Abstract
We consider the effects of dipoledipole interactions on a nonlinear interferometer with spin1 BoseEinstein condensates. Compared with the traditional atomic SU(1,1) interferometer, the shotnoise phase sensitivity can be beaten with respect to the input total average number of particles; and the improved sensitivity depends on the effective strength of the dipolar interaction via modifying the trapping geometry. It indicates that the best performance of the interferometer is achieved with highly oblate trap potential. The Bayesian phase estimation strategy is explored to extract the phase information. We show that the CramérRao phase uncertainly bound can saturate, when the ideal disentangle scheme is applied. The phase average of the phase sensitivity is also discussed.
Similar content being viewed by others
Introduction
Interferometers as the extremely useful and flexible precise measuring tool, play a key role in the field of quantum metrology^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21}. Recently, there are mainly two classes of interferometers^{2,3}: passive [e.g., MachZehnder interferometer (MZI)] and active [e.g., SU(1,1)] interferometers. The ultimate goal of both these setups is to beyond the shotnoise limit (SNL) for phase estimation. It is well known that, for the MZI to beat SNL, i.e., \(\mathrm{1/}\sqrt{N}\) with N being the total particles number, nonclassicality of the input states are necessary. While for SU(1,1) interferometer the situation is different, because it applies the nonlinear opticalparametric amplifier (OPA), which mixes the optical beams and then converts the classical input states into photon pairs^{2,3,4}. A potential advantage of the SU(1,1) interferometer is that even for classical sources of the input states the SNL can also be surpassed.
Spinor BoseEinstein condensates due to their unique coherence properties and the controlled nonlinearity are viewed as the ideal sources for an atomic interferometer. The coherent spinmixing dynamics (SMD) in the spin1 BECs can generate entangled states^{22,23,24,25,26} by converting two atoms in the m_{ f } = 0 state into one atom in the m_{ f } = 1 state and the other in the m_{ f } = −1 state, which is the atomic analogue of OPA. Experimentally, ref.^{27} has used the spinchanging collisions in a spinor BEC as the nonlinear mechanism to realize a atomic SU(1,1) interferometer. In this scheme the interferometer operations belong to the SU(1,1) group and the phase sensitivity can be obtained analytically by using meanfield approximation, but the number of particles used for phase estimation inside the interferometer is very small. To obtain a relatively large number of particles for probe states, in ref.^{28}, the authors considered a full quantum analysis and found that the subshotnoise(SSN) phase sensitivity can be obtained with respect to the total particles inside the interferometer.
Up to now, studies of nonlinear atomic interferometer with spinor BECs have focused mainly on swave contact interaction^{27,28,29}. According to the recent experimental and theoretical observation in ^{23}Na and ^{87}Rb atoms, the magnetic dipoledipole interactions (MDDIs) are indeed not negligible for these spinor condensates^{26,30,31,32,33,34,35,36,37,38,39}. For example, in ^{87}Rb atoms, the magnitude of the dipolar energy can be as large as 10% of the spinexchange energy^{35,36}. In particular, the longrange and anisotropic nature of the dipolar interaction may further enhance its effects^{36,38,40}. Thus, the effects of the MDDI should be considered in a reliable SSN sensitivity interferometer based on spinor BECs.
In this paper, we study the effects of MDDI on the phase sensitivity in a spinmixing interferometer based on ^{87}Rb condensates. In the quantum metrology field, the quantum Fisher information (QFI)^{41,42,43} has been widely used to characterize the phase sensitivity. In this work, we will also describe the phase sensitivity for our spinmixing interferometer with the QFI. By calculating the QFI, we find that, the QFI depends on both the evolution time of SMD and the trapping geometry. Our results indicate that, for certain evolution time the enhancement SSN sensitivities can be reached with respect to the total input number of particles N by using the highly oblate trap potential. Finally, we also explore the Bayesian phase estimation strategy to extract the optimal phase information and phase average sensitivity with different disentangle methods.
Results
Model and Hamiltonian
Similar to the optical SU(1,1) interferometer, the spinmixing interferometer can also divide into three steps: (I) entangled states preparation with spinexchange collisions, (II) phase encoding, and (III) disentangling and measurement.
To realize the atom interferometer scheme as shown in Fig. 1, we consider N spin1 Rb atoms confined in a threedimensional potential with ferromagnetic spinexchange collisional interaction. Assuming all spin components share a common spatial mode ϕ(r), under the singlemode approximation, the Bose condensate can be described by following Hamiltonian
The first term originates from the swave contact interaction, which contains the SMD and has been considered for atom SU(1,1) interferometers^{27,28}. The last two terms are induced by the dipolar interaction. Notice, here we have used the absolute value of the spin exchange strength \(c=({c}_{2}\mathrm{/2)}\,\int \,dr\varphi (r{)}^{4}\) as energy unit (the corresponding unit for time is ħ/c), where c_{2} = 4πħ^{2}(a_{2} − a_{0})/(3 M) with M being the mass of the atom and a_{0,2} the swave scattering length for two spin1 atoms in the symmetric channel of the total spin 0 and 2, respectively. For a Gaussian mode function with characteristic lengths q_{x,y,z} in x, y, z directions, the rescaled dipolar interaction strengths can be read as^{36}
where (κ_{ x }, κ_{ y }) ≡ (q_{ x }/q_{ z }, q_{ y }/q_{ z }) characterizes the shape of the condensate, \({c}_{d}={\mu }_{0}{\mu }_{B}^{2}{g}_{F}^{2}\mathrm{/(4}\pi )\) is the strength of the MDDI with μ_{ B } the Bohr magneton, and g_{ F } the Landé gfactor for ^{87}Rb atoms, we have c_{ d }/c_{2} ≈ 0.1. In the above equations, I_{0,1}(x) is the modified Bessel functions of the first kind, and erfc(x) is the complementary error function. The value of d_{s,n} can be positive, 0, or negative, depending on the values of κ_{x,y}. In particular, d_{s,n} = 0 (d_{ n } = 0) if κ_{ x } = κ_{ y } = 1 (κ_{ x } = κ_{ y }).
In Eq. (1), the manybody angular momentum operators are given by
with \({\hat{a}}_{\alpha =0,\pm 1}\) being the annihilation operator of the αth spin state, and \({\hat{N}}_{\alpha }\) is the number operator of the α spin component. The term \(({\hat{a}}_{1}^{\dagger }{\hat{a}}_{1}^{\dagger }{\hat{a}}_{0}^{2}+h.c)\) include in \({\hat{S}}^{2}\) is identical to the OPA in nonlinear optics, which is the main factors influencing the spinmix process.
As shown in step (I), we start with a source of N = N_{0} particles in the m_{ f } = 0 pure state, namely Ψ(0)〉 = 0, N, 0〉 with Fock basis N_{1}, N_{0}, N_{−1}〉. Governing by Hamiltonian (1), at time t_{evo} we can obtain an entangled state \({\rm{\Psi }}({t}_{{\rm{evo}}})\rangle ={\sum }_{m,k}\,{\bar{g}}_{mk}({t}_{{\rm{evo}}})k,N2k+m,km\rangle \), where \({\bar{g}}_{mk}({t}_{{\rm{evo}}})\) can be obtained numerically (see Methods).
And then the phase information is encoded into this probe state [step (II)]. To eliminate the effects of atomic nonlinear interaction on the phase accumulation and measurement, we need the accurately controllable spinchanging collisions. In the present of strong enough external magnetic field, the spinmixing process would be stopped due to the socalled quadratic Zeeman effect. It shifts the the levels of the f = 1 down, and induces the energy difference between the m_{ f } = 0 and m_{ f } = ±1 modes via the supplemented Hamiltonian \({H}_{{B}^{2}}=q({\hat{N}}_{1}+{\hat{N}}_{1})\)^{27,28}. In ref.^{27}, the authors obtained energy difference q = (2π)72 Hz when B = 0.9 G. Note that the linear Zeeman effect does not affect the spinchanging collisions since the energy gained by one particle has to be spent by the other. Although spinchanging might be turned on by quenching the magnetic field down to zero, ramping up and down magnetic fields lacks the necessary control and speed. Experimentally, instead we can use microwave dressing to compensate the magnetic field during t_{evo}, by applying a faroffresonate detuned πpolarized microwave field to coupe 1, 0〉 to 2, 0〉 with the Rabi frequency Ω and the detuning Δ^{27}. Microwave dressing supplements the Hamiltonian with \({H}_{{\rm{\Omega }}}=\frac{{{\rm{\Omega }}}^{2}}{4{\rm{\Delta }}}({\hat{N}}_{1}+{\hat{N}}_{1})\), which can shift up and down the energy level by either red or blue detuning. Making \({{\rm{\Omega }}}^{2}\mathrm{/4}{\rm{\Delta }}=\mathrm{(1}+{d}_{s})\mathrm{(2}{\hat{N}}_{0}\mathrm{1)}\) during t_{PS}, the effective interaction of linear phase shift reads:
with phase shift θ = 2qt_{PS}.
In step (III), to estimate the phase shift θ, we should first disentangling the m_{ f } = ±1 modes, and then measure the number of particles in them. An ideal method to disentangling is to make Ht_{evo} = −H′t′_{evo}. Thanks to both the sign and the strength of the nonlinear coupling are experimentally adjustable, which indicates that we maybe make \(c{t}_{{\rm{evo}}}\simeq \tilde{c}{t^{\prime} }_{{\rm{evo}}}\) and \({c}_{d}{t}_{{\rm{evo}}}\simeq {\tilde{c}}_{d}{t}_{{\rm{evo}}}\) to realize time reversal readout scheme.
We can see in all the above three steps, the MDDI plays a important role, later we shall study the effects of the MDDI on the precision of phase estimation in the dipolar atomics interferometer.
Quantum Fisher information in the present of dipolar interaction
Now, we investigate the effect of MDDI on the phase estimation by calculating the QFI. It gives a theoretically achievable limit on the precision of an unknown parameter θ by the quantum CramérRao theorem Δ^{2}θ ≥ Δ^{2}θ_{ QCR } = 1/(mF_{ Q }), where m represents the number of independent measurements. In our interferometer the QIF can be obtained as
with \({\hat{N}}_{s}={\hat{N}}_{1}+{\hat{N}}_{1}\). For convenience, we define the mean quantum Fisher information \({\bar{F}}_{Q}={F}_{Q}/N\), where \(\bar{F}=1\) means the SNL and \({\bar{F}}_{Q} > 1\) means the SSN phase sensitivity.
In Fig. 2(a), we have plotted the maximal mean QFI \({\bar{F}}_{Q}^{{\rm{\max }}}\) with long enough evolution time for different κ_{ x }, κ_{ y }. As it is shown, we have \({\bar{F}}_{Q}^{{\rm{\max }}} > 1\), which depends on the trapping geometry (κ_{ x }, κ_{ y }); and the best QFI is appeared in the regimes of κ_{ x } = κ_{ y }, corresponding to the axial symmetry with d_{ n } = 0. It indicates that we can obtain the SSN sensitivity with respects to the total atom number N. However, in Fig. 2(a) the maximal QFI is attained for long enough evolution times, and hence the mechanism of decoherence in the condensate cannot be neglected. In fact, to reach the SSN sensitivities long evolution time is not necessary. In Fig. 2(b), we have shown the shortest evolution time to reach the SSN sensitivities as a function of κ_{ x }, κ_{ y }. It is clearly shown that the MDDI can reduce the evolution time to obtain the SSN.
To avoid the mechanism of decoherence and obtain the relatively large QFI, in the spinmixing interferometer the maximum SSD evolution time we considered is mainly focus on the scale of \(\sim \hslash /(c\sqrt{N})\), which is much shorter than the lifetime of a spin1 BECs. And hence, for sufficiently large N and fast phase encoding the nonlinear interferometer, we can safely neglect the decoherence processes of the condensate.
In Fig. 3(a), we have plotted the mean QFI as a function of evolution time for different κ_{ x }, κ_{ y }. As it is shown, we can find that for short time scale the QFI increase with time evolution, it means that proper period of evolution time can improve the QFI, and the values can surpass the SNL. Figure 3(a) indicates that due to the MDDI we can obtain better QFI than the case without it (i.e., κ_{ x } = κ_{ y } = 1). In particular, we can obtain the best QFI when \({\mathrm{log}}_{10}\,{\kappa }_{x}={\mathrm{log}}_{10}\,{\kappa }_{y}=1\), i.e., pancakeshaped condensate. In Fig. 3(b) we plot the dependence of the mean QFI on the trapping geometry (κ_{ x }, κ_{ y }) covering the parameter regime 0.1 ≤ κ_{x,y} ≤ 10 with evolution time \(c\sqrt{N}\hslash {t}_{{\rm{evo}}}/\hslash =1\). It is shown large QFI can find in the regime κ_{ x } = κ_{ y }, corresponding to the axial symmetry with d_{ n } = 0. The diagonal lines in Fig. 3 illustrate the changes of the QFI when the condensate changes from the elongated trap (cigarshaped) to oblate trap (pancakeshaped). In both Fig. 3(a,b), the best QFIs are found in the region with highly oblate \({\mathrm{log}}_{10}\,{\kappa }_{x}={\mathrm{log}}_{10}\,{\kappa }_{y}=1\). This means that we can obtain the best SSN sensitivities with respect to the total input number of particles N, which is larger than the case without the MDDI, by initially setting the shape of the condensate.
The mechanism of the MDDI improves the phase sensitivity in short time scales can be understood from Hamiltonian (1). After rescaling Eq. (1), we have
The anisotropic constants are given by
which depend on the MDDI. The values of A_{ x }(κ_{ x }, κ_{ y }), A_{ y }(κ_{ x }, κ_{ y }) and E(κ_{ x }, κ_{ y }) covering the parameter regime 0.1 ≤ κ_{x,y} ≤ 10 are shown in Fig. 4. According to Fig. 4, we have A_{ x } ≥ 0 (A_{ y } ≥ 0) if κ_{ x } ≥ 1 (κ_{ y } ≥ 1). In particular, if κ_{ x } = κ_{ y } we have E = 0 and A_{ x } = A_{ y }, then Eq. (1) further reduces to \( {\mathcal H} ^{\prime} ={\hat{S}}^{2}{A}_{1}{\hat{S}}_{z}^{2}\) with \({A}_{1}=\frac{3{d}_{s}}{1+{d}_{s}}\) and \({\hat{S}}_{z}={\hat{a}}_{1}^{\dagger }{\hat{a}}_{1}{\hat{a}}_{1}^{\dagger }{\hat{a}}_{1}\).
Now we will investigate the effects of anisotropic constants A_{ x }(κ_{ x }, κ_{ y }), A_{ y }(κ_{ x }, κ_{ y }), E(κ_{ x }, κ_{ y }) and A_{1}(κ_{ x }, κ_{ y }) on the QFI with rescaled times, respectively. From Fig. 5, we can find that the values of the mean QFIs almost has no influence on the parameter E, but it significantly depends on the values of A_{x,y}. That is, the positive values of A_{x,y} can improve the QFI for short time scales, while decrease it for negative values of A_{x,y}. Figure 5(c) shows the QFI in an atomic interferometer based on axialsymmetry condensate, from it we can find that the term including \({S}_{z}^{2}\) nearly do not affect the QFI within short time scales, but it can enhance the QFI for relatively long time time scales.
It is well known that the SNL of phase sensitivity can be surpassed using squeezed states. Later, we will investigate the effects of the condensate shape on the phase sensitivity by calculating the spin squeezing in the system considered. Unlike the spin1/2 systems which can be uniquely specified by different components of the total spin vector S ≡ (S_{ x }, S_{ y }, S_{ z }), the state of spin1 atomic BoseEinstein condensates is specified in terms of both the spin vector and nematic tensor Q_{ ij } = S_{ i }S_{ j } + S_{ j }S_{ i } − (4/3)δ_{ ij } which constitutes SU(3) Lie algebra, with δ_{ ij } being the Kronecker delta and ({i, j} ∈ {x, y, z})^{44,45}. For the initial state we considered, we can always numerical check that \(\langle {\bf{S}}\rangle \simeq 0\), but the quadrupole elements 〈Q_{ ii }〉 ≠ 0. And in both of the subspaces, {S_{ x }, Q_{ yz }, Q_{+}} and {S_{ y }, Q_{ xz }, Q_{–}} that exhibit squeezing, where Q_{+} and Q_{−} are defined Q_{+} = Q_{ zz } − Q_{ yy } and Q_{−} = Q_{ xx } − Q_{ zz }, respectively. Then the two different spinnematic squeezing parameters in an SU(2) subspace are defined by \({\xi }_{x(y)}^{2}={{\rm{\min }}}_{\theta }\,\langle {[{\rm{\Delta }}(\cos \theta {S}_{x(y)}+\sin \theta {Q}_{yz(xz)})]}^{2}\rangle /\langle {Q}_{\pm }\mathrm{/2}\rangle \), with θ being the quadrature angle^{45}. If \({\xi }_{x(y)}^{2} < 1\) indicates spinnematic squeezing.
In Fig. 6, we plot the evolution of spinnematic squeezing parameter \({\xi }_{x}^{2}\) for different trapping geometry. From Fig. 6, we can see that the strong spinnematic squeezing can be obtained in the regimes of κ_{ x } = κ_{ y }, which displays the same changing trend as the QFI. Figure 6 also shows that the highly oblate trap can shorten the optimal squeezing time before it “over squeezing”. In ref.^{45}, the authors proposed a scheme to store the best spinnematic squeezing for quantum metrology by applying periodic microwave pulses.
Optimal Fisher information in the presence of dipolar interaction
Below, we focus on the atomic interferometer with axialsymmetry condensate, κ_{ x } = κ_{ y }. To demonstrate the feasibility of SSN phase sensitivity given by the QFI, we employ a protocol based on a Bayesian analysis of the measurement results with atomnumber N_{±1} after step (III). Then the classical Fisher information (CFI) is^{14}
where \(P({N}_{\pm 1}\theta )={\langle {N}_{\pm 1}{{\rm{\Psi }}}_{{\rm{out}}}^{(\theta )}\rangle }^{2}\) is the conditional probability that particle N_{±1} is measured for given phase shift θ. And \({{\rm{\Psi }}}_{{\rm{out}}}^{(\theta )}\rangle ={e}^{iH^{\prime} {t^{\prime} }_{{\rm{evo}}}}{e}^{i{H}_{{\rm{PS}}}{t}_{{\rm{PS}}}}{e}^{iH{t}_{{\rm{evo}}}}{\rm{\Psi }}\mathrm{(0)}\rangle \). Then the saturable lower bound of phase sensitivity is given by the CR bound, \({\rm{\Delta }}{\theta }_{CR}=1/\sqrt{m{F}_{C}(\theta )}\), m denotes the number of independent measurements. Unlike the QFI, the CFI depends on the phase θ, and by this definition, we have Δθ_{ QCR } ≤ Δθ_{ CR }.
Figure 7 illustrates the optimal CFI \({F}_{C}^{{\rm{opt}}}\equiv {{\rm{\max }}}_{\theta }\,{F}_{C}(\theta )\) as a function of trapping geometry κ_{x,y}. Here, we consider κ_{ x } = κ_{ y }, then d_{ n } = 0. In Fig. 6, we compare the optimal CFI with two different disentangle methods. The first approach to disentangling, which considered in ref.^{28}, is to apply a π/2 phase shift to the m_{ f } = 0 mode, namely \({\hat{a}}_{0}\to i{\hat{a}}_{0}\). After this operation, the manybody angular momentum operators in Eq. (1) become \({S}^{2}\to \tilde{{\hat{S}}^{2}}\) and \({\hat{S}}_{x}^{2}{\hat{S}}_{y}^{2}\to \tilde{{\hat{S}}_{x}^{2}}\tilde{{\hat{S}}_{y}^{2}}\) with
The advantage of this scheme is that it experimentally implement easily. As shown in Fig. 7, we can see that the MDDI can induce better CFI. And we can obtain the SSN phase uncertainties with respect to the total input number of atoms N. However, under this scheme, the optical CFI cannot reach the values given by QFI, due to the imperfect disentangle. If one want to obtain the optimal phase sensitivity, we can implement the ideal disentangle method by changing the sign of c and c_{ d }, i.e., \(c\to \tilde{c}\) and \({c}_{d}\to {\tilde{c}}_{d}\). The values of c can be controlled via Feshbach resonance, and the value of c_{ d } can be tuned from \({c}_{d}\to {\tilde{c}}_{d}\in [\mathrm{1/2},1]{c}_{d}\) by using a rotating orienting field to the dipole moments^{46}. Note that for perfected disentangle scheme, we can also apply the Loschmidt echo protocol to get the optimal phase sensitivity given by the QFI^{43}.
To describe well the behaviors of phase estimation for different phase with θ ∈ [0, 2π], we use the phase average of the FI, which is given by
Figure 7(c,d) indicate that for these two disentangle methods the phase average of the CFI \(\bar{I}\) both can surpass the SQL. From Fig. 7, we can see that in the dipolar atomic spinmixing interferometer, both the optimal and phase average CFI can reach the SNN limit as long as choose proper evolution time. The phase sensitivities depends on the trapping geometry, and highly oblate trap potential can further improve the average phase estimation precision.
Discussion
In summary, we have studied a nonlinear interferometer with the dipolar spin1 BoseEinstein condensate. By calculating the QFI, we found that the phase sensitivity of the interferometer depends on both the SMD evolution time and the MDDI. It is indicated that proper period of evolution time can improve the QFI, and that the subshotnoise phase sensitivity with respect to the total input number of particles N can achieve, due to the high transfer rates of particles in the spinchanging process. Moreover, for fixed SSD evolution time, the optimal phase estimation precision is mainly determined by the strength and the sign of the effective dipolar interaction. Our results shown that, the enhancement phase sensitivity can be achieved by tuning the effective MDDI via modifying the trapping geometry. It is indicated that the best performance of the interferometer is achieved with highly oblate trap potential. We also explored the Bayesian phase estimation strategy to extract the phase information. It is shown that the CramérRao phase uncertainly bound can saturate, when the ideal disentangle scheme is applied within the time scales that the particle loss effects can be neglected^{28}. The phase average uncertainly is discussed, which can also achieve the SSN sensitivity.
Finally, it should be pointed out that the results we have obtained in this paper are based on spin1 ^{87}Rb BoseEinstein condensate. Indeed, the larger the dipole moment is, the greater the effect is on the nonlinear interferometer. In experiments, dipolar BECs have been realized for atoms with large magnetic dipole moments, such as ^{164}Dy with dipole moment 10 μ_{ B }, which is much larger than ^{87}Rb’s moment equal μ_{ B }^{47}. Therefore, it will result in the strength of MDDI comparable with the swave contact interaction in Dy atomic condensate. In ref.^{48}, we have investigated the improved spin squeezing induced by MDDI of scalar Dy atomic condensate trapped in a doublewell potential, which is useful resource for quantum metrology. In its spinor counterpart, the ground state of Dy atom is ^{5}I_{8} with zero nuclear spin, which is spin8 dipolar condensate. Exploring such complex collisional behavior of Dy atom requires further investigation, but it may be greatly aid attempts in spinmixing interferometry.
Methods
The derivation of Hamiltonian (1)
In the second quantized form, the total Hamiltonian of the system, including swave collisions and the MDDI, reads as
where
is the Hamiltonian excluding MDDI. And the dipoledipole interaction term is
with e = (r − r′)/r − r′ an unit vector.
Substituting ψ_{ α }(r) = a_{ α }ϕ(r) into the Hamiltonian, we get
where \(c=({c}_{2}\mathrm{/2)}\,\int \,dr\varphi (r{)}^{4}\) is the spinexchange interaction strength, and \({\bf{S}}={a}_{\alpha }^{\dagger }{{\bf{F}}}_{\alpha \beta }{a}_{\beta }\) is the total manybody angular momentum operator. And the dipoledipole interaction can reduce to
Here, we have used the relations e_{±} = e_{ x } ± ie_{ y }, \({e}_{\pm }=\,\sin \,{\theta }_{e}{e}^{\pm i{\phi }_{e}}\), and e_{ z } = cos θ_{ e }.
For the Gaussian mode function
the last two terms of H_{ d } vanish. After introducing the two paramters
we obtain
Then the total Hamiltonian reduces to
Dynamics of spin
Hamiltonian H can be expanded in Fock state basis \({N}_{1},{N}_{0},{N}_{1}\rangle \) with N_{ α } ≥ 0 and N_{1} + N_{0} + N_{−1} = N. Numerically, it is more convenient to express Fock state basis as \(m,k\rangle \) where m = N_{1} − N_{−1} and k = N_{1} is the mumber of atoms in m_{ F } = 1 component. Since N_{−1} = k − m and N_{0} = N − 2k + m, we find
Then the matrix elements of Hamiltonian H become \({H}_{mk,m^{\prime} k^{\prime} }\equiv \langle m,kHm^{\prime} ,k^{\prime} \rangle \), and the dimension is D × D with D = (N + 1)(N + 2)/2. The index r of state \(m,k\rangle \) i.e., r(m, k) is stored in a 1D array as
After diagonalizing H, we obtain the eigenstates as ψ_{ s }〉
if we define \({\varphi }_{r}\rangle \equiv m,k\rangle \) with r = r(m, k), we have \({\psi }_{s}\rangle ={\sum }_{r}\,{u}_{r,s}{\varphi }_{r}\rangle \) with \({u}_{r,s}=\langle {\varphi }_{r}{\psi }_{s}\rangle \).
Assuming that the initial state takes the form \({\rm{\Psi }}\mathrm{(0)}\rangle ={\sum }_{r}\,{f}_{r}{\varphi }_{r}\rangle \) which is a superposition of number states, it can be expanded in the \(\{{\psi }_{s}\rangle \}\) basis as \({\rm{\Psi }}\mathrm{(0)}\rangle ={\sum }_{s}\,{g}_{s}{\psi }_{s}\rangle \), the time evolution of this state is
When the initial state is actually a number state \({\rm{\Psi }}\mathrm{(0)}\rangle ={\varphi }_{{r}_{0}}\rangle \), i.e., \({f}_{r^{\prime} }={\delta }_{r^{\prime} ,{r}_{0}}\), therefore
References
Caves, C. M. Quantummechanical noise in an interferometer. Phys. Rev. D 23, 1693 (1981).
Yurke, B., McCall, S. L. & Klauder, J. R. SU(2) and SU(1,1) interferometers. Phys. Rev. A 33, 4033 (1986).
Sanders, B. C., Milburn, G. J. & Zhang, Z. J. Mod. Opt. 44, 1309 (1997).
Hudelist, F., Kong, J., Liu, C., Jing, J., Ou, Z. Y. & Zhang, W. Quantum metrology with parametric amplifier based photon correlation interferometers. Nature Commun. 5, 3049 (2014).
Plick, W. N., Dowling, J. P. & Agarwal, G. S. Coherentlightboosted, subshot noise, quantum interferometry. New J. Phys. 12, 083014 (2010).
Marino, A. M., Corzo Trejo, N. V. & Lett, P. D. Effect of losses on the performance of an SU(1,1) interferometer. Phys. Rev. A 86, 023844 (2012).
Holland, M. J. & Burnett, K. Interferometric detection of optical phase shifts at the Heisenberg limit. Phys. Rev. Lett. 71, 1355 (1993).
Dorner, U. et al. Optimal Quantum Phase Estimation. Phys. Rev. Lett. 102, 040403 (2009).
Giovannetti, V., Lloyd, S. & Maccone, L. Quantum Metrology. Phys. Rev. Lett. 96, 010401 (2006).
Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nature Photon. 5, 222 (2011).
Sanders, B. C. & Milburn, G. J. Optimal Quantum Measurements for Phase Estimation. Phys. Rev. Lett. 75, 2944 (1995).
Ma, J., Wang, X., Sun, C. P. & Nori, F. Quantum spin squeezing. Phys. Rep. 509, 89 (2011).
Humphreys, P. C., Barbieri, M., Datta, A. & Walmsley, I. A. Quantum Enhanced Multiple Phase Estimation. Phys. Rev. Lett. 111, 070403 (2013).
Pezzé, L., Smerzi, A., Khoury, G., Hodelin, J. F. & Bouwmeester, D. Phase Detection at the Quantum Limit with Multiphoton MachZehnder Interferometry. Phys. Rev. Lett. 99, 223602 (2007).
Pezzé, L. & Smerzi, A. Ultrasensitive TwoMode Interferometry with SingleMode Number Squeezing. Phys. Rev. Lett. 110, 163604 (2013).
Dowling, J. P. Contemp. Phys. 49, 125 (2008).
Lvovsky, A. I., Sanders, B. C. & Tittel, W. Optical quantum memory. Nature Photon. 3, 706 (2009).
Ma, J., Huang, Y. X., Wang, X. & Sun, C. P. Quantum Fisher information of the GreenbergerHorneZeilinger state in decoherence channels. Phys. Rev. A 84, 022302 (2011).
Huelga, S. F., Macchiavello, C., Pellizzari, T., Ekert, A. K., Plenio, M. B. & Cirac, J. I. Improvement of Frequency Standards with Quantum Entanglement. Phys. Rev. Lett. 79, 3865 (1997).
Escher, B. M., de Matos Filho, R. L. & Davidovich, L. General framework for estimating the ultimate precision limit in noisy quantumenhanced metrology. Nature Phys. 7, 406 (2011).
Gao, Y. Quantum optical metrology in the lossy SU(2) and SU(1,1) interferometers. Phys. Rev. A 94, 023834 (2016).
Law, C. K., Pu, H. & Bigelow, N. P. Quantum Spins Mixing in Spinor BoseEinstein Condensates. Phys. Rev. Lett. 81, 5257 (1998).
Ho, T.L. Spinor Bose Condensates in Optical Traps. Phys. Rev. Lett. 81, 742 (1998).
StamperKurn, D. M. et al. Optical Confinement of a BoseEinstein Condensate. Phys. Rev. Lett. 80, 2027 (1998).
Chang, M.S., Qin, Q., Zhang, W., You, L. & Chapman, M. S. Coherent spinor dynamics in a spin1 Bose condensate. Nature Phys. 1, 111 (2005).
StamperKurn, D. M. & Ueda, M. Spinor Bose gases: Symmetries, magnetism, and quantum dynamics. Rev. Mod. Phys. 85, 1191 (2013).
Linnemann, D. et al. QuantumEnhanced Sensing Based on Time Reversal of Nonlinear Dynamics. Phys. Rev. Lett. 117, 013001 (2016).
Gabbrielli, M., Pezzé, L. & Smerzi, A. SpinMixing Interferometry with BoseEinstein Condensates. Phys. Rev. Lett. 115, 163002 (2015).
Szigeti, S. S., LewisSwan, R. J. & Haine, S. A. PumpedUp SU(1,1) Interferometry. Phys. Rev. Lett. 118, 150401 (2017).
Stenger, J., Inouye, S., StamperKurn, D. M., Miesner, H.J., Chikkatur, A. P. & Ketterle, W. Spin domains in groundstate BoseEinstein condensates. Nature 396, 345 (1998).
Vengalattore, M., Leslie, S. R., Guzman, J. & StamperKurn, D. M. Spontaneously Modulated Spin Textures in a Dipolar Spinor BoseEinstein Condensate. Phys. Rev. Lett. 100, 170403 (2008).
Barrett, M. D., Sauer, J. A. & Chapman, M. S. AllOptical Formation of an Atomic BoseEinstein Condensate. Phys. Rev. Lett. 87, 010404 (2001).
Santos, L., Shlyapnikov, G. V., Zoller, P. & Lewenstein, M. BoseEinstein Condensation in Trapped Dipolar Gases. Phys. Rev. Lett. 85, 1791 (2000).
Yi, S. & You, L. Trapped atomic condensates with anisotropic interactions. Phys. Rev. A 61, 041604(R) (2000).
Yi, S., You, L. & Pu, H. Quantum Phases of Dipolar Spinor Condensates. Phys. Rev. Lett. 93, 040403 (2004).
Huang, Y., Zhang, Y., Lü, R., Wang, X. & Yi, S. Macroscopic quantum coherence in spinor condensates confined in an anisotropic potential. Phys. Rev. A 86, 043625 (2012).
Pu, H., Zhang, W. & Meystre, P. Ferromagnetism in a Lattice of BoseEinstein Condensates. Phys. Rev. Lett. 87, 140405 (2001).
Zhang, W., Yi, S., Chapman, M. S. & You, J. Q. Coherent zerofield magnetization resonance in a dipolar spin1 BoseEinstein condensate. Phys. Rev. A 92, 023615 (2015).
Xing, H., Wang, A., Tan, Q. S., Zhang, W. & Yi, S. Heisenbergscaled magnetometer with dipolar spin1 condensates. Phys. Rev. A 93, 043615 (2016).
Kajtoch, D. & Witkowska, E. Spin squeezing in dipolar spinor condensates. Phys. Rev. A 93, 023627 (2016).
Helstrom, C. W. Quantum Detection and Estimation Theory. (Academic Press, New York, 1976).
Holevo, A. S. Probabilistic and Statistical Aspects of Quantum Theory. (NorthHolland, Amsterdam, 1982).
Macrì, T., Smerzi, A. & Pezzé, L. Loschmidt echo for quantum metrology. Phys. Rev. A 94, 010102 (2016).
Hamley, C. D. et al. Spinnematic squeezed vacuum in a quantum gas. Nat. Phys. 8, 305 (2012).
Huang, Y., Xiong, H. N., Sun, Z. & Wang, X. Generation and storage of spinnematic squeezing in a spinor BoseEinstein condensate. Phys. Rev. A 92, 023622 (2015).
Giovanazzi, S., Görlitz, A. & Pfau, T. Tuning the Dipolar Interaction in Quantum Gases. Phys. Rev. Lett. 89, 130401 (2002).
Lu, M., Burdick, N. Q., Youn, S. H. & Lev, B. L. Strongly Dipolar BoseEinstein Condensate of Dysprosium. Phys. Rev. Lett. 107, 190401 (2011).
Tan, Q. S., Lu, H. Y. & Yi, S. Spin squeezing of a dipolar Bose gas in a doublewell potential. Phys. Rev. A 93, 013606 (2016).
Acknowledgements
Q.S.T. thanks Professor Su Yi for valuable discussions. This work was supported by the National Natural Science Foundation of China under Grants Nos 11375060, 11775075, and 11434011 (Kuang), Nos 11547159, 11375059, and 11565011 (Xie) and 11665010 (Tan).
Author information
Authors and Affiliations
Contributions
Q.S.T. contributed to the initial idea and performed the calculation. Q.S.T. and L.M.K. cowrote the paper. Q.S.T., Q.T.X. and L.M.K. analyzed and discussed the results. All authors reviewed the manuscript and agreed with the submission.
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Tan, QS., Xie, QT. & Kuang, LM. Effects of dipolar interactions on the sensitivity of nonlinear spinorBEC interterometry. Sci Rep 8, 3218 (2018). https://doi.org/10.1038/s41598018215669
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598018215669
This article is cited by

Controlling quantum coherence of a twocomponent Bose–Einstein condensate via an impurity atom
Quantum Information Processing (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.