Abstract
In a quantum world, a watched arrow never moves. This is the quantum Zeno effect^{1}. Repeatedly asking a quantum system ‘are you still in your initial state?’ blocks its coherent evolution through measurement backaction. Quantum Zeno dynamics (QZD; refs 2, 3) gives more freedom to the system. Instead of pinning it to a single state, it sets a border in its evolution space. Repeatedly asking the system ‘are you beyond the border?’ makes this limit impenetrable. As the border can be designed by choosing the measured observable, QZD allows one to dynamically tailor the system’s Hilbert space. Recent proposals, particularly in the cavity quantum electrodynamics context^{4,5}, highlight the interest of QZD for quantum state engineering tasks^{6,7,8,9,10,11}, which are the key to quantumenabled technologies and quantum information processing. We report the observation of QZD in the 51dimensional Hilbert space of a large angular momentum J = 25. Continuous selective interrogation limits the evolution of this angular momentum to an adjustable multidimensional subspace. This confined dynamics leads to the production of nonclassical ‘Schrödinger cat’ states^{12,13}, quantum superpositions of angular momenta pointing in different directions. These states are promising for sensitive metrology of electric and magnetic fields. This QZD approach could also be generalized to cavity and circuit quantum electrodynamics experiments^{4,5,13} by replacing the angular momentum with a photonic harmonic oscillator.
Main
Quantum Zeno dynamics modifies the classical motion of a system by introducing a impenetrable barrier in the Hilbert space^{4,5,6,7,8,9,10,11}. This barrier can be equivalently induced by repeating a projective quantum measurement, performing a selective pulsed unitary acting on the states at the border (‘Bang Bang’ control) or even by applying a strong continuous coupling to these states^{3}, as verified in a recent experiment^{14}. In that experiment, however, the evolution of the system was restricted to a twodimensional subspace. The dynamics was simply that of a spin 1/2, and did not exhibit the most striking features of QZD (ref. 4).
In this Letter we implement QZD in a large atomic angular momentum J = 25 (‘spin’ or top), represented as an arrow pointing on a generalized Bloch sphere. In the 51dimensional Hilbert space, we isolate tailorable multidimensional manifolds. We show how QZD induces a very nonclassical dynamics inside the Zeno subspace, leading to the generation of Schrödinger cat spin states^{12}, in which the arrow points at the same time in two different directions. As spinsqueezed states^{15}, which are the focus of intense attention, these cat states lead to quantumenabled metrological applications^{13}.
The angular momentum projection on the polar axis of the generalized Bloch sphere is quantized, taking the values J − k, with k = 0…2J (the corresponding eigenstates being J, J − k〉). The dynamical evolution from the initial state J, J〉 (north pole of the Bloch sphere) is induced by a resonant field driving transitions between these eigenstates. In classical terms, it corresponds to a rotation of the arrow along a meridian from the north to the south pole and back. In quantum terms, at each stage of the rotation, the system is in a spin coherent state^{16}, superposition of J, J − k〉s, the average value of J − k coinciding with the projection of the arrow on the polar axis.
Repeatedly measuring the value of this projection would freeze the rotation, merely realizing the quantum Zeno effect. Here, instead, we implement the QZD by applying continuously a selective unitary evolution addressing only one of the J, J − k〉 states. This state corresponds to a welldefined ‘limiting latitude’ on the Bloch sphere. The spin is forbidden to cross the limiting latitude and its motion remains confined on the north polar cap^{5}.
This confined motion is nontrivial. As the rotating spin reaches the limiting latitude crossing point, it vanishes suddenly and reappears at a point on the limiting latitude with opposite longitude (inversion of the spin’s azimuthal phase). The rotation then resumes towards the north pole. The complete, smooth rotation of the classical dynamics is interrupted by sudden phase inversions and replaced by a confined motion on the polar cap bounded by the limiting latitude. Caught at the phase inversion time, the spin is transiently in a quantum superposition of two spin coherent states pointing along opposite longitudes—a cat state.
This confined evolution is similar to that predicted for QZD in the cavity or circuit quantum electrodynamics context^{4,5}. The dynamics of an angular momentum near the north pole of the Bloch sphere is analogous to that of a onedimensional field oscillator, with k playing the role of the photon number^{12}. In this analogy, the polar cap of the Bloch sphere becomes the phase plane spanned by the field quadratures. Our experiment can thus be viewed as a quantum simulation of the cavity quantum electrodynamics version of QZD.
The spin J = 25 is implemented in a subspace of the Stark manifold of a Rydberg atom. The interest of coherent manipulations of Rydberg manifolds has already been demonstrated in pioneering experiments on coherent wave packet dynamics^{17,18,19}. We take advantage of the versatility of this system to demonstrate here a new quantum feature. Figure 1a sketches parts of three adjacent Rydberg manifolds^{20} (principal quantum numbers n_{f} = 52, n_{e} = 51 and n_{g} = 50) in a static electric field F, defining the quantization axis Oz. The eigenstates are sorted in columns according to their magnetic quantum number m (selected to be positive). The circular state^{21,22} in the n_{e} manifold (thickest line) has the maximum allowed m = n_{e} − 1 value. A σ_{+}polarized radiofrequency (RF) field couples it to a ladder of nearly equidistant levels (thick lines). The transitions between adjacent ladder states are at the Stark angular frequency ω_{a} = (3/2)n_{e}ea_{0}F/ℏ within small secondorder corrections in F (a_{0}: Bohr radius, e: charge quantum). Because the atom is prepared initially in the circular state, the other levels in the manifold are not populated by the RFinduced dynamics and are ignored. The atom evolves within a ladder of 51 levels, n_{e}, k = 0〉…  n_{e}, k = 50〉, where n_{e}, k = 0〉 is the circular state.
The coherent evolution induced by the RF field is ruled by the Hamiltonian^{23}: This Hamiltonian describes the rotation of a J = 25 angular momentum at a Rabi frequency Ω_{rf} (ref. 16), with the correspondence n_{e}, k〉 →  J, J − k〉.
The atomic state, driven by the RF, moves down and up the ladder, whereas the equivalent angular momentum rotates around a meridian of its Bloch sphere. We observe this rotation by applying the resonant RF for a time t_{1} and then measuring the populations of n_{e}, k〉 as a function of t_{1} by field ionization (Methods). Figure 2a shows P(k, t_{1}) (k = 0…5) versus t_{1}, for F = 2.35 V cm^{−1}, corresponding to ω_{a}/2π = 230.15 MHz and Ω_{rf}/2π = 152 ± 4 kHz. The conspicuous cascade down the state ladder reveals the spin rotation. The insets show snapshots of the population distribution in the ladder levels. Data are in excellent agreement with the theoretical expectations for a rotating spin coherent state^{24}.
To induce the QZD, we continuously interrogate the atom by selectively addressing one of the spin states with a ‘Zeno’ continuous wave microwave field (MW) resonant on the transition n_{e}, k_{z}〉 →  n_{g}, k_{z}〉 (red arrow on Fig. 1a). For levels n_{e}, k ≠ k_{z}〉, this MW is nonresonant and produces only small light shifts. For k = k_{z}, the Zeno MW admixes n_{g}, k_{z}〉 with n_{e}, k_{z}〉 replaced by a pair of dressed states, ±〉, separated by Ω_{mw} (dynamical Stark splitting).
The resulting level ladder is sketched in Fig. 1b. The σ_{+} transitions within the subspace {  n_{e}, 0〉,  n_{e}, 1〉, …  n_{e}, k_{z} − 1〉, +〉} (arrows in Fig. 1b) are nearly degenerate at the frequency ω_{a}. The Zeno MW dressing opens, between +〉 and −〉, a gap wider than the coupling matrix element of between the spin states. It makes it nearly impossible for the RF drive to induce, in an evolution from n_{e}, 0〉, transitions towards states below +〉. The population of −〉 is negligible and this state can be disregarded in the discussion. Moreover, after an appropriate adiabatic switchingoff of the Zeno MW (Supplementary Information), +〉 is mapped onto n_{e}, k_{z}〉 (−〉, being mapped onto n_{g}, k_{z}〉). The QZD thus splits the angular momentum Hilbert space into , made up of the k_{z} + 1 levels with k ≤ k_{z} close to the north pole of the Bloch sphere, and the complementary southern subspace (k > k_{z}).
After a RFinduced QZD lasting a time t_{1}, we probe the level populations in . We adiabatically switch off the Zeno MW and measure P(k, t_{1}). Figure 2b presents the results of this procedure for k_{z} = 5 and Ω_{mw}/2π = 3.4 MHz. The state distribution now bounces off a ‘wall’ at k = k_{z} + 1 and nearly returns into the initial state after 1.6 μs. This dynamics is drastically different from the runaway process observed without Zeno MW (Fig. 2a). It is in excellent agreement with a complete numerical simulation of the experiment based on the independently measured experimental parameters (Supplementary Information).
The top frame in Fig. 2b shows the total population detected in . It drops by 25% at the bouncing time. This loss is mainly due to a residual transfer into through the Zeno barrier. The insets show the histograms of P(k, t_{1}) at four different times. As t_{1} increases, the P(k, t_{1}) values become radically different from those obtained without QZD (Fig. 1a). We can clearly see that the level population at the bouncing time is no longer that of a coherent spin state. Not only does QZD restrict the evolution to a subspace of five states instead of 51, but the dynamics itself exhibits striking nonclassical features.
We get a clearer picture of this dynamics by a direct measurement of the spin’s Qfunction^{25}, transposing to spin systems the quantum optics Husimi distribution. It is defined on the Bloch sphere as Q(θ, ϕ) = (2J + 1)/(4π) 〈n_{e}, 0  R^{†}(θ, ϕ)ρR(θ, ϕ)  n_{e}, 0〉, where ρ is the angular momentum density operator and R the rotation along a meridian of the Bloch sphere bringing the north pole to the direction defined by the polar angles θ and ϕ.
Determining Q thus amounts to measuring the population in n_{e}, 0〉 after rotating the state by means of a resonant RF pulse whose duration t_{2} controls θ and whose adjustable phase controls ϕ. We perform this rotation with a RF power much larger than that used for the QZD (coupling ). It couples and even in the presence of the Zeno MW (Supplementary Information).
Figure 3a shows six snapshots of Q for k_{z} = 4 and Ω_{mw}/2π = 3.08 ± 0.11 MHz. The initial n_{e}, 0〉 state has a Gaussian Qfunction centred at the north pole, which first moves, upwards in Fig. 3, towards the limiting latitude (dashed red line). It then splits into two components with opposite azimuthal phases. The upper component rapidly decreases, whereas the lower component grows. At t_{1} = 0.76 μs the two peaks are balanced (fourth frame). After the phase inversion, the Qfunction is mainly located in the lower part of the limiting latitude and resumes its motion towards the north pole, reached again at t_{1} = 1.46 μs (last frame). Figure 3b presents the results of the full numerical simulation. The excellent agreement between simulation and experiment confirms our understanding of the system Zeno dynamics and of the spin state measurement process. The observed evolution, especially the phase inversion, is very similar to the prediction of ref. 4 in the cavity quantum electrodynamics context. The fact that here the atom can populate the barrier state +〉 does not qualitatively change the dynamics.
At t_{1} = 0.76 μs, we expect the system to be in a superposition of two spin coherent states with opposite azimuthal phases. However, the coherence of this superposition is not conspicuous in the Qfunction. To get this information, we reconstruct the full angular momentum density matrix, ρ, at this time, through a maximum likelihood method^{26}. It is based on the measurement of the population of several levels after adjustable RFinduced rotations and adiabatic switchingoff of the Zeno MW (Methods).
Figure 4a shows, on the Bloch sphere, the corresponding angular momentum Wigner function^{27} W(θ, ϕ) at t_{1} = 0.76 μs. As in the quantum optics context, negative values for this quasiprobability distribution are an unambiguous indication of the state nonclassicality. We observe two positive maxima near the limiting latitude. They correspond to the two spin coherent statelike components pointing towards opposite azimuthal phases at the phase inversion time. In between, the interference fringes with their negativite values give vivid evidence that we prepare a genuine quantum superposition of two distinct mesoscopic spin states—a cat state. These interference patterns cannot be observed when the Zeno subspace is only of dimension two (ref. 14). Figure 4b presents the simulated Wigner function, taking into account the exact Hamiltonian of the system and all the known imperfections. Experiment and simulation are in excellent agreement (mutual fidelity 0.93). The measured state has a purity Trρ^{2} = 0.75 (simulation: 0.91). It is limited by inhomogeneities of the static electric field.
This experiment demonstrates the implementation of QZD in a Hilbert space large enough to allow us to generate mesoscopic superposition states. This is a significant step towards quantum control through Hilbert space engineering. It has been shown that the quantum control of the massively multilevel Rydberg states structure leads to important applications in state tailoring^{17,19} and quantum information^{18}. The QZD opens an easily tailorable route towards the generation of such states. Moreover, the concepts and techniques used here are of general interest^{6,7,8,9,10,11} and could be applied, for instance, to superconducting qubits in circuit quantum electrodynamics (ref. 13), with direct applications to quantum information processing^{28}.
Improving the homogeneity of the electric field would allow us to observe QZD on a longer timescale. We could perform the experiment for a smaller value Ω_{rf}, which would reduce the leakage through the Zeno barrier, improving the fidelity of the cat state generation, and extend these experiments to larger k_{z} values to prepare larger Schrödinger cat states. This opens the way to metrology beyond the standard quantum limit. The fast oscillations of the Wigner function near the north pole makes the measurement of its value a signal which is very sensitive to small rotations^{13}. Such states could therefore be used as a very sensitive probe of small static magnetic or electric fields.
We also plan to investigate engineered decoherence^{29}, through the application of a controlled electric field noise. The rich level structure of the Rydberg manifolds opens the way to the implementation of decoherencefree qubits through leveldressing schemes^{30}. Furthermore, the atomic state could be mapped onto that of a highQ cavity by tuning selected transitions in resonance by means of the Stark effect. The realization of a fewqubits processor with a single multilevel atom in a cavity is within reach.
Methods
The atoms are produced by excitation of a thermal rubidium beam. Two electrodes A and B facing each other (diameter 60 mm) produce the directing electric field F along Oz. The gap between A and B is surrounded by four independent electrodes, on which we apply RF signals to produce σ_{+} fields with tunable phase and amplitude.
The experimental sequence duration is 35 μs. First, the atomic sample (1.5 atoms on the average) is prepared in the circular Rydberg state by pulsed laser excitation followed by an RFinduced adiabatic rapid passage through the spin ladder. The whole process is completed in 5.6 μs. Dopplerselective laser excitation addresses atoms with a velocity v = 254 ± 4 m s^{−1}.
The residual static field inhomogeneities and the atomic motion limit the useful observation time. Therefore the QZD itself lasts at most 3 μs, with a first RF pulse of duration t_{1}, eventually followed by a second RF rotation for the state reconstruction experiments. The Zeno MW is then adiabatically switched off, and the atoms fly towards the fieldionization detector D outside the electrode structure.
Our detector resolves states in adjacent Rydberg manifolds, but does not resolve directly the n_{e}, k〉 states. Hence the population P(k_{p}, t_{1}) of n_{e}, k_{p}〉 is measured by applying, before field ionization, a resonant πmicrowave (MW) pulse tuned to the n_{e}, k_{p}〉 →  n_{f}, k_{p}〉 transition (green arrow on Fig. 1a). This pulse does not address the levels n_{e}, k ≠ k_{p}〉, owing to the difference between the linear Stark frequencies in adjacent manifolds. Field ionization selectively measures the population of n_{f}, k_{p}〉, equal to P(k_{p}, t_{1}) within the πpulse transfer efficiency, ${\eta}_{{k}_{p}}$ 0.9 (Supplementary Information).
All parameters of the experiment are independently measured or extracted from fits between the data and a numerical simulation of the experiment, taking into account the complete level structure.
The Wigner function measurement is based on a complete reconstruction of the atomic state ρ_{t} in the n_{g} and n_{e} manifolds. Measurements of the populations of the n_{g}, k〉 and n_{e}, k〉 (k < 6) levels after a rotation of the spin states are used to fit ρ_{t} using a maximum likelihood procedure. The final result is then projected onto the spin state ladder. This approach leads to a direct calibration of the experimental imperfections: approximately 9.3% of the population is spuriously transferred in levels outside the union of and .
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Acknowledgements
We thank S. Pascazio and P. Facchi for many discussions and fruitful exchanges. We acknowledge funding by ANR under the project ‘QUSCOINCA’ and by the EU under the ERC project ‘DECLIC’.
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Author notes
 Adrien Signoles
 & Adrien Facon
These authors contributed equally to this work.
Affiliations
Laboratoire Kastler Brossel, Collège de France, CNRS, École Normale Supérieure, UPMC Univ Paris 06, 11, place Marcelin Berthelot 75005 Paris, France
 Adrien Signoles
 , Adrien Facon
 , Dorian Grosso
 , Igor Dotsenko
 , Serge Haroche
 , JeanMichel Raimond
 , Michel Brune
 & Sébastien Gleyzes
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Contributions
All authors contributed to the experimental setup. A.S., A.F. and D.G. collected the data and analysed the results. JM.R., S.H. and M.B. supervised the research. S.G. led the experiment and performed the numerical simulations. All authors discussed the results and the manuscript.
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The authors declare no competing financial interests.
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Correspondence to Serge Haroche.
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