We provide a protocol to measure out-of-time-order correlation functions. These correlation functions are of theoretical interest for diagnosing the scrambling of quantum information in black holes and strongly interacting quantum systems generally. Measuring them requires an echo-type sequence in which the sign of a many-body Hamiltonian is reversed. We detail an implementation employing cold atoms and cavity quantum electrodynamics to realize the chaotic kicked top model, and we analyze effects of dissipation to verify its feasibility with current technology. Finally, we propose in broad strokes a number of other experimental platforms where similar out-of-time-order correlation functions can be measured.Comment: 12 pages, 5 figures; v3: introduction revised for greater clarity and accessibilit

We squeeze unconditionally the collective spin of a dilute ensemble of laser-cooled 87 Rb atoms using their interaction with a driven optical resonator. The shape and size of the resulting spin uncertainty region are well described by a simple analytical model [M.H.S., I.D.L., V.V., arXiv:0911.3936] through two orders of magnitude in the effective interaction strength, without free parameters. We deterministically generate states with up to 5.6(6) dB of metrologically relevant spin squeezing on the canonical 87 Rb hyperfine clock transition.Squeezed spin states [1][2][3][4][5][6], where a component of the total angular momentum of an ensemble of spins has less uncertainty [7,8] than is possible without quantum mechanical correlations [9][10][11][12], attract interest for both fundamental and practical reasons. Fundamentally, they allow the study of many-body entanglement but retain a simple description in terms of a single collective angular-momentum variable [4,5]. Practically, they may be a means to overcome the projection noise limit on precision [2,3,13,14]. Spin squeezing has been demonstrated using entanglement of ions via their shared motional modes [9], repulsive interactions in a Bose-Einstein condensate [10], or partial projection by measurement [11,12].In a companion paper [15] we propose a cavity feedback method for deterministic production of squeezed spin states using light-mediated interactions between distant atoms in an optical resonator. This approach generates spin dynamics similar to those of the one-axis twisting Hamiltonian H ∝ S 2 z in Kitagawa and Ueda's original proposal [1]. Cavity squeezing scales to much higher particle number than direct manipulation of ions [9] (but see Ref.[16] for a potentially scalable approach) and employs dilute ensembles rather than dense condensates of interacting atoms [10]. Unlike measurement-based squeezing [11,12], it unconditionally produces a known squeezed state independent of detector performance.Here we implement cavity squeezing for the canonical |F = 1, m F = 0 ↔ |F = 2, m F = 0 hyperfine clock transition in 87 Rb atoms, achieving a 5.6(6) dB improvement in signal-to-noise ratio [2,3]. To our knowledge, this is the largest such improvement to date. Moreover, the shape and orientation of the uncertainty regions we observe agree with a straightforward analytical model [15], without free parameters, over two orders of magnitude in effective interaction strength.Our scheme, similar in spirit to the proposal of Ref. [17], relies on the repeated interaction of the atomic ensemble with light circulating in an optical resonator, as illustrated in Fig. 1. We label the two relevant eigenstates (clock states) of each one of N 0 atoms as the spin-up and spin-down states of a spin-1/2 s i , and define a total spin S = i s i . Its z component corresponds to the popu- lation difference between clock states and its azimuthal angle corresponds to their relative phase. For a given total spin magnitude S = |S| ≤ S 0 = N 0 /2 and a given permutation symmetry of the en...

The geometric structure of an energy band in a solid is fundamental for a wide range of many-body phenomena in condensed matter and is uniquely characterized by the distribution of Berry curvature over the Brillouin zone. In analogy to an Aharonov-Bohm interferometer that measures the magnetic flux penetrating a given area in real space, we realize an atomic interferometer to measure Berry flux in momentum space. We demonstrate the interferometer for a graphene-type hexagonal lattice, where it has allowed us to directly detect the singular π Berry flux localized at each Dirac point. We show that the interferometer enables one to determine the distribution of Berry curvature with high momentum resolution. Our work forms the basis for a general framework to fully characterize topological band structures and can also facilitate holonomic quantum computing through controlled exploitation of the geometry of Hilbert space.More than thirty years ago, Berry [1] delineated the effects of the geometric structure of Hilbert space on the adiabatic evolution of quantum mechanical systems. These ideas have found widespread applications in science [2] and are routinely used to calculate the geometric phase shift acquired by a particle moving along a closed path-a phase shift that is determined only by the geometry of the path and is independent of the time spent en route. Geometric phases provide an elegant description of the celebrated Aharonov-Bohm effect [3], where a magnetic flux in a confined region of space influences the eigenstates everywhere via the magnetic vector potential. In condensed-matter physics, an analogous Berry flux in momentum space is responsible for various anomalous velocities and Hall responses [4] and lies at the heart of many-body phenomena ranging from quantum Hall physics [5] to topological insulators [6]. The Berry flux density (Berry curvature) is indeed essential to the characterization of an energy band and determines its topological invariants. However, mapping out the geometric structure of an energy band [7][8][9] has remained a major unresolved challenge for experiments.Here, we demonstrate a versatile technique for measuring geometric phases in reciprocal space using spin-echo interferometry with ultracold atoms [9, 10]. In contrast to typical solid state experiments, where all geometric effects are averaged over the Fermi sea, the use of a Bose-Einstein condensate (BEC) enables measurements with high momentum resolution. We exploit this resolution to directly detect the singular topological properties of an individual Dirac cone [11] in a graphene-type hexagonal optical lattice (see Fig. 1). Concentrated at the Dirac point is a π Berry flux, which is analogous to a magnetic flux generated by an infinitely narrow solenoid [12]. This localized flux gives rise to several striking properties of graphene, including the half-integer shift in the positions of quantum Hall plateaus [13,14], the phase of Shubnikov-de Haas oscillations [13,14], and the polarization dependence in photoemission spec...

We generate entangled states of an ensemble of 5x10{4} 87Rb atoms by optical quantum nondemolition measurement. The resonator-enhanced measurement leaves the atomic ensemble, prepared in a superposition of hyperfine clock levels, in a squeezed spin state. By comparing the resulting reduction of quantum projection noise [up to 8.8(8) dB] with the concomitant reduction of coherence, we demonstrate a clock input state with spectroscopic sensitivity 3.0(8) dB beyond the standard quantum limit.