Abstract
Quantum metrology promises highprecision measurements of classical parameters with far reaching implications for science and technology. So far, research has concentrated almost exclusively on quantumenhancements in integrable systems, such as precessing spins or harmonic oscillators prepared in nonclassical states. Here we show that large benefits can be drawn from rendering integrable quantum sensors chaotic, both in terms of achievable sensitivity as well as robustness to noise, while avoiding the challenge of preparing and protecting largescale entanglement. We apply the method to spinprecession magnetometry and show in particular that the sensitivity of stateoftheart magnetometers can be further enhanced by subjecting the spinprecession to nonlinear kicks that renders the dynamics chaotic.
Introduction
Quantumenhanced measurements (QEM) use quantum effects in order to measure physical quantities with larger precision than what is possible classically with comparable resources. QEMs are therefore expected to have large impact in many areas, such as improvement of frequency standards^{1,2,3,4,5}, gravitational wave detection^{6,7}, navigation^{8}, remote sensing^{9}, or measurement of very small magnetic fields^{10}. A wellknown example is the use of socalled NOON states in an interferometer, where a state with N photons in one arm of the interferometer and zero in the other is superposed with the opposite situation^{11}. It was shown that the smallest phase shift that such an interferometer could measure scales as 1/N, a large improvement over the standard \(1{\mathrm{/}}\sqrt N\) behavior that one obtains from ordinary laser light. The latter scaling is known as the standard quantum limit (SQL), and the 1/N scaling as the Heisenberg limit (HL). So far the SQL has been beaten only in few experiments, and only for small N (see e.g., ^{3,12,13}), as the required nonclassical states are difficult to prepare and stabilize and are prone to decoherence.
Sensing devices used in quantum metrology so far have been based almost exclusively on integrable systems, such as precessing spins (e.g., nuclear spins, NV centers, etc.) or harmonic oscillators (e.g., modes of an electromagnetic field or mechanical oscillators), prepared in nonclassical states (see ref. ^{14} for a recent review). The idea of the present work is to achieve enhanced measurement precision with readily accessible input states by disrupting the parameter coding by a sequence of controlled pulses that renders the dynamics chaotic. At first sight this may appear a bad idea, as measuring something precisely requires welldefined, reproducible behavior, whereas classical chaos is associated with unpredictible longterm behavior. However, the extreme sensitivity to initial conditions underlying classically chaotic behavior is absent in the quantum world with its unitary dynamics in Hilbert space that preserves distances between states. In turn, quantumchaotic dynamics can lead to exponential sensitivity with respect to parameters of the system^{15}.
The sensitivity to changes of a parameter of quantumchaotic systems has been studied in great detail with the technique of Loschmidt echo^{16}, which measures the overlap between a state propagated forward with a unitary operator and propagated backward with a slightly perturbed unitary operator. In the limit of infinitesimally small perturbation, the Loschmidt echo turns out to be directly related to the quantum Fisher information (QFI) that determines the smallest uncertainty with which a parameter can be estimated. Hence, a wealth of known results from quantum chaos can be immediately translated to study the ultimate sensitivity of quantumchaotic sensors. In particular, linear response expressions for fidelity can be directly transfered to the exact expressions for the QFI.
Ideas of replacing entanglement creation by dynamics were proposed previously^{17,18,19,20,21}, but focussed on initial state preparation, or robustness of the readout^{22,23}, without introducing or exploiting chaotic dynamics during the parameter encoding. They are hence comparable to spinsqueezing of the input state^{24}. Quantum chaos is also favorable for state tomography of random initial states with weak continuous time measurement^{25,26}, but no attempt was made to use this for precision measurements of a parameter. A recent review of other approaches to quantumenhanced metrology that avoid initial entanglement can be found in ref. ^{27}.
We study quantumchaotic enhancement of sensitivity at the example of the measurement of a classical magnetic field with a spinprecession magnetometer. In these devices that count amongst the most sensitive magnetometers currently available^{28,29,30,31,32}, the magnetic field is coded in a precession frequency of atomic spins that act as the sensor. We show that the precision of the magneticfield measurement can be substantially enhanced by nonlinearly kicking the spin during the precession phase and driving it into a chaotic regime. The initial state can be chosen as an essentially classical state, in particular a state without initial entanglement. The enhancement is robust with respect to decoherence or dissipation. We demonstrate this by modeling the magnetometer on two different levels: firstly as a kicked top, a wellknown system in quantum chaos to which we add dissipation through superradiant damping; and secondly with a detailed realistic model of a spinexchangerelaxationfree atomvapor magnetometer including all relevant decoherence mechanisms^{28,33}, to which we add nonlinear kicks.
Results
Physical model of a quantumchaotic sensor
As a sensor we consider a kicked top (KT), a wellstudied quantumchaotic system^{34,35,36} described by the timedependent Hamiltonian
where J_{ i } (i = x, y, z) are components of the (pseudo)angular momentum operator, J ≡ j + 1/2, and we set ħ = 1. J_{ z } generates a precession of the (pseudo)angular momentum vector about the zaxis with precession angle α which is the parameter we want to estimate. “Pseudo” refers to the fact that the physical system need not be an actual physical spin, but can be any system with 2j + 1 basis states on which the J_{ i } act accordingly. For a physical spinj in a magnetic field B in zdirection, α is directly proportional to B. The \(J_y^2\)term is the nonlinearity, assumed to act instantaneously compared to the precession, controlled by the kicking strength k and applied periodically with a period τ that leads to chaotic behavior. The system can be described stroboscopically with discrete time t in units of τ (set to τ = 1 in the following),
with the unitary Floquetoperator
that propagates the state of the system from right after a kick to right after the next kick^{34,35,36}. T denotes timeordering. The total spin is conserved, and 1/J can be identified with an effective ħ, such that the limit j → ∞ corresponds to the classical limit, where X = J_{ x }/J, Y = J_{ y }/J, Z = J_{ z }/J become classical variables confined to the unit sphere. (Z, ϕ) can be identified with classical phase space variables, where ϕ is the azimuthal angle of X = (X, Y, Z)^{36}. For k = 0, the dynamics is integrable, as the precession conserves Z and increases ϕ by α for each application of U_{ α }(0). Phase space portraits of the corresponding classical map show that for k ≲ 2.5, the dynamics remains close to integrable with large visible Kolmogorov–Arnold–Moser tori, whereas for k ≳ 3.0 the chaotic dynamics dominates^{36}.
States that correspond most closely to classical phase space points located at (θ, ϕ) are SU(2)coherent states (“spincoherent states”, or “coherent states” for short), defined as
in the usual notation of angular momentum states \(\left {jm} \right\rangle\) (eigenbasis of J^{2} and J_{ z } with eigenvalues j(j + 1) and m, \(2j \in {\Bbb N}\), m = −j, −j + 1, …, j). They are localized at polar and azimuthal angles θ, ϕ with smallest possible uncertainty of all spinj states (associated circular area ~1/j in phase space). They remain coherent states under the action of U_{ α }(0), i.e., just get rotated, \(\phi \mapsto \phi + \alpha\). For the KT, the parameter encoding of α in the quantum state breaks with the standard encoding scheme (initial state preparation, parameterdependent precession, measurement) by periodically disrupting the coding evolution with parameterindependent kicks that generate chaotic behavior (see Fig. 1).
An experimental realization of the kicked top was proposed in ref. ^{37}, including superradiant dissipation. It has been realized experimentally^{38} in cold cesium vapor using optical pulses (see Supplementary Note 1 for details).
Quantum parameter estimation theory
Quantum measurements are most conveniently described by a positiveoperator valued measure (POVM) {Π_{ ξ }} with positive operators Π_{ ξ } (POVM elements) that fulfill ∫dξΠ_{ ξ } = 1. Measuring a quantum state described by a density operator ρ_{ α } yields for a given POVM and a given parameter α encoded in the quantum state a probability distribution p_{ α }(ξ) = tr(Π_{ ξ }ρ_{ α }) of measurement results ξ. The Fisher information I_{Fisher,α} is then defined by
The minimal achievable uncertainty, i.e., the variance of the estimator Var(α_{est}), with which a parameter α of a state ρ_{ α } can be estimated for a given POVM with M independent measurements is given by the Cramér–Rao bound, Var(α_{est}) ≥ 1/(MI_{Fisher,α}). Further optimization over all possible (POVM)measurements leads to the quantumCramér–Rao bound (QCRB),
which presents an ultimate bound on the minimal achievable uncertainty, where I_{ α } is the quantum Fisher information (QFI), and M the number of independent measurements^{39}.
The QFI is related to the Bures distance \({\rm d}{s}_{{\mathrm{Bures}}}^2\) between the states ρ_{ α } and ρ_{α+dα}, separated by an infinitesimal change of the parameter α, \({\rm d}s_{{\mathrm{Bures}}}^2(\rho ,\sigma )\) ≡ \(2\left( {1  \sqrt {F\left( {\rho ,\sigma } \right)} } \right)\). The fidelity F(ρ, σ) is defined as \(F(\rho ,\sigma ) = \left\Vert {\rho ^{1/2}\sigma ^{1/2}} \right\Vert_1^2\), and \(\left\Vert A \right\Vert_1 \equiv {\mathrm{tr}}\sqrt {AA^\dagger }\) denotes the trace norm^{40}. With this^{41},
For pure states \(\rho = \left \psi \right\rangle \left\langle \psi \right\), \(\sigma = \left \phi \right\rangle \left\langle \phi \right\), the fidelity is simply given by \(F(\rho ,\sigma ) = \left {\left\langle {\psi \phi } \right\rangle } \right^2\). A parameter coded in a pure state via the unitary transformation \(\left {\psi _\alpha } \right\rangle = e^{  i\alpha G}\left {\psi (0)} \right\rangle\) with hermitian generator G gives the QFI^{42}
which holds for all α, and where \(\left\langle \cdot \right\rangle \equiv \left\langle {\psi _\alpha  \cdot \psi _\alpha } \right\rangle\).
Loschmidt echo
The sensitivity to changes of a parameter of quantumchaotic systems has been studied in great detail with the technique of Loschmidt echo^{16}, which measures the overlap \(F_\epsilon\)(t) between a state propagated forward with a unitary operator U_{ α }(t) and propagated backward with a slightly perturbed unitary operator \(U_{\alpha + \epsilon }(  t) = U_{\alpha + \epsilon }^\dagger (t)\), where \(U_\alpha (t) = T{\kern 1pt} {\mathrm{exp}}\left( {  \frac{i}{\hbar }{\int}_0^t {\mathrm{d}}t{\prime}H_\alpha (t{\prime})} \right)\) with the time ordering operator T, the Hamiltonian, \(H_{\alpha + \epsilon }\)(t) = H_{ α }(t) + \(\epsilon\)V(t) and the perturbation V(t),
\(F_\epsilon\) is exactly the fidelity that enters via the Bures distance in the definition Eq. (7) of the QFI for pure states, such that \(I_\alpha (t) = {\mathrm{lim}}_{\epsilon \to 0}4\frac{{1  F_\epsilon (t)}}{{\epsilon ^2}}\).
Benchmarks
In order to assess the influence of the kicking on the QFI, we calculate as benchmarks the QFI for the (integrable) top with Floquet operator U_{ α }(0) without kicking, both for an initial coherent state and for a Greenberger–Horne–Zeilinger (GHZ) state \(\left {\psi _{{\mathrm{GHZ}}}} \right\rangle = \left( {\left {j,j} \right\rangle + \left {j,  j} \right\rangle } \right){\mathrm{/}}\sqrt 2\). The latter is the equivalent of a NOON state written in terms of (pseudo)angular momentum states. The QFI for the time evolution Eq. (2) of a top with Floquet operator U_{ α }(0) is given by Eq. (8) with G = J_{ z }. For an initial coherent state located at θ, ϕ it results in a QFI
As expected, I_{ α }(t) = 0 for θ = 0 where the coherent state is an eigenstate of U_{ α }(0). The scaling ∝ t^{2} is typical of quantum coherence, and I_{ α }(t) ∝ j signifies a SQLtype scaling with N = 2j, when the spinj is composed of N spin−\(\frac{1}{2}\) particles in a state invariant under permutations of particles. For the benchmark, we use the optimal value θ = π/2 in Eq. (10), i.e., I_{top,CS} ≡ 2t^{2}j. For a GHZ state, the QFI becomes
which clearly displays the HLtype scaling ∝ (2j)^{2} ≡ N^{2}.
Results for the kicked top without dissipation
In the fully chaotic case, known results for the Loschmidt echo suggest a QFI of the KT ∝ tj^{2} for times t with t_{E} < t < t_{H}, where \(t_{\mathrm{E}} = \frac{1}{\lambda }{\mathrm{ln}}\left( {\frac{{{\mathrm{\Omega }}_V}}{{h^d}}} \right)\) is the Ehrenfest time, and t_{H} = ħ/Δ the Heisenberg time; λ is the Lyapunov exponent, Ω_{ V } the volume of phasespace, h^{d} with d the number of degrees of freedom the volume of a Planck cell, and Δ the mean energy level spacing^{16,36,43}. For the kicked top, \(h^d \simeq {\mathrm{\Omega }}_V{\mathrm{/}}(2J)\). More precisely, we find for \(t \simeq t_{\mathrm E}\) a QFI I_{ α } ∝ tj^{2} and for \(t \gg t_{\mathrm{H}}\) (see Methods)
where s denotes the number of invariant subspaces s of the Hilbert space (s = 3 for the kicked top with α = π/2, see page 359 in ref. ^{15}), and σ_{cl} is a transport coefficient that can be calculated numerically. The infinitesimally small perturbation relevant for the QFI makes that one is always in the perturbative regime^{44,45}. The Gaussian decay of Loschmidt echo characteristic of that regime becomes the slower the smaller the perturbation and goes over into a power law in the limit of infinitesimally small perturbation^{16}.
The numerical results for the QFI in Fig. 2 illustrate a crossover of powerlaw scalings in the fully chaotic case (k = 30) for an initial coherent state located on the equator (θ, ϕ) = (π/2, π/2). The analytical Loschmidt echo results are nicely reproduced: a smooth transition in scaling from tj^{2} → t^{2}j for t = t_{E} → t ≳ t_{H} can be observed and confirms Eqs. (15) and (16) in the Methods for t > t_{E} = ln(2J)/λ, with the numerically determined Lyapunov exponent \(\lambda \simeq 2.4733\), and Eq. (12) for t ≳ t_{H} ≃ J/3^{16}. We find for relatively large j (j ≳ 10^{2}) a scaling I_{ α } ∝ j^{1.08} in good agreement with Eq. (12) predicting a linear jdependence for large t ≳ t_{H}. During the transient time t < t_{E}, when the state is spread over the phase space, QFI shows a rapid growth that can be attributed to the generation of coherences that are particularly sensitive to the precession.
The comparison of the KT’s QFI I_{α,KT} with the benchmark I_{top,CS} of the integrable top in Fig. 2c shows that a gain of more than two orders of magnitude for j = 4000 can be found at t ≲ t_{E}. Around t_{E} the state has spread over the phase space and has developed coherences while for larger times t > t_{E} the top catches up due to its superior time scaling (t^{2} vs. t). The longtime behavior yields a constant gain <1, which means that the top achieves a higher QFI than the KT in this regime. The gain becomes constant as both top and KT exhibit a t^{2} scaling of the QFI.
Whereas in the fully chaotic regime the memory of the initial state is rapidly forgotten, and the initial state can therefore be chosen anywhere in the chaotic sea without changing much the QFI, the situation is very different in the case of a mixed phase space, in which stability islands are still present. Figure 3 shows phase space distributions for k = 3 of QFI and gain Γ = I_{α,KT}/I_{top,CS} exemplarily for large t and j (t = 2^{15}, j = 4000) in comparison with the classical Lyapunov exponent, where ϕ, Z signify the position of the initial coherent state.
The QFI nicely reproduces the essential structure in classical phase space for the Lyapunov exponent λ (see Methods for the calculation of λ). Outside the regions of classically regular motion, the KT clearly outperforms the top by more than two orders of magnitude. Remarkably, the QFI is highest at the boundary of nonequatorial islands of classically regular motion. Coherent states located on that boundary will be called edge states.
The diverse dynamics for different phase space regions calls for dedicated analyses. Figure 4 depicts the QFI with respect to j and t. The blue area is lower and upper bounded by the benchmarks I_{top,CS} and I_{top,GHZ}, Eq. (11), respectively.
We find that initial states in the chaotic sea perform best for small times while for larger times (t ≳ 300 for j = 4000) edge states perform best. Note that this is numerically confirmed up to very high QFI values (>10^{14}). The superiority of edge states holds for j ≳ 10 for large times (t ≳ 10^{3}, see Fig. 4a). Large values of j allow one to localize states essentially within a stability island. A coherent state localized within a nonequatorial stability island shows a quadratic tscaling analog to the regular top. For a state \( {\psi _{{\mathrm{eq}}}} \rangle\) localized around a point within an equatorial island of stability the QFI drastically decays with increasing j (brown triangles). The scaling with t for j = 4000 reveals that QFI does not increase with t in this case, it freezes. One can understand the phenomenon as arising from a freeze of fidelity due to a vanishing time averaged perturbation^{16,46}: the dynamics restricts the states to the equatorial stability island with time average \(\overline {\langle {\psi _{{\mathrm{eq}}}} ( {U_\alpha ^t} )^\dagger (k)J_zU_\alpha ^t(k) {\psi _{{\mathrm{eq}}}} \rangle }\) = 0. This can be verified numerically, and contrasted with the dynamics when initial states are localized in the chaotic sea or on a nonequatorial island.
Results for the dissipative kicked top
For any quantumenhanced measurement, it is important to assess the influence of dissipation and decoherence. We first study superradiant damping^{47,48,49,50,51} as this enables a proofofprinciple demonstration with an analytically accessible propagator for the master equation with spins up to j ≃ 200 and correspondingly large gains. Then, in the next subsection, we show by detailed and realistic modeling including all the relevant decoherence mechanisms that sensitivity of existing stateoftheart alkalivaporbased spinprecession magnetometers in the spinexchangerelaxationfree (SERF) regime can be enhanced by nonlinear kicks.
At sufficiently low temperatures (\(k_{\rm B}T \ll \hbar \omega\), where ħω is the level spacing between adjacent states \(\left {jm} \right\rangle\)) superradiance is described by the Markovian master equation for the spindensity matrix ρ(t) with continuous time,
where J_{±} ≡ J_{ x } ± iJ_{ y }, with the commutator [A, B] = AB − BA, and γ is the dissipation rate, with the formal solution ρ(t) = exp(Λt)ρ(0) ≡ D(t)ρ(0). The full evolution is governed by dρ(t)/dt = Λρ(t) − iħ[H_{KT}(t), ρ(t)]. Dissipation and precession about the zaxis commute, Λ(J_{ z }ρJ_{ z }) = J_{ z }(Λρ)J_{ z }. Λ can therefore act permanently, leading to the propagator P of ρ from discrete time t to t + τ for the dissipative kicked top (DKT)^{36,52}
For the sake of simplicity, we again set the period τ = 1, and t is taken again as discrete time in units of τ. Then, γτ ≡ γ controls the effective dissipation between two unitary propagations. Classically, the DKT shows a strange attractor in phase space with a fractal dimension that reduces from d = 2 at γ = 0 to d = 0 for large γ, when the attractor shrinks to a point attractor and migrates towards the ground state \(\left {j,  j} \right\rangle\)^{52}. Quantum mechanically, one finds a Wigner function with support on a smeared out version of the strange attractor that describes a nonequilibrium steady state reached after many iterations. Such a nontrivial state is only possible through the periodic addition of energy due to the kicking. Because of the filigrane structure of the strange attractor, one might hope for relatively large QFI, whereas without kicking the system would decay to the ground state, where the QFI vanishes. Creation of steady nonequilibrium states may therefore offer a way out of the decoherence problem in quantum metrology, see also section V.C in ref. ^{27} for similar ideas.
Vanishing kicking strength, i.e., the dissipative top (DT) obtained from the DKT by setting k = 0, will serve again as benchmark. While in the dissipationfree regime, we took the top’s QFI and with it its SQLscaling (∝ jt^{2}) as reference, SQLscaling no longer represents a proper benchmark, because damping typically corrupts QFI with increasing time. To illustrate the typical behavior of QFI, we exemplarily choose certain spin sizes j and damping constants γ here and in the following, such as j = 40 and γ = 0.5 × 10^{−3} in Fig. 5a, while computational limitations restrict us to j ≲ 200.
Figure 5 shows the typical overall behavior of the QFI of the DT and DKT as function of time: after a steep initial rise ∝ t^{2}, the QFI reaches a maximum whose value is the larger the smaller the dissipation. Then the QFI decays again, dropping to zero for the DT, and a plateau value for the DKT. The time at which the maximum value is reached decays roughly as 1/(jγ) for the DT, and as 1/j^{0.95} and 1/γ^{1.94} for the DKT. The plateau itself is in general relatively small for the limited values of j that could be investigated numerically, but it should be kept in mind that (i) for the DT the plateau does not even exist (QFI always decays to zero for large time, as dissipation drives the system to the ground state \(\left {j,  j} \right\rangle\) which is an eigenstate of J_{ z } and hence insensitive to precession); and (ii) there are exceptionally large plateau values even for small j, see e.g., the case of j = 2 in Fig. 5b. There, for γ = 1.58 × 10^{−3}, the plateau value is larger by a factor 2.35 than the DT’s QFI optimized over all initial coherent states for all times. Note that since Λ(J_{ z }ρJ_{ z }) = J_{ z }(Λρ)J_{ z }, for the DT an initial precession about the zaxis that is part of the state preparation can be moved to the end of the evolution and does not influence the QFI of the DT. Optimizing over the initial coherent state can thus be restricted to optimizing over θ.
When considering dynamics, it is natural also to include time as a resource. Indeed, experimental sensitivities are normally given as uncertainties per square root of Hertz: longer (classical) averaging reduces the uncertainty as \(1{\mathrm{/}}\sqrt {T_{{\mathrm{av}}}}\) with averaging time t = T_{av}. For fair comparisons, one multiplies the achieved uncertainty with \(\sqrt {T_{{\mathrm{av}}}}\). Correspondingly, we now compare rescaled QFI and Fisher information, namely \(I_\alpha ^{(t)} \equiv I_\alpha {\mathrm{/}}T_{{\mathrm{av}}}\), \(I_{{\mathrm{Fisher}},\alpha }^{(t)} \equiv I_{{\mathrm{Fisher}},\alpha }{\mathrm{/}}T_{{\mathrm{av}}}\). A protocol that reaches a given level of QFI more rapidly has then an advantage, and best precision corresponds to the maximum rescaled QFI or Fisher information, \(\hat I_\alpha ^{(t)} \equiv {\mathrm{max}}_tI_\alpha ^{(t)}\) or \(\hat I_{{\mathrm{Fisher}},\alpha }^{(t)} \equiv {\mathrm{max}}_t{\kern 1pt} I_{{\mathrm{Fisher}},\alpha }^{(t)}\).
Figure 6 shows that in a broad range of dampings that are sufficiently strong for the QFI to decay early, the maximum rescaled QFI of the DKT beats that quantity of the DT by up to an order of magnitude. Both quantities were optimized over the location of the initial coherent states.
Figure 7 shows \(\hat I_\alpha ^{(t)}\) and the gain in that quantity compared to the nonkicked case as function of both the damping and the kicking strength. One sees that in the intermediate damping regime (γ ≃ 10^{−3}) the gain increases with kicking strength, i.e., increasingly chaotic dynamics.
For exploiting the enhanced sensitivity shown to exist through the large QFI, one needs also to specify the actual measurement of the probe. In principle, the QCRB formalism allows one to identify the optimal POVM measurement if the parameter is known, but these may not always be realistic. In Fig. 8, we investigate J_{ y } as a feasible example for a measurement for a spin size j = 200 after t = 2 time steps. We find that there exists a broad range of kicking strengths where the reference (stateoptimized but k = 0) is outperformed in both cases, with and without dissipation. In a realistic experiment, control parameters such as the kicking strength are subjected to variations. A 5% variance in k, which was reported in ref. ^{38}, reduces the Fisher information only marginally and does not challenge the advantage of kicking. This can be calculated by rewriting the probability that enters in the Fisher information in Eq. (5) according to the law of total probability, p_{ α }(ξ) = ∫dkp(k)p_{ α }(ξk) where p(k) is an assumed Gaussian distribution of k values with 5% variance and p_{ α }(ξk) = tr[Π_{ ξ } ρ_{ α }(k)] with Π_{ ξ } a POVM element and ρ_{ α }(k) the state for a given k value. The advantage from kicking remains when investigating a rescaled and timeoptimized Fisher information (not shown in Fig. 8).
Improving a SERF magnetometer
We finally show that quantumchaotically enhanced sensitivity can be achieved in stateoftheart magnetometers by investigating a rather realistic and detailed model of an alkalivaporbased spinprecession magnetometer acting in the SERF regime. SERF magentometers count amongst the most sensitive magnetometers for detecting small quasistatic magnetic fields^{28,29,30,31,32}. We consider a cesiumvapor magnetometer at room temperature in the SERF regime similar to experiments with rubidium in ref. ^{53}. Kicks on the single cesiumatom spins can be realized as in ref. ^{38} by exploiting the spindependent rank2 (ac Stark) lightshift generated with the help of an offresonant laser pulse. Typical SERF magnetometers working at higher temperatures with high buffergas pressures exhibit an unresolved excited state hyperfine splitting due to pressure broadening, which makes kicks based on rank2 lightshifts ineffective. Dynamics are modeled in the electronic ground state 6^{2}S_{1/2} of ^{133}Cs that splits into total spins of f = 3 and f = 4, where kicks predominantly act on the f = 3 manifold.
The model is quite different from the foregoing superradiance model because of a different decoherence mechanism originating from collisions of Cs atoms in the vapor cell: We include spinexchange and spindestruction relaxation, as well as additional decoherence induced by the optical implementation of the kicks. With this implementation of kicks one is confined to a small spin size f = 3 of single atoms, such that the large improvements in sensitivity found for the large spins discussed above cannot be expected. Nevertheless, we still find a clear gain in the sensitivity and an improved robustness to decoherence due to kicking. Details of the model described with a master equation^{33,54} can be found in the Supplementary Note 2.
Spins of cesium atoms are initially pumped into a state spinpolarized in zdirection orthogonal to the magnetic field B = B\(\hat {\bf{y}}\) in ydirection, whose strength B is the parameter α to be measured. We let spins precess in the magnetic field, and, by incorporating small kicks about the xaxis, we find an improvement over the reference (without kicks) in terms of rescaled QFI and the precision based on the measurement of the electronspin component S_{ z } orthogonal to the magnetic field. The best possible measurement precision ΔB in units of T/\(\sqrt {{\mathrm{Hz}}}\) per 1 cm^{3} vapor volume is \({\mathrm{\Delta }}B = 1{\mathrm{/}}\sqrt {nI_B^{(t)}}\) where n ≃ 2 × 10^{10} is the number of cesium atoms in 1 cm^{3}. For a specific measurement, \(I_B^{(t)}\) must be replaced by the corresponding rescaled Fisher information \(I_{{\mathrm{Fisher,}B}}^{(t)}\). We compare the models with and without kicks directly on the basis of the Fisher information rather than modeling in addition the specific optical implementation and the corresponding noise of the measurement of S_{ z }. Neglecting this additional readoutspecific noise leads to slightly better precision bounds than given in the literature, but does not distort the comparison.
The magnetic field was set to B = 4 × 10^{−14} T in ydirection, such that the condition for the SERF regime is fulfilled, i.e., the Larmor frequency is much smaller than the spinexchange rate, and the period is set to τ = 1 ms. Since kicks induce decoherence in the atomic spin system, we have to choose a very small effective kicking strength of k ≃ 6.5 × 10^{−4} for the kicks around the xaxis (with respect to the f = 3 groundstate manifold), generated with an offresonant 2 μs light pulse with intensity I_{kick} = 0.1 mW/cm^{2} linearly polarized in xdirection, to find an advantage over the reference.
The example of Fig. 9 shows about 31% improvement in measurement precision ΔB for an optimal measurement (QFI, upper right inset) and 68% improvement in a comparison of S_{ z } measurements (inset), which is impressive in view of the small system size. The achievable measurement precision of the kicked dynamics exhibits an improved robostness to decoherence: rescaled QFI for the kicked dynamics continues to increase and sets itself apart from the reference around the coherence time associated with spindestruction relaxation. The laser light for these pulses can be provided by the laser used for the read out, which is typically performed with an offresonant laser. A further improvement in precision is expected from additionally measuring kick pulses for readout or by applying kicks not only to the f = 3 but also to the f = 4 groundstate manifold of ^{133}Cs. Further, it might be possible to dramatically increase the relevant spinsize by applying the kicks to the joint spin of the cesium atoms, for instance, through a doublepass Faraday effect^{55}.
Discussion
Rendering the dynamics of quantum sensors chaotic allows one to harvest a quantum enhancement for quantum metrology without having to rely on the preparation or stabilization of highly entangled states. Our results imply that existing magnetic field sensors^{31,56} based on the precession of a spin can be rendered more sensitive by disrupting the timeevolution by nonlinear kicks. The enhancement persists in rather broad parameter regimes even when including the effects of dissipation and decoherence. Besides a thorough investigation of superradiance damping over large ranges of parameters, we studied a cesiumvaporbased atomic magnetomter in the SERF regime based on a detailed and realistic model^{28,29,30,31,53}. Although the implementation of the nonlinearity via a rank2 light shift introduces additional decoherence and despite the rather small atomic spin size f ≤ 4, a considerable improvement in measurement sensitivity is found (68% for a readout scheme based on the measurement of the electronic spincomponent S_{ z }). The required nonlinearity that can be modulated as function of time has been demonstrated experimentally in ref. ^{38} in cold cesium vapor.
Even higher gains in sensitivity are to be expected if an effective interaction can be created between the atoms, as this opens access to larger values of total spin size for the kicks. This may be achieved e.g., via a cavity as suggested for pseudospins in ref. ^{57}, or the interaction with a propagating light field as demonstrated experimentally in refs. ^{55,58} with about 10^{12} cesium atoms. More generally, our scheme will profit from the accumulated knowledge of spinsqueezing, which is also based on the creation of an effective interaction between atoms. Finally, we expect that improved precision can be found in other quantum sensors that can be rendered chaotic as well, as the underlying sensitivity to change of parameters is a basic property of quantumchaotic systems.
Methods
QFI for kicked timeevolution of a pure state
The QFI in the chaotic regime with large system dimension 2J and times larger than the Ehrenfest time, t > t_{E}, is given in linearresponse theory by an autocorrelation function \(C(t) \equiv \left\langle\tilde V(t)\tilde V(0)\right\rangle \left\langle \tilde V(t)\right\rangle\left\langle\tilde V(0)\right\rangle\) of the perturbation of the Hamiltonian in the interaction picture, \(H_{\alpha + \epsilon }\)(t) = H_{ α }(t) + \(\epsilon\)V(t), \(\tilde V(t) = U_\alpha (  t)V(t)U_\alpha (t)\):
In our case, the perturbation V(t) = J_{ z } is proportional to the parameterencoding precession Hamiltonian, and the first summand in Eq. (15) can be calculated for an initial coherent state,
giving a tj^{2}scaling starting from t_{E}. Due to the finite Hilbertspace dimension of the kicked top, the autocorrelation function decays for large times to a finite value \(\bar C\), leading to a term quadratic in t from the sum in Eq. (15) that simplifies to \(I_\alpha = 4\overline C t^2\) for \(t \gg t_{\rm H}\). If one rescales J_{ z } → J_{ z }/J such that it has a well defined classical limit, random matrix theory allows one to estimate the average value of C(t) for large times: \(\bar C = 2Js\sigma _{{\mathrm{cl}}}\), and σ_{cl} is a transport coefficient that can be calculated numerically^{16}. This yields Eq. (12).
Lyapunov exponent
A data point located at (Z, ϕ) for the Lyapunov exponent in Fig. 2c was obtained numerically by averaging over 100 initial conditions equally distributed within a circular area of size 1/j (corresponding to the coherent state) centered around (Z, ϕ).
Data availability
Numerical simulation data from this work have been submitted to figshare.com with DOI 10.6084/m9.figshare.5901640. Relevant data are also available from the authors upon request.
References
 1.
Huelga, S. F. et al. Improvement of frequency standards with quantum entanglement. Phys. Rev. Lett. 79, 3865–3868 (1997).
 2.
Meyer, V. et al. Experimental demonstration of entanglementenhanced rotation angle estimation using trapped ions. Phys. Rev. Lett. 86, 5870–5873 (2001).
 3.
Leibfried, D. et al. Toward Heisenberglimited spectroscopy with multiparticle entangled states. Science 304, 1476–1478 (2004).
 4.
Wasilewski, W. et al. Quantum noise limited and entanglementassisted magnetometry. Phys. Rev. Lett. 104, 133601 (2010).
 5.
Koschorreck, M., Napolitano, M., Dubost, B. & Mitchell, M. W. Subprojectionnoise sensitivity in broadband atomic magnetometry. Phys. Rev. Lett. 104, 093602 (2010).
 6.
Goda, K. et al. A quantumenhanced prototype gravitationalwave detector. Nat. Phys. 4, 472–476 (2008).
 7.
Aasi, J. et al. Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light. Nat. Photonics 7, 613–619 (2013).
 8.
Giovannetti, V., Lloyd, S. & Maccone, L. Quantumenhanced positioning and clock synchronization. Nature 412, 417–419 (2001).
 9.
Allen, E. H. & Karageorgis, M. Radar systems and methods using entangled quantum particles. US Patent 7,375,802 (2008).
 10.
Taylor, J. et al. Highsensitivity diamond magnetometer with nanoscale resolution. Nat. Phys. 4, 810–816 (2008).
 11.
Boto, A. N. et al. Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit. Phys. Rev. Lett. 85, 2733–2736 (2000).
 12.
Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglementfree Heisenberglimited phase estimation. Nature 450, 393–396 (2007).
 13.
Nagata, T., Okamoto, R., O’Brien, J. L. & Takeuchi, K. S. S. Beating the standard quantum limit with fourentangled photons. Science 316, 726–729 (2007).
 14.
Pezzè, L., Smerzi, A., Oberthaler, M. K., Schmied, R. & Treutlein, P. Quantum metrology with nonclassical states of atomic ensembles. Preprint at https://arxiv.org/abs/1609.01609v2 (2016).
 15.
Peres, A. in Quantum Chaos (eds Cerdeira, H. A., Ramaswamy, R., Gutzwiller, M. C., Casati, G.) (World Scientific, Singapore, 1991).
 16.
Gorin, T., Prosen, T., Seligman, T. H. & Žnidarič, M. Dynamics of Loschmidt echoes and fidelity decay. Phys. Rep. 435, 33–156 (2006).
 17.
Boixo, S. et al. Quantum metrology: dynamics versus entanglement. Phys. Rev. Lett. 101, 040403 (2008).
 18.
XiaoQian, W., Jian, M., XiHe, Z. & XiaoGuang, W. Chaos and quantum Fisher information in the quantum kicked top. Chin. Phys. B 20, 050510 (2011).
 19.
Song, L., Ma, J., Yan, D. & Wang, X. Quantum Fisher information and chaos in the Dicke model. Eur. Phys. J. D 66, 201 (2012).
 20.
Weiss, C. & Teichmann, N. Signatures of chaosinduced mesoscopic entanglement. J. Phys. B 42, 031001 (2009).
 21.
Fröwis, F., Sekatski, P. & Dür, W. Detecting large quantum Fisher information with finite measurement precision. Phys. Rev. Lett. 116, 090801 (2016).
 22.
Macrì, T., Smerzi, A. & Pezzè, L. Loschmidt echo for quantum metrology. Phys. Rev. A 94, 010102 (2016).
 23.
Linnemann, D. et al. Quantumenhanced sensing based on time reversal of nonlinear dynamics. Phys. Rev. Lett. 117, 013001 (2016).
 24.
Ma, J., Wang, X., Sun, C. & Nori, F. Quantum spin squeezing. Phys. Rep. 509, 89–165 (2011).
 25.
Madhok, V., Riofrío, C. A., Ghose, S. & Deutsch, I. H. Information gain in tomography–a quantum signature of chaos. Phys. Rev. Lett. 112, 014102 (2014).
 26.
Madhok, V., Riofrío, C. A. & Deutsch, I. H. Review: characterizing and quantifying quantum chaos with quantum tomography. Pramana 87, 65 (2016).
 27.
Braun, D. et al. Quantum enhanced measurements without entanglement. Preprint at https://arxiv.org/abs/1701.05152 (2017).
 28.
Allred, J., Lyman, R., Kornack, T. & Romalis, M. Highsensitivity atomic magnetometer unaffected by spinexchange relaxation. Phys. Rev. Lett. 89, 130801 (2002).
 29.
Kominis, I. K., Kornack, T. W., Allred, J. C. & Romalis, M. V. A subfemtotesla multichannel atomic magnetometer. Nature 422, 596–599 (2003).
 30.
Savukov, I. & Romalis, M. Effects of spinexchange collisions in a highdensity alkalimetal vapor in low magnetic fields. Phys. Rev. A 71, 023405 (2005).
 31.
Budker, D. & Romalis, M. Optical magnetometry. Nat. Phys. 3, 227–234 (2007).
 32.
Sheng, D., Li, S., Dural, N. & Romalis, M. Subfemtotesla scalar atomic magnetometry using multipass cells. Phys. Rev. Lett. 110, 160802 (2013).
 33.
Appelt, S. et al. Theory of spinexchange optical pumping of ^{3}He and ^{129}Xe. Phys. Rev. A 58, 1412–1439 (1998).
 34.
Haake, F., Kuś, M. & Scharf, R. Classical and quantum chaos for a kicked top. Z. Phys. B Condens. Matter 65, 381–395 (1987).
 35.
Haake, F., Kuś, M. & Scharf, R. in Coherence, Cooperation, and Fluctuations (eds Haake, F., Narducci, L., Walls, D.) (Cambridge University Press, Cambridge, 1986).
 36.
Haake, F. Quantum Signatures of Chaos, Vol. 54 (Springer Science & Business Media, Berlin, 2013).
 37.
Haake, F. Can the kicked top be realized? J. Mod. Opt. 47, 2883–2890 (2000).
 38.
Chaudhury, S., Smith, A., Anderson, B., Ghose, S. & Jessen, P. S. Quantum signatures of chaos in a kicked top. Nature 461, 768–771 (2009).
 39.
Helstrom, C. W. Quantum detection and estimation theory. J. Stat. Phys. 1, 231–252 (1969).
 40.
Miszczak, J. A., Puchała, Z., Horodecki, P., Uhlmann, A. & Życzkowski, K. Sub and superfidelity as bounds for quantum fidelity. Quantum Inf. Comput. 9, 0103–0130 (2009).
 41.
Braunstein, S. L. & Caves, C. M. Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72, 3439–3443 (1994).
 42.
Braunstein, S. L. & Caves, C. M. Wringing out better Bell inequalities. Ann. Phys. 202, 22–56 (1990).
 43.
Zaslavsky, G. M. Stochasticity in quantum systems. Phys. Rep. 80, 157–250 (1981).
 44.
Jacquod, P., Silvestrov, P. & Beenakker, C. Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo. Phys. Rev. E 64, 055203 (2001).
 45.
Benenti, G. & Casati, G. Quantumclassical correspondence in perturbed chaotic systems. Phys. Rev. E 65, 066205 (2002).
 46.
Sankaranarayanan, R. & Lakshminarayan, A. Recurrence of fidelity in nearly integrable systems. Phys. Rev. E 68, 036216 (2003).
 47.
Dicke, R. H. Coherence in spontaneous radiation processes. Phys. Rev. 93, 99–110 (1954).
 48.
Bonifacio, R., Schwendiman, P. & Haake, F. Quantum statistical theory of superradiance I. Phys. Rev. A. 4, 302–313 (1971).
 49.
Haake, F. Statistical Treatment of Open Systems by Generalized Master Equations, Springer Tracts in Modern Physics, Vol. 66, 98168 (Springer, Berlin, 1973).
 50.
Gross, M. & Haroche, S. Superradiance: an essay on the theory of collective spontaneous emission. Phys. Rep. 93, 301–396 (1982).
 51.
Kaluzny, Y., Goy, P., Gross, M., Raimond, J. & Haroche, S. Observation of selfinduced Rabi oscillations in twolevel atoms excited inside a resonant cavity: the ringing regime of superradiance. Phys. Rev. Lett. 51, 1175–1178 (1983).
 52.
Braun, D. Dissipative Quantum Chaos and Decoherence, Springer Tracts in Modern Physics, Vol. 172 (Springer, Berlin, 2001).
 53.
Balabas, M., Karaulanov, T., Ledbetter, M. & Budker, D. Polarized alkalimetal vapor with minutelong transverse spinrelaxation time. Phys. Rev. Lett. 105, 070801 (2010).
 54.
Deutsch, I. H. & Jessen, P. S. Quantum control and measurement of atomic spins in polarization spectroscopy. Opt. Commun. 283, 681–694 (2010).
 55.
Takeuchi, M. et al. Spin squeezing via oneaxis twisting with coherent light. Phys. Rev. Lett. 94, 023003 (2005).
 56.
Ledbetter, M., Savukov, I., Acosta, V., Budker, D. & Romalis, M. Spinexchangerelaxationfree magnetometry with Cs vapor. Phys. Rev. A 77, 033408 (2008).
 57.
Agarwal, G., Puri, R. & Singh, R. Atomic Schrödinger cat states. Phys. Rev. A 56, 2249–2254 (1997).
 58.
Julsgaard, B., Kozhekin, A. & Polzik, E. S. Experimental longlived entanglement of two macroscopic objects. Nature 413, 400–403 (2001).
Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft (DFG), Grant No. BR 5221/11. Numerical calculations were performed in part with resources supported by the Zentrum für Datenverarbeitung of the University of Tübingen.
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D.B. initiated the idea and L.F. made the calculations and numerical simulations. Both authors contributed to the interpretation of data and the writing of the manuscript.
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Fiderer, L.J., Braun, D. Quantum metrology with quantumchaotic sensors. Nat Commun 9, 1351 (2018). https://doi.org/10.1038/s4146701803623z
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