ourselves to the subspace of even-particle number, in terms of which the atom number is always even in the hyperfine state m F = 0.