Planar squeezing by quantum non-demolition measurement in cold atomic ensembles

Quantum squeezing, the reduction of uncertainty in quantum observables below 'classical' or standard quantum limits (SQLs) [1–3], is an area of active research with applications in quantum information processing [4–8], quantum communications [9–11] and quantum metrology [12–14]. Here we consider a practical, measurement-based strategy to produce a new kind of squeezing in atomic spin ensembles. By way of introduction, we note that a spin F obeys the spin uncertainty relations

$$\begin{equation} \Delta F_{z} \Delta F_{x} \geqslant \textstyle\frac{1}{2} {\vert\langle F_y \rangle\vert} \end{equation} \tag{ 1 }$$

and permutations (throughout we take ℏ = 1). These are distinctly different from the Heisenberg relation $\Delta X \Delta P{\geqslant }\frac {1}{2}$ that governs canonical variables X,P [15]. For canonical variables, the constant right-hand side (rhs) of the Heisenberg relation implies that reduction of the variance in one quadrature inevitably increases the variance of the other. In contrast, in spin systems the rhs of the uncertainty relation may vanish, e.g. if 〈F_y〉 = 0, with the consequence that two spin components, e.g. F_z and F_x, may be simultaneously squeezed, with the uncertainty being absorbed by the third component. We note that this is subtly different from the Einstein–Podolsky–Rosen (EPR) entangled state studied in [16] where the sum and difference of canonical variables of two different spin systems were simultaneously squeezed. An analogous situation involving the Stokes parameters of light, which themselves obey angular momentum commutation relations, has also been studied [17–19]. For atomic spin ensembles, the suggestion of 'intelligent' spin states [20] predates even the concept of optical squeezing, and planar quantum squeezing (PQS) has recently been analyzed and proposed for quantum metrological applications [21, 22].

As described in [21, 22], PQS has the prospect of improving the precision of atomic interferometers at arbitrary phase angles, as it allows both orthogonal variances to be below the shot-noise level [23, 24]. This is opposed to squeezing on a single direction, which is only beneficial to refine the estimate of a phase angle already known with some precision (see figure 1). High-bandwidth atomic magnetometry [25, 26], in which the precession angle may not be predictable in advance, would be a natural application. At the same time, there is interest in the entanglement properties of PQS states, which can be detected by spin-squeezing inequalities [21, 22, 27].

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He et al [21] proposed an implementation of PQS using a two-well Bose condensate with tunable and attractive interactions. Here, we propose an alternative strategy in which the PQS state is prepared by sensitive [28] non-destructive [29] spin measurements. Sufficiently good measurements satisfy quantum non-demolition (QND) criteria [30] and produce (conditional) squeezing [14] of the measured spin variable. Stroboscopic probing a polarized ensemble of cold ⁸⁷Rb atoms, precessing in a constant external magnetic field, allows sequential measurement of two orthogonal components. As we show, this can squeeze two spin components simultaneously. We work with the collective total angular momentum variable F for spin-1 systems. The collective atomic spin is measured using paramagnetic Faraday rotation with an off-resonant probe. The ensemble of spins, polarized along a fixed direction by optical pumping, interacts with an optical pulse of duration τ and polarization described by the Stokes vector S = (S_x,S_y,S_z), through an effective Hamiltonian.

The paper is structured as follows: in section 2, we describe the formalism for light–atom interactions in polarized atomic ensembles and define spin-squeezing parameters. In section 3, we report numerical studies of PQS by QND measurement. In section 4, we discuss entanglement detection using spin-squeezing inequalities for the generated PQS states. In section 5 we provide an example for the applications of these ideas in high bandwidth (single shot) atomic magnetometry.

Our modeling is based on covariance matrix techniques introduced by Madsen and Mølmer [31–33], extended to include inhomogeneities by Koschorreck and Mitchell [34], to three spin components by Toth and Mitchell [35] and to spin-1 atoms by Colangelo et al [36]. In brief, we consider a magnetic field B oriented along the y-axis, in interaction with a spin-1 ensemble, e.g. ⁸⁷Rb in the F = 1 ground state, and probed using Faraday rotation probing by near-resonant light pulses. The ensemble is described by collective spin observables, for example $\skew3\hat {F}_x \equiv \sum _i \skew3\hat {f}_x^{(i)}$, where $\skew3\hat {F}_x^{(i)}$ indicates the x-component of the spin of the ith atom. Here F_i correspond to the macroscopic magnetization components, which precess about the external magnetic field B at the Larmor frequency. Both spin orientation variables $\skew3\hat {F}_{\!x}, \skew3\hat {F}_{\!y},\skew3\hat {F}_{\!z}$ and also spin alignment variables $\skew4\hat {J}_{\!x},\skew4\hat {J}_{\!y},\skew4\hat {J}_{\!k},\skew4\hat {J}_{\!l},\skew4\hat {J}_{\!m}$ defined using $\skew4\hat {j}_{\!x} \equiv \skew3\hat {f}_x^2 - \skew3\hat {f}_y^2$, $\skew4\hat {j}_y \equiv \skew3\hat {f}_x \skew3\hat {f}_y + \skew3\hat {f}_y \skew3\hat {f}_x$, $\skew4\hat {j}_{\!k} \equiv \skew4\hat {j}_{\!k} =\skew3\hat {f}_{\!x}\skew3\hat {f}_{\!z}+\skew3\hat {f}_{\!z} \skew3\hat {f}_{\!x}, \skew4\hat {j}_{\!l} =\skew3\hat {f}_{\!y} \skew3\hat {f}_{\!z}+\skew3\hat {f}_{\!z} \skew3\hat {f}_{\!y}, \skew4\hat {j}_{\!m}=\frac {1}{\sqrt {3}}(2\skew3\hat {f}_{\!z}^2-\skew3\hat {f}_{\!x}^2-\skew3\hat {f}_{\!y}^2)$, respectively, are required to describe the spin-1 system. The optical pulses are described in terms of Stokes operator components $\hat {S}_x,\hat {S}_y, \hat {S}_z$, defined as

$$\begin{equation} \skew3\hat{S}_x\equiv\textstyle\frac{1}{2}\hat{\bf{a}}^{\dagger}\boldsymbol{\sigma}_{\!x}\hat{\bf{a}}, \quad \skew3\hat{S}_y\equiv\textstyle\frac{1}{2}\hat{\bf{a}}^{\dagger}\boldsymbol{\sigma}_{\!y}\hat{\bf{a}}, \quad \skew3\hat{S}_z\equiv\textstyle\frac{1}{2}\hat{\bf{a}}^{\dagger}\boldsymbol{\sigma}_{\!z}\hat{\bf{a}}, {\quad \skew3\hat{S}_0\equiv\textstyle\frac{1}{2}\hat{\bf{a}}^{\dagger}{\mathbbm{1}}\hat{\bf{a}}}, \end{equation} \tag{ 2 }$$

where $\hat {\bf{a}}\equiv (\hat a_+,\hat a_-)^{\mathrm {T}}$ and $\hat a_+, \hat a_-$ are the annihilation operators for the left and right circular polarization, σ_x, σ_y, σ_z the Pauli matrices and ${\mathbbm{1}}$ is the identity matrix. The $\hat {S}_x$, $\hat {S}_y$ and $\hat {S}_z$, $\hat {S}_0$ Stokes operators represent polarized light in the horizontal or vertical direction, polarized light in the ±45° direction, right-hand or left-hand circularly polarized light and the pulse energy respectively. As mentioned, they obey the same commutation relations as angular momentum operators.

The full computational machinery is described in detail in [36]. In brief, we use a phase-space vector to describe the state of the whole system

$$\begin{equation} \hat{\bf{V}} = {\bf B} \oplus {\hat{\bf{F}}} \oplus {\hat{\bf{J}}} \oplus \bigoplus_{{m=1}}^{N_{\rm pulses}} {\hat{\bf{S}}}^{(m)}, \end{equation} \tag{ 3 }$$

where ⊕ indicates the direct sum, the superscript ^(m) indicates the mth optical pulse and B is the magnetic field vector at the location of the ensemble. B is here a classical field, whereas the other components of $\hat {\bf {V}}$ are operators. We work within the Gaussian approximation, i.e. we assume that $\hat {\bf {V}}$ is fully characterized by its average $\langle {\hat {\bf {V}}}\rangle$ and by its covariance matrix Γ_V :

$$\begin{equation} \Gamma_{\bf{V}}\equiv\textstyle\frac{1}{2}\langle\hat{\bf{V}}\wedge\hat{\bf{V}}+(\hat{\bf{V}}\wedge\hat{\bf{V}})^{\mathrm{T}}\rangle-\langle\hat{\bf{V}}\rangle\wedge\langle\hat{\bf{V}}\rangle, \end{equation} \tag{ 4 }$$

where ∧ indicates the outer product. The phase space vector $\hat {\bf {V}}$ evolves under the effect of the Hamiltonian

$$\begin{equation} H_{\mathrm{eff}} = -g_{\mathrm{F}} \mu_{\mathrm{B}} {\bf B}{\cdot} {\hat{\bf F}}+\frac{1}{\tau}[G_{1}\skew3\hat{S}_{z} \hat{F}_{z} + G_{2}(\skew3\hat{S}_{x}\hat{J}_{x}+\skew3\hat{S}_{y}\hat{J}_{y}{+\frac{1}{\sqrt{3}} \skew3\hat{S}_{0}\hat{J}_{m}})], \end{equation} \tag{ 5 }$$

where μ_B is the Bohr magneton and g_F is the gyromagnetic factor, τ is the pulse duration and G_1,2 are vectorial and tensorial coupling coefficients, respectively. Also included in the evolution are the decohering effects of incoherent scattering of probe photons and inhomogeneous magnetic fields [34].

The strategy to generate planar squeezing is illustrated in figure 2, and can be described as follows: an initial state is prepared and made to precess in the x–z plane by a magnetic field oriented along y (the ${\bf{ B}}{\cdot } {\hat {\bf{ F}}}$ term in H_eff). Pulses of light, short on the time-scale of the precession, and with initial polarization along S_x, are sent through the ensemble at quarter-cycle intervals. Through the $\hat {S}_{z} \skew3\hat {F}_{z}$ term, the light–atom interaction rotates the polarization from S_x toward S_y by an angle proportional to $\skew3\hat {F}_z$. A measurement of S_y then gives an indirect measurement of F_z, reducing its uncertainty. If this measurement is sufficiently precise it leaves F_z squeezed. A quarter-cycle later, the state has precessed so that the F_x component can be measured, and possibly squeezed. The terms proportional to G₂ couple the orientation (F) components to the alignment (J) components, and for this application are simply an inconvenience. We suppress the effect of the G₂ term by dynamical decoupling, as described in [29].

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While a given spin component can be squeezed by non-demolition measurement of that component, it is perhaps not obvious that planar squeezing, which requires simultaneous squeezing of non-commuting observables, should be producible by measurement. There is, after all, no projective measurement that indicates both F_x and F_z. Nevertheless, if 〈F_y〉 ≈ 0, the uncertainty relation of equation (1) does not impose a quantum back-action on F_x when F_z is measured, nor vice versa. This suggests that sequentially measuring F_x and F_z should squeeze both spin components and thus give planar squeezing.

The operational definition we adopt for planar squeezing follows the approach by He et al [21, 22], and can be summarized as follows. First, from equation (1) and permutations, we take Δ²F_x = Δ²F_z = F_||/2 as the SQL, where $F_{\vert \vert } \equiv \sqrt {F_x^2 + F_z^2}$, so that F_|| is the magnitude of the in-plane spin components. Considering then the noise in the plane, we define the planar variance Δ²F_|| ≡ Δ²F_x + Δ²F_z, with SQL Δ²F_|| = F_||, and the planar squeezing parameter

$$\begin{equation} \xi_{\vert\vert}^2 \equiv \frac{\Delta^2 F_{\vert\vert}}{F_{\vert\vert}}. \end{equation} \tag{ 6 }$$

A planar squeezed state has ξ²_|| < 1, and also has individual component variances below the SQL, i.e. ξ²_x < 1, and ξ²_z < 1 , where ξ²_i ≡ 2Δ²F_i/F_||, so that ξ²_|| = (ξ²_x + ξ²_z)/2. We can also define a metrological squeezing parameter similar to the Wineland criterion [2] (see also equation (33) of et al [22])

$$\begin{equation} \xi_{{\vert\vert},\mathrm{M}}^2 \equiv \frac{F \Delta^2 F_{\vert\vert}}{F_{\vert\vert}^2}, \end{equation} \tag{ 7 }$$

where F = 〈N_A〉 is the number of spins. A PQS state with ξ²_||,M < 1 gives enhanced metrological sensitivity to arbitrary phase shifts. In order to achieve this kind of state, it is convenient to have 〈F_y〉 = 0 so that the uncertainties on the plane are only constrained by ΔF_xΔF_z ⩾ 0. The uncertainty reduction in the two planar components comes at the expense of increasing the noise in the third component, as required by ΔF_z(x)ΔF_y ⩾ 〈F_x(z)〉/2.

We simulate an experiment illustrated in figure 2. An average of 〈N_A〉 = 1.25 × 10^{6 87}Rb atoms are cooled in the F = 1 ground state and held in a weakly focused single beam optical dipole trap with an effective optical depth (OD) α₀ ≈ 65 [38]. To produce a PQS, we apply a magnetic field B_y to coherently rotate an initially F_x polarized coherent spin state in the x,z plane, and stroboscopically probe the spins at four times the Larmor frequency. We probe the atoms with μs pulses of linearly polarized off-resonant light detuned by Δ = −2π × 10 GHz on the D₂ line, detected by a shot-noise limited polarimeter. The pulses are sent through the atoms in pairs of alternating h- and v-polarization in order to cancel the effect of tensorial light shifts [29]. Each pulse is τ = 1 μs long and contains 10⁹ photons, and the two pulses in a pair are separated by 1 μs. Each stroboscopic measurement consists of a single pair of pulses with a total duration of 3 μs.

The coupling constant between the light and the atoms is G₁ = 1.2 × 10⁻⁸, calculated for this probe detuning using the experimental parameters described in [14]. The atomic ensemble is initially polarized in the x-direction by optical pumping, i.e. into a state | + f_x〉^⊗N_A, where $\vert +f_x\rangle \equiv \frac {1}{2} (\vert m=-1\rangle +\sqrt {2}\vert m=0\rangle + \vert m=+1\rangle )$ and where N_A is subject to Poissonian fluctuations. This last condition is not imposed by quantum physics, which would allow N_A to be sharp, but rather represents the practical fact that trap loading is a stochastic process. We refer to this kind of state as a PCSS. The x-polarized PCSS has average values 〈F_x〉 = 〈N_A〉, 〈F_y〉 = 〈F_z〉 = 0 and variances Δ²F_y = Δ²F_z = 〈N_A〉/2 (due to quantum fluctuations) and Δ²F_x = 〈N_A〉 (due to ΔN_A). Further details of numerical simulations and the initial state are described in [36]. The external magnetic field along y, B_y = 14.3 mG, B_x = B_z = 0, produces Larmor precession in the (x,z)-plane, with a period of T_L = 100 μs, considerably longer than the ∼ 3 μs time required for the probing.

Results of numerical simulations for a typical condition are shown in figures 3 and 4. The ensemble of atoms is initially polarized in x-direction, and precesses in the (x,z)-plane due to the B-field at the Larmor frequency. As shown in figure 3(a), the mean value of the out-of-plane component remains near zero, i.e. 〈F_y〉 ≪ 〈N_A〉, so that the relevant uncertainty relation is ΔF_xΔF_z ⩾ |〈F_y〉|/2 imposes little back-action and the uncertainties in the (x,z) plane can be simultaneously reduced. The observed deviation from 〈F_y〉 = 0 is due to residual effects of the tensorial coupling term $G_{2}(\hat {S}_{x}\skew4\hat {J}_{x}+\hat {S}_{y}\skew4\hat {J}_{y}+{+\frac {1}{\sqrt {3}} \hat {S}_{0}\skew4\hat {J}_{m}})$ in the effective Hamiltonian.

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Figure 3(b) shows the variances of the spin components during the probing procedure. Because the initial state is a F = 1 PCSS, the individual (x,z) variances are initially unequal, but quickly take on similar values through measurement. They oscillate as a consequence of the Larmor precession, and are sequentially reduced due to the stroboscopic probing, which can be seen in the step-like jumps in the various curves. At the same time, the variance in the orthogonal direction Δ²F_y increases well above the SQL, as expected for a PQS state.

Figure 4(a) shows the planar squeezing parameter ξ²_||, which drops below unity indicating the production of planar squeezing by QND measurement.

3.1. Two-spin-component measurement-induced squeezing

3.2. Achievable squeezing versus optical depth

In the absence of scattering noise and decoherence, a QND measurement reduces the variance of the measured parameter by a factor 1/(1 + κ), where κ is the signal-to-noise ratio of the measurement, proportional to the number of photons used in the measurement. At the same time, spontaneous scattering events add noise (variance) and reduce F_||, by amounts approximately linear in the number of photons used. An often-used estimate of the trade-off between these effects [31, 32] finds a minimum squeezing parameter ξ² = 1/(1 + α₀η) + 2η, where η is the probability for any given atom to suffer a spontaneous emission event. This expression has a minimum $\xi ^{2}_{\mathrm { min}}=2\sqrt {(2/\alpha _0)}$ for the optimal η, which is $\eta _{0}=1/\sqrt {2 \alpha _0}$. For a typical system with optical density α₀ ≈ 25 this optimal value gives a lower bound on the amount of squeezing given by roughly ξ²_min ≈ 0.5 (3 dB of noise reduction) [39].

The predicted planar squeezing by this simple model is shown in figure 5(a) (red dashed line), for an OD ranging from 0 < α ⩽ 150. Also shown in figure 5(a) are full simulation results (inset), from which we can extract a more accurate estimate of the achievable planar squeezing, shown as blue symbols in the main graph. The difference at large α₀ (and thus large squeezing) may be attributable to finite precession angle between the h- and v-polarized parts of the probing. In the next section, we characterize the entanglement content of the generated planar squeezed states.

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A number of spin-squeezing inequalities can in principle detect entanglement in planar squeezed states. He et al [21] give a simple inequality, obeyed by all separable states:

$$\begin{equation} \frac{\Delta^2 F_{\vert\vert}}{\langle N_{\mathrm{A}} \rangle} \geqslant C_{F}, \end{equation} \tag{ 8 }$$

where C₁ = 7/16. As shown in figure 5(a), our procedure for producing planar squeezing can violate this inequality for ODs α₀ ≳ 35.

A generalized spin squeezing parameter from Vitagliano et al [27] (equation 5(d))

$$\begin{equation} \xi_d^2\equiv \frac{(\langle N_{\mathrm{A}} \rangle-1)[(\tilde{\Delta}F_x)^2+(\tilde{\Delta}F_z)^2]}{\langle\tilde{F}_y^2\rangle-\langle N_{\mathrm{A}} \rangle(\langle N_{\mathrm{A}} \rangle-1)F^2}, \end{equation} \tag{ 9 }$$

where $\langle \tilde {F}_i^2\rangle \equiv \langle F_i^2\rangle -\langle \sum _n (f_i^{(n)})^2\rangle$ and $(\tilde {\Delta }F_i)^2=(\Delta F_i)^2-\langle \sum _n (f_i^{(n)})^2\rangle$ are modified second order moments and variances, also detects entanglement (ξ²_d < 1) in planar squeezed states, as shown in figure 5(b).

Regarding the difficulty of detecting entanglement in PQS states, we note a significant difference relative to one-component spin squeezing. With just one component, the border between squeezed and un-squeezed states coincides with the border between separable and entangled states. That is, a CSS is a pure product state and also has SQL noise. In contrast, by the planar squeezing criterion of He et al [22] (given as equation (6)) a CSS (with a definite N_A), is already a planar squeezed state with ξ_|| < 1. In this sense, planar squeezing is easier to achieve than entanglement.

We now describe a possible application of PQS in optical magnetometry, for determination of arbitrary angles with precision below the SQL. States that are squeezed in only one component can give a metrological advantage over a limited range of angles. Protocols to employ these states either require prior knowledge of the phase (enough to place them within the range of improved sensitivity), or adaptive procedures to determine the phase during the measurement. In contrast, planar squeezed states can give improved sensitivity for any precession angle [21, 22]. Here we show that QND-generated PQS states are effective for this purpose.

We have in mind a measurement scheme in which we initially prepare a PCSS in the x–z plane and measure for a few Larmor precession cycles, preparing a PQS state as described above. The PQS state is then allowed to freely evolve for as long as the coherence time of the state allows, before we again measure to determine the phase shift accumulated during this free evolution. The conditionally squeezed PQS state allows us to predict the outcome of the second measurement with a sensitivity better than the SQL for arbitrary cumulative phase shifts.

We consider a state, initially oriented in the F_x direction, and allowed to precess in response to a field B along the y-direction. The precession angle versus time t will be ϕ = ω_Lt, where ω_L is the Larmor frequency, which in turn is proportional to B_y. After precession, the measured component F_z is

$$\begin{equation} F_z^{({\rm out})} = F_x^{({\rm in})} \sin \phi + F_z^{({\rm in})} \cos \phi, \end{equation} \tag{ 10 }$$

where superscripts ⁽ⁱⁿ⁾,^(out) indicate the operators before and after the precession, respectively. We can calculate the uncertainty in ϕ as Δ²ϕ = Δ²F^(out)_z/|d〈F^(out)_z〉/dϕ|² or, given that 〈F⁽ⁱⁿ⁾_z〉 = 0,

$$\begin{equation} \Delta^2 \phi = \frac{{\Delta^2 F(\phi)}} { \vert F\vert^2 \cos^2 \phi }, \end{equation} \tag{ 11 }$$

where Δ²F(ϕ) ≡ Δ²F⁽ⁱⁿ⁾_xsin²ϕ + Δ²F⁽ⁱⁿ⁾_zcos²ϕ + cov(F⁽ⁱⁿ⁾_x,F⁽ⁱⁿ⁾_z)sin 2ϕ, and ${\mathrm { cov}}(A,B) \equiv \frac {1}{2} \langle A B + BA \rangle - \langle A \rangle \langle B \rangle$ is the covariance. We note that PQS states reduce the planar variance for arbitrary angles on a finite interval, with the exception of the specific singular values which make the denominator in equation (11) equal to zero. This feature is depicted in figure 6(a), for realistic PQS states with different levels of technical noise, generated via QND measurements.

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Figure 6(b) shows a comparison of the phase variances, relative to the PCSS. The results show that while a single-component squeezed state (SSS) is more precise than the PCSS (and also the PQS) over a range of angles, the PQS gives a near-uniform advantage relative to the CSS and can be used for phase reconstruction below the SQL without prior knowledge of ϕ.

We have studied numerically the possibility to generate the recently proposed 'planar quantum squeezing', in which the variances of two orthogonal spin components are simultaneously squeezed, via quantum non-demolition measurement of cold atomic ensembles. We find that significant planar squeezing can be generated under realistic conditions and that this squeezing implies entanglement detectable with spin-squeezing inequalities. Considering the use of planar squeezed states in an optical magnetometry context, we find that planar squeezing can give a metrological advantage for estimation of arbitrary phase angles, whereas single-component spin squeezing is only advantageous for specific angle ranges. This is promising for high-bandwidth atomic magnetometry, in which a changing precession frequency may make it difficult or impossible to anticipate the precession phase.

The authors gratefully acknowledge Géza Tóth, Peter D Drummond and Giuseppe Vitagliano for fruitful discussions. They also would like to thank Naeimeh Behbood, Ferran Martin Ciurana and Mario Napolitano for helpful discussions. This work was supported by the Spanish MINECO under the project MAGO (ref. no. FIS2011-23520), by the European Research Council under the project AQUMET and by Fundació Privada Cellex. GP acknowledges financial support from Marie-Curie International Fellowship COFUND. All authors contributed equally to this work.