Squeezed spin states, which were first introduced by Kitagawa and Ueda,^[1] are quantum correlated states with reduced fluctuations in one of the collective spin components.^[2,3] Such states have attracted considerable interest because they not only play significant roles in investigating many-body entanglement,^[4,5] but also have important applications in atom interferometers and high-precision atom clocks.^[6,7] In general, there are two methods to produce spin squeezing. One is based on a one-axis twisting Hamiltonian , where q is the nonlinear spin–spin interaction strength and J_x is the collective spin operator in the x direction. When the initial state is prepared as ∣J_z = −j⟩ for q > 0 (∣J_z = j⟩ for q < 0), the maximal squeezing factor for this one-axis twisting Hamiltonian scales as N^−2/3,^[1] where N is the total atomic number and J = N/2. In contrast, for the other two-axis twisting Hamiltonian , the maximal squeezing factor scales as N⁻¹ with the same initial state.^[1] Since the scaling of N⁻¹ approaches the Heisenberg limit, implementing the two-axis twisting Hamiltonian in current experimental setups is very important and necessary.^[8–15] A possible scheme is to transform the one-axis twisting Hamiltonian into the two-axis twisting Hamiltonian by applying pulse sequences or continuous driving in the two-component Bose–Einstein condensates.^[16–19] However, the experimental achievement of such two-axis twisting Hamiltonian is still challenging.

The ultrahigh-finesse cavities interacting with two-level particles (such as atoms, nitrogen vacancy centers, superconducting qubits, etc) are also powerful platforms to produce effective spin squeezing, since the photon of cavity mode can induce nonlinear spin–spin interaction and thus generate the required one-axis twisting Hamiltonian.^[20–30] Recently, multi-mode cavities^[31] have attracted much attention both experimentally and theoretically.^[32–37] On one hand, these setups can be used to explore novel physics, such as the spin-orbit-induced anomalous Hall effect,^[38,39] the crystallization and frustration,^[40,41] the spin glass,^[42–45] and the gapless Nambu–Goldstone-type mode without rotating-wave approximation.^[46] Moreover, two-mode field squeezing^[47] and unconditional preparation of a two-mode squeezed state of effective bosonic modes^[48] have also been achieved by introducing two cavities.

In this paper, inspired by recent experiments of cavity-assisted Raman transitions,^[49,50] we mainly realize a generalized two-axis spin Hamiltonian , where χ is an anisotropic parameter and ω₀ is an effective atomic resonant frequency, when an ensemble of ultracold six-level atoms interacts with two quantized cavity fields and two pairs of Raman lasers. The Hamiltonian achieved has a distinct property that the interaction strength q, the anisotropic parameter χ, and the effective atomic resonant frequency ω₀ can be tuned independently. For reasonable parameters, the well-studied one- and two-axis twisting Hamiltonians are recovered. Numerical results reveal that for the standard two-axis twisting Hamiltonian (χ = −1 and ω₀ = 0), the corresponding maximal squeezing factor scales as N⁻¹, as expected. On the other hand, in the two-axis twisting Hamiltonian H_TAT, spin squeezing is usually reduced when increasing the effective atomic resonant frequency ω₀. Surprisingly, we find that by combining with the anisotropic parameter χ (> −1), this effective atomic resonant frequency ω₀ can enhance spin squeezing greatly. These results are of benefit for achieving the required spin squeezing in experiments.

Figure 1(a) shows our proposal, in which an ensemble of ultracold six-level atoms interacts with two quantized cavity fields and two pairs of Raman lasers. These six levels consist of two stable ground states (|0⟩ and |1⟩) and four excited states (|r₁⟩, |r₁⟩, |s₁⟩, and |s₂⟩) (see Fig. 1(b)). For ⁸⁷Rb atom, the detailed levels can be chosen as |0⟩ → 5S_1/2 |F = 1,m_F = 0⟩, |1⟩ → 5S_1/2 |F = 2,m_F = 0⟩, |r₁⟩ → 5P_1/2 |F = 1,m_F = 0⟩, |s₁⟩ → 5P_3/2 |F = 1,m_F = 0⟩, |r₂⟩ → 5P_1/2 |F = 1,m_F = − 1⟩, and |s₂⟩ → 5P_3/2 |F = 1,m_F = −1⟩. Two independent photon modes, whose creation and annihilation operators are a^† (b^†) and a (b), mediate the |0⟩ ↔ |s₁⟩ and |1⟩ ↔ |r₁⟩ (|0⟩ ↔ |s₂⟩ and |1⟩ ↔ |r₂⟩) transitions (red and blue solid lines) with atom–photon coupling strengths g_s1 (g_s2) and g_r1 (g_r2), respectively. Whereas two pairs of Raman lasers govern the other transitions {|0⟩ ↔ |r₁⟩, |1⟩ ↔ | s₁⟩} and {|0⟩ ↔ | r₂⟩, |1⟩ ↔ | s₂⟩} (red and blue dashed lines) with Rabi frequencies {Ω_r1, Ω_s1} and {Ω_r2, Ω_s2}, respectively. Δ_r1,2 and Δ_s1,2 are the detunings from the excited states.

Fig. 1.

Figure Option ViewDownloadNew Window

Fig. 1. (a) Proposed experimentally-feasible setup that an ensemble of ultracold six-level atoms interacts with two quantized cavity fields and two pairs of Raman lasers. A couple of orthogonally placed cavities Ca and Cb are arranged in a specific plane, and a guided magnetic field B is applied along one of the two cavities (see, for example, Ca) to fix a quantized axis and split the Zeeman sublevels of the atomic ensemble. Both the cavity modes and Raman lasers propagated in Ca (Cb) are assumed to be right-handed circularly polarized (linearly polarized along the guided magnetic field B). (b) Energy-level structure and their transitions induced by two photon modes and two pairs of Raman lasers. For 87Rb atom, the detailed levels can be chosen as |0⟩ → 5S1/2 |F = 1,mF = 0⟩, |1⟩ → 5S1/2 |F = 2,mF = 0⟩, |r1⟩ → 5P1/2 |F = 1,mF = 0⟩, |s1⟩ → 5P3/2 |F = 1,mF = 0⟩, |r2⟩ → 5P1/2 |F = 1,mF = −1⟩, and |s2⟩ → 5P3/2 |F = 1,mF = −1⟩. In such an arrangement, the transitions labeled by the red lines (blue lines) are thus π transitions (σ− transitions).

The total Hamiltonian illustrated in Fig. 1 can be written, under the rotating wave approximation, as

By means of the Hamiltonian (1), we first realize a two-mode Dicke model with independently-tunable parameters. In the interaction picture with respect to the free Hamiltonian

We now introduce two new unitary transformations, U₁ = exp [G₁J_x(a^† − a)] and U₂ = exp [G₂J_y(b^†−b)] with G₁ = λ₁/ω_A and G₂ = λ₂/ω_B, to rewrite the Hamiltonian (5) as

For different parameters, the generalized two-axis spin Hamiltonian (6) can reduce to some well-studied Hamiltonians. For example, when χ > 0, the Hamiltonian (6) becomes a standard Lipkin–Meshkov–Glick model.^[52–58] When ω₀ = 0, the Hamiltonian (6) reduces to a generalized two-axis twisting Hamiltonian . If further setting χ = −1, a standard two-axis twisting Hamiltonian is derived. Finally, when χ = 0, the Hamiltonian (6) turns into a generalized one-axis twisting Hamiltonian ,^[59–63] which reduces to the standard one-axis twisting Hamiltonian for ω₀ = 0. These results imply that the Hamiltonian (6) has an important application in achieving the required spin squeezing.

Notice that when ω_A < 0, the effective spin–spin interaction strength q > 0. Thus, we can use the initial state |J_z = −j⟩ to discuss spin squeezing of the generalized two-axis spin Hamiltonian (6). This initial state |J_z = −j⟩ can be easily prepared in experiments.

In order to investigate spin squeezing, it is very necessary to consider the following time-dependent squeezing factor:^[1]

In the spherical coordinates, n₀ = (sin θ cos φ, sinθ sin φ, cos θ), where θ = arccos (⟨J_z⟩ /| J|), and φ = arccos (⟨J_x⟩/|J| sin θ) for ⟨J_y⟩ > 0 or φ = 2π − arccos (⟨ J_x⟩/| J| sin θ) for ⟨J_y⟩ ≤ 0. Two orthogonal bases are given by n₁ = (−sin φ,cos φ,0) and n₂ = (− cos θ cos φ,cos θ sin φ,−sin θ). Thus, J_n1,2 = J· n_1,2, J_n⊥ = J_n1 cos ϕ + J_n2 sin ϕ, and could be achieved, when ϕ varies from 0 to 2π in the plane that is perpendicular to the mean-spin direction n₀. It should be noticed that in experiments, the maximal squeezing factor

Before proceeding, we check the validity of the Hamiltonian (6), when the initial state is chosen as |J_z = −j⟩. For the generalized two-axis spin Hamiltonian, it is very hard to obtain the analytical result of the spin squeezing factor .^[2] In Fig. 2, we plot the corresponding spin squeezing factors of the Hamiltonians (5) and (6). It can be seen clearly that the results of the Hamiltonian (6) are almost identical to those of the Hamiltonian (5). Therefore, we will apply the Hamiltonian (6) to discuss the experimentally measurable maximal squeezing factor in the rest of this paper.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Numerical plot of the time-dependent spin squeezing factors of both the Hamiltonians (5) and (6), with the same initial states |Jz = −j⟩. In panels (a) and (b), ωA/ω0 = 200, ωB/ω0 = ± 200 [“+” for (a) and “−” for (b)], λ1/ω0 = 2, λ2/ω0 = 1, and N = 15. In panels (c) and (d), ωA/ω0 = 200, ωB/ω0 = ± 200 [“+” for (c) and “−” for (d)], λ1/ω0 = 1, λ2/ω0 = 2, and N = 20.

We first address a simple case without the effective atomic resonant frequency (ω₀ = 0), in which the generalized two-axis spin Hamiltonian (6) reduces to the generalized two-axis twisting Hamiltonian . In Fig. 3, we numerically plot the maximal squeezing factor of the Hamiltonian H_GTAT as a function of the atomic number N for the different anisotropic parameters χ, when the initial state is chosen as |J_z = −j⟩. This figure shows clearly that when χ = 0, the generalized two-axis twisting Hamiltonian H_GTAT becomes the standard one-axis twisting Hamiltonian , whose maximal squeezing factor scales as N^−2/3.^[1] When increasing the anisotropic parameter χ, the maximal squeezing factor decreases, i.e., spin squeezing is enhanced. In particular, when χ = −1, the Hamiltonian H_GTAT turns into the standard two-axis twisting Hamiltonian , whose maximal squeezing factor scales as N⁻¹,^[1] as expected. In addition, the maximal squeezing factor as a function of the anisotropic parameter χ is also plotted in the insert part of Fig. 3. This figure shows that χ = −1 is an optimal point to achieve the maximal squeezing factor of the generalized two-axis twisting Hamiltonian H_GTAT.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. Numerical plot of the maximal squeezing factors as a function of the atomic number N for the different anisotropic parameters χ. Insert: numerical plot of the maximal squeezing factor as a function of the anisotropic parameter χ, when the atomic number is chosen as N = 100. In both subfigures, q = 1 and the initial states are chosen as |Jz = −j⟩.

In the real experiments, the effective atomic resonant frequency ω₀ always exists. In Fig. 4, we numerically plot the maximal squeezing factor of the Hamiltonian as a function of the effective atomic resonant frequency ω₀, when the the initial state is chosen as |J_z = −j⟩. It can be seen from this figure that with the increasing of the effective atomic resonant frequency ω₀ in the standard two-axis twisting Hamiltonian H_TAT, the maximal squeezing factor increases, i.e., spin squeezing is reduced.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. Numerical plot of the maximal squeezing factor as a function of the effective atomic resonant frequency ω0. Insert: numerical plot of the time-dependent squeezing factor , when ω0/q = 0.3. In both subfigures, the initial states, the anisotropic parameter, and the atomic number are chosen as |Jz = −j⟩, χ = −1, and N = 100, respectively.

According to the above discussion, we argue that when increasing the anisotropic parameter χ (from χ = −1) or introducing the effective atomic resonant frequency ω₀ in the standard two-axis twisting Hamiltonian H_TAT, the maximal squeezing factor increases, i.e., spin squeezing is reduced. Surprisingly, when we control these two parameters simultaneously, spin squeezing can be enhanced greatly. To see this clearly, in Fig. 5 we numerically plot the maximal squeezing factors of the generalized two-axis spin Hamiltonian (6), i.e., , as a function of the effective atomic resonant frequency ω₀ for different anisotropic parameters χ, when the initial state is chosen as |J_z = −j⟩. This figure shows that in the case of χ = −0.05 or χ = −0.5, when increasing the effective atomic resonant frequency ω₀, the maximal squeezing factor first decreases greatly and then increases, i.e., spin squeezing is first enhanced greatly and then reduced. For the generalized one-axis twisting model , the maximal squeezing factor has a similar behavior (see green dashed-dotted line in Fig. 5),^[59] but its magnitude cannot arrive at the order of our considered two-axis spin Hamiltonian (6), because these two Hamiltonians have different scalings with respect to the atomic number N. These results are of benefit for achieving the required spin squeezing in experiments.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. Numerical plot of the maximal squeezing factors as a function of the effective atomic resonant frequency ω0 for the different anisotropic parameters χ. The initial state and the atomic number are chosen as |Jz = −j⟩ and N = 100, respectively.

Finally, we address the experimental feasibility by considering ⁸⁷Rb atom, with the six levels depicted in Fig. 1. In this case, the decay rates of the atomic excited states and photons are about γ/2π = 6.0 MHz and κ/2π = 1.3 MHz (the ground states are stable and we do not need to consider the corresponding decays).^[49,50] On the other hand, based on the transitions between the different levels, the atom–photon coupling strengths can reach g_s1/2π = 20 MHz and g_s2/2π = 15 MHz, respectively, both of which are much larger than the photon decay rate κ. As a consequence, the system is dominated by the Hamiltonian dynamics. When Ω_s1/2π = 20 MHz and Δ_s1/2π = 100 MHz, together with the above estimated parameter values, the adiabatic condition for deriving Eq. (A2) is reasonable, and moreover, we have λ₁/2π = 4 MHz and q/2π = 0.8 MHz for ω_A/2π = −20 MHz. This choice of the detuning ω_A also satisfies the dispersive condition, which is a key condition to realize our generalized two-axis spin Hamiltonian (6). For the anisotropic parameter χ, it can be easily controlled by tuning both the detuning ω_B and the Rabi frequency Ω_s2.

When N = 100 with above parameters, the numerical result shows that the shortest time for generating the maximal squeezing factor is about t_m ≃ 25 ns (see, for example, in the insert part of Fig. 4), which is shorter than the photon lifetime κ⁻¹. With the increase of the atomic number N, t_m becomes shorter and shorter, since t_m = 1/N^0.8, as shown in Fig. 6. In the real experiments with N = 10⁴ or 10⁵,^[49,50] t_m = 0.6 ns or 0.1 ns. This indicates that the maximal spin squeezing can be created very quickly in experiments. Notice that for more rigorous consideration of the effect inducing the photon decay κ, we should use the method of quantum master equation. However, here we mainly take the first step to realize the two-axis twisting Hamiltonian and achieve spin squeezing (approaching the Heisenberg limit) in the two cavities.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. Numerical plot of the shortest time tm for generating the maximal squeezing factor as a function of the atomic number N. The initial state, the effective atomic resonant frequency, and the anisotropic parameter are chosen as |Jz = −j⟩, ω0/q = 0.3, and χ = −1, respectively.

In summary, we have proposed an experimentally feasible system, in which an ensemble of ultracold six-level atoms interacts with two quantized cavity fields and two pairs of Raman lasers, to realize a generalized two-axis spin Hamiltonian . We have numerically calculated the experimentally measurable maximal squeezing factor and revealed that when ω₀ = 0 and χ = −1, the maximal squeezing factor scales as N⁻¹. More importantly, we have found that by combining with the anisotropic parameter χ (> −1), the effective atomic resonant frequency ω₀ can enhance spin squeezing greatly. Our results are of benefit for achieving the required spin squeezing in experiments, and have a potential application in quantum information and quantum metrology.