Spin Squeezing

Spin squeezing is the redistribution of fluctuations in a collective spin so that one transverse component is quieter than the coherent-spin reference. For spectroscopy and sensing, reduced noise is useful only if enough mean-spin length remains to convert a small rotation into a detectable signal. This is why two common squeezing parameters answer different questions:

• the Kitagawa–Ueda parameter asks whether a transverse spin variance is below coherent-spin noise;
• the Wineland parameter asks whether that variance, together with the signal response, enables a sub-standard-quantum-limit linear readout.

For $N$ effective spin-$1/2$ probes with a mean spin along $x$, the operational parameter is

$$\xi_R^2 = \frac{ N(\Delta J_\perp)^2 }{ |\langle\mathbf J\rangle|^2 }.$$

When the squeezed component is aligned with a Ramsey readout, one ideal trial has local phase variance

$$(\Delta\phi)^2 = \frac{\xi_R^2}{N}.$$

Thus $\xi_R^2<1$ means both a metrological gain over the matched coherent-spin state and, under the stated spin-$1/2$ model, a sufficient witness of multipartite entanglement. It does not establish a complete clock advantage: preparation time, atom loss, local-oscillator noise, detection noise, dynamic range, and accepted-shot rules still belong in the comparison.

Required background. Squeezing explains covariance and signal response; the standard quantum limit supplies the independent-spin comparison; Fisher information relates readout to precision.

Helpful background. Spin coherent states gives the rotated product-state construction, and covariance explains correlated fluctuations.

Collective Spin and the Coherent Reference

Take $N$ two-level systems and define dimensionless collective operators

$$J_k = \frac12 \sum_{i=1}^{N} \sigma_k^{(i)}, \qquad k\in\{x,y,z\}.$$

They obey

$$[J_i,J_j] = i\epsilon_{ijk}J_k.$$

Factors of $\hbar$ can be restored by replacing each $J_k$ with $\hbar J_k$. The collective language applies whenever the two modes of each probe can be treated as an effective qubit. They may be internal clock states, two momentum paths, two wells, or two interferometer arms. It does not imply that all physical spins or all couplings are literally identical.

The product state polarized along $+x$ is

$$|\mathrm{CSS}_x\rangle = |{+x}\rangle^{\otimes N}.$$

It lies in the symmetric total-spin sector $J=N/2$ and has

$$\begin{aligned} \langle J_x\rangle&=\frac N2, & \langle J_y\rangle&=\langle J_z\rangle=0,\\ (\Delta J_y)^2&=\frac N4, & (\Delta J_z)^2&=\frac N4. \end{aligned}$$

The $N/4$ transverse variance is quantum projection noise. It follows from adding $N$ independent binary fluctuations, not from technical uncertainty. The corresponding angular uncertainty is

$$\Delta\phi_{\mathrm{CSS}} = \frac{\Delta J_\perp}{|\langle\mathbf J\rangle|} = \frac1{\sqrt N}.$$

This is the independent-probe benchmark developed in Standard Quantum Limit. A fair squeezing claim compares against a coherent state with the same atom number, interrogation, detector efficiency, and total repetition budget, including unsuccessful preparation attempts.

Transverse Covariance Geometry

Suppose the mean spin points along $x$. In the transverse $y$–$z$ plane define

$$V = \begin{pmatrix} (\Delta J_y)^2 & \operatorname{Cov}(J_y,J_z)\\ \operatorname{Cov}(J_y,J_z) & (\Delta J_z)^2 \end{pmatrix},$$

where

$$\operatorname{Cov}(A,B) = \frac12\langle AB+BA\rangle - \langle A\rangle\langle B\rangle.$$

A transverse component at angle $\vartheta$ is

$$J_\vartheta = J_y\cos\vartheta + J_z\sin\vartheta.$$

Its variance is the quadratic form

$$(\Delta J_\vartheta)^2 = \mathbf u_\vartheta^{\mathsf T} V \mathbf u_\vartheta, \qquad \mathbf u_\vartheta = \begin{pmatrix} \cos\vartheta\\ \sin\vartheta \end{pmatrix}.$$

The minimum and maximum transverse variances are the eigenvalues

$$V_\mp = \frac{V_{yy}+V_{zz}}2 \mp \frac12 \sqrt{ (V_{yy}-V_{zz})^2 + 4V_{yz}^2 }.$$

For unequal eigenvalues, an unambiguous minimum-axis angle is

$$\vartheta_{\min} =\frac12\operatorname{atan2}(-2V_{yz},V_{zz}-V_{yy}) \pmod{\pi}.$$

If $V_{yy}=V_{zz}$ and $V_{yz}=0$, every transverse axis has the same variance. A nonlinear interaction often first creates a tilted ellipse. A collective rotation must then align its narrow axis with the component used by the estimator.

The Robertson–Schrödinger relation gives

$$V_{yy}V_{zz}-V_{yz}^2 \ge \frac14 |\langle J_x\rangle|^2.$$

Reducing $V_-$ therefore generally enlarges $V_+$, creates covariance, reduces the mean spin, or combines these effects. Spin squeezing redistributes quantum uncertainty; it does not remove the angular-momentum uncertainty relation.

Noise Squeezing and Metrological Squeezing

For $N$ spin-$1/2$ probes, the Kitagawa–Ueda parameter is

$$\xi_S^2 = \frac{4V_-}{N}.$$

It compares the narrowest transverse variance with the coherent-spin value $N/4$. Therefore

$$\xi_S^2<1$$

means transverse noise squeezing. This definition does not penalize loss of mean-spin length.

Define the Ramsey contrast or normalized spin length by

$$C = \frac{2|\langle\mathbf J\rangle|}{N}, \qquad 0\le C\le1.$$

The Wineland parameter is

$$\xi_R^2 = \frac{NV_-}{|\langle\mathbf J\rangle|^2} = \frac{\xi_S^2}{C^2}.$$

It is possible to have $\xi_S^2<1$ but $\xi_R^2\ge1$. The state is quieter than a coherent spin, yet its shortened mean spin provides too little signal slope for an improved linear Ramsey measurement.

For variance ratios, a positive quoted gain is conventionally

$$G_{\mathrm{dB}} = -10\log_{10}\xi_R^2.$$

Thus $\xi_R^2=0.25$ is $6.02\ \mathrm{dB}$ of metrological gain. The phase standard deviation improves only by

$$\xi_R = \sqrt{\xi_R^2} = 0.5.$$

Authors also use $\xi^2$, $\zeta^2$, or a signed decibel value for several inequivalent quantities. A reproducible result states the formula, reference, contrast treatment, and whether detection noise was included.

Ramsey Readout

Prepare a state with mean spin along $x$ and align the squeezed component with $J_y$. Let the unknown phase be encoded by

$$U_\phi = e^{-i\phi J_z}.$$

In the Heisenberg picture,

$$U_\phi^\dagger J_y U_\phi = J_y\cos\phi + J_x\sin\phi.$$

If $\langle J_y\rangle=0$, then near $\phi=0$,

$$\left. \frac{\partial\langle J_y\rangle_\phi}{\partial\phi} \right|_{\phi=0} = \langle J_x\rangle.$$

Error propagation gives

$$(\Delta\phi)^2 = \frac{(\Delta J_y)^2}{\langle J_x\rangle^2} = \frac{\xi_R^2}{N}.$$

After $\nu$ independent accepted repetitions,

$$(\Delta\phi)^2_\nu = \frac{\xi_R^2}{\nu N}.$$

The last two equations are local statements. They assume a calibrated midfringe, an approximately unbiased estimator, a known squeezing angle, and a phase prior narrow enough to avoid fringe aliases. Anti-squeezed noise can enter rapidly when the readout axis or local-oscillator phase is uncertain.

For a clock transition of frequency $f_0$, Ramsey dark time $T$, cycle time $T_c$, and averaging duration $\tau$, the atom-noise-only estimate is

$$\sigma_y(\tau) \simeq \frac{\xi_R}{2\pi f_0T\sqrt N} \sqrt{\frac{T_c}{\tau}}.$$

This expression omits the Dick effect, local-oscillator phase noise, dead-time changes caused by squeezing preparation, state-detection errors, and systematic frequency shifts. Those omissions are why a measured $\xi_R^2<1$ is not by itself a complete clock demonstration.

Tangent-plane approximation and Bloch-sphere curvature

Near a well-polarized state with total spin $J\simeq N/2$, define tangent-plane variables

$$X = \frac{J_y}{\sqrt J}, \qquad P = \frac{J_z}{\sqrt J}.$$

Because $[J_y,J_z]=iJ_x$,

$$[X,P] \simeq i$$

when $J_x\simeq J$. The coherent spin has

$$(\Delta X)^2 = (\Delta P)^2 = \frac12.$$

This Holstein–Primakoff regime explains why weak spin squeezing resembles oscillator quadrature squeezing. The approximation has a boundary: the spin phase space is a sphere, not an infinite plane. Strong nonlinear evolution can wrap or fold the distribution around the sphere. Then second moments and a linear $J_y$ readout may cease to capture the available phase information.

This oversqueezed regime can have poor $\xi_R^2$ even while the state has large quantum Fisher information. Nonlinear observables, parity, full distribution fitting, or an interaction-based echo may recover some of that information. It is therefore wrong to identify all metrologically useful many-body states with narrow Gaussian ellipses.

Entanglement and Fisher Information

For $N$ distinguishable spin-$1/2$ particles, every fully separable state obeys the collective-spin inequality

$$\xi_R^2 \ge 1.$$

Consequently,

$$\xi_R^2<1 \quad\Longrightarrow\quad \text{particle entanglement}.$$

One way to prove the bound uses the Bloch components $(x_i,y_i,z_i)$ of each constituent in a product state:

$$\operatorname{Var}(J_y)=\frac14\sum_i(1-y_i^2) \ge\frac14\sum_i x_i^2 \ge\frac{(\sum_i x_i)^2}{4N} =\frac{\langle J_x\rangle^2}{N}.$$

Variance is concave under classical mixing, while the squared mean is convex, so the bound extends to all fully separable mixed states. It does not require permutation symmetry.

The implication is one-way. GHZ and symmetric Dicke states can have zero mean spin and large QFI for an appropriate rotation axis, so the Wineland parameter is then undefined or uninformative. A total-spin singlet is different: $J_k|\psi\rangle=0$ for every collective component, giving zero QFI for a uniform collective rotation. Its entanglement can instead help detect a spatially nonuniform field, as Exercise 4 illustrates.

After aligning the squeezed readout with $J_y$ and the mean with $x$, a phase generated by $J_z$ has QFI bounded by the available linear readout:

$$F_Q[\rho,J_z] \ge \frac{N}{\xi_R^2}.$$

Thus $\xi_R^2<1$ certifies

$$F_Q[\rho,J_z]>N,$$

which is metrologically useful entanglement for that collective rotation. The reverse need not hold: $F_Q>N$ says that some measurement can beat the independent-probe bound locally, not that a first-moment spin readout does so.

Entanglement depth is the smallest $k$ for which a state is $k$-producible. A state is $k$-producible if it is a mixture of pure states whose entangled blocks contain at most $k$ particles. For $N$ qubits and a collective linear generator, such states satisfy

$$F_Q \le s k^2+r^2,$$

where

$$s = \left\lfloor\frac Nk\right\rfloor, \qquad r = N-sk.$$

Violating this bound rules out $k$-producibility. Spin-moment data can also be compared with the tighter polarization-dependent Sørensen–Mølmer curves. The often quoted estimate $1/\xi_R^2$ is not, by itself, an exact finite-$N$ entanglement depth. The witness must account for fluctuating atom number, inhomogeneous coupling, internal spin larger than $1/2$, and confidence intervals.

One-Axis Twisting

The canonical unitary mechanism is one-axis twisting,

$$H_{\mathrm{OAT}} = \hbar\chi J_z^2.$$

Starting from $|\mathrm{CSS}_x\rangle$, the evolution is

$$|\psi(t)\rangle = e^{-i\chi tJ_z^2} |\mathrm{CSS}_x\rangle.$$

Each $J_z$ sector acquires a phase proportional to $m_z^2$. On the tangent plane this produces a shear: slices with different $J_z$ rotate at different rates. The shear creates a nonzero $y$–$z$ covariance, so the minimum-noise axis is initially tilted. A linear spin rotation aligns that axis with the later readout.

In the ideal all-to-all collective model at large $N$,

$$\xi_{S,\min}^2 \propto N^{-2/3}, \qquad \chi t_{\mathrm{opt}} \propto N^{-2/3}.$$

Here the normalized noise scales as $N^{-2/3}$; with contrast near one, the phase variance scales as $N^{-5/3}$. An ideal $\xi_R^2\propto N^{-1}$ would instead give phase variance proportional to $N^{-2}$. The asymptotic law does not survive automatically when $\chi$ changes with system size or when scattering, finite-range interactions, loss, and control overhead are included.

For a finite-$N$ calculation, expand the initial product state in the symmetric $J_z$ basis:

$$|\psi(\mu)\rangle =2^{-N/2}\sum_{k=0}^N \sqrt{\binom Nk}\, e^{-i\mu(k-N/2)^2}|J=N/2,m=k-N/2\rangle, \qquad \mu=\chi t.$$

The matrix element $\langle J,m+1|J_+|J,m\rangle=\sqrt{J(J+1)-m(m+1)}$ then gives the mean spin and transverse covariance directly. In particular, $C=|\cos\mu|^{N-1}$. This calculation includes the spherical curvature that a tangent-plane shear eventually misses.

Exact collective-spin evolution for $N=100$ under $e^{-i\mu J_z^2}$, without loss or detection noise. The transverse noise $\xi_S^2$ falls initially, but contrast also decays. The useful linear-readout parameter $\xi_R^2=\xi_S^2/C^2$ eventually exceeds one even while the minimum transverse variance remains below the coherent-spin reference. The vertical axis is logarithmic.

Past the optimum, the distribution bends around the Bloch sphere. The minimum linear variance eventually worsens even though non-Gaussian correlations continue to develop. An echo

$$U_{\mathrm{echo}} = U_{\mathrm{OAT}}^\dagger e^{-i\phi J_n} U_{\mathrm{OAT}}$$

can map a small phase into an amplified collective signal. Such interaction-based readout trades strict time reversal and control accuracy for relaxed detector resolution.

Two-Axis Countertwisting and Unstable Dynamics

With the mean spin along $x$, one form of two-axis countertwisting is

$$H_{\mathrm{TACT}} = \hbar\chi (J_yJ_z+J_zJ_y).$$

After a $45^\circ$ rotation in the transverse plane this is proportional to the difference of two squared spin components. One transverse direction is contracted while the orthogonal direction expands. In the local Gaussian regime the squeezing is approximately exponential, and the ideal collective model can reach

$$\xi_{R,\min}^2 \propto \frac1N.$$

This is Heisenberg-like variance scaling under the specified collective-spin resource model. It is not a universal noisy scaling law. Exact two-axis interactions are harder to realize than one-axis twisting, so experiments may synthesize them with pulse sequences, continuous drives, Floquet engineering, or alternating interaction axes.

A related twist-and-turn model is

$$H_{\mathrm{TnT}} = \hbar\chi J_z^2 - \hbar\Omega J_x.$$

For $\chi>0$ and $\Omega>0$, the $+x$ fixed point is stable. The $-x$ fixed point is unstable when $0<\Omega<2\chi J=\chi N$: linearization gives

$$\dot J_y\simeq(\Omega-2\chi J)J_z,\qquad \dot J_z\simeq-\Omega J_y,$$

so perturbations grow at rate $\sqrt{\Omega(2\chi J-\Omega)}$. Near this unstable fixed point, fluctuations separate along stable and unstable directions. This can generate strong squeezing or non-Gaussian states faster than simple shearing, but it also increases sensitivity to initialization, timing, and Hamiltonian calibration.

Measurement-Induced Squeezing

A quantum-nondemolition measurement can acquire information about one collective component without directly randomizing that same component. For an atomic spin $J_z$ coupled to an optical Stokes component $S_z$, a schematic interaction is

$$H_{\mathrm{QND}} = \hbar gS_zJ_z.$$

With a strongly polarized probe, an outgoing light quadrature carries an estimate of $J_z$. In a Gaussian model, combining prior variance $V_0$ with an independent measurement of equivalent variance $V_m$ gives conditional variance

$$\frac1{V_{\mathrm{cond}}} = \frac1{V_0} + \frac1{V_m}.$$

The QND interaction creates backaction on the conjugate spin component. Real dispersive probes can also cause spontaneous scattering, differential light shifts, and coupling inhomogeneity. The useful quantity is therefore

$$\xi_R^2 = \frac{ NV_{\mathrm{cond}} }{ |\langle\mathbf J\rangle_{\mathrm{after}}|^2 },$$

not the conditional variance alone.

Without feedback, the conditional mean depends on the random measurement record. The ensemble of records can be broad even though each conditioned state is narrow. Real-time feedback rotates or displaces each conditioned state toward a common target and makes the preparation deterministic. Retaining every record, selecting a subset, and discarding the record produce different conditional or unconditional states. The preparation time and any rejection probability affect the number of usable measurements.

Collisional condensate interactions and cavity feedback can approximate one-axis twisting; dispersive optical probing implements the conditional measurement just described. Finite-range Rydberg, dipolar, or trapped-ion couplings need not reduce to a single collective-spin Hamiltonian.

Imperfections and Calibration

Contrast loss

If $\xi_S^2$ is held fixed while the contrast falls, then

$$\xi_R^2 = \frac{\xi_S^2}{C^2}$$

grows quadratically. For example, $6\ \mathrm{dB}$ of transverse variance reduction corresponds to $\xi_S^2\simeq0.25$. At contrast $C=0.5$, the Wineland parameter returns to approximately one.

Squeezing-angle error

Let $V_-$ and $V_+$ be the principal variances. A readout error $\delta$ sees

$$V_{\mathrm{read}} = V_-\cos^2\delta + V_+\sin^2\delta.$$

For small $\delta$,

$$V_{\mathrm{read}} \simeq V_- + \delta^2(V_+-V_-).$$

Stronger anti-squeezing therefore demands tighter phase control. Reporting only the narrow variance hides this vulnerability.

Detection noise

A calibrated linear detector often has

$$V_{\mathrm{obs}} = G^2V_{\mathrm{atom}} + V_{\mathrm{det}},$$

where $G$ is the atom-to-signal gain. Subtracting $V_{\mathrm{det}}$ can infer the atomic variance, but a sensor using that detector experiences $V_{\mathrm{obs}}$. Reports should distinguish raw performance from detection-noise-subtracted state characterization.

Inhomogeneous coupling

If probe $i$ couples with weight $w_i$, the measured operator is

$$J_z^{(w)} = \frac12 \sum_iw_i\sigma_z^{(i)}.$$

For an independent coherent reference,

$$\left(\Delta J_z^{(w)}\right)^2_{\mathrm{CSS}} = \frac14 \sum_iw_i^2,$$

not necessarily $N/4$. For a uniform small rotation of an initially $x$-polarized product state, the slope is $(\sum_iw_i)/2$, giving

$$(\Delta\phi)^2_{\mathrm{CSS},w} =\frac{\sum_iw_i^2}{(\sum_iw_i)^2} =\frac1{N_{\mathrm{eff}}}, \qquad N_{\mathrm{eff}}=\frac{(\sum_iw_i)^2}{\sum_iw_i^2}\le N.$$

This assumes a nonzero summed response. A spatially varying signal changes the slope and therefore requires a different effective-number calculation.

Loss and fluctuating atom number

Random loss removes correlated particles, shortens the mean spin, and can add binomial noise. Fluctuating $N$ changes both the coherent reference and the mapping from population difference to phase. A robust analysis conditions on or models the measured $N$ for each shot, then propagates uncertainty in that calibration into $\xi_R^2$ and any entanglement-depth statement.

Decoherence during interrogation

Independent dephasing typically reduces

$$|\langle\mathbf J\rangle| \longrightarrow e^{-\Gamma T}|\langle\mathbf J\rangle|$$

while also modifying transverse covariances. Because the Wineland denominator contains the squared spin length, even modest decoherence can erase a state preparation gain. Common-mode noise may be rejected by a differential comparison, but that does not prove the same gain for two independent remote sensors.

Clocks and Many-Body Sensors

Spin squeezing is attractive for clocks because a collective population measurement is already the natural Ramsey output. Moderate squeezing can improve phase precision without requiring single-particle-resolved readout. A smaller source-state variance does not by itself demonstrate a better clock: the Allan-deviation formula above must use the actual cycle time and the noise of the running apparatus. Atomic Clocks examines those effects; Sensing Case Studies compares representative experiments.

Many-body sensors extend the same logic beyond clocks. A collective spin may encode magnetic field, rotation, acceleration, interaction strength, or a spatially weighted signal. The best squeezed component must be aligned with the generator and measured mode. Local correlations that do not improve the chosen collective response may be scientifically interesting without improving that sensor.

Spin squeezing is also a many-body diagnostic. Collective first and second moments can witness entanglement in samples too large for tomography. This economy comes with limited resolution: different density operators can share the same moments, and failure to violate a squeezing inequality does not prove separability. Entanglement Witnesses develops the general witness logic.

Worked Example

Consider $N=10^4$ effective qubits. The coherent-spin transverse variance is

$$V_{\mathrm{CSS}} = \frac N4 = 2500.$$

Suppose tomography finds

$$V_- = 500, \qquad C = 0.80.$$

The noise-squeezing parameter is

$$\xi_S^2 = \frac{4V_-}{N} = 0.20.$$

The metrological parameter is

$$\xi_R^2 = \frac{0.20}{0.80^2} = 0.3125.$$

The corresponding gain is

$$G_{\mathrm{dB}} = -10\log_{10}(0.3125) \simeq 5.05\ \mathrm{dB}.$$

An ideal coherent state has one-shot local phase uncertainty $0.010\ \mathrm{rad}$. The squeezed state gives

$$\Delta\phi = \frac{\sqrt{0.3125}}{100} \simeq 5.59\times10^{-3}\ \mathrm{rad}.$$

The variance improved by a factor $3.2$, while the standard deviation improved by only $\sqrt{3.2}\simeq1.79$. A clock-level claim would still require cycle time, local-oscillator noise, and raw detector performance.

Exercises

1. Coherent-spin projection noise

For $|\mathrm{CSS}_x\rangle=|{+x}\rangle^{\otimes N}$, derive $\langle J_x\rangle=N/2$ and $(\Delta J_y)^2=(\Delta J_z)^2=N/4$.

Solution

For each probe,

$$\langle\sigma_x\rangle=1, \qquad \langle\sigma_y\rangle = \langle\sigma_z\rangle = 0.$$

Therefore

$$\langle J_x\rangle = \frac12\sum_i\langle\sigma_x^{(i)}\rangle = \frac N2.$$

Independence removes cross-covariances. Since $(\sigma_y^{(i)})^2=(\sigma_z^{(i)})^2=I$,

$$(\Delta J_y)^2 = \frac14\sum_i(\Delta\sigma_y^{(i)})^2 = \frac N4,$$

and similarly for $J_z$.

2. Find the squeezed axis

A transverse covariance matrix is

$$V = \begin{pmatrix} 10 & 6\\ 6 & 19 \end{pmatrix}.$$

Find its principal variances.

Solution

The eigenvalues are

$$V_\mp = \frac{29}2 \mp \frac12\sqrt{(-9)^2+4(6)^2}.$$

Because

$$\sqrt{81+144} = 15,$$

the principal variances are

$$V_-=7, \qquad V_+=22.$$

The off-diagonal covariance means neither laboratory axis is a principal axis; a rotation is required before readout.

3. Contrast can erase noise squeezing

A state has $\xi_S^2=0.16$. What minimum contrast is required for $\xi_R^2<1$? Evaluate $\xi_R^2$ at $C=0.30$.

Solution

Since

$$\xi_R^2 = \frac{\xi_S^2}{C^2},$$

metrological squeezing requires

$$C > \sqrt{0.16} = 0.40.$$

At $C=0.30$,

$$\xi_R^2 = \frac{0.16}{0.09} \simeq 1.78.$$

The transverse noise is squeezed, but the state does not beat the coherent linear-readout benchmark.

4. A singlet senses gradients, not uniform rotations

For $|s\rangle=(|01\rangle-|10\rangle)/\sqrt2$, compare the pure-state QFI for generators $J_z=(Z_1+Z_2)/2$ and $G=(Z_1-Z_2)/2$.

Solution

Both components have total $J_z=0$, so $J_z|s\rangle=0$ and $F_Q[|s\rangle,J_z]=4\operatorname{Var}(J_z)=0$. For the difference generator,

$$G|s\rangle=\frac{|01\rangle+|10\rangle}{\sqrt2}, \qquad \langle G\rangle=0,\qquad \langle G^2\rangle=1.$$

Thus $F_Q[|s\rangle,G]=4$. A gradient creates a relative phase even though a spatially uniform field does not. Entanglement alone does not specify which signal a state can detect.

5. Why one-axis twisting creates covariance

For $H=\hbar\chi J_z^2$, compute the Heisenberg equation for $J_y$ and give its leading behavior near a state polarized along $x$.

Solution

Using $\dot J_y=(i/\hbar)[H,J_y]$ and $[J_z,J_y]=-iJ_x$,

$$\dot J_y = \chi(J_zJ_x+J_xJ_z).$$

Near a strongly polarized state, replace $J_x$ by its mean value $J$:

$$\dot J_y \simeq 2\chi J J_z.$$

Thus different $J_z$ slices acquire different $J_y$ displacements. This is a shear in the transverse plane and creates $y$–$z$ covariance.

6. A Fisher-information depth bound

For $N=100$ and $\xi_R^2=0.20$, use $F_Q\ge N/\xi_R^2$. Can the state be four-producible?

Solution

The squeezing result implies

$$F_Q \ge \frac{100}{0.20} = 500.$$

For $k=4$,

$$s = \left\lfloor\frac{100}{4}\right\rfloor = 25, \qquad r=0.$$

Every four-producible state obeys

$$F_Q \le 25(4^2) = 400.$$

The observed lower bound exceeds 400, so the state is not four-producible and has entanglement depth at least five. The same argument does not certify depth six because the five-producible bound is 500 and the inferred QFI may equal that value.

7. Raw and inferred detector noise

A detector has gain $G=2$, observed variance $V_{\mathrm{obs}}=44$, and independently calibrated detector variance $V_{\mathrm{det}}=20$. Find the inferred atomic variance. Which variance belongs in a raw sensor claim?

Solution

From

$$V_{\mathrm{obs}} = G^2V_{\mathrm{atom}} + V_{\mathrm{det}},$$

we obtain

$$V_{\mathrm{atom}} = \frac{44-20}{4} = 6.$$

The value 6 characterizes the inferred atomic state under the detector model. The raw instrument experiences the full observed variance 44 in detector units, unless an additional protocol suppresses or calibrates that noise in the estimator.

8. Averaging-time gain

A squeezing protocol achieves $\xi_R^2=0.25$ without changing cycle time or other noise. By what factor does one-shot phase standard deviation improve? By what factor does the averaging time for a fixed variance decrease?

Solution

The standard-deviation ratio is

$$\xi_R = \sqrt{0.25} = 0.5.$$

The one-shot standard deviation is halved. Variance averages as $1/\nu$, so a factor of four fewer independent trials reaches the same target variance:

$$\frac{\tau_{\mathrm{sq}}}{\tau_{\mathrm{CSS}}} = \xi_R^2 = 0.25.$$

More generally, with unchanged interrogation time and other noise, $\tau_{\mathrm{sq}}/\tau_{\mathrm{CSS}}=\xi_R^2T_{c,\mathrm{sq}}/T_{c,\mathrm{CSS}}$. Doubling the cycle time changes the ratio from $0.25$ to $0.50$.

References

• K. C. Cox, G. P. Greve, J. M. Weiner, and J. K. Thompson, “Deterministic squeezed states with collective measurements and feedback,” Physical Review Letters 116, 093602 (2016), doi:10.1103/PhysRevLett.116.093602.
• P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzè, and A. Smerzi, “Fisher information and multiparticle entanglement,” Physical Review A 85, 022321 (2012), doi:10.1103/PhysRevA.85.022321.
• M. Kitagawa and M. Ueda, “Squeezed spin states,” Physical Review A 47, 5138–5143 (1993), doi:10.1103/PhysRevA.47.5138.
• I. D. Leroux, M. H. Schleier-Smith, and V. Vuletić, “Implementation of cavity squeezing of a collective atomic spin,” Physical Review Letters 104, 073602 (2010), doi:10.1103/PhysRevLett.104.073602.
• J. Ma, X. Wang, C. P. Sun, and F. Nori, “Quantum spin squeezing,” Physics Reports 509, 89–165 (2011), doi:10.1016/j.physrep.2011.08.003.
• L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, “Quantum metrology with nonclassical states of atomic ensembles,” Reviews of Modern Physics 90, 035005 (2018), doi:10.1103/RevModPhys.90.035005.
• M. Schulte, C. Lisdat, P. O. Schmidt, U. Sterr, and K. Hammerer, “Prospects and challenges for squeezing-enhanced optical atomic clocks,” Nature Communications 11, 5955 (2020), doi:10.1038/s41467-020-19403-7.
• A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, “Many-particle entanglement with Bose–Einstein condensates,” Nature 409, 63–66 (2001), doi:10.1038/35051038.
• A. S. Sørensen and K. Mølmer, “Entanglement and extreme spin squeezing,” Physical Review Letters 86, 4431–4434 (2001), doi:10.1103/PhysRevLett.86.4431.
• G. Tóth, “Multipartite entanglement and high-precision metrology,” Physical Review A 85, 022322 (2012), doi:10.1103/PhysRevA.85.022322.
• G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” Journal of Physics A 47, 424006 (2014), doi:10.1088/1751-8113/47/42/424006.
• D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, “Spin squeezing and reduced quantum noise in spectroscopy,” Physical Review A 46, R6797–R6800 (1992), doi:10.1103/PhysRevA.46.R6797.