Two-axis spin squeezing of two-component BEC via a continuous driving
In two-component BEC, the one-axis twisting Hamiltonian leads to spin squeezing with the limitation that scales with the number of atoms as . We propose a scheme to transform the one-axis twisting Hamiltonian into a two-axis twisting Hamiltonian, resulting in enhanced spin squeezing approaching the Heisenberg limit. Instead of pulse sequences, only one continuous driving field is required to realizing such transforming, thus the scheme is promising for experiment realizations, to an one-axis twisting Hamiltonian. Quantum information processing and quantum metrology may benefit from this method in the future.
pacs:42.50.Dv , 03.75.Gg
Introduction. Squeezed spin states (SSSs) Kitagawa93 (); Wineland94 (); Ma11 (), whose concept was firstly established by Kitagawa and Ueda Kitagawa93 (), are entangled quantum states of an ensemble of spin systems. The SSS attracted considerable attention due to their significant roles in studying many-particle entanglement Sorensen01 (); Bigelow01 (); Sorensen01L (); Amico08 (); Horodecki09 (); Guhne09 () and applications for high-precision measurements Wineland94 (); Wineland92 (); Bollinger96 (); Polzik08 (); Cronin09 (); Agarwal96 (); Meiser08 (); Andre04 (). In the original proposal Kitagawa93 (), there are two distinguished ways to produce SSS, one interaction in the form as is known as one-axis twisting (OAT), the other one in the form as is known as two-axis twisting (TAT). The OAT scheme just can reduce the noise limit to the scale as where is atom number, while the TAT can produce the SSS with the squeezing parameter scaling with Kitagawa93 (). Both in theory and experiment, most schemes can only produce effective OAT-type spin-spin interactions, such as direct atom collisions in Bose-Einstein condensates (BEC) Gross10 (); Riedel10 (); Julia12 (), indirect spin-spin interaction by quantum nondemolition measurement Chaudhury07 (); Takano09 (); Inoue13 (); Leroux12 (); Julsgaard01 (); Louchet10 () and cavity feedback Schleier10 (); Leroux10 ().
The two-component BEC is a very promising system for OAT SSSs Poulsen01 (); Raghavan01 (); Jenkins02 (), and it has been demonstrated in experiments recently Orzel01 (); Esteve08 (); Gross10 (); Riedel10 (). It holds two main advantages, including the considerable long coherence time and the strong atom-atom interaction, is very potential for future applications. Therefore, various efforts are dedicated to realizing the TAT type Hamiltonian to enhance the squeezing in such system Helmerson01 (); Liu11 (); Duan13 (); Zhang14 (). One of the proposals Liu11 () transforms an OAT Hamiltonian into an effective TAT Hamiltonian by applying a large number of repeated Rabi pulses, which would be sensitive to the accumulation of control errors. In another scheme Duan13 (), one or two global rotation pulses are applied at an appropriate evolution time and with optimized rotation angles, which reduces the number of pulses greatly, but requires a long evolution time to achieve the optimal squeezing and the control pulse is spin number dependent.
In this paper, we propose a scheme to transform the OAT into the effective TAT spin squeezing in BEC by continuous coherent driving. Under the driving, the spin state is rotating along the direction perpendicular to the twisting axis, then generate the effect Hamiltonian as mixed OAT and TAT. By carefully choosing and tuning the amplitude and and frequency of the driving field, pure TAT can be realized and a Heisenberg limited noise reduction is obtained. Compared with the previous scheme Liu11 (), our proposal uses a continuous field instead of pulse sequences, which is more friendly for experiments. What’s more, our scheme is spin number independent and needs a shorter evolution time compared with Duan13 (). The principle of continuous driving transformed the OAT to the TAT can also be applied to other systems, such as the cavity feedback Schleier10 (); Leroux10 () and spin state dependent geometry phase Yan14 () induced OAT.
Here in terms of the Pauli matrices is the collective angular momentum operator for the spin ensemble consisting of atoms. The first term of the Hamiltonian is the OAT induced by atom-atom collisions, with the nonlinear interaction strength. The second term is the external classical laser driving with magnetic field along the -axis. For the continuous driving, we assume , where and are the strength and frequency of the driving field, respectively.
Transform the Hamiltonian (1) into the interaction representation, we get
where and . According to the Jacobi-Anger expansion where is the -th Bessel function of the first kind, the terms in Eq. (2) can be expanded as
When is quite large (), the high-order terms with are neglected due to the rotating wave approximation. Then, the Hamiltonian becomes
where the constant . Therefore, the external driving field leads to the twisting effect along both and directions. This can be interpreted intuitively as the rotation of spins perpendicular to the axis of OAT (-axis) diverted the twisting axis.
Rewriting the Hamiltonian by adding a constant (which is conserved during the dynamics), we obtain a mixture of an OAT Hamiltonian and a TAT Hamiltonian as
Tune the values of and to be , then and the effective Hamiltonian of the system becomes
Obviously, exhibits the well-known TAT Hamiltonian. Alternatively, we can also adjust the parameters to satisfy , then we obtain another TAT Hamiltonian
Therefore, the OAT Hamiltonian can be transformed into the TAT Hamiltonian by tuning the amplitude and frequency of the driving field. Similar ideas have been studied by Law et al. Law (), where the underlying physics is the same with the continuous driving method studied here. In that work, a steady field are applied for coherent controlling of the SSS, which is consist with our model with , effectively generate a mixture of OAT and TAT. It’s worth noting that the effective nonlinear interaction strength reduces to , which is due to the cancellation of part of spin squeezing when rotating of the squeezing direction.
Numerical results. To verify our idea above, we study the spin squeezing numerically by solving the evolution of spin state. The initial state is chosen to be a coherent spin state (CSS) Kitagawa93 () along the axis, which is satisfying with , where are the eigenstates of . We choose squeezing parameter Kitagawa93 () to quantify the squeezing, where refers to the direction perpendicular to the mean spin direction and the minimization is taken over all such directions.
In Fig. 1, we plot the spin squeezing parameter as a function of the evolution time for different driving frequency but fixed the ratio that to obtain optimized TAT. The results for both the (a) and (b) are agrees well with the effective TAT Hamiltonian [Eq. 6]. There are fast oscillations of the for small , which is attributed to the high-order terms in the Jacobi-Anger expansion [Eq. 3] when is not satisfied. For example, the oscillation period for is about , corresponding to which consist with the period of high order terms. Therefore, higher frequency is favorable for larger number of atoms. In addition, we find that when the number of atoms increases, it needs a shorter time to reach the optimal squeezing, and the time is already much shorter than the scheme Duan13 ().
Next, we investigate how the optimal squeezing of without approximation scales with . We plot the optimal spin squeezing (minimum value of ) as a function of the number of atoms in Fig. 2. The red solid line corresponding to shows the optimal spin squeezing parameter which is the well-known Heisenberg limited noise reduction, and it agrees well with (the blue dashed line). For comparison, we also present the scaling of the OAT Hamiltonian .
Although the optimal TAT should satisfy , we could expect the enhanced spin squeezing by continuous driving field is robust against imperfection parameters, since the external driving field could lead to mixture of OAT and TAT effectively [Eq. 5]. In Fig. 3(a), we show the squeezing parameter as a function of the evolution time for different with , the dynamics under is also presented for comparison. We can find that at the optimal squeezing generated by is 0.02805. It is better than that generated by (0.0479) while worse than that generated by (0.0177). But at , the squeezing is even worse than , which is owing to approaching , then the Hamiltonian is close to . If we change the initial state correspondingly along -axis, which is satisfying , the effect of this Hamiltonian approaches the idea TAT, as shown in Fig. 3(b). Therefore, our scheme can always enhance the OAT Hamiltonian to achieve better SSS even though the achievable is deviated from optimal value.
Finally, we plot the optimal spin squeezing parameter of the Hamiltonian as a function of in Fig. 4(a) with the initial state being a CSS along the axis and in Fig. 4(b) with the initial state being a CSS along the axis. In Fig. 4(a), the minimum value equals to the optimal squeezing of the TAT Hamiltonian appears at , which agrees with the optimal condition. The rapid growth of for is due to the Hamiltonian changing to . In Fig. 4(b), it shows a section of gentle variance which almost equals to the optimal squeezing of the TAT Hamiltonian. There are two points which make , and the minimum value of the Bessel function between the two points is about [inset of Fig. 4(b)] which has no large variance comparing with . Thus, there is a quite large range approaching the TAT squeezing, which is favorable for experiments.
Conclusions. We have proposed a scheme to transform an OAT Hamiltonian into a TAT type by applying a continuous driving field. We find that a TAT Hamiltonian can be obtained by tuning the ratio of the driving field amplitude to the frequency, and even though at other more achievable values of , the squeezing performance of our scheme is more better than the OAT scheme. Compared with the previous proposals Liu11 ()Duan13 (), our scheme is more friendly for experiments and faster. Since the continuous driving field can be manipulated relatively easily, we believe it is realizable with current techniques as reported in Ref. Gross10 (); Riedel10 ().
Acknowledgments. This work was supported by National Fundamental Research Program, National Natural Science Foundation of China (No. 11274295, 2011cba00200) and Doctor Foundation of Education Ministry of China (No. 20113402110059).
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