Near Heisenberg limited parameter estimation precision by a dipolar Bose gas reservoir engineering

One of main goals of quantum metrology is to achieve parameter (or phase) estimation precision beyond the shot-noise limit Caves1981 ; Yurke1985 ; Holland1993 ; Giovannetti2011 ; Dorner2009 ; Sanders1995 ; Ma2011 ; Ma2011B ; Humphreys2013 ; Dowling2008 ; Lvovsky2009 . Atomic spin squeezed states (SSSs) play an important role in quantum phase estimation and have been widely studied  Kitagawa1993 ; Wineland1994 ; Momer ; Sorensen2001 ; Poulsen2001 ; Jin2009 ; Jin2010 ; Guehne2009 ; Wang2003 ; Liu2011 in the past few decades ever since the pioneer work of Kitagawa and Ueda Kitagawa1993 , who showed that the SSSs can be dynamically generated from the so-called one-axis twisting interaction (OAT) among spin-$1/2$ particles bou ; kas . As useful quantum resource, the SSSs have been proposed to achieve such a sub-shot-noise limited phase sensitivity Wineland1994 ; Momer ; Sorensen2001 ; Poulsen2001 ; Jin2009 ; Jin2010 ; Guehne2009 ; Wang2003 ; Liu2011 . Recently, Stroble et al. strobel experimentally demonstrated that the OAT can also generate entangled non-Gaussian states (ENGSs), which can outperform the spin-squeezed state. The OAT interaction have been proposed and demonstrated using ion traps Momer , Rydeberg atoms macri , nitrogen-vacancy centers Bennett and atomic Bose-Einstein condensates (BECs).

BECs due to their unique coherence properties and the controllable nonlinearity bou ; kas , have attracted much attention for quantum metrology. Experimental realizations of the OAT model has been proposed and demonstrated through Feshbach resonances gros or spatially separating the components of BECs rie . Besides, the atomic BECs also often as the reservoirs suitable for engineering is considered widely Palzer ; Will ; Cirone ; Spethmann ; Scelle ; Zipkes ; Schmid ; Balewski ; Recati ; Ng ; bargill ; Bru . For instance, one can drastically enhance the OAT interaction by placing a two-state condensate in a completely different special BEC reservoir bargill .

So far, the studies of bosonic atoms for metrology are mainly focused on $s$-wave contact interaction. However, for ultracold atoms there also exists long-rang magnetic dipole-dipole interaction (MDDI) yi2000 ; Goral2002 ; yi ; yi2007 ; lu2 . In experiments, dipolar BECs have been realized for atoms with large magnetic dipole moments Lu ; Aikawa . Furthermore, both the sign and the strength of the effective dipolar interaction can be tuned via a fast rotating orienting field Giovanazzi ; Griesmaier2006 ; Lahaye . Very recently, Yuan et al. have used the quasi-2D dipolar BEC as reservoir engineering to study the non-Markovian dynamics of an impurity atom yuan . Therefore, the effects of the MDDI should be considered in the realizations of the OAT model based on dipolar BEC.

In this paper, we realize the OAT model induced by the reservoir dephasing noise, which has been widely viewed as one of the main obstacles for quantum metrology. We consider the dynamics of two-mode BEC consisting of $N$ atoms coupled to a one-dimensional (1D) dipolar Bose gas reservoir. It show that, the collisions interaction between the dipolar BEC reservoir and the immersed atoms can be described by a spin-boson model. Through calculating the SS $\xi_{R}$, and the quantum Fisher information (QFI) $F_{Q}$, we find that the dephasing noise can produce SSSs and ENGSs. And the degree of the SS and entanglement both depends on the relative strength and sign of the dipolar and contact interaction. In other words, the repulsive dipolar interaction reservoir can induce better SS and ENGSs. Compared with spin squeezed states, ENGSs can last for a very long time under the dephasing noise. It can monotonically increase in the regimes without SS ($\xi_{R}>1$), next successively undergoes metastable entangled states and entanglement suddenly increase, corresponding to $F_{Q}\simeq N^{2}/2$ and $F_{Q}\sim N^{2}$, respectively. According to Cramér-Rao theorem, $\Delta\theta\geq 1/\sqrt{F_{Q}}$, we know that $F_{Q}>N$ means that the states are entangled and useful for sub-shot-noise-limited phase-estimation precision; and $F_{Q}=N^{2}$ is the maximal entangled states, corresponding to the Heisenberg limit. This confirms that the phase estimation sensitivity can approach to Heisenberg limit, when using ENGSs for metrology.

The paper is organized as follows. In Sec. II, we give the model of immersed atoms interacting with the thermally equilibrated quasi-1D dipolar BEC reservoir. In Sec. III, we study the dynamics evolution of the atoms due to the dephasing noise. The SS and entanglement dynamical behaviors are discussed in Sec. IV and Sec. V. Finally, we draw our conclusion in Sec. VI.

We consider a system of $N$ two states ${}^{87}\mathrm{Rb}$ atoms with up $|\uparrow\rangle\equiv|F=2,m_{F}=-1\rangle$ and down states $|\downarrow\rangle\equiv|F=1,m_{F}=1\rangle$ immersed in a quasi-1D dipolar gas reservoir. And the system atoms are confined in a harmonic trap that is independent of the internal states [see Fig.1(a)]. In general, the interaction between the system atoms and the reservoir is described by the Hamiltonian

where $H_{A}$ is the two-state atomic Hamiltonian, $H_{B}$ is the dipolar gas reservoir Hamiltonian, and $H_{AB}$ describes their interaction.

In the interaction picture with respect to ${H}_{B}^{\prime}$, the total Hamiltonian is

which is a non-Hermitian dephasing spin-boson model with $N_{\uparrow}=\left(J_{z}+N/2\right)$ being the up state number operator. Here the non-Hermitian term $\Gamma_{\mathrm{loss}}$ is phenomenally introduced to describe the one-boby particle loss rate, owing to inelastic collisions between the system atoms and the noncondensed thermal atoms. It results in the particles be kicked out from the system. Such a kind of loss is a typical dissipation effect and has been widely studied strobel ; pawlowski ; hao ; spehner ; huang .

By using of Magnus expansion bargill , the time evolution operator can be read as $U(t)=e^{-itH_{\mathrm{eff}}},$ where the effective Hamiltonian is (see Appendix B for details)

with $\lambda^{\prime}\equiv\lambda+\delta_{\uparrow}-N\Delta(t)$ and $\alpha_{k}\equiv g_{k}(1-e^{i\omega_{k}t})/(t{\omega_{k}})$. From the above equation, we can find that the collisional interaction between the atoms and reservoir induce a nonlinear term $\propto J_{z}^{2}$, corresponding to the OAT Hamiltonian, and the noise induced nonlinear strength is Breuer

Here, the reservoir spectral density $J(\omega)$ defined as $J(\omega)=\sum_{k\neq 0}\left|g_{k}\right|^{2}\delta(\omega-\varepsilon_{k}/\hbar)$. In the continuum limit, $L^{-1}\sum_{k}\rightarrow(2\pi)^{-1}\int dk$, we have

where $f(k)\equiv k^{2}e^{-k^{2}\ell_{A}^{2}/2}$ with $k_{i}(\omega)$ being the roots of the equation $\varepsilon(k)=\hbar\omega,$ and the dimensionless parameter is

Assuming that the initial state of the total system is given by

where $\left|\Phi(0)\right\rangle_{A}\equiv\frac{1}{2^{N/2}}\left(\left|\uparrow\right\rangle+\left|\downarrow\right\rangle\right)^{\otimes N}=\sum_{m}c_{m}(0)\left|j,m\right\rangle$ is CSS, with the probability amplitudes $c_{m}=2^{-j}\left(C_{2j}^{j+m}\right)^{1/2}$ and total spin $j=N/2$ for a system consisting of $N$ condensated atoms. And the density matrix of reservoir read as

with $\beta$ the inverse temperature. With the help of Eq. (18), the time-evolution reduced matrix elements of the atom system at any later time $t$ is found by tracing over the reservoir degrees of freedom

with decoherence function

which is a function of temperature $T$.

In Eq. (24), $\Delta(t)$ and $\gamma(t)$, respectively, correspond to the unitary and non-unitary evolution due to the effects of the reservoir. Equations (19) and (25) show that both $\Delta(t)$ and $\gamma(t)$ depend on the reservoir spectral density $J(\omega)$, which can be controlled by tuning the MDDI of the dipolar Bose gas reservoir  Giovanazzi ; Griesmaier2006 ; Lahaye . This is the main difference from Feshbach resonances method whose control is on the system atoms directly and will strongly enhance three-body loss near resonances regime beside the one-body loss. Therefore, in our model the one-body particle loss due to the inelastic collisions of the noncondensed atoms maybe the main loss mechanism that need considering. When temperature is low enough the values of one-body loss rate $\Gamma_{\mathrm{loss}}$ will be very small, since there is only a small number of thermal atoms with sufficient energy to knock atoms out of condensate Dalfovo . For simplicity, hereafter we only consider the case of $T\rightarrow 0$, and choose $\Gamma_{\mathrm{loss}}$ as a free parameter.

As a concrete example, we consider a BEC reservoir of ${}^{162}\mathrm{Dy}$ atoms, for which we have $\mu_{m}=9.9\mu_{B}$  and $a_{B}=112a_{0}$ with $\mu_{B}$ the Bohr magneton. This means that $a_{dd}\simeq 131a_{0}\$ and dipolar interaction is mainly attractive since $\epsilon_{dd}>1,$ then the attraction is stronger than the short-range repulsion. However, not only the contact interaction strength can be tunable via Feshbach resonance, but also both the sign and the strength of the effective dipolar interaction can be tuned by the use of a fast rotating orienting field. Later, we will consider the values of $\epsilon_{dd}\in[-1,1]$, which is repulsive (attractive) MDDI for $\epsilon_{dd}<0$ ($\epsilon_{dd}>0$).

Numerically, it is convenient to introduce the dimensionless units: $\hbar\omega_{\perp}$ for energy, $\omega_{\perp}^{-1}$ for time, and $\ell_{B}=[\hbar/(m\omega_{\perp})]^{1/2}$ for length. To obtain the values of dimensionless parameter $\eta$ and $\Theta$, we assume a quasi-1D trap with $\omega_{x}=2\pi\times 20\mathrm{Hz}$ and $\omega_{\perp}=2\pi\times 10^{3}\mathrm{Hz}$; and the corresponding harmonic oscillator width is $\ell_{B}=\ell_{A}\simeq 2.5\times 10^{-7}\mathrm{m}$. We assume that the linear density of the quasi-1D condensate is $n_{0}=10^{8}\mathrm{m^{-1}}$, and the $s$-wave scattering length between Rb and Dy atoms is $a_{AB}\sim\mathrm{5nm}$ yuan . Then, we shall take $\eta=5$ and $\Theta=1.5\times 10^{-2}$ in the results presented below.

Since both the SS and QFI depend on $\gamma(t)$ and $\Delta(t),$ let us first investigate the time dependence of dephasing factor. From Fig. 2, we can clearly see that the squeezing rate $\Delta(t)$ is nearly constant, $\Delta(\infty)$. Whereas $\gamma(t)$ is decreasing with time, what is more we can get very small $\gamma(t)$ (e.g.$10^{-3}-10^{-4}$) when $\epsilon_{dd}<0$. Comparing $\Delta(t)$ and $\gamma(t)$, we can also find that the values of $\Delta(t)$ (squeezing rate) is larger than $\gamma(t)$ (dispersive rate) for $\epsilon_{dd}<0$, which means that we can obtain strong squeezing and large QFI by the reservoir’s engineering with repulsive MDDI.

In Fig. 3, we plot the SS $\xi^{2}_{R}$ dynamics of two-mode BEC consisting $N$ atoms coupled to a 1D dipolar Bose gas reservoir for various $\epsilon_{dd}$ values. As is shown in Fig. 3(a), the optimal squeezing can be achieved within short time scale, and after a transient time, it is lost ($\xi_{R}^{2}>1$) and then ENGSs are produced. However, for repulsive-dipolar-interaction reservoir, we can obtain stronger SS, due to their smaller dispersive rate $\gamma(t)$. In Fig. 3(b), we further plot the time evolution of $\xi^{2}_{R}$ for various atom loss rates. It indicates that the squeezing degree degraded with the increasing of the atom loss rate.

Figure 4 illustrates the QFI amplification rates with respect to the initial state (CSS) ${F}_{Q}/N$. In Fig. 4(a), we present the time dependence of QFI amplification rates with repulsive interaction. We see that differ from the case of SS, the amplified QFI can last for a very long time, which means that the ENGSs can be produced and achieve the maximal even in the regimes without squeezing. The optimal QFI first monotonically increasing and then reach a metastable $\sim N/2$ in the $yz$ plane. Subsequently, the QFI suddenly increasing in the $x$-axis direction at the optimal interrogation time $t_{\mathrm{opt}}$. As shown in Fig. 4(b), the maximal amplification rate $F^{\mathrm{max}}_{Q}/N$ is proportional to the atom number $N$, and the scale factor is $\sim N$, which is the Heisenberg scale. What is more, compared with the attractive MDDI reservoir the repulsive interaction can induce larger QFI, this result is presented in Fig. 4(c).

Comparison of Figs. 4(a) and 5(a), we can see that for not too large atom loss rates $\Gamma_{\mathrm{loss}}$ the optimal evolution time is $t_{\mathrm{opt}}=\pi/[2\Delta(t)]$, which is the same as the case of $N=2$.

Figures 5(b) and (c) illustrate the QFI with respect to atom number $N$ for different values of loss parameters. As shown in Fig. 5(b), under the values of $\Gamma_{\mathrm{loss}}$ we considered, we can obtain near-Heisenberg scaling. Figures 5(c) plots the rates of the Heisenberg limit $F^{\mathrm{max}}_{Q}/N^{2}$ with respect to atom number $N$ for different values of loss parameters $\Gamma_{\mathrm{loss}}$. It indicates that for not too large $\Gamma_{\mathrm{loss}}$, we can obtain the Heisenberg scaling only with a prefactor. We can also see, when increasing the loss rate the values of $F^{\mathrm{max}}_{Q}/N^{2}$ degrades with the increasing of number of atoms $N$, since the collective dissipate rare $N\Gamma_{\mathrm{loss}}$ depends on $N$. Fortunately, for the low temperature limit we considered the values of $\Gamma_{\mathrm{loss}}$ is not too large, hence we can still obtain the robust sub-shot-noise-limited phase sensitivity.

To demonstrate the near Heisenberg-limited sensitivity with the ENGSs realized by our model, we calculate the fidelity between the optimal ENGSs and spin cat states Momer , which are the maximal entangled states and have the Heisenberg-limited sensitivity for metrology. It is given by

where $|\theta_{0},\phi_{0}\rangle\equiv e^{i\theta_{0}(J_{x}\sin\phi_{0}-J_{y}\cos\phi_{0})|j,j\rangle}$ is the CSS. With the definition of fidelity, we have

where $\varrho_{\mathrm{cat}}=|\Psi\rangle_{\mathrm{cat}}\langle\Psi|$.

Form Fig. 6, we can find that the fidelity depends on both the $\epsilon_{dd}$ and particle number $N$. The maximal values occurs at $\epsilon_{dd}=-1$. Because of the dissipative rate $\gamma(t)$, the fidelity decrease with the increase of $N$. As shown in Fig. 6(a), when $\Gamma_{\rm loss}=0$ we get $\mathcal{F}_{\varrho}\simeq 0.94,0.85$, and $0.80$ for $N=10,30$, and $50$. And the effect of $\Gamma_{\rm loss}$ on fidelity also is shown in Fig. 6(b).

In summary, we have realized the SSSs and ENGSs by immersing atoms in a thermally equilibrated quasi-1D dipolar BEC reservoir. It has demonstrated that the repulsive-dipolar-interaction reservoir can induce better SS and entanglement. We have shown that owing to the dephasing noise, even in the regimes without SS the ENGSs can successively undergo highly metastable entangled states and entanglement suddenly increase. To explain the highly sensitivity for metrology, we calculated the fidelity between the optimal ENGSs and spin cat states, and found that the optimal ENGS is similar to the spin cat state. It has confirmed that by the use of ENGSs for metrology, the phase estimation sensitivity can surpass that by SS and even approach to Heisenberg limit for neglectable atom loss rates. The effect of the atom loss rate as a free parameter has also been considered.

Finally, we give two remarks on the above obtained results: First, these results we have obtained in this paper are based on the negligible spatial evolution of the immersed condensate wave functions. Usually this assumption is enough to capture the basic processes and physics, and detailed consideration of the impact of spatial dynamics can be investigated by adopting the multi-configurational time-dependent Hartree for bosons method stre2006 . Second, the scheme we proposed in this work can also suit the case that the system atoms are weak or no interaction two-level impurity atoms which are not condensate.