Abstract
An ensemble of atoms can operate as a quantum sensor by placing atoms in a superposition of two different states. Upon measurement of the sensor, each atom is individually projected into one of the two states. Creating quantum correlations between the atoms, that is entangling them, could lead to resolutions surpassing the standard quantum limit^{1,2,3} set by projections of individual atoms. Large amounts of entanglement^{4,5,6} involving the internal degrees of freedom of lasercooled atomic ensembles^{4,5,6,7,8,9,10,11,12,13,14,15,16} have been generated in collective cavity quantumelectrodynamics systems, in which many atoms simultaneously interact with a single optical cavity mode. Here we report a matterwave interferometer in a cavity quantumelectrodynamics system of 700 atoms that are entangled in their external degrees of freedom. In our system, each individual atom falls freely under gravity and simultaneously traverses two paths through space while entangled with the other atoms. We demonstrate both quantum nondemolition measurements and cavitymediated spin interactions for generating squeezed momentum states with directly observed sensitivity \(3\,.\,{4}_{0.9}^{+1.1}\) dB and \(2\,.\,{5}_{0.6}^{+0.6}\) dB below the standard quantum limit, respectively. We successfully inject an entangled state into a Mach–Zehnder lightpulse interferometer with directly observed sensitivity \(1\,.\,{7}_{0.5}^{+0.5}\) dB below the standard quantum limit. The combination of particle delocalization and entanglement in our approach may influence developments of enhanced inertial sensors^{17,18}, searches for new physics, particles and fields^{19,20,21,22,23}, future advanced gravitational wave detectors^{24,25} and accessing beyond meanfield quantum manybody physics^{26,27,28,29,30}.
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Main
Lightpulse matterwave interferometers exploit the quantized momentum kick given to atoms during absorption and emission of light to split atomic wave packets so that they traverse distinct spatial paths at the same time. Additional momentum kicks then return the atoms to the same point in space to interfere the two matterwave wave packets. The key to the precision of these devices is the encoding of information in the phase ϕ that appears in the superposition of the two quantum trajectories within the interferometer. This phase must be estimated from quantum measurements to extract the desired information. For N atoms, the phase estimation is fundamentally limited by the independent quantum collapse of each atom to an r.m.s. angular uncertainty \(\Delta {\theta }_{{\rm{SQL}}}=1/\sqrt{N}\) rad, known as the standard quantum limit (SQL)^{2}.
Here we demonstrate a matterwave interferometer^{31,32} with a directly observed interferometric phase noise below the SQL, a result that combines two of the most striking features of quantum mechanics: the concept that a particle can appear to be in two places at once and entanglement between distinct particles. This work is also a harbinger of future quantum manybody simulations with cavities^{26,27,28,29} that will explore beyond meanfield physics by directly modifying and probing quantum fluctuations or in which the quantum measurement process induces a phase transition^{30}.
Quantum entanglement between the atoms allows the atoms to conspire together to reduce their total quantum noise relative to their total signal^{1,3}. Such entanglement has been generated between atoms using direct collisional^{33,34,35,36,37,38,39} or Coulomb^{40,41} interactions, including relative atom number squeezing between matter waves in spatially separated traps^{33,35,39} and mapping of internal entanglement onto the relative atom number in different momentum states^{42}. A trapped matterwave interferometer with relative number squeezing was realized in ref. ^{35}, but the interferometer’s phase was antisqueezed and thus the phase resolution was above the SQL.
We report the generation of cavity quantumelectrodynamics entanglement between the external momentum states of different atoms using two distinct approaches that both rely on the strong collective coupling between the atoms and an optical cavity. In the first approach, we realize cavityenhanced quantum nondemolition (QND) measurements^{4,5,7,8} to essentially measure and subtract out the quantum noise. In the second approach, we use the cavity to mediate unitary interactions between the atoms to realize socalled oneaxis twisting (OAT)^{1,14,15,16,43} or an alltoall Ising interaction. Both approaches have been realized for generating as much as 18.5 dB of entanglement^{4,5}, but only between internal states of atoms and with only the realization of directly observed enhancements in entangled microwave clocks^{12,13} and magnetometers^{44}. Cavity approaches to OAT^{45} and QND^{46} entanglement of purely Bragg interferometers have also been proposed.
Strong collective coupling to the cavity NC ≫ 1 is the key requirement for both approaches to generate entanglement, where C is the single particle cooperativity parameter^{43,47,48}. Previously, an interferometer was operated in a low finesse cavity^{49,50}, to provide power buildup, spatial mode filtering and precise beam alignment. Here we achieve matterwave interferometric control^{31,32} simultaneously with strong collective coupling NC ≈ 500 by operating inside a cavity with high finesse \({\mathcal{F}}=1.3\times 1{0}^{5}\) with small mode waist w_{0} = 72 μm.
Our twomirror cavity is vertically oriented along \(\hat{{\bf{Z}}}\) (Fig. 1). The cavity has a power decay rate κ = 2π × 56(3) kHz at 780 nm, a mirror separation L = 2.2 cm and a free spectral range ω_{FSR} = 2π × 6.7879 GHz (all error bars reported are 1σ uncertainties). Rubidium atoms are laser cooled inside the cavity and then allowed to fall under gravity for a duration of T_{fall}, guided tightly along the cavity axis by a hollow (Laguerre–Gauss LG_{01}like) bluedetuned optical dipole guide^{51} with thermal r.m.s. cloud transverse radius of r_{r.m.s.} = 4.7(8) μm ≪ w_{0} (Methods).
Manipulating matter waves
We manipulate matterwave wave packets using velocitysensitive twophoton transitions with wavelength λ = 780 nm. The combined absorption and stimulated emission of photons imparts 2ħk momentum kicks oriented along the cavity axis, where k = 2π/λ and ħ is the reduced Planck constant.
For Raman transitions in which both momentum and spin states are changed, we use the magnetically insensitive ^{87}Rb clock states, \(\left\downarrow \right\rangle \equiv \leftF=1,{m}_{F}=0\right\rangle \) and \(\left\uparrow \right\rangle \equiv \leftF=2,{m}_{F}=0\right\rangle \), separated by the hyperfine transition frequency ω_{HF} ≈ 2π × 6.835 GHz. The driving laser’s frequency is stabilized between two TEM_{00} longitudinal modes approximately Δ = 2π × 85 GHz bluedetuned of \(\left\uparrow \right\rangle \to \lefte\right\rangle \equiv \left{5}^{2}{{\rm{P}}}_{3/2},F=3\right\rangle \) (Fig. 2a). As shown in Fig. 2b, the cavity free spectral range is tuned such that two sidebands at ±ω_{R} are approximately ±2π × 23 MHz from resonance with the closest TEM_{00} mode when 2ω_{R} = ω_{HF}. This configuration allows enough light to nonresonantly enter the cavity for a twophoton Rabi frequency Ω_{TwoPh} = 2π × 10 kHz. By injecting the Raman tones nonresonantly and with opposite detunings, we greatly suppress laser frequency noise from being converted into phase and amplitude noise inside the cavity. Such noise manifests as noise in the Raman rotations and undesired Bragg scattering to other momentum states. The frequency difference of the sidebands is linearly ramped at a rate of 25 kHz ms^{−}^{1} to compensate for the acceleration of the atoms by gravity (Methods).
In Fig. 2c, we show the initial axial velocity spectrum of the atoms as mapped out by inducing velocitydependent spin flips. We use this same process to select atoms within a narrow range of initial velocities for coherent manipulation of matter waves, resulting in approximately N_{0} = 800−1,200 atoms in \(\downarrow \rangle \) with r.m.s. momentum spread Δp = 0.1ħk set by choice of the twophoton Rabi frequency Ω_{TwoPh} = 2π × 1.4 kHz (Methods).
In Fig. 2d, we demonstrate the quantized nature of the momentum kicks imparted by the intracavity Raman transitions. After velocity selection, a π/2 pulse is followed by a second Raman π pulse to place the atoms into a superposition of \(\left0\hbar k,\downarrow \right\rangle \) and \(\left4\hbar k,\downarrow \right\rangle \) in the falling frame of reference. We observe this as two distinct peaks separated in the subsequent velocity spectrum. Future interferometers might evolve in such superpositions so as to minimize systematic errors and dephasing due to differential environmental couplings to \(\left\uparrow \right\rangle \) and \(\left\downarrow \right\rangle \).
Complementary to hyperfine spinstate changing Raman transitions, we also demonstrate intracavity Bragg transitions in this highfinesse and highcooperativity cavity. The Bragg coupling (Methods) connects states \(\leftn\hbar k\right\rangle \leftrightarrow \left(n+2)\hbar k\right\rangle \) with no change in the spin degree of freedom, as shown in Fig. 2e. After velocity selection, the wave packet is coherently split by a Bragg π/2 pulse, followed by successive π pulses to transfer momentum to one of the wave packet components for a momentum difference of up to 10ħk. Access to Bragg transitions opens the door to both large momentum transfer operations for greater sensitivity and to improved coherence times in future work.
Squeezing on momentum states
We now turn our attention to creating entanglement between atoms that includes this external degree of freedom. We describe the collective state of our matterwave interferometer using a Bloch sphere with average Bloch vector \({\bf{J}}=\langle \,{\hat{J}}_{x}\hat{{\bf{x}}}+{\hat{J}}_{y}\hat{{\bf{y}}}+{\hat{J}}_{z}\hat{{\bf{z}}}\rangle \) of length \(J\equiv {\bf{J}}\le {N}_{0}/2\) in a fictitious coordinate space (Fig. 1b). The collective pseudospin projection operators are defined as \({\hat{J}}_{z}\equiv \frac{1}{2}({\hat{N}}_{\uparrow }{\hat{N}}_{\downarrow })\) with collective population projection operators \({\hat{N}}_{\uparrow }={\sum }_{i}^{{N}_{0}}{a\rangle }_{i}{}_{i}\langle a\) and \({\hat{N}}_{\downarrow }={\sum }_{i}^{{N}_{0}}{b\rangle }_{i}{}_{i}\langle b\), and similarly for other pseudospin projections, where \({\lefta\right\rangle }_{i}={\left2\hbar k,\uparrow \right\rangle }_{i}\) and \({\leftb\right\rangle }_{i}={\left0\hbar k,\downarrow \right\rangle }_{i}\) for the ith atom. We use a Raman π/2 pulse to nominally prepare all atoms in an unentangled coherent pseudospin state described by the Bloch vector \({\bf{J}}=J\hat{{\bf{x}}}\). The SQL arises from the nonzero variance of the spin projection operators \({(\Delta {J}_{z})}^{2}=\langle \,{\hat{J}}_{z}^{2}\,\rangle {\langle {\hat{J}}_{z}\rangle }^{2}\ne 0\), and so on, and is visualized on the Bloch sphere as a quasiprobability distribution of the orientation of the Bloch vector from trial to trial. We prepare squeezed momentum states using both QND measurements^{4,5,47} and OAT^{1,14,43} in which the quantum noise is reduced in one spinmomentum projection at the expense of increased quantum noise along the orthogonal projection.
The Wineland parameter W characterizes the phase enhancement of a squeezed state with phase uncertainty Δθ that is certified to arise from entanglement between the atoms^{3},
Physically, W is the reduction in the angular noise variance of the phase estimation relative to the SQL, \(\Delta {\theta }_{{\rm{SQL}}}=1/\sqrt{N}\), one would have for a pure state with a Bloch vector length J_{c} = N/2 equal to that of the actual mixed or partially decohered state prepared without the squeezing operation (Methods) .
Collective QND measurements of the free falling atomic samples are used to estimate the number of atoms in different spinmomentum states without revealing singleparticle information^{47,51}. The two momentum states interact differently with the optical cavity because they carry distinct spin labels. We tune a TEM_{00} cavity mode with resonance frequency ω_{c} to the blue of the \(\left\uparrow \right\rangle \to \lefte\right\rangle \) transition ω_{a} by δ_{c} = ω_{c} − ω_{a} (Fig. 2a). After adiabatically eliminating the excited state \(\lefte\right\rangle \) and ignoring meanfield light shifts that will be spinechoed away, the effective Hamiltonian^{43} describing the atom–cavity QND interaction can be expressed in a rotating frame at the atomic transition frequency as
where the cavity field is described by creation and annihilation operators \({\hat{c}}^{\dagger }\) and \(\hat{c}\). The cavity resonance shifts by an amount χ_{QND} = 2π × 335(4) Hz per atom in \(\left\uparrow \right\rangle \) at a detuning δ_{c} = 2π × 175 MHz (Methods). The population N_{↑} of atoms in the momentum state with spin label \(\left\uparrow \right\rangle \) can be estimated by measuring the cavity frequency shift, which is estimated by detecting the probe light reflected from the cavity input mirror as the laser frequency is swept across resonance (Figs. 1a and 3a,b). A typical measurement lasts 150 μs. The population N_{↓} of atoms in the momentum state with spin label \(\left\downarrow \right\rangle \) is measured with the same technique after transferring the atoms to \(\left\uparrow \right\rangle \) using a Raman π pulse. The Raman π pulse serves the additional functions of reoverlapping the wave packets and cancelling the average light shift of the probe.
Collective QND measurements are used in creating conditional spin squeezing. The spinmomentum projection in the population basis is measured once with the premeasurement outcome \({J}_{z{\rm{p}}}=\frac{1}{2}\left({N}_{\uparrow }{N}_{\downarrow }\right){ }_{{\rm{pre}}}\), which localizes the state to below the initial coherent spinstate level, producing a squeezed state. The same projection is then measured a second time with the final measurement outcome labelled \({J}_{z{\rm{f}}}=\frac{1}{2}\left({N}_{\uparrow }{N}_{\downarrow }\right){ }_{{\rm{fin}}}\). The quantum fluctuation is common to both measurements and can be partially subtracted by considering the difference J_{zd} = J_{zf} − J_{zp}, but any rotation of the state (that is, signal) that occurs in the interim appears only in the final measurement outcome. Each final population measurement is made after first optically pumping atoms in \(\left\uparrow \right\rangle \) to \(\leftF=2,{m}_{F}=2\right\rangle \) to achieve lower readout noise (estimated at more than 15 dB below the projection noise level) by using the optical cycling transition to \(\leftF=3,{m}_{F}=3\right\rangle \).
The length of the Bloch vector J_{s} after the premeasurement is measured by inserting a π/2 pulse between the pre and final measurements (Methods). Specifically, J_{s} is estimated from the fringe amplitude of J_{zf} versus the azimuthal phase ϕ of the π/2 pulse as it is varied between 0 to 2π. The initial length of the Bloch vector J_{c} needed for estimating the spectroscopic enhancement is estimated in the same manner, but without the premeasurement applied.
Figure 3c shows the spectroscopic enhancement W versus the strength of the QND interaction as parameterized by M_{i}, the average number of incident photons that enter the cavity during each population premeasurement window. At low M_{i}, the probe’s vacuum noise limits the spectroscopic enhancement, whereas at high M_{i}, the spectroscopic enhancement is limited by freespace scattering of the probe light that leads to a reduction in J_{s} and transitions to other ground states that decorrelate the pre and final measurements. Near M_{i} = 600, N = 1,170(30) atoms, and δ_{c} = 2π × 175 MHz, we achieve W = 0.46(11) or \({3.4}_{0.9}^{+1.1}\) dB of directly observed squeezing in the momentumspin basis.
We also realize entanglement by cavitymediated interactions^{14,43,48}. The OAT Hamiltonian^{1}
is generated by applying a fixed frequency drive tone offset from the average dressed cavity resonance by δ_{p} ≳ κ/2 (ref. ^{52}). In brief, the populations in each momentumspin state tune the cavity closer to or further from resonance with the fixed frequency drive tone, allowing more or less light into the cavity such that \({\hat{c}}^{\dagger }\hat{c}\propto {\hat{N}}_{\uparrow }\). To a first approximation, the spinlight QND Hamiltonian is thus transformed into a spinonly Hamiltonian with a relevant term proportional to \({\hat{N}}_{\uparrow }^{2}\). A repeated application of the dynamics after a π pulse realizes the Hamiltonian dynamics of equation (3).
The unitary OAT interactions drive shearing of the atomic quantum noise distribution with a resulting squeezed state minimum noise projection oriented at a small angle α_{0} from \(\hat{{\bf{z}}}\) (Fig. 3d and Fig. 4b(inset)). The state is rotated so that the minimum noise projection is along \(\hat{{\bf{z}}}\). The momentumspin populations are destructively read out as before with measurement outcome labelled J_{zf}. The Bloch vector lengths J_{s} (J_{c}) with (without) OAT squeezing are also measured just as for the QND squeezing. We directly observe a spectroscopic enhancement from OAT of W = 0.56(8) or \({2.5}_{0.6}^{+0.6}\) dB. The optimal configuration was realized with M_{i} ≈ 700 photons, δ_{c} = 2π × 350 MHz, δ_{p} = 2.7 × κ/2, χ_{OAT} ≈ 2π × 10 Hz and N = 730(10) atoms.
Entangled matterwave interferometry
We now turn to injecting the prepared entangled state into a matterwave interferometer with the sequence shown in Fig. 4a. After preparing a squeezed state with OAT, a Raman beam splitter rotation orients the squeezing along \(\hat{{\bf{y}}}\). The spin projection J_{y} will change if a small signal phase ϕ is applied. The orienting of the squeezing is accomplished via a (π/2 + α_{0}) pulse aligned to the atomic Bloch vector along \(\hat{{\bf{x}}}\). A relative phase accumulates between the wave packets during a free evolution time T_{evol}, a Raman π ‘mirror’ pulse is applied, followed by another free evolution time T_{evol}. Finally, a readout π/2 pulse transfers the signal ϕ and the squeezing into a displacement in the momentumspin population basis \(\hat{{\bf{z}}}\) with a measurement outcome J_{zf}. The Bloch vector lengths J_{s} and J_{c} are measured in separate experiments with and without OAT applied by scanning the azimuthal phase of the final π/2 pulse of the interferometer and measuring the fringe amplitude as before (Fig. 4c).
We achieve a directly observed spectroscopic enhancement \(W={1.7}_{0.5}^{+0.5}\) dB beyond the SQL with N = 660(15) atoms as shown in Fig. 4b. Without OAT, the performance of our interferometer is worse than the SQL because of imperfect interferometer contrast C_{i} = 2J_{c}/N_{0} ≈ 0.9. We note that the actual phase variance of the squeezed interferometer is improved by \({3.4}_{1.2}^{+0.9}\) dB compared with this unsqueezed interferometer (Methods).
Phase sensitivity beyond the SQL was limited to evolution times T_{evol} < 0.7 ms (Fig. 4d). A comparable level of decrease in phase sensitivity was observed in an identical sequence in which all optical Raman pulses were replaced by equivalent microwave pulses, suggesting that the spin degrees of freedom may be responsible for the observed loss in sensitivity. We also observe that if the squeezed spin projection is left in the population basis J_{z} during the interferometer, then the squeezing persists for several milliseconds. From this, we conclude that the entangled state persists for longer than we can directly confirm.
In the future, the combination of Raman and Bragg techniques demonstrated here would enable the most delicate portion of the interferometer to be operated fully with the two portions of the superposition possessing the same spin label.
To further improve interferometer sensitivity, the entanglement can be combined with large momentum transfer sequences or one could inject the squeezed state into a lattice interferometer to hold the atoms for longer^{50}. One could also prepare the entanglement in the cavity and allow the atoms to undergo free fall outside of the cavity with readout by fluorescence measurement^{12}, another promising path for scaling to larger momentum transfers and longer interferometer times. The amount of momentum squeezing could be improved with larger collective cooperativity NC. The need for velocity selection limits our final number of atoms, so higher atom density in momentum space through improved axial cooling or the use of a Bose–Einstein condensate could lead to significant improvements^{34,53,54,55}. As the atom number is increased, it will be necessary to reduce the level of classical rotationadded noise or to make the added noise common mode as is done for gravity gradiometers and for proposed gravity wave and dark matter detectors^{23,24,25,56}.
In this work, the OATsqueezed states were successfully used to realize a squeezed matterwave interferometer, whereas the QNDsqueezed states were not. The OAT produced states were generated at lower atom number and associated smaller momentum spread, leading to less classical added rotation noise relative to the SQL and reduced shortening of the Bloch vector during the rotations. The QNDsqueezed states would be enhanced by improving the total effective quantum efficiency from q ≈ 0.1 here to, for instance, q ≈ 0.4 in previous work^{5}.
It may also be possible to generate spinsqueezed states using optical cycling transitions in rubidium, strontium and ytterbium^{5,10,16,47,51} and then use Raman transitions to map the entanglement to purely momentum states^{42,57}. The fundamental scaling of the achievable Wineland parameter would improve to W ∝ 1/NC from the current scaling \(W\propto 1/\sqrt{NC}\) (ref. ^{47}). Indeed, the combination of larger atom number and probing on a cycling transition are the primary reasons for the larger amounts of squeezing achieved in previous work^{4,5} compared to the present results.
This proofofprinciple lightpulse matterwave interferometer paves the way for using cavitygenerated entanglement as a quantum resource, enabling the next generation of interferometers with higher precision, enhanced measurement bandwidth, higher accuracy and smaller size. Such devices will advance the frontiers of both practical applications and discoveries in fundamental science.
Methods
Bluedetuned doughnut dipole guide
The blue dipole guide laser is a 760 nm interference filter external cavity diode laser (ECDL) locked to a reference cavity for improved longterm stability. The laser is modulated by a fibre electrooptic phase modulator (EOM) with modulation index β ≈ 1.3 at the cavity free spectral range ω_{FSR}. By exciting adjacent longitudinal modes of the cavity with opposite spatial parity with respect to the centre of the cavity, one creates an axially uniform blue dipole guide near the centre of the cavity^{51}. The doughnutmode LG_{01} profile is constructed from the ± first diffraction orders of a forkpattern phase plate. Stressinduced birefringence of the cavity mirrors breaks cylindrical symmetry and splits the Hermite–Gaussian HG_{10} and HG_{01} modes up to δ_{HG} = 2π × 100–500 kHz, depending on the cavity piezo voltage, to be compared with the 157(5) kHz fullwidth halfmaximum cavity linewidth for these modes. For the data presented here, δ_{HG} = 2π × 350 kHz. Before entering the cavity, the two LG modes are sent along separate paths. One path enters a freespace EOM to generate sidebands for locking the cavity to the blue dipole guide laser. The other path passes through two acoustooptic modulators (AOMs) with a δ_{HG} frequency difference such that the projected HG modes combine within the cavity to approximate an LG_{01} mode’s radial intensity distribution by LG_{01} = HG_{01} + iHG_{10}. Because the frequency splitting δ_{HG} is much greater than the radial trap frequency, the atoms effectively experience the timeaveraged radial trapping potential of an LG_{01} mode.
Laser cooling
The experimental sequence is repeated every 750 ms. Each trial begins with a twodimensional magnetooptical trap (MOT) loading a threedimensional MOT with 10^{8} atoms near the cavity centre for approximately 0.5 s. The MOT coils are turned off, and around 2 × 10^{5} atoms are cooled by polarization gradient cooling to 15 μK and loaded into an 813.5 nm reddetuned intracavity lattice with fullwidth halfmaximum cavity linewidth 166(5) kHz. Additional radial confinement is provided by the blue dipole guide. The red lattice depth is ramped down to a depth of 80 μK or 250E_{l}, where E_{l} is the recoil energy of the lattice. We then apply Λenhanced grey molasses cooling. Each of the six molasses beams has 2.5 mW and 1 cm beam waist. The light is detuned 2π × 42 MHz blue of \(\leftF=2\right\rangle \to \left{F}^{{\prime} }=2\right\rangle \). A fibre EOM generates a 100 μW sideband to coherently form the Λ system as \(\leftF=1\right\rangle \leftrightarrow \left{F}^{{\prime} }=2\right\rangle \leftrightarrow \leftF=2\right\rangle \). After 5 ms, the temperature of the ensemble is reduced to 6 μK.
We then perform twodimensional degenerate Raman sideband cooling (RSBC) to further cool the radial temperature^{58}. Three RSBC beams form a triangular lattice in a plane perpendicular to the cavity axis \(\hat{{\bf{Z}}}\), with trapping frequency ω_{tri} = 2π × 75 kHz. The blue dipole guide and red lattice continue to provide a background radial trap. The RSBC laser is bluedetuned 50 GHz from the \(\leftF=1\right\rangle \leftrightarrow \left{F}^{{\prime} }=2\right\rangle \) transition so that atoms are trapped at the nodes of the triangular lattice, suppressing scattering off the cooling beams. A bias magnetic field of 0.11 G along \(\hat{Z}\) is applied to match the firstorder Zeeman splitting to the trap frequency ω_{tri}. The polarizations of the three beams are twisted 10° from the vertical to create the Raman coupling for driving the vibrational mode transition \(\leftF=1,{m}_{F},{n}_{{\rm{tri}}}\right\rangle \to \leftF=1,{m}_{F}1,{n}_{{\rm{tri}}}1\right\rangle \) that reduces the vibrational quantum number n_{tri} in the local traps. During RSBC, atoms are continuously repumped back to \(\leftF=1,{m}_{F}=1\right\rangle \) by a separate laser.
To improve the coupling of the atoms to the cavity we apply multiple cooling cycles each lasting 2 ms. The RSBC light is ramped on over 0.3 ms, cooling occurs for 1.2 ms and then RSBC light is ramped off over 0.3 ms. After 225 μs, the atoms have oscillated back to the centre of the cavity, at which point we repeat the cooling cycle. After three cooling cycles, we slowly turn off the remaining red lattice and the RSBC lattice over 3 ms so that the atoms start to free fall. Atoms are then optically pumped to \(\left\uparrow \right\rangle \) using a pair of πpolarized laser beams on resonance with the \(\leftF=1\right\rangle \to \left{F}^{{\prime} }=2\right\rangle \) and \(\leftF=2\right\rangle \to \left{F}^{{\prime} }=2\right\rangle \) transitions applied transverse to the cavity axis. Just before the interferometer sequence, the radial temperature is 1.4(5) μK. State transfer with a microwave pulse may be used for future improvement to reduce heating associated with optical pumping.
Atomic and cavity probe lasers
To stabilize the frequencies of the Raman lasers and the atomic probe relative to the cavity, we frequency lock a separate cavity probe laser to the cavity and then perform offset frequency phase locks to this laser. The cavity probe is locked to a cavity TEM_{00} mode approximately 160 GHz to the blue of the atomic transition frequency ω_{a} such that this mode is essentially unperturbed by the presence of atoms. The locking of the cavity probe to the cavity is done by a Pound–Drever–Hall lock at very low phase modulation index for a single sideband to carrier power ratio of 10^{−4}. Rather than locking to the carrier, we lock to the weak sideband. This enables us to reduce the amount of power entering the cavity to only 400 pW (half from the sideband and half nonresonantly from the carrier) while still operating above the technical noise floor of the photodiode. This lock is always engaged. To allow phase locking of other lasers to the cavity probe with relative beat notes of less than 2 GHz, some of the laser light is passed through a fibre EOM driven strongly at 13.6 GHz to generate very high order sidebands.
The atomic probe laser is phaselocked with an offset frequency of approximately 13.6 × 12 = 163.2 GHz to the red of the cavity probe, placing it close to ω_{a}. The offset phaselock frequency is adjusted to maintain the atomic probe laser approximately δ_{c} /2π + 80 MHz blue of ω_{a}. We derive three important tones from this laser: a homodyne reference beam, a path length stabilization beam used for removing path length noise and drift, and the actual atomic probe tone used for OAT and QND measurements. The path length stabilization beam is passed through an EOM that is modulated at 80 MHz to create a weak sideband that will serve as the atomic probe tone. The combined path length stabilization and atomic probe tones are reflected from the cavity and detected on a single homodyne detector. The homodyne reference beam is shifted by an 80 MHz AOM to have the same frequency as the atomic probe tone. The quadrature of the atomic probe tone that we detect in the homodyne is actively stabilized by adjusting the phase of the homodyne reference tone. This is achieved by detecting the phase of the path length stabilization tone appearing in the homodyne detector at 80 MHz and then holding this phase constant by feedback on the frequency of the 80 MHz AOM used to shift the homodyne reference beam.
The laser could be actively locked to the dressed resonance as in ref. ^{5} or the linear part of the dispersive could be used to estimate small frequency shifts, but for this work we sweep the atomic probe laser, and all derived beams, so that the atomic probe tone sweeps through cavity resonance at 1.5 MHz ms^{−1}. Although this simplifies the experiment, it results in a 6 dB loss of quantum efficiency for a fixed amount of freespace scattering when compared to performing homodyne detection on the line centre. Including this loss of efficiency, the net effective quantum efficiency is approximately 10%.
When using the atomic probe to drive OAT, it is ideal to operate with the driving laser detuned from cavity resonance by δ_{p} = κ/2 to suppress freespace scattering. However, we work at larger detunings for two reasons. First, an increased detuning reduces deleterious QND interactions (or, equivalently, photon shot noise from the applied drive tone) that were neglected in our description of the emergence of the unitary dynamics^{52}. Second, this enables operation in a linearized regime even in the presence of shottoshot total atom number fluctuations. We empirically find an optimum detuning of δ_{p} = 2.7 × κ/2 with χ_{OAT} ≈ 2π × 10 Hz.
The cavity probe, atomic probe and Raman lasers are distributedBraggreflector lasers with freerunning linewidths of approximately 500 kHz. We use external optical feedback to narrow their linewidths^{59}. A small fraction of the power from each laser is picked off and then retroreflected back into the laser with a round trip length of 3 m in free space. The frequency of each laser is primarily determined by the length of the optical feedback path length which is stabilized using a piezoelectric actuator to move the retroreflection mirror and a freespace phase modulator EOM for fast actuation with unity gain frequency of 500 kHz. By optimizing the optical feedback fraction typically between 10^{−4} and 10^{−3}, we achieve Lorentzian linewidths of less than 1 kHz.
Microwave source
Highfidelity Raman pulse sequences require agile control of lowphase noise microwaves. Our microwave source is based on that in ref. ^{60}. A lowphasenoise 100 MHz crystal oscillator (Wenzel ULN 50116843) is multiplied to 6.800 GHz using a nonlinear transmission line frequency comb generator (Picosecond Pulse Labs LPN7110SMT). The stable 6.800 GHz is provided as the local oscillator for a single sideband modulator (Analog Devices HMC496).
The required inphase I and quadrature Q modulation inputs to the single sideband modulator are created using three radiofrequency (RF) tones from an Analog Devices AD9959 DDS. Two RF tones are at the same frequency near 135 MHz and are 90° out of phase. The phase, frequency and amplitude of these two tones can be jumped for arbitrary rotations on the Bloch sphere, for selecting different momentumchanging transitions, velocimetry and so on. The third RF tone starts near 100 MHz but is continuously ramped in frequency at a rate of 2kg_{∥} ≈ 2π × 25.1 kHz ms^{−1} to match the time variation of the twophoton Doppler shift as the atoms fall under gravity. Each of the two initial RF tones are mixed with this third signal to generate tones near 35 MHz for the I and Q inputs to the single sideband modulator.
Finally, the modulator output near 6.835 GHz is divided in frequency by two using a lownoise divider (Analog Devices HMC862A) and applied to a fibrecoupled EOM to generate the desired Raman tones as the ±firstorder sidebands. We estimate that the noise contributed by this frequency source is at least 30 dB below the SQL for 1,000 atoms.
Raman transitions and velocity selection
The laser that drives the Raman transitions is detuned Δ = 2π × 85 GHz blue of ω_{a}. As is done for the atomic probe, the Raman laser is stabilized with respect to the cavity by an offset frequency phase lock to the cavity probe. The offset frequency is set to centre the Raman laser between two adjacent longitudinal TEM_{00} cavity modes. The two Raman tones, whose generation is described above, are symmetrically detuned from the cavity resonances by approximately (ω_{HF} − ω_{FSR})/2 = 2π × 23 MHz. With 2.5 mW of total σ^{+}polarized light incident on the cavity, the EOM modulation index allows a maximum observed twophoton Rabi frequency of Ω_{TwoPh} = 2π × 15 kHz, with the Rabi frequency tuned to smaller values by adjusting the total incident power using an AOM. For the large momentum transfers shown in Fig. 2e, Bragg transitions are driven by two laser tones derived from the same laser with difference frequency \({\omega }_{{\rm{B}}}={\delta }_{{\rm{vs}}}b\left(t{t}_{{\rm{vs}}}\right)\), where b is the chirp rate defined below.
As atoms fall under gravity, the relative Doppler shift for light propagating upwards versus downwards chirps linearly in time. We compensate for this effect by linearly ramping the instantaneous frequency of the sidebands as 2ω_{R} = ω_{HF} + δ − b(t − t_{vs}) with chirp rate b = 2π × 25.11 kHz ms^{−1} ≈ 2kg_{∥}. Here g_{∥} = 9.8 m s^{−}^{2} is the projection onto the cavity axis of the local acceleration due to gravity, δ is the twophoton detuning in the falling frame of reference and t_{vs} is the time at which we will apply the first π pulse for velocity selection described below. We also note that during rotation pulses, we adjust the twophoton detuning δ by approximately 4 kHz (in a phase coherent manner) to compensate for differential a.c. Stark shifts of the two pseudospin states induced by the Raman beams. In the accelerating reference frame, the phase of the interferometer fringe evolves quadratically with T nominally as ϕ = (2kg_{∥} − b)T^{2}, from which we extract a value of g_{∥} consistent with the known local value of gravitational acceleration to within the uncertainty of the angular orientation of the cavity axis with respect to local gravity. The chirp rate b is nominally tuned so that much less than 1 rad of phase evolves in the accelerating frame but, in the laboratory frame, the accumulated phase evolves as ϕ = 2kg_{∥}T^{2}, which is approximately 2,500 rad for the largest T = 4 ms explored in this work.
After being released from the 813 nm lattice and falling for T_{fall} = 15 ms, the atoms are optically pumped to \(\left\uparrow \right\rangle \), and the twophoton Raman detuning is set to δ_{vs} = −2π × 400 kHz ≈ bT_{fall} to transfer a group of atoms to \(\left\downarrow \right\rangle \) from the centre of the axial velocity distribution^{61}. Atoms in \(\left\uparrow \right\rangle \) are removed by a transverse radiation pressure force. The velocity selection is then repeated to further narrow the momentum width of the selected atoms down to Δp < 0.1ħk set by the twophoton Rabi frequency Ω_{TwoPh }= 2π × 1.4 kHz.
The Raman laser is a distributedBraggreflector laser with a freerunning linewidth of approximately 500 kHz. We observed that the cavity converted laser frequency noise to intracavity amplitude noise near δ_{vs} that can resonantly drive undesired Bragg transitions, leading to a loss of nearly 50% of the population to other momentum states outside of the desired twolevel basis for all the Raman pulses involved in the interferometer sequence combined. We note that in the symmetric detuning configuration here the Raman transitions are firstorder insensitive to conversion of laser frequency noise to both amplitude (AM) and phase (PM) noise on the intracavity Raman tones. However, the Bragg transitions are firstorder sensitive because of the opposite parity of the standing wave modes being driven.
After narrowing the laser to a Lorentzian linewidth of less than 1 kHz, we found the fraction of total atoms lost out of the desired twolevel manifold is less than 3(3)% for all the Raman pulses involved in the interferometer sequence combined. We also observed residual offresonance transitions to other momentum states if the turn on and off of the Raman beams was too rapid. The fraction of atoms lost per pulse was reduced to 0.2(1.0)% per pulse by using an RF switch with 3 μs rise time to gate the Raman tones. Without the shortening of the Bloch vector J_{c} from the two effects, we estimate that the observed spectroscopic enhancement could be improved by 0.2(2) dB for the full interferometer sequence.
Wineland criterion
The Wineland criterion is often presented in the form^{5}
where the initial C_{i }and final C_{f }contrasts are related to Bloch vector lengths here by C_{i} ≡ 2J_{c}/N_{0} and C_{f} ≡ 2J_{s}/N_{0} for total atom number N_{0}. By rearranging terms, it can also be expressed in a more physically meaningful form as the ratio \(W={(\Delta \theta /\Delta {\theta }_{{\rm{SQL}}})}^{2}\) between the observed angular resolution \(\Delta \theta =\frac{\Delta {J}_{z}}{{J}_{{\rm{s}}}}\) with entanglement and the SQL \(\Delta {\theta }_{{\rm{SQL}}}=1/\sqrt{N}\equiv 1/\sqrt{2{J}_{{\rm{c}}}}\) for a pure state with the same Bloch vector length J_{c} as that of the actual mixed state when entanglement is not created.
We now establish the connection between the spin operators and actual experimental measurements. We define the cavity frequency shifts induced by a single atom in \(\leftF=2,{m}_{F}=2\right\rangle ,\leftF=2,{m}_{F}=0\right\rangle \) and \(\leftF=1,{m}_{F}=0\right\rangle \) as χ_{2}, χ_{0} ≡ χ_{QND} and χ_{↓}, respectively.
For OAT squeezing, we estimate the angular resolution Δθ after the squeezing generation or the full squeezed interferometer sequence as follows. To measure the final spin projection J_{zf}, we optically pump the atoms in \(\left\uparrow \right\rangle \) to \(\leftF=2,{m}_{F}=2\right\rangle \), measure the cavity frequency shift with outcome labelled ω_{1f}, blow away atoms in \(\leftF=2\right\rangle \), apply a Raman π pulse, optically pump the atoms in \(\left\uparrow \right\rangle \) to \(\leftF=2,{m}_{F}=2\right\rangle \) and measure a second cavity frequency shift with outcome labelled ω_{2f}. We estimate the final spin projection J_{zf} from the difference between the two cavity frequency shifts \({J}_{z{\rm{f}}}=\frac{{\omega }_{1{\rm{f}}}{\omega }_{2{\rm{f}}}}{2{\chi }_{2}}\frac{\epsilon }{{\chi }_{2}}{\omega }_{2{\rm{f}}}\), where \(\epsilon =\frac{{\chi }_{\downarrow }\,/\,2}{{\chi }_{2}}\). To convert the spin projection J_{zf} into an estimate of the Bloch vector polar angle θ_{f}, we measure the length of the Bloch vector J_{s} by scanning the azimuthal phase ϕ of the readout π/2 pulse. In the case of the squeezed interferometer, this is the final π/2 pulse of the interferometer and just before the measurement J_{zf}. In the case of OATsqueezed state generation, this is an added π/2 pulse after the squeezing and just before the measurement J_{zf}. We fit the resulting differential cavity frequency shifts \(\left({\omega }_{1{\rm{f}}}{\omega }_{2{\rm{f}}}\right){ }_{\varphi }\) to the function \({y}_{0}+{A}_{{\rm{f}}}\sin \left(\varphi {\varphi }_{0}\right)\) with fitted offset y_{0}, amplitude A_{f} and phase offset ϕ_{0}. The Bloch vector length is then estimated by \({J}_{{\rm{s}}}=\frac{{A}_{{\rm{f}}}}{2{\chi }_{2}{\chi }_{\downarrow }}\). The Bloch vector polar angle θ_{f} from the final measurement is thus estimated by \({\theta }_{{\rm{f}}}=\frac{{J}_{z{\rm{f}}}}{{J}_{{\rm{s}}}}=\frac{{\omega }_{1{\rm{f}}}{\omega }_{2{\rm{f}}}}{{A}_{{\rm{f}}}}{\epsilon }\frac{{\omega }_{1{\rm{f}}}+{\omega }_{2{\rm{f}}}}{{A}_{{\rm{f}}}}+2{{\epsilon }}^{2}\frac{{\omega }_{2{\rm{f}}}}{{A}_{{\rm{f}}}}\). The angular resolution Δθ is approximated as \(\Delta \theta =\Delta {\theta }_{{\rm{f}}}\approx \frac{\Delta \left({\omega }_{1{\rm{f}}}{\omega }_{2{\rm{f}}}\right)}{{A}_{{\rm{f}}}}\), where we note the scale factors χ_{2}, and so on, are cancelled at the order of ϵ^{0}. With a typical value of ∣ϵ∣ < 1/50 and the fractional total number fluctuation \(\Delta \left(\frac{{\omega }_{1f}+{\omega }_{2f}}{{A}_{f}}\right)\) being less than 0.03, the corrections of order ϵ^{1} would need to be included for squeezing 30 dB below the SQL.
For the QND measurements, we perform premeasurements to localize the quantum state and use the final measurements to verify the squeezing generated by the premeasurements as described before. The phase resolution is defined as the phase fluctuation between the pre and final measurements \(\Delta \theta =\Delta \left({\theta }_{{\rm{p}}}{\theta }_{{\rm{f}}}\right)\). The Bloch vector polar angle of the final measurements θ_{f} is estimated as in the OAT measurement with the atomic population optically pumped to \(\leftF=2,{m}_{F}=2\right\rangle \). For the premeasurements, we measure pairs of cavity frequency shifts ω_{1p} and ω_{2p} separated by π pulses but without the optical pumping so the atomic population is in \(\left\uparrow \right\rangle \) during the cavity frequency shift measurements. The spin projection J_{zp} in the premeasurements is estimated from the differential frequency shift \({J}_{z{\rm{p}}}=\frac{{\omega }_{1{\rm{p}}}{\omega }_{2{\rm{p}}}}{2\left({\chi }_{0}{\chi }_{\downarrow }\right)}\). The length of the Bloch vector J_{s} just after the premeasurement is measured by adding a π/2 pulse just after the premeasurement and scanning its azimuthal phase ϕ, after which we perform a single cavity frequency shift measurement with outcome labelled ω_{1f}_{ϕ}. We then fit the resulting fringe to the function \({y}_{0}+{A}_{{\rm{p}}}\sin \left(\varphi {\varphi }_{0}\right)\) and estimate the Bloch vector length \({J}_{{\rm{s}}}=\frac{{A}_{{\rm{p}}}}{{\chi }_{0}{\chi }_{\downarrow }}\). The Bloch vector polar angle θ_{p} is evaluated \({\theta }_{{\rm{p}}}=\frac{{J}_{z{\rm{p}}}}{{J}_{{\rm{p}}}}=\frac{{\omega }_{1{\rm{p}}}{\omega }_{2{\rm{p}}}}{2{A}_{{\rm{p}}}}\). As before, the angular phase resolution is sufficiently approximated by keeping only to the order of ϵ^{0} as \(\Delta \theta =\Delta \left({\theta }_{{\rm{p}}}{\theta }_{{\rm{f}}}\right)\approx \Delta \left(\frac{{\omega }_{1{\rm{p}}}{\omega }_{2{\rm{p}}}}{2{A}_{{\rm{p}}}}\frac{{\omega }_{1{\rm{f}}}{\omega }_{2{\rm{f}}}}{{A}_{{\rm{f}}}}\right)\) with no dependence on scale factors χ_{2}, χ_{0} or χ_{↓}.
For estimating the SQL Δθ_{SQL}, we measure the length of the Bloch vector \({J}_{{\rm{c}}}={J}_{{\rm{s}}}{ }_{{M}_{{\rm{i}}}=0}=\frac{{A}_{{\rm{p}}}\,{ }_{{M}_{{\rm{i}}}=0}}{{\chi }_{0}{\chi }_{\downarrow }}\) using the same sequence for measuring J_{s} in the QND premeasurements described just above but setting the photon number M_{i} to zero during the premeasurements or squeezing for QND measurement or OAT, respectively. To estimate the SQL \(\Delta {\theta }_{{\rm{SQL}}}=1/\sqrt{N}=1/\sqrt{2{J}_{{\rm{c}}}}\) we therefore need to know accurate values of χ_{0} and χ_{↓}. To sufficient approximation \({\chi }_{0}={g}^{2}\left(\frac{{B}_{3}}{{\delta }_{{\rm{c}}}}+\frac{{B}_{2}}{{\delta }_{{\rm{c}}}+{\delta }_{2}}+\frac{{B}_{1}}{{\delta }_{{\rm{c}}}+{\delta }_{1}}\right)\) with atom–cavity coupling g discussed below, hyperfine splittings δ_{2} = 2π × 266.7 MHz, δ_{1} = 2π × 423.6 MHz and branching ratios \({B}_{3}=\frac{6}{15},{B}_{2}=\frac{3}{12},{B}_{1}=\frac{1}{60}\) of the excited states \(\left{F}^{{\prime} }=3,2,1,{m}_{F}=1\right\rangle \) to the groundstate \(\left\uparrow \right\rangle \) transition that interact with the probe light. To an sufficient approximation, \({\chi }_{\downarrow }={g}^{2}\left(\frac{{B}_{2,\downarrow }}{{\delta }_{{\rm{c}}}+{\delta }_{2}{\omega }_{{\rm{HF}}}}+\frac{{B}_{1,\downarrow }}{{\delta }_{{\rm{c}}}+{\delta }_{1}{\omega }_{{\rm{HF}}}}\right)\) with branching ratios \({B}_{2,\downarrow }=\frac{3}{12},{B}_{1,\downarrow }=\frac{5}{12}\) of the \(\left{F}^{{\prime} }=2,1,{m}_{F}=1\right\rangle \) to the groundstate \(\left\downarrow \right\rangle \) transition. Although not used in the calculation, the cavity frequency shift from a single atom in \(\leftF=2,{m}_{F}=2\right\rangle \) is approximated by \({\chi }_{2}=\frac{{g}^{2}}{{\delta }_{{\rm{c}}}}\) for the cycling transition between the excited state \(\leftF=3,{m}_{F}=3\right\rangle \) and the ground state \(\leftF=2,{m}_{F}=2\right\rangle \).
The maximum singleatom vacuum Rabi splitting \(2{g}_{0}=2\sqrt{\frac{2{D}^{2}{\omega }_{{\rm{c}}}}{{\rm{\pi }}L{w}_{0}^{2}{{\epsilon }}_{0}\hbar }}=\) \(2\times 2{\rm{\pi }}\times 0.4853(5)\,{\rm{MHz}}\) (ref. ^{7}), with fractional uncertainty dominated by the fractional uncertainty (1.1 × 10^{−3}) on the dipole matrix element D for the \(\leftF=2,{m}_{F}=2\right\rangle \) to \(\leftF=3,{m}_{F}=3\right\rangle \) transition, and ϵ_{0} the vacuum permeability. The cavity length L and mode waist w_{0} are determined very precisely by measuring the free spectral range and transverse mode frequency splitting. As the atoms traverse many standing waves of the cavity during the measurement windows, we can average over the standing waves to arrive at a timeaveraged spatially dependent coupling \({g}_{t}(r,z)=\frac{{g}_{0}}{\sqrt{2}}\frac{{{\rm{e}}}^{{r}^{2}/{w}_{0}^{2}}}{\sqrt{1+{\left(\frac{z}{{Z}_{{\rm{R}}}}\right)}^{2}}}\), where Z_{R} = 2.1 cm is the Rayleigh range of the cavity^{51}. The effective singleatom–cavity coupling frequency is given by the ensemble averaged moments of the spatially dependent \({g}_{t}\left(r,z\right)\) as \(g=\sqrt{\frac{\left\langle {g}_{t}{\left(r,z\right)}^{4}\right\rangle }{\left\langle {g}_{t}{\left(r,z\right)}^{2}\right\rangle }}=\frac{{g}_{0}}{\sqrt{2}}\left(1{f}_{{\rm{cor}}}\right)=2{\rm{\pi }}\times 0.341(2)\,{\rm{MHz}}\) (ref. ^{7}). The final fractional uncertainty (6 × 10^{−3}) on g is dominated by the uncertainty on the correction factor \({f}_{{\rm{c}}{\rm{o}}{\rm{r}}}\approx \frac{{z}_{0}^{2}+{\sigma }_{z}^{2}}{2{Z}_{{\rm{R}}}^{2}}+\frac{{r}_{{\rm{r}}.{\rm{m}}.{\rm{s}}.}^{2}}{{w}_{0}^{2}}\), where \({z}_{0}=1\left(2\right)\) mm is the axial position of the cloud relative to the cavity centre, σ_{z} = 0.5(3) mm is the r.m.s. axial spread of the cloud and r_{r.m.s.} is the r.m.s. cloud radius of the atoms. The fractional uncertainty on g contributed from z_{0}, σ_{z} and r_{r.m.s.} are 5 × 10^{−3}, 4 × 10^{−3} and 2 × 10^{−3}, respectively.
The uncertainties on the cavity detuning δ_{c} = 175(2) MHz or 350(2) MHz lead to fractional uncertainties of at most 0.01 on \(\left({\chi }_{{\rm{QND}}}{\chi }_{\downarrow }\right)\). Because the atoms move along the cavity axis, the probe light is Doppler shifted by the order of δ_{vs}/2; however, here δ_{c} ≫ δ_{vs} so that there is only a negligible fractional correction to χ_{QND} of order \({({\delta }_{{\rm{vs}}}/2{\delta }_{{\rm{c}}})}^{2}\lesssim 1{0}^{6}\). The effect of spread in momentum states is even more negligible.
Combining uncertainties from g and δ_{c}, the fractional uncertainty on \(({\chi }_{{\rm{QND}}}{\chi }_{\downarrow })\) is at most 1.4 × 10^{−2}. This uncertainty combined with the fractional uncertainty on the fitted fringe amplitude A_{p} of 9 × 10^{−3} yields a total fractional uncertainty on the SQL variance \({(\Delta {\theta }_{{\rm{SQL}}})}^{2}\) of 1.7 × 10^{−2}. To estimate the angular resolutions \({(\Delta \theta )}^{2}\), we typically use 100 to 200 experimental trials, which leads to a typical statistical fractional uncertainty on \({(\Delta \theta )}^{2}\) of 0.1 to 0.2. The final reported uncertainties on the Wineland parameters are thus dominated by the statistical uncertainties on the phase resolution \({\left(\Delta \theta \right)}^{2}\).
Without the QND premeasurements or OAT, the mixed state actually performs worse than the SQL, conceptually due to the spin noise from the dephased or decohered fraction of the atoms that contribute noise but no signal. This is why the observed improvement in the interferometer sensitivity is larger than the Wineland parameter; however, the Wineland parameter captures what fraction of the improvement can be certified to arise because of entanglement between the atoms and not just because of cancellation of spin noise alone.
Vibration noise
Mechanical vibrations of the cavity mirrors are equivalent to a fluctuating phase reference for the atoms. A commercial vibrometer was used to measure the spectral density S_{a}(ω) of acceleration noise at a location on the optical table close to the portion that supports the vacuum chamber. In the limit of zeroduration pulses, the transfer function for a Mach–Zehnder interferometer \( T(\omega ){ }^{2}=\frac{64{k}^{2}}{{\omega }^{4}}\sin {\left(\frac{\omega {T}_{{\rm{evol}}}}{2}\right)}^{4}\) converts accelerations to an integrated phase noise \({\varphi }^{2}={\int }_{0}^{\infty } T(\omega ){ }^{2}\) \({S}_{{\rm{a}}}(\omega )\,{\rm{d}}\omega \). For a sequence with T_{evol} = 0.3 ms, we estimate that the phase noise caused by vibrations is 20 dB lower than the phase resolution set by the SQL of 1,000 atoms.
Data availability
The datasets generated during and/or analysed during the current study are available in the CU Scholar repository, with the identifier https://doi.org/10.25810/t39mc562.
Change history
06 December 2022
A Correction to this paper has been published: https://doi.org/10.1038/s41586022055824
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Acknowledgements
We acknowledge funding support from the National Science Foundation under grant numbers 1734006 (Physics Frontier Center) and OMA2016244 (QLCI), DOE Quantum Systems Accelerator, NIST and DARPA. We acknowledge helpful feedback on the manuscript from D. Z. Anderson and A. M. Rey, helpful discussions with M. Jaffe and N. Poli, and laser locking development by D. Wu.
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G.P.G., C.L. and B.W. contributed to the building of the experiment. G.P.G. and C.L. conducted the experiments and data analysis. J.K.T. conceived and supervised the experiments. G.P.G., C.L. and J.K.T. wrote the manuscript. All authors discussed the experiment implementation and results and contributed to the manuscript.
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Greve, G.P., Luo, C., Wu, B. et al. Entanglementenhanced matterwave interferometry in a highfinesse cavity. Nature 610, 472–477 (2022). https://doi.org/10.1038/s41586022051979
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DOI: https://doi.org/10.1038/s41586022051979
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