1. Introduction
Spin squeezed state [1–3] has attracted considerable attention, both theoretically and experimentally due to the role in the fundamental study of particle correlation and many-particle entanglement [4–6]. In the pioneering work [1], a scheme was proposed for the dynamical generation of spin-squeezed quantum states in an ensemble of spin-1/2 particles via nonlinear interaction between particles. Such a scheme is widely modeled as the so-called one-axis twisting which was implemented experimentally by utilizing interatomic interactions in two-component Bose–Einstein condensates (BECs) [7–19] and cavity-assisted setups [20, 21]. Recently, the spin squeezing was generated in a spin-1 BECs, which is induced through spin–exchange interaction [22–36]. For the spin-1 system, it possesses additional degrees of freedom associated with the nematic tensor operator ${\hat{Q}}_{{ij}}={\hat{S}}_{i}{\hat{S}}_{j}+{\hat{S}}_{j}{\hat{S}}_{i}-(4/3){\delta }_{{ij}}$ with δij being the Kronecker delta and {i, j} ∈ {x, y, z}. The Hamiltonian of the system is conveniently written in terms of the spin and nematic-tensor operators that constitute the SU(3) Lie algebra, and the squeezing can be created via spin-nematic correlation [37–43]. However, the previous works mainly focused on the generation of the squeezed state itself. In practice, it is desirable to maintain and even enhance the squeezing due to its application to improve the precision of measurements in precise devices such as atomic clocks [44, 45], interferometers [46], magnetometers [47], etc.
Recently, nonlinear three-mode interferometers were realized in spinor BECs via spin-mixing dynamics (SMD) [48–55]. The SMD consists of binary collisions that create or annihilate pairs of correlated particles and thus the squeezed state is generated within the interferometer. It allows phase measurements reaching a relatively high-precision. However, the previous protocols for the high-precision spin-mixing interferometer suffered one drawback because they did not make use of all the atomic particles within the phase measurement. A recent work proposed an alternative ‘pumped-up’ approach to use all particles to participate in the phase measurement and realized a quantum-enhanced measurement [56]. However, the higher phase sensitivity that can be achieved in spinor interferometer is unknown. How to further improve the phase sensitivity is an important goal in the experimental frontier.
In this work, we propose a scheme for storage and enhancement of the spin-nematic squeezing in a spinor condensate with an external magnetic field. We consider all of the atoms initially prepared in the mf = 0 mode. The SMD generates a spin-nematic squeezed state. We rapidly turn off the external magnetic field when the spin-nematic squeezing parameter attains its minimum. We observe the optimal spin-nematic squeezing can be maintained in a nearly fixed direction along x-axis. We also show that the optimal squeezing can be enhanced when the initial magnetic field is larger than a critical value. For a proper initial magnetic field, optimal squeezing can even be increased by 4.5 dB. We further propose a spin-mixing interferometer in which the squeezed state’s quantum correlations is fully utilized, and thus the phase sensitivity of the interferometer has a great enhancement. Finally, we discuss the effect of the detection noise on the phase sensitivity. We show that the sensitivity of the output with the initial squeezed state generated by our protocol is better than that which is generated by the free evolution.
2. Model
We consider a spinor condensate with an external magnetic field, which has been realized in the experiment with cold atoms in an optical trap [25]. The spin-dependent collisional interaction is typically much smaller than the independent one, and thus the single-mode approximation is expected to be valid which allows the different spin components of the condensate take the same spatial wave function [31]. Under the single-mode approximation, the Hamiltonian for the spinor condensate can be written as [22, 57, 58]
$\begin{eqnarray}\hat{H}=\lambda {\hat{S}}^{2}+\displaystyle \frac{q}{2}{\hat{Q}}_{{zz}},\end{eqnarray}$
where λ is the spin-dependent interaction energy, $q=\tfrac{{\mu }_{B}^{2}{B}^{2}}{4{\rm{\Delta }}{E}_{\mathrm{hf}}}$ is the quadratic Zeeman energy with B being the magnetic field and ΔEhf is the ground state hyperfine energy splitting [25, 59]. ${\hat{S}}^{2}={\hat{S}}_{x}^{2}+{\hat{S}}_{y}^{2}+{\hat{S}}_{z}^{2}$ is the total spin operator and ${\hat{Q}}_{{zz}}=(2/3){\hat{a}}_{1}^{\dagger }{\hat{a}}_{1}-(4/3){\hat{a}}_{0}^{\dagger }{\hat{a}}_{0}+(2/3){a}_{-1}^{\dagger }{\hat{a}}_{-1}$ is an element of spin-1 nematic tensor with ${\hat{a}}_{i}$ being the annihilation operator of the ith spin mode. The nonlinear collisional spin interaction $\lambda {\hat{S}}^{2}$ contains term $2\lambda ({\hat{a}}_{0}^{{\dagger }^{2}}{\hat{a}}_{1}{\hat{a}}_{-1}+{\hat{a}}_{1}^{\dagger }{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{0}^{2})$ and thus the spin mixing is generated in the dynamical evolution. In addition, spin squeezing would be also generated due to such a nonlinear interaction.3. Spin-nematic squeezing
Unlike spin-1/2 atoms, in which the spin squeezing results from spin-spin correlation, the squeezing in the integer-spin systems can be induced by the spin-nematic correlation. The relative spin and nematic operators of the spin-1 BECs are defined as [22, 23]
$\begin{eqnarray}{\hat{S}}_{x}=\displaystyle \frac{1}{\sqrt{2}}({\hat{a}}_{1}^{\dagger }{\hat{a}}_{0}+{\hat{a}}_{0}^{\dagger }{\hat{a}}_{-1}+{\hat{a}}_{0}^{\dagger }{\hat{a}}_{1}+{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{0}),\end{eqnarray}$
$\begin{eqnarray}{\hat{S}}_{y}=\displaystyle \frac{{\rm{i}}}{\sqrt{2}}(-{\hat{a}}_{1}^{\dagger }{\hat{a}}_{0}-{\hat{a}}_{0}^{\dagger }{\hat{a}}_{-1}+{\hat{a}}_{0}^{\dagger }{\hat{a}}_{1}+{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{0}),\end{eqnarray}$
$\begin{eqnarray}{\hat{Q}}_{{yz}}=\displaystyle \frac{{\rm{i}}}{\sqrt{2}}({\hat{a}}_{0}^{\dagger }{\hat{a}}_{-1}-{\hat{a}}_{1}^{\dagger }{\hat{a}}_{0}+{\hat{a}}_{0}^{\dagger }{\hat{a}}_{1}-{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{0}),\end{eqnarray}$
$\begin{eqnarray}{\hat{Q}}_{{xz}}=\displaystyle \frac{1}{\sqrt{2}}({\hat{a}}_{1}^{\dagger }{\hat{a}}_{0}-{\hat{a}}_{0}^{\dagger }{\hat{a}}_{-1}+{\hat{a}}_{0}^{\dagger }{\hat{a}}_{1}-{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{0}),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\hat{Q}}_{{xx}}=-\displaystyle \frac{1}{3}{a}_{1}^{\dagger }{\hat{a}}_{1}-\displaystyle \frac{1}{3}{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{-1}\\ \quad +\,\displaystyle \frac{2}{3}{\hat{a}}_{0}^{\dagger }{\hat{a}}_{0}+{\hat{a}}_{1}^{\dagger }{\hat{a}}_{-1}+{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{1},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\hat{Q}}_{{yy}}=-\displaystyle \frac{1}{3}{a}_{1}^{\dagger }{\hat{a}}_{1}-\displaystyle \frac{1}{3}{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{-1}\\ \quad +\,\displaystyle \frac{2}{3}{\hat{a}}_{0}^{\dagger }{\hat{a}}_{0}-{\hat{a}}_{1}^{\dagger }{\hat{a}}_{-1}-{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{1}.\end{array}\end{eqnarray}$
We notice $\{{\hat{S}}_{x}$, ${\hat{Q}}_{{yz}}$, ${\hat{Q}}_{+}$} and $\{{\hat{S}}_{y}$, ${\hat{Q}}_{{xz}}$, ${\hat{Q}}_{-}\}$ constitute two SU(2) subspaces of SU(3) group with ${\hat{Q}}_{+}={\hat{Q}}_{{zz}}-{\hat{Q}}_{{yy}}$ and ${\hat{Q}}_{-}={\hat{Q}}_{{xx}}-{\hat{Q}}_{{zz}}$. Based on the generalized uncertainty relation ${\rm{\Delta }}\hat{A}{\rm{\Delta }}\hat{B}\geqslant 1/2| [\hat{A},\hat{B}]| $, the squeezing exists in the SU(2) subspace of SU(3) and occurs when the operator pairs possess non-zero expectation values for their commutation relations. Assume all of the atoms initially condensates in the mf = 0 mode, only two of the SU(3) commutators have non-zero expectation values, i.e. $\langle {\hat{Q}}_{\pm }\rangle \ne 0$. Such a case reminds us of the relevant uncertainty relations ${\rm{\Delta }}{\hat{S}}_{x}{\rm{\Delta }}{\hat{Q}}_{{yz}}\geqslant | \langle {\hat{Q}}_{+}\rangle /2| $ and ${\rm{\Delta }}{\hat{S}}_{y}{\rm{\Delta }}{\hat{Q}}_{{xz}}\geqslant | \langle {\hat{Q}}_{-}\rangle /2| $. Thus the spin-nematic squeezing induces the redistribution of quantum noise in the subspaces $\{{\hat{S}}_{x}$, ${\hat{Q}}_{{yz}}$, ${\hat{Q}}_{+}$} and $\{{\hat{S}}_{y}$, ${\hat{Q}}_{{xz}}$, ${\hat{Q}}_{-}\}$. Focus on the first one of the above subspaces, the spin-nematic squeezing parameter is defined as [22, 60] $\begin{eqnarray}{\xi }_{x}^{2}={({\rm{\Delta }}(\cos \theta {\hat{S}}_{x}+\sin \theta {\hat{Q}}_{{yz}}))}^{2}/| \left\langle {\hat{Q}}_{+}/2\right\rangle | ,\end{eqnarray}$
where θ is the quadrature angle and ${\xi }_{x}^{2}$ is obtained by minimizing the above expression over the angle θ. The parameter ${\xi }_{x}^{2}\lt 1$ indicates spin-nematic squeezing.We consider sodium atoms, which have positive λ, and assume the system with initial atoms are condensated in the mode mf = 0. Because the Hamiltonian conserves the total number of atoms $\hat{N}={\hat{a}}_{1}^{\dagger }{\hat{a}}_{1}+{\hat{a}}_{0}^{\dagger }{\hat{a}}_{0}+{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{-1}$ and the magnetization ${\hat{S}}_{z}={\hat{a}}_{1}^{\dagger }{\hat{a}}_{1}-{\hat{a}}_{-1}^{\dagger }{\hat{a}}_{-1}$, we can denote the initial state as $\left|N,0\right\rangle $, where $\left|N,k\right\rangle $ is so-called pairs basis with N the total particle number and k being the number of pairs of atom in the mf = ±1 modes. The SMD generates the squeezed state as
$\begin{eqnarray}\left|\psi (t)\right\rangle =\sum _{k=0}^{N/2}{a}_{k}(t)\left|N,k\right\rangle .\end{eqnarray}$
In figure 1, we plot the spin-nematic squeezing parameter $10{\mathrm{log}}_{10}({\xi }_{x}^{2})$ as a function of λt with different q. When q = 0, the spin-nematic squeezing rapidly decreases to its minimum and disappears quickly. When q ≠ 0, the squeezing exhibits collapsed oscillations as shown by the dashed and solid curves in figure 1. It is shown the squeezing will not disappear which means the presence of the quadratic Zeeman energy protects the squeezing. In addition, the squeezing can also be enhanced with a proper quadratic Zeeman energy, such as q = 25, as shown in figure 1 with the red solid line. However, the maximal squeezing cannot be maintained for a long time.
Figure 1. Time evolution of the spin-nematic squeezing parameter $10{\mathrm{log}}_{10}({\xi }_{x}^{2})$ for N = 103 with different quadratic Zeeman energy q. |
To store the maximal squeezing, we consider a two-step squeezing protocol. First let the initial state free dynamic evolution under the Hamiltonian $\hat{H}$, as shown in equation (1 ). Next turn off the quadratic Zeeman energy at a maximal-squeezing time, denoted as tm, that maximal spin squeezing occurs, and then allow the squeezed state to evolve under the Hamiltonian ${\hat{H}}_{0}=\lambda {\hat{S}}^{2}$. So the effective evolution operator of our protocol can be written as ${{\rm{e}}}^{-{\rm{i}}{\hat{H}}_{0}{{t}}_{2}}{{\rm{e}}}^{-{\rm{i}}\hat{H}{{t}}_{m}}$. In the following, we set the total squeezing time t = t2 + tm. Experimentally, the SMD can be accurately controlled via microwave dressing [25]. In figure 2(a), we plot the spin-nematic squeezing parameter as a function of λt with our protocol. It is shown the maximal squeezing can be stored for a long time. Moreover, it is stored in a fixed direction, i.e. θ ≈ 0, which means the squeezing direction is along the x axis, as shown in figure 2(b). For such a case, measuring the squeezing becomes easy. It only requires the squeezed state tomography involving one SU(3) rotations. A π/2 rotation about the Sy axis that rotates the fluctuations in Sx into the measurement basis Sz.
Figure 2. Time evolution of (a) the spin-nematic squeezing parameter $10{\mathrm{log}}_{10}({\xi }_{x}^{2})$ and (b) the squeezing angle ${\theta }_{\min }$ for the quadratic zeeman term q = 18.734 with N = 103. Dashed red curves correspond to the free dynamic evolution under the Hamiltonian $\hat{H}$ and solid blue curves denote the case for turning-off the quadratic zeeman term at the time tm which is indicated by the vertical dotted line. |
In addition to achieve the persistent squeezing, it is desirable to enhance the squeezing. Using the above method and choose a proper initial q, the squeezing can be enhanced. In figure 3(a), we plot the spin-nematic squeezing parameter $10{\mathrm{log}}_{10}({\xi }_{x}^{2})$ as a function of time with q = 53. We clearly see the squeezing can be nearly enhanced to −25 dB. In figure 3(b), we plot the optimal spin-nematic squeezing, defined as ${\xi }_{\mathrm{opt}}^{2}={\min }_{t}({\xi }_{x}^{2})$, as a function of q. Comparing the result of ${\xi }_{\mathrm{opt}}^{2}$ which is obtained by our proposed protocol with that through the free evolution, ${\xi }_{\mathrm{opt}}^{2}$ can be nearly increased by 4.5 dB. We notice that there is a sudden transition of ${\xi }_{\mathrm{opt}}^{2}$ at qc = 18.734. Consider a spinor dynamical energy 2Nλ = 15πℏ Hz [58] and the quadratic Zeeman effect q = 2πℏ × 276.5B2 Hz/G2, the magnetic field for the critical point qc is B = 22 mG. Varying q from the critical point qc to 100, we can realize the increasement of the spin-nematic squeezing. Hence, the chosen of the parameter q for the enhancement of optimal squeezing is not unique. Interestingly, when we choose the initial q = qc, the maximal spin-nematic squeezing is stored with a fixed direction, see figure 2. Here, we emphasize that qc is different from the critical point which is a phase transition point shown in [53, 57, 61]. In fact qc has a close relation to the particle number of the system. As shown in the set of figure 3(c), qc is plotted as a function of particle number, it clearly shows that qc increases with N increasing.
Figure 3. (a) Evolution of the spin-nematic squeezing parameter $10{\mathrm{log}}_{10}({\xi }_{x}^{2})$ with q = 53. (b) The optimal spin-nematic squeezing parameter $10{\mathrm{log}}_{10}({\xi }_{\mathrm{opt}}^{2})$ as a function q. The optimal squeezing exhibits a sudden transition at a critical point qc (indicated by the vertical dotted line). (c) The squeezing angle θopt of the optimal squeezing as a function of q. The solid blue line represents the case of turning-off the external magnetic field at tm, while the dashed red line denotes free evolution case. The set shows qc as a function of N. |
The above result for the sudden transition of the optimal squeezing at the critical point can be understood by the squeezing angle of the optimal squeezing, denote as θopt, under the free evolution. In figure 3(c), we plot θopt as a function of q at tm. We find when q ≥ qc, θopt = 0, and thus the optimal squeezing is always along the x axis. It indicates that the minimal variance on the (${\hat{S}}_{x}$, ${\hat{Q}}_{{yz}}$) plane is ${\rm{\Delta }}{\hat{S}}_{x}$. When we turn off the quadratic Zeeman energy, ${\rm{\Delta }}{\hat{S}}_{x}$ would be fixed in the dynamical process due to that the operator ${\hat{S}}_{x}$ commutates with ${\hat{H}}_{0}$. Thus we can realize the storage and enhancement of the optimal squeezing.
The spin squeezing has been proved that it improved proof-of-principle interferometric measurements. Here we show the spin-nematic squeezing parameter gives the metrological precision relative to the standard quantum limit for interferometer. In the following, we propose an interferometer with the spinor condensate and discuss the limit of the phase sensitivity.
4. Quantum metrology with spinor condensate
Now we propose a feasible protocol to characterize and exploit the spin-nematic squeezed state for metrological applications. The protocol starts with a state of all atoms condensate in the mf = 0 mode, then the particle entanglement is created dynamically by the spin mixing which is dominated by a unitary evolution ${\hat{U}}_{1}={{\rm{e}}}^{-{\rm{i}}\hat{H}{{t}}_{m}}$ with $\hat{H}$ is the Hamiltonian shown in equation (1 ). At tm, we turn off the magnetic field and let the squeezed state evolve under ${\hat{U}}_{2}={{\rm{e}}}^{-{\rm{i}}{\hat{H}}_{0}{{t}}_{2}}$. The state after these two processes is denoted as the inside state of the interferometer and given by ${\left|\psi \right\rangle }_{\mathrm{in}}={\hat{U}}_{2}{\hat{U}}_{1}\left|N,0\right\rangle $. Finally, a phase is encoded by a unitary transformation ${{\rm{e}}}^{-{\rm{i}}\hat{Q}\phi },$ where $\hat{Q}=\cos {\theta }_{\mathrm{opt}}{\hat{Q}}_{{yz}}-\sin {\theta }_{\mathrm{opt}}{\hat{S}}_{x}$. Here we note that the squeezing is along the axis with θ = θopt, and thus the phase is encoded along the antisqueezing direction. The above phase encoding process can be realized by three rotations. The first process is a rotation about ${\hat{Q}}_{{zz}}$ with an angle π/4 − θopt/2, which is given by the operator ${{\rm{e}}}^{{\rm{i}}(\pi /4-{\theta }_{\mathrm{opt}}/2){\hat{Q}}_{{zz}}}$. The second is a φ rotation about the ${\hat{S}}_{x}$ axis, which can be understood as a radio-frequency (RF) pulse is employed to imprint or encode a phase. The final step is a reversion of the first one and given by ${{\rm{e}}}^{-{\rm{i}}(\pi /4-{\theta }_{\mathrm{opt}}/2){\hat{Q}}_{{zz}}}$ . The output state after the full protocol becomes ${\left|\psi \right\rangle }_{\mathrm{out}}={{\rm{e}}}^{-{\rm{i}}\hat{Q}\phi }{\left|\psi \right\rangle }_{{\rm{in}}}$.
The phase is estimated by a measurement signal $\hat{J}$ which has a precision11 ) reduces to
$\begin{eqnarray}{\rm{\Delta }}{\phi }^{2}=({\rm{\Delta }}\hat{J}{)}_{\mathrm{out}}^{2}/| \partial \langle \hat{J}{\rangle }_{\mathrm{out}}/\partial \phi {| }^{2},\end{eqnarray}$
where $\langle \hat{J}{\rangle }_{\mathrm{out}}=\left\langle \psi \right|\hat{J}{\left|\psi \right\rangle }_{\mathrm{out}}$ and $({\rm{\Delta }}\hat{J}{)}_{\mathrm{out}}^{2}$ is the correspondence variance. In this work, we choose a single generator to consider: $\hat{J}={\hat{Q}}_{+}$, then the phase sensitivity is obtained as $\begin{eqnarray}{\rm{\Delta }}{\phi }^{2}=\displaystyle \frac{{\sin }^{2}(2\phi )({\rm{\Delta }}{\hat{Q}}_{\perp }{)}_{\mathrm{in}}^{2}+{\cos }^{2}(2\phi )({\rm{\Delta }}{\hat{Q}}_{+}{)}_{\mathrm{in}}^{2}}{{\sin }^{2}(2\phi )| \langle {\hat{Q}}_{+}{\rangle }_{\mathrm{in}}{| }^{2}},\end{eqnarray}$
with $\langle {\hat{Q}}_{+}{\rangle }_{\mathrm{in}}=\left\langle \psi \right|{\hat{Q}}_{+}{\left|\psi \right\rangle }_{\mathrm{in}}$ and $({\rm{\Delta }}{\hat{Q}}_{\perp }{)}_{\mathrm{in}}^{2}={\rm{\Delta }}(\cos {\theta }_{\mathrm{opt}}{\hat{S}}_{x}+\sin {\theta }_{\mathrm{opt}}{\hat{Q}}_{{yz}}{)}_{\mathrm{in}}^{2}$ which corresponds to the minimal variance on the plane $\{{\hat{S}}_{x},{\hat{Q}}_{{yz}}\}$. The phase uncertainty Δφ2 reaches its minimum at φ = π/4. In this case, equation ( $\begin{eqnarray}{\rm{\Delta }}{\phi }_{\min }^{2}={\xi }_{x}^{2}/(2| \langle {\hat{Q}}_{+}{\rangle }_{\mathrm{in}}| ).\end{eqnarray}$
It shows that the metrology is closely related to the squeezing. In figure 4(a), we plot ${\rm{\Delta }}{\phi }_{\min }^{2}$ as a function of t with q = 53. As expected, the phase sensitivity is enhanced with the method which we have proposed for the increasement of spin-nematic squeezing. To exploit the scaling of the phase sensitivity to the particle number of the system. In figure 4(b), we show the optimal phase sensitivity, ${\rm{\Delta }}{\phi }_{\mathrm{opt}}^{2}\equiv {\min }_{(q,t)}({\rm{\Delta }}{\phi }_{\min }^{2})$, as a function of particle number and we see clearly that ${\rm{\Delta }}{\phi }_{\mathrm{opt}}^{2}\propto 1/{N}^{2}$. Therefore, the phase sensitivity of the interferometer with our method can be greatly improved. Here we denote that the Heisenberg limit for the spin-1 system is 1/(4N2).
Figure 4. (a) Phase sensitivity ${\rm{\Delta }}{\phi }_{\min }^{2}$ as a function of λt for q = 53 with N = 103 . (b) Scaling of ${\rm{\Delta }}{\phi }_{\mathrm{opt}}^{2}$ as a function of N. The cases for the dashed and solid lines are the same with that in figure 2. The black dotted line is the Heisenberg limit. |
In fact, there are many other schemes of interferometer which also has a high precision, such as construct an interferometer via adiabatic transitions through quantum phase transitions and heralded mechanisms [53, 62]. The advantage for our produced scheme is that the preparation time of the squeezed state is much shorter than that generated by the adiabatic transition [61, 62].
5. Detection noise
Quantum-enhanced measurements typically require single-particle resolution. However, the correlated state is fragility to the noise sources and thus limited the sensitivity of entanglement-enhanced metrology with squeezed state. A typical noise is detection noise which makes n and n + σ particles indistinguishable and leads the counting efficiency decrease. In the practical experiment, the detection noise in the spinor condensate arises mainly from the photon shot noise of the probing light [61]. To describe finite detection efficiency we consider a Gaussian noise [63, 64]. The variance of ${\hat{Q}}_{+}$ at the output state is then given by1 ). To clearly show the effect of the detection noise on the phase sensitivity, we plot the optimal sensitivity ${\rm{\Delta }}{\phi }_{\mathrm{opt}}^{2}$, with q to be fixed as q = 53, as a function of σ2 in figure 5(c). In typical experiments σ2 ≈ 100 for the system with 104 particles and a high detection sensitivity has been discussed with σ = 1. Here, we vary σ2 from 0 to 20. Our result shows that the ${\rm{\Delta }}{\phi }_{\mathrm{opt}}^{2}$ with our method is much better than that under the free evolution when σ2 is small.
$\begin{eqnarray}\Space{0ex}{0.2ex}{0ex}({\rm{\Delta }}{\hat{Q}}_{+}^{{\rm{d}}{n}}{\Space{0ex}{1.0ex}{0ex})}_{\mathrm{out}}^{2}=({\rm{\Delta }}{\hat{Q}}_{+}{)}_{\mathrm{out}}^{2}+{\sigma }^{2}.\end{eqnarray}$
It results in the phase sensitivity $\begin{eqnarray}{\rm{\Delta }}{\phi }_{\min \ }^{2}={\xi }_{x}^{2}/2| \langle {\hat{Q}}_{+}{\rangle }_{\mathrm{in}\ }| +{\sigma }^{2}/| \langle {\hat{Q}}_{+}{\rangle }_{\mathrm{in}}{| }^{2}.\end{eqnarray}$
In figures 5(a) and (b), we plot the ${\rm{\Delta }}{\phi }_{\min \ }^{2}$ as a function of λt with different σ2. Comparing the result of the sensitivity shown in figure 4(a), the presence of the detection noise suppresses the phase sensitivity. However, the ${\rm{\Delta }}{\phi }_{\min \ }^{2}$ with our protocol is still better than that with the free evolution under Hamiltonian shown in equation (
Figure 5. Phase sensitivity ${\rm{\Delta }}{\phi }_{\min }^{2}$ as a function of λt with (a) σ2 = 10 and (b) σ2 = 5. (c) Dependence of ${\rm{\Delta }}{\phi }_{\mathrm{opt}}^{2}$ on the detection noise σ2. The cases for dashed and solid lines are the same with that in figure 2. The parameters are chosen as q = 53 and N = 103. |
6. Conclusion
In summary, we have proposed a simple method to store and improve the spin-nematic squeezing in a spinor BEC condensate with an external magnetic field. By rapidly turning-off the magnetic field at the time when the maximal squeezing occurs, the spin-nematic squeezing is stored for a long time with a fixed direction along x-axis. Choosing a proper initial quadratic Zeeman energy, the optimal spin-nematic squeezing can even be enhanced by 4.5 dB. We also propose an interferometer with the spinor condensate, and show that the phase sensitivity can be enhanced with our method which is presented for the improvement of spin-nematic squeezing. Even in the presence of detection noise, the phase sensitivity of the interferometer with our method still has advantage compared to the free evolution. Our scheme for the enhancement of spin-nematic squeezing and phase sensitivity is quite robust for wide range of parameters. We hope our scheme will pave the way to atomic ultrasensitive spin-mixing interferometry in experiment near future.