Quantum metrology with nonclassical states of atomic ensembles

Quantum technologies exploit entanglement to revolutionize computing, measurements, and communications. This has stimulated the research in different areas of physics to engineer and manipulate fragile many-particle entangled states. Progress has been particularly rapid for atoms. Thanks to the large and tunable nonlinearities and the well-developed techniques for trapping, controlling, and counting, many groundbreaking experiments have demonstrated the generation of entangled states of trapped ions, cold, and ultracold gases of neutral atoms. Moreover, atoms can strongly couple to external forces and fields, which makes them ideal for ultraprecise sensing and time keeping. All these factors call for generating nonclassical atomic states designed for phase estimation in atomic clocks and atom interferometers, exploiting many-body entanglement to increase the sensitivity of precision measurements. The goal of this article is to review and illustrate the theory and the experiments with atomic ensembles that have demonstrated many-particle entanglement and quantum-enhanced metrology.

Figure 1

Building blocks of phase estimation: (i) preparation of the probe state ${\widehat{\rho }}_0$, (ii) encoding of a phase shift $\theta$ that depends on the physical quantity of interest, (iii) readout, where $\mu$ indicates a generic measurement result, and (iv) estimation, where the estimator $\mathrm{\Theta }(\mu )$ is a function of the measurement result(s). The uncertainty $\mathrm{\Delta }\theta$ of the estimation depends crucially on all of these operations.

Figure 2

Gain of phase sensitivity over the standard quantum limit $\mathrm{\Delta }{\theta }_{\mathrm{SQL}}=1/\sqrt{N}$ achieved experimentally with trapped ions (black symbols), Bose-Einstein condensates (red), and cold thermal ensembles (blue). The gain is shown on logarithmic [left, dB, $10{\log }_{10}(\mathrm{\Delta }{\theta }_{\mathrm{SQL}}/\mathrm{\Delta }\theta {)}^2$] and linear [right, $(\mathrm{\Delta }{\theta }_{\mathrm{SQL}}/\mathrm{\Delta }\theta {)}^2$] scales. The solid thick line is the Heisenberg limit $\mathrm{\Delta }{\theta }_{\mathrm{HL}}=1/N$. Stars refer to directly measured phase sensitivity gains, performing a full phase estimation experiment. Circles are expected gains based on a characterization of the quantum state, e.g., calculated as $\mathrm{\Delta }\theta ={\xi }_{\mathrm{R}}/\sqrt{N}$, where ${\xi }_{\mathrm{R}}$ is the spin-squeezing parameter or as $\mathrm{\Delta }\theta =1/\sqrt{F_{\mathrm{Q}}}$, where $F$ is the Fisher information. Filled (open) circles indicate results obtained (inferred) without (with) subtraction of technical and/or imaging noise. Every symbol is accompanied by a number (in chronological order) corresponding to the reference reported in the side table. Here $N$ is the total number of particles or, in the presence of fluctuations, the mean total.

Figure 3

Two-mode interferometers. In a Mach-Zehnder interferometer (a) two spatial modes $|a⟩$ and $|b⟩$ are combined on a balanced beam splitter, followed by a relative phase shift $\theta ={\theta }_a-{\theta }_b$ between the two arms, and finally recombined on a second balanced beam splitter. In a Ramsey interferometer (b) a resonant Rabi rotation creates a balanced superposition between two internal states $|a⟩$ and $|b⟩$, followed by a relative phase shift $\theta =(\mathrm{\Delta }E/\hslash )T_{\mathrm{R}}$ given by the energy difference $\mathrm{\Delta }E$ between these states multiplied by the interrogation time $T_{\mathrm{R}}$. Finally a second resonant Rabi rotation recombines the two modes. (c) Equivalent representation of Mach-Zehnder and Ramsey interferometer operations as rotations of the collective spin on the generalized Bloch sphere. The initial state here ${|a⟩}^{\otimes N}$ is pointing toward the north pole. The full sequence is equivalent to the rotation of an angle $\theta$ around the $y$ axis.

Figure 4

Schematic representation of different phase estimation protocols. (a) Maximum likelihood estimation. ${\mathrm{\Theta }}_{\mathrm{MLE}}(\mathit{\mu })$ is the absolute maximum (dashed line) of the likelihood function $P(\mathit{\mu }|\phi )$ (solid line) corresponding to the observed sequence of results $\mathit{\mu }$. As schematically shown in the inset, the phase sensitivity $\mathrm{\Delta }{\theta }_{\mathrm{MLE}}$ is identified as the root-mean-square fluctuation (shaded region) of the statistical distribution of ${\mathrm{\Theta }}_{\mathrm{MLE}}$, obtained by repeating the measurements several times (at a fixed value of $\theta$). (b) Method of moments. The blue line is $\overline{\mu }$ as a function of the parameter $\phi$. Via this functional monotonic behavior, it is possible to associate to the measured mean value ${\overline{\mu }}_{\nu }$ the phase estimate ${\mathrm{\Theta }}_{\mathrm{mom}}(\mathit{\mu })$. The phase uncertainty $\mathrm{\Delta }{\theta }_{\mathrm{mom}}$ follows from the statistical uncertainty $\mathrm{\Delta }\mu /\sqrt{\nu }$ of ${\overline{\mu }}_{\nu }$ (gray region). For $\nu \gg 1$, $\mathrm{\Delta }{\theta }_{\mathrm{mom}}$ has a simple expression, Eq. (23), obtained by error propagation, locally approximating $\overline{\mu }$ by the tangent to the curve (thin black line). (c) Bayesian estimation. Posterior probability distribution $P(\phi |\mathit{\mu })$ (solid line). The phase estimate ${\mathrm{\Theta }}_{\mathrm{B}}(\mathit{\mu })$ can be chosen as the weighted averaged phase. The integral ${\int }_{\mathrm{\Delta }{\theta }_{\mathrm{B}}}d\phi P(\phi |\mathit{\mu })$ gives the probability that the true phase value falls into the confidence interval $\mathrm{\Delta }{\theta }_{\mathrm{B}}$ (gray region) around ${\mathrm{\Theta }}_{\mathrm{B}}$.

Figure 5

Rotation of different quantum states. Wigner distribution normalized by its maximum value (left column), see Sec. 2d, and spin probability $P_y(m)={|⟨m_y|\psi ⟩|}^2$ along the $y$ direction (thick red histograms, right column) of different quantum states $|\psi ⟩$: (a) a coherent spin state pointing along the positive $x$ axis, $|\pi /2,0,N⟩$ (Sec. 2c3); (b) a spin-squeezed state with ${\xi }_{\mathrm{R}}^2=0.1$ (Sec. 2c5); (c) a twin-Fock state (Sec. 2c6); and (d) a NOON state (Sec. 2c7). The thin blue histogram is $P_y(m)={|⟨m_y|e^{-i\theta {\widehat{J}}_z}|\psi ⟩|}^2$ obtained after a rotation of the state by an angle $\theta =2/\sqrt{N}$ in (a), $\theta =2{\xi }_{\mathrm{R}}/\sqrt{N}$ in (b), $\theta =2/N$ in (c), and $\theta =\pi /N$ in (d). Here $N=100$.

Figure 6

Useful $k$-particle entanglement for quantum metrology. $k$-separable states have a quantum Fisher information bounded by the solid line, Eq. (38). The dashed line is $F_{\mathrm{Q}}/N=k$. Here $N=100$. Adapted from [237].

Figure 7

Relation between spin squeezing and quadrature squeezing. For $⟨{\widehat{J}}_z⟩\approx N/2\gg 1$, the Bloch sphere can be approximated locally as a plane orthogonal to the mean spin direction (here the $z$ axis). Spin squeezing along the ${\widehat{J}}_x$ axis is equivalent to squeezing of the $\widehat{X}$ quadrature. A rotation around the $y$ axis is equivalent to a displacement in the $X$ direction.

Figure 8

${\xi }_{\mathrm{S}}^2$ vs ${\xi }_{\mathrm{R}}^2$. The metrological spin-squeezing parameter ${\xi }_{\mathrm{R}}^2$ is given by the ratio between the Kitagawa-Ueda spin-squeezing parameter ${\xi }_{\mathrm{S}}^2$ and the squared spin length (Ramsey contrast) $2|⟨{\widehat{J}}_{\mathit{s}}⟩|/N$. ${\xi }_{\mathrm{R}}^2<1$ is found in the blue (dark) region: the upper limit is the standard quantum limit (SQL), while the lower limit is the bound obtained for optimal spin-squeezed states; see Sec. 2c5a. The dashed line is Eq. (43), which is tight in the limit $⟨{\widehat{J}}_{\mathit{s}}⟩\to 0$. The lower thick black line is the Heisenberg limit (HL). The orange region above the SQL highlights the regime of parameters showing ${\xi }_{\mathrm{S}}^2<1$ but no metrological spin squeezing (i.e., ${\xi }_{\mathrm{R}}^2\ge 1$). The white lines are lower bounds to ${\xi }_{\mathrm{R}}^2$ obtained for $k$-particle entangled states ([515]). The gray region is not accessible by metrological spin-squeezed states. Here $N=1000$.

Figure 9

Quasiprobability representations. Upper row: $P$, $W$, and $Q$ representations of a coherent state pointing in the $+x$ direction (see Table 1), with $\mathit{s}=\{1,0,0\}$. Lower row: $P$, $W$, and $Q$ representations of a spin-squeezed state reached by one-axis twisting with $\chi t=0.01\pi$ (see Sec. 3b). The color scale for each panel is normalized to its maximum value. Here $N=100$.

Figure 10

(a) External bosonic Josephson junction realized by a Bose-Einstein condensate trapped in a double-well potential (thick black line). The thin colored lines are the amplitudes of the mean-field wave functions (see text) localized on the left and right wells. (b) Internal bosonic Josephson junction made by a trapped Bose-Einstein condensate in two different hyperfine states. For each state, the thick black line is the harmonic trap and the thin colored line is the condensate wave function.

Figure 11

Entanglement in the ground states of the bosonic Josephson junction. Upper panels: Normalized Wigner distributions of the ground state $|{\psi }_{\mathrm{\Lambda }}⟩$ of the Hamiltonian (69) in different regimes and for $\delta =0$. The histograms are probability distributions $P_y(m)={|⟨m_y|{\psi }_{\mathrm{\Lambda }}⟩|}^2$ and $P_z(m)={|⟨m_z|{\psi }_{\mathrm{\Lambda }}⟩|}^2$. Crossing from the Rabi to the Fock regime (for positive nonlinearities), $P_z$ narrows while $P_y$ broadens. In the Fock regime, $P_y$ vanishes for odd values of $m$. In the transition from the disordered to the ordered phase (for negative nonlinearities), $P_y$ narrows while $P_z$ broadens and splits when crossing the critical value $\mathrm{\Lambda }=-1$. Lower panels: Normalized quantum Fisher information ($F_{\mathrm{Q}}/N$, red line) and inverse spin-squeezing parameter ($1/{\xi }_{\mathrm{R}}^2$, blue line) as a function of $\mathrm{\Lambda }$. For $\mathrm{\Lambda }>0$, the solid black line is Eq. (83). For $\mathrm{\Lambda }<0$, the solid black lines are Eq. (87) in the disordered regime and Eq. (88) in the ordered regime. Here $N=100$ and the color scale of the Wigner distributions is as in Fig. 5.

Figure 12

Spin squeezing in the ground states of the bosonic Josephson junction. Symbols report number squeezing (with photon shot-noise subtracted) and phase coherence obtained by splitting a Bose-Einstein condensate. Open and filled symbols correspond to different barrier heights (larger for the open symbols). The shaded areas show systematic error bounds. Solid lines are reference values for ${\xi }_{\mathrm{R}}^2$, the orange (0 dB) line being the standard quantum limit. Measurements are shown for the two main well pairs of a six-well lattice [red (dark) and blue (light) circles] and for a double-well potential (green diamonds). The total atom number in each pair is approximately $N=2200$ in the six-well case and $N=1600$ in the double-well case. The inset shows single-site-resolving absorption images of atoms trapped in an optical lattice superposed on an atomic dipole trap. Adapted from [133].

Figure 13

Number and phase distributions after the adiabatic splitting of a Bose-Einstein condensate in a double-well potential. (a) Distribution of photon imbalance $s=s_{\mathrm{L}}-s_{\mathrm{R}}$ between fluorescence images of the left and right clouds (inset), proportional to the imbalance of atom numbers in the two wells. The solid black line is the normal distribution corresponding to the measured number-squeezing factor, the dotted red line is the expected distribution when detection noise is subtracted, and the dashed blue line is the distribution expected for a coherent spin state in the absence of detection noise. (b) The curves indicate a normal distribution with the measured $\mathrm{\Delta }\phi$ (solid black line) and the distributions expected for a coherent state in the absence (dashed blue line) and in the presence (dash-dotted green line) of detection noise. The inset shows a typical matter-wave interference pattern from which the phase is extracted. Adapted from [34].

Figure 14

One-axis twisting dynamics. The left panel reports the inverse spin-squeezing parameter ($1/{\xi }_{\mathrm{R}}^2$, blue, lower line) and normalized quantum Fisher information ($F_{\mathrm{Q}}/N$, red, upper line) as a function of $\chi t/\pi$. The right panels show snapshots of the Wigner distribution at different times. For $\chi t=\pi /2$ we plot both the Wigner distributions for $N=100$ (left) and $N=101$ (right). In all plots $N=100$ (unless specified) and the color scale is as in Fig. 5.

Figure 15

Spin-noise tomography and reconstructed Wigner function of a spin-squeezed Bose-Einstein condensate. Top: Reconstructed Wigner distribution of a spin-squeezed state of $N\approx 1250$ atoms. The black contour line indicates where the Wigner distribution has fallen to $1/e$ of its maximum. For comparison, the circular $1/e$ contour of an ideal coherent spin state is shown. Bottom: Observed spin fluctuations of a spin-squeezed state (solid circles) and of a coherent spin state (open circles), as a function of the turning angle of the Bloch sphere. Solid lines are results of dynamical simulations including losses and technical noise: blue (lowest), spin-squeezed state with losses but without technical noise; red (highest), spin-squeezed state with losses and technical noise; and black, coherent state with losses and technical noise. Adapted from [452].

Figure 16

Direct experimental demonstration of a phase sensitivity beyond the standard quantum limit using an atomic spin-squeezed state. Left: Ramsey fringe scanned over a full $2\pi$ period. The blue (lighter) data are obtained with a coherent spin state, the red (darker) data with a spin-squeezed state showing a visibility of 92%. Solid lines are $\pm N/2$, measured for each phase setting, as a reference. Right: Ramsey fringe around the maximum-slope optimal point. The solid lines are fits through the lower and upper ends of the 2 standard deviation error bars. The gray shaded areas are the uncertainty regions for an ideal coherent spin state; they correspond to the standard quantum limit. Red (darker) data show a phase error 15% below the standard quantum limit. Adapted from [179].

Figure 17

Bell correlations in a Bose-Einstein condensate. Measurement of the Bell correlation witness $\mathcal{W}$ of Eq. (63) on a spin-squeezed state as a function of the angle $\vartheta$ between the squeezing axis $a$ and the axis $n$ lying in the squeezing plane, with $1\sigma$ error bars. The continuous red line is the value of $\mathcal{W}$ computed from the measurement of ${\zeta }_a^2=4⟨{\widehat{J}}_a^2⟩/N$ and the fitted Rabi oscillation ${\mathcal{C}}_n={\mathcal{C}}_b\sin (\vartheta )$ with ${\mathcal{C}}_b=2⟨{\widehat{J}}_b⟩/N$. Bell correlations are present in the blue shaded region. Right inset: Representation of the same data (${\zeta }_a^2$ and ${\mathcal{C}}_b$) as a single black point, with $1\sigma$ error bars. Red shaded region: Entanglement detected by spin squeezing ${\xi }_{\mathrm{R}}^2<1$, Eq. (41). Blue shaded region: Bell correlations detected by violation of the witness inequality (64). Adapted from [479].

Figure 18

Twist-and-turn dynamics. Top: Normalized quantum Fisher information ($F_{\mathrm{Q}}/N$, red upper line) and inverse spin-squeezing parameter ($1/{\xi }_{\mathrm{R}}^2$, blue lower line) as a function of ${\omega }_{\pi }t/\pi$, where ${\omega }_{\pi }=\mathrm{\Omega }\sqrt{\mathrm{\Lambda }-1}$. The inset shows the effective potential $W_{\pi }(z)$ initially (dotted line, $\mathrm{\Lambda }=0$) and after a quench to a finite $\mathrm{\Lambda }$ (solid line, $\mathrm{\Lambda }=1.5$). The initial coherent spin state corresponds to a Gaussian wave packet located at the top of the barrier. Bottom: Snapshots of the Wigner distribution at different times. The solid black lines are the mean-field separatrices and the dots are fixed points of the semiclassical model. Here $N=100$ and the color scale is as in Fig. 5.

Figure 19

Entanglement of non-spin-squeezed states. (a) Experimental Husimi distributions at different evolution times of the twist-and-turn dynamics. (b) Normalized Fisher information ($F/N$, red diamonds) and inverse spin-squeezing parameter ($1/{\xi }_{\mathrm{R}}^2$, blue circles) as a function of the tomography angle [identified by the rotation angle $\alpha$ in (a)]. The gray shaded area is accessible only by entangled states. At 26 ms, spin squeezing cannot witness the entanglement that is detected by the Fisher information. (c) Squared Hellinger distance as a function of the rotation angle $\theta$ [see inset for the rotation of the initial (blue) state to a final (green) one] at three evolution times: $t=0$ corresponding to a separable state (black points), $t=15\mathrm{ms}$ corresponding to an optimal spin-squeezed state (light gray points), and $t=26\mathrm{ms}$ corresponding to a state that is entangled but not spin squeezed (red points). The curvature of the quadratic fits (solid lines) is proportional to $F/N$. Adapted from [525].

Figure 20

Binary $s$-wave collision of two spin-$F$ bosons. (a) When two spin-$F$ bosons in internal states $|F,m_F⟩$ and $|F,m_F^{\prime }⟩$ approach each other (here $F$ is the hyperfine spin and $m_F$, $m_F^{\prime }=-F,-F+1,\dots ,F$ the magnetic quantum number), they couple to form a total spin $G=F+F^{\prime }$. (b) The combined internal state is given by $|G,m_G⟩$. For bosons undergoing elastic $s$-wave scattering, $G$ is restricted to even values satisfying $0\le G\le 2F$, e.g., $G=0$, 2 for two $F=1$ particles. (c) After the collision the atoms fly apart and their internal states are again well described by the hyperfine spins $F$. However, the magnetic quantum numbers may have changed: $|F,m_F⟩|F,m_F^{\prime }⟩\mapsto |F,m_F^{\prime \prime }⟩|F,m_F^{\prime \prime \prime }⟩$. The conservation of total angular momentum imposes $m_F+m_F^{\prime }=m_F^{\prime \prime }+m_F^{\prime \prime \prime }$.

Figure 21

Entanglement in the ground state of the spin-mixing Hamiltonian. The upper panel reports a schematic representation of the different phases obtained as a function of $q$ (linear Zeeman shifts are not shown). The lower panel shows $F_Q[|{\psi }_{\mathrm{gs}}⟩,{\widehat{J}}_x]/N^2$ (solid line) and $F_Q[|{\psi }_{\mathrm{gs}}⟩,{\widehat{\mathcal{S}}}_x]/N^2$ (dashed line) as a function of $q/4N|\lambda |$. The left inset shows the energy gap $\mathrm{\Delta }E$ between the ground state and the first excited state, closing at $q_c=\pm 4N|\lambda |$. The right inset shows the normalized population of the $m_F=0$ mode $N_0/N$. Here $\lambda <0$ and $N=1000$.

Figure 22

Population correlations after spin-mixing dynamics. Left panel: $⟨{\widehat{N}}_{+1}+{\widehat{N}}_{-1}⟩$ (red, light circles) and ${\mathrm{\Delta }}^2({\widehat{N}}_{+1}-{\widehat{N}}_{-1})$ (blue, dark circles) as a function of the total atom number $N$. Error bars are fluctuations (1 standard deviation) and the gray area corresponds to the sub-Poisson regime. The black line is a theoretical model including particle loss due to spin relaxation. The inset shows the distribution of $N_{+1}+N_{-1}$ for $250<N<300$, where the black line is a fitted squeezed-vacuum distribution corresponding to $r\approx 2$. Adapted from [178]. Right panel: Standard deviation of the population difference $({\widehat{N}}_{+1}-{\widehat{N}}_{-1})/2$ (red, lower line), compared to the projection noise $\sqrt{N_{+1}+N_{-1}}/2$ (dashed line), and $\sqrt{(N_{+1}+N_{-1})/4+{\sigma }_{\mathrm{dn}}^2}$ (dot-dashed line) taking into account a number-independent detection noise ${\sigma }_{\mathrm{dn}}=20$ (dotted line). The blue, upper line is the experimental result for unentangled atoms. The shaded area indicates the standard deviation. Adapted from [332].

Figure 23

Twin-Fock interferometry. The inset of (a) shows the experimental operations: (1) spin dynamics, (2) a resonant microwave $\pi$ pulse between $|2,-1⟩$ and $|1,0⟩$, (3) a pulse of variable duration, defining $\theta$, and (4) a second $\pi$ pulse. (a) Second moment $⟨{\widehat{J}}_z^2⟩$ of the population imbalance ${\widehat{J}}_z=({\widehat{N}}_{+1}-{\widehat{N}}_{-1})/2$ as a function of $\theta$ (dots with error bars) for postselected numbers of atoms between 6400 and 7600. The solid line is a polynomial fit (with gray uncertainty region); the dotted line is the theoretical prediction including detection noise. (b) Same for $(\mathrm{\Delta }{\widehat{J}}_z^2{)}^2=⟨{\widehat{J}}_z^4⟩-{⟨{\widehat{J}}_z^2⟩}^2$. (c) Phase estimation uncertainty obtained via error propagation $\mathrm{\Delta }\theta =(\mathrm{\Delta }{\widehat{J}}_z^2)/|d⟨{\widehat{J}}_z^2⟩/d\theta |$ (solid orange line with gray uncertainty region) compared to the theoretical prediction (dashed orange line) and Cramér-Rao bound (dotted orange line) including detection noise only. Around $\theta =0.015\mathrm{rad}$ the phase variance lies 1.61 dB below the standard quantum limit (black dashed line). Adapted from [332].

Figure 24

Top: Quadrature variance $4{\mathrm{\Delta }}^2({\widehat{\mathcal{S}}}_x\cos \varphi +{\widehat{\mathcal{S}}}_y\sin \varphi )/N$ of the state generated after variable duration of the spin-mixing dynamics (colored lines and symbols), as functions of the quadrature angle $\varphi$. Symbols are experimental results obtained via state tomography, whereas the solid lines are theoretical predictions. Bottom: Reconstructed phase space distributions at $t=15\mathrm{ms}$ (left) and $t=45\mathrm{ms}$ (right); the black ellipses are the $1/\sqrt{e}$ uncertainty ellipse predicted theoretically. Adapted from [195].

Figure 25

Einstein-Podolsky-Rosen entanglement with spinor condensates. (a) Experimental variances $V_{X(\varphi )}^{+}$ and $V_{X(\varphi )}^{-}$ as functions of the local oscillator phase $\varphi$. The dashed line is the quadrature variance for the vacuum state. The lowest measured quadrature variance is $V_X^{-}=0.42$, corresponding to a squeezing of 3.77 dB below the vacuum limit. (b) Product $V_{X(\varphi )}^{-}\times V_{X(\varphi +\pi /2)}^{+}$: data points below the dashed line violate the inequality (62) and thus signal Einstein-Podolsky-Rosen entanglement. Data reach $V_{X(\varphi )}^{-}\times V_{X(\varphi +\pi /2)}^{+}=0.18(3)$, which is 2.4 standard deviations below the limit of $1/4$. (c) Sum $V_{X(\varphi )}^{+}+V_{X(\varphi +\pi /2)}^{-}$: Data points below the dotted line violate the inequality (61) and thus signal entanglement between the $m_F=\pm 1$ modes, and data below the dashed line signal Einstein-Podolsky-Rosen entanglement. Data reach $V_{X(\varphi )}^{+}+V_{X(\varphi +\pi /2)}^{-}=0.85(8)$. Adapted from [412].

Figure 26

Conditional spin squeezing via QND measurements. A light beam (blue arrows) propagates through the atomic ensemble. By measuring the output light field, one gains information about the atomic state. The initial coherent spin state (orange circle) changes after the detection of the light beam (green ellipse). It becomes squeezed in ${\widehat{J}}_z$ and shifted (conditioned by the measurement result), according to Eqs. (111a) and (111b).

Figure 27

Spin squeezing via quantum nondemolition measurements in free space. (a) A cloud of cold Cs atoms is confined in an elongated dipole trap placed in one arm of an optical Mach-Zehnder interferometer and aligned with the largest resonant optical depth. The relative populations of the clock levels $|a⟩$ and $|b⟩$ are measured via the phase shift $\varphi$ accumulated by two detuned probe beams propagating through the atomic cloud. (b) Red diamonds are $\text{Var}(\mathrm{\Phi })$ (with $\mathrm{\Phi }={\varphi }_1-\zeta {\varphi }_2$) obtained after two consecutive QND measurements of covariance $\zeta$. Blue symbols are $\text{Var}(\mathrm{\Phi })$ obtained for a coherent spin state ($\mathrm{\Phi }={\varphi }_1$, dots, and $\mathrm{\Phi }={\varphi }_2$, stars). The solid line is a quadratic fit and the dashed line is the expected linear scaling with the atom number $N$ due to projection noise. The dot-dashed line is the projection noise scaled down by the factor $(1-\eta {)}^2=0.64$, corresponding to the reduction by the measured observed loss of atomic coherence. Different color regions are the optical shot noise (light blue), detector noise (dark blue), and projection noise (green, light). The inset illustrates the trade-off between spin squeezing ${\xi }_{\mathrm{R}}^2$ and loss of coherence $\eta$ due to spontaneous emission. Adapted from [15].

Figure 28

Spin squeezing via cavity-based quantum nondemolition measurements in the dispersive regime. (a) Laser-cooled atoms are optically trapped in a standing wave (red, lighter line) inside an optical resonator. The cavity resonance is shifted in proportion to the relative population of two clock levels. The shift is measured from the transmission of a probe beam (blue, darker line). Uniform atom-light coupling is achieved using trapping and probe beams of commensurate frequencies. (b) Measured spin-noise reduction $4(\mathrm{\Delta }{\widehat{J}}_z{)}^2/N$ (dots) normalized to the coherent spin state projection noise. The solid line is a model fit. (c) Allan deviation of a clock that uses the generated squeezed states (black dots; the solid line indicates $9.7\times {10}^{-11}{\mathrm{s}}^{1/2}/\sqrt{\tau }$) or coherent spin states (blue circles; the dashed line is the standard quantum limit). Adapted from [224].

Figure 29

Entangled states generated by the detection of a single photon. (a) An atomic ensemble confined in a cavity interacts with vertically polarized light. The detection of an outgoing horizontally (vertically) polarized single photon projects the atoms into a state with a negative (positive) Wigner distribution. (b) Experimental Wigner distribution of the heralded atomic state obtained from the least-squares-fitted atomic density matrix via Eq. (66). It shows negative parts. The dashed line is the contour at which the Wigner distribution of an $N$-atom coherent spin state is equal to $1/\sqrt{e}$ of its maximum value. Adapted from [354].

Figure 30

Spin squeezing obtained via quantum nondemolition measurements in the normal mode splitting regime. (a) Energy-level diagram showing the cavity shift due to atoms in level $|b⟩$. (b) ${\xi }_{\mathrm{R}}^2/N$ as a function of atom number $N$ for uncorrelated particles (black dots) and spin-squeezed states [red (gray) dots]. The inset shows $1/{\xi }_{\mathrm{R}}^2$ of the spin-squeezed state (red dots) as a function of $N$. The black line is a fit to the data according to a $1/N^2$ scaling. Adapted from [46].

Figure 31

State engineering via quantum Zeno dynamics. (a) $|a⟩$ and $|b⟩$ are coupled by a microwave field at Rabi frequency $\mathrm{\Omega }$, while the optical cavity transition $|b⟩\leftrightarrow |e⟩$ is probed with a coupling rate much faster than $\mathrm{\Omega }$. (b) Starting with all atoms in $|b⟩$, microwave Rabi coupling and simultaneous measurement of the cavity transmission leads to a coherent evolution restricted to the subspace highlighted by the orange (right shaded) area. The preparation of the state with no atoms in $|b⟩$ (green, left shaded area) is forbidden by the quantum Zeno effect. The upper row shows the Husimi distributions of symmetric Dicke states. (c) Experimental Husimi distributions for different evolution times $t/T$, where $T=\pi /\mathrm{\Omega }$ is the time for a Rabi $\pi$ pulse in the absence of cavity probing. The upper row shows direct measurements, while the lower row shows the reconstructed density matrix. Adapted from [26].

Figure 32

Squeezing via light-mediated effective interaction between atoms. (a) The atoms are trapped in a standing-wave dipole trap inside an optical resonator. (b) The probe laser is detuned from cavity resonance by half a linewidth, so that atom-induced shifts of the cavity frequency change the transmitted power by an amount proportional to $J_z$. (c) Effective spin squeezing ${\xi }_{\mathrm{eff}}^2={\sigma }^2(Q)\mathcal{C}(0)/\mathcal{C}(Q{)}^2$ giving the gain with respect to the sensitivity experimentally achieved in the absence of entanglement, as a function of the shearing strength $Q$ (proportional to the photon number in the cavity). Here ${\sigma }^2(Q)=(\mathrm{\Delta }{\widehat{J}}_z{)}^2/(N_{\mathrm{eff}}/4)$, and $\mathcal{C}(Q)$ is the contrast of Rabi oscillations of the sheared state, shown in the inset. The effective spin squeezing is related to the metrological spin squeezing Eq. (41) as ${\xi }_{\mathrm{eff}}^2=\mathcal{C}(0){\xi }_{\mathrm{R}}^2$. An effective squeezing ${\xi }_{\mathrm{eff}}^2=-5.6\mathrm{dB}$ is reached, corresponding to ${\xi }_{\mathrm{R}}^2=-4.6\mathrm{dB}$. Adapted from [311].

Figure 33

Common ion-trapping setups. (a) Linear Paul trap. Ions are trapped using radio-frequency oscillating electric fields combined with static electric control fields. In tight radial confinement, laser-cooled ions form a linear string (see the inset image for eight ions) with a spacing determined by the balance between external confining fields and ion-ion Coulomb repulsion. From [41]. (b) Penning trap. Ions are confined in the $x\text{−}y$ plane with a strong homogeneous magnetic field and axially by a quadrupole electric field. The green ODF arrows indicate bichromatic light fields used to generate entanglement. The inset shows an image of a 2D crystal of 91 ions. From [47]. (c) Each trapped ion can be modeled as an effective internal two-level system, $|a⟩$ and $|b⟩$ interacting with a radiation characterized by Rabi frequency $\mathrm{\Omega }$ and decay rate $\gamma$, and a shared external harmonic oscillator potential with equally spaced energy levels of mode frequency ${\omega }_m$.

Figure 34

Phase sensitivity of ion Schrödinger cat states. (a) Typical parity oscillations obtained with cat states, Eq. (127). These oscillations have a characteristic period $2\pi /N$ (here for $N=8$). From [369]. (b) Summary of the experimental achievements (symbols). Here we show the Fisher information $F=V^2{N}^2$ as a function of the number of qubits $N$, obtained from the extracted experimental visibilities $V$. The upper thick line is the Heisenberg limit $F=N^2$, and the lower thick line is the standard quantum limit $F=N$. The thin lines are bounds for useful $k$-particle entanglement, Eq. (38): they delimit from below a shaded region corresponding to ($k+1$)-particle entanglement. In particular, the darker red region stands for useful genuine $N$-particle entanglement. The next lighter red region stands for useful ($N-1$)-particle entanglement, and so on. For instance, the point at $N=10$ reveals useful 4-particle entanglement. The inset summarizes experimental results obtained with Schrödinger cat states of $N=2$ ions. Adapted from [422].

Figure 35

Metrological spin-squeezing parameter as a function of the number of ions. The black points are obtained for a coherent spin state. The purple symbols are obtained with spin-squeezed states (optimized over probe preparation time): with (open squares) and without (solid squares) subtraction of photon shot noise. The dashed (solid) line is a theoretical prediction (including spontaneous emission). From [47].

Figure 36

Differential interferometry scheme (a): two Mach-Zehnder interferometers working in parallel are coupled to the shot-to-shot fluctuating phase noise $\phi$. The signal to be estimated is $\theta$. (b) Normalized Fisher information ($F/N$, gain over the shot noise) as a function of the phase noise, modeled according to Eq. (139) with $P(\phi |\theta )\sim e^{-(\phi -\theta {)}^2/(2{\sigma }_{\mathrm{pn}}^2)}$. The green (darker) lines refer to NOON states in each interferometer, while the orange (lighter) lines refer to spin-squeezed states obtained via Kitagawa-Ueda evolution $e^{-i\chi t{\widehat{J}}_z^2}|0,\pi /2⟩$ of a coherent spin state $|0,\pi /2⟩$, with $\chi t=0.01\pi$. The thick solid lines are obtained with a differential interferometer, while the dashed lines are obtained with a single interferometer. The thin black solid line is Eq. (142). The horizontal dashed line is $N^2/4$. Here $N=100$.

Figure 37

Interaction-based readout. (a) Normalized Fisher information ($F/N$, gain over the shot noise) as a function of the detection noise, modeled according to Eq. (145) with a Gaussian $P(\mu |\tilde{\mu })$ of width ${\sigma }_{\mathrm{dn}}$ and $\tilde{\eta }=1$. The green (darker) lines refer to a NOON state, and the orange (lighter) lines to a spin-squeezed state obtained via Kitagawa-Ueda evolution $e^{-i\tau {\widehat{J}}_z^2}|0,\pi /2⟩$ of a coherent spin state $|0,\pi /2⟩$, with $\tau =0.01\pi$. The solid lines are obtained with an interaction-based readout, and the dashed lines with a standard particle number detection. (b) Probability distributions of a NOON state $P(\mu |\theta )={|⟨{\mu }_z|e^{i(\pi /2){\widehat{J}}_x}{e}^{-i\theta {\widehat{J}}_z}|\mathrm{NOON}⟩|}^2$ for $\theta =0$ (red, lighter histogram) and $\theta =\pi /N$ (blue, darker histogram). Substructures that characterize the probability distributions are washed out by a detection noise of just one atom ($\sigma =1$, black line). (c) Probability distribution for an interaction-based readout $P(\mu |\theta )={|⟨{\mu }_z|e^{-i(\pi /2){\widehat{J}}_x^2}{e}^{-i[\theta +\pi /(2N)]{\widehat{J}}_z}|\mathrm{NOON}⟩|}^2$. The distribution for $\theta =0$ [$P(\mu |\theta )={\delta }_{\mu ,-N/2}$, red, lighter histogram] and that for $\theta =\pi /N$ [$P(\mu |\theta )={\delta }_{\mu ,N/2}$, blue, darker histogram] are maximally distinguishable and thus robust against detection noise: the thick and thin black lines are $P_{\mathrm{dn}}(\mu |\theta )$ for ${\sigma }_{\mathrm{dn}}=\sqrt{N}$ for $\theta =0$ and $\pi /N$, respectively, showing that the two distributions do not overlap even in the presence of large detection noise. Here $N=100$.

Figure 38

Stability of an atomic clock beating the standard quantum limit. Allan deviation of an atomic clock with a spin-squeezed input state (red data with error bars, the solid red line being a guide to the eye), with $T_{\mathrm{R}}=200\mu \mathrm{s}$, $N=3.5\times {10}^4$, and $T_{\text{cycle}}=9\mathrm{s}$. The red dotted line $\sigma (\tau )=1.1\times {10}^{-9}{\mathrm{s}}^{1/2}/\sqrt{\tau }$ is a factor of 2.8 in variance (corresponding to 4.5 dB) below the standard quantum limit [black dashed line $\sigma (\tau )=1.85\times {10}^{-9}{\mathrm{s}}^{1/2}/\sqrt{\tau }$]. Open black circles: reference measurements with uncorrelated atoms. From [312].

Figure 39

Entanglement-assisted optical magnetometry. (a) Measurement sensitivity obtained in the absence (squares) and the presence (open circles) of entanglement between atomic ensembles in a two-cell radio-frequency magnetometer. An improvement was demonstrated for short radio-frequency pulse durations $\tau$, where ${\delta }_{\mathrm{RF}}=1/\tau$. From [574]. (b) Measurement sensitivity as a function of the number of atoms with a coherent spin state (blue circles) and spin-squeezed (red diamonds) probe. Solid lines are predictions obtained from an independent analysis of the probe state, while dashed lines are different noise contributions to the sensitivity. From [488].

Figure 40

Performance of a spin-squeezed Bose-Einstein condensate magnetometer expressed in terms of ${\xi }_{\mathrm{R}}^2$ as a function of varying interrogation times $T_{\mathrm{R}}$ for squeezed (blue diamonds) and coherent (red circles) input states. Dashed lines model the constant performance of ${\xi }_{\mathrm{R}}^2=-4.0\mathrm{dB}$ (squeezed state) and $+0.2\mathrm{dB}$ (coherent state) plus technical noise due to shot-to-shot frequency fluctuations. Inset: Scanning probe measurements of microwave fields. The Bose-Einstein condensate (blue) is translated near the surface of an atom chip to measure the spatial dependence of microwave magnetic near fields. Adapted from [396].