Creating massive entanglement of Bose-Einstein condensed atoms

Heisenberg-limited spin squeezing in a hybrid system with silicon-vacancy centers

In this paper, we investigate the spin squeezing in a hybrid quantum system consisting of a Silicon-Vacancy (SiV) center ensemble coupled to a diamond acoustic waveguide via the strain interaction. Two sets of non-overlapping driving fields, each contains two time-dependent microwave fields, are applied to this hybrid system. By modulating these fields, the one-axis twist (OAT) interaction and two-axis two-spin (TATS) interaction can be independently realized. In the latter case the squeezing parameter scales to spin number as ξ R2∼1.61N -0.64 with the consideration of dissipation, which is very close to the Heisenberg limit. Furthermore, this hybrid system allows for the study of spin squeezing generated by the simultaneous presence of OAT and TATS interactions, which reveals sensitivity to the parity of the number of spins Ntot, whether it is even or odd. Our scheme enriches the approach for generating Heisenberg-limited spin squeezing in spin-phonon hybrid systems and offers the possibility for future applications in quantum information processing.

Order Indices and Entanglement Production in Quantum Systems

The review is devoted to two important quantities characterizing many-body systems, order indices and the measure of entanglement production. Order indices describe the type of order distinguishing statistical systems. Contrary to the order parameters characterizing systems in the thermodynamic limit and describing long-range order, the order indices are applicable to finite systems and classify all types of orders, including long-range, mid-range, and short-range orders. The measure of entanglement production quantifies the amount of entanglement produced in a many-partite system by a quantum operation. Despite that the notions of order indices and entanglement production seem to be quite different, there is an intimate relation between them, which is emphasized in the review.

Cavity-Assisted Single-Mode and Two-Mode Spin-Squeezed States via Phase-Locked Atom-Photon Coupling

We propose a scheme to realize the two-axis countertwisting spin-squeezing Hamiltonian inside an optical cavity with the aid of phase-locked atom-photon coupling. By careful analysis and extensive simulation, we demonstrate that our scheme is robust against dissipation caused by cavity loss and atomic spontaneous emission, and it can achieve significantly higher squeezing than one-axis twisting. We further show how our idea can be extended to generate two-mode spin-squeezed states in two coupled cavities. Because of its easy implementation and high tunability, our scheme is experimentally realizable with current technologies.

Spin squeezing in a quadrupolar nuclei NMR system

We have produced and characterized spin-squeezed states at a temperature of 26 °C in a nuclear magnetic resonance quadrupolar system. The experiment was carried out on 133Cs nuclei of spin I=7/2 in a sample of lyotropic liquid crystal. The source of spin squeezing was identified as the interaction between the quadrupole moment of the nuclei and the electric field gradients present within the molecules. We use the spin angular momentum representation to describe formally the nonlinear operators that produce the spin squeezing on a Hilbert space of dimension 2I+1=8. The quantitative and qualitative characterization of this spin-squeezing phenomenon is expressed by a squeezing parameter and squeezing angle developed for the two-mode Bose-Einstein condensate system, as well as by the Wigner quasiprobability distribution function. The generality of the present experimental scheme points to potential applications in solid-state physics.

Orbit-induced spin squeezing in a spin-orbit coupled Bose-Einstein condensate

In recent pioneer experiment, a strong spin-orbit coupling, with equal Rashba and Dresselhaus strengths, has been created in a trapped Bose-Einstein condensate. Moreover, many exotic superfluid phenomena induced by this strong spin-orbit coupling have been predicted. In this report, we show that this novel spin-orbit coupling has important applications in quantum metrology, such as spin squeezing. We first demonstrate that an effective spin-spin interaction, which is the heart for producing spin squeezing, can be generated by controlling the orbital degree of freedom (i.e., the momentum) of the ultracold atoms. Compared with previous schemes, this realized spin-spin interaction has advantages of no dissipation, high tunability, and strong coupling. More importantly, a giant squeezing factor (lower than −30 dB) can be achieved by tuning a pair of Raman lasers in current experimental setup. Finally, we find numerically that the phase factor of the prepared initial state affects dramatically on spin squeezing.

REDUCED DENSITY MATRIX AND ENTANGLEMENT ENTROPY OF PERMUTATIONALLY INVARIANT QUANTUM MANY-BODY SYSTEMS

In this paper we discuss the properties of the reduced density matrix of quantum many body systems with permutational symmetry and present basic quantification of the entanglement in terms of the von Neumann (VNE), Renyi and Tsallis entropies. In particular, we show, on the specific example of the spin 1/2 Heisenberg model, how the RDM acquires a block diagonal form with respect to the quantum number k fixing the polarization in the subsystem conservation of Sz and with respect to the irreducible representations of the Sn group. Analytical expression for the RDM elements and for the RDM spectrum are derived for states of arbitrary permutational symmetry and for arbitrary polarizations. The temperature dependence and scaling of the VNE across a finite temperature phase transition is discussed and the RDM moments and the Rényi and Tsallis entropies calculated both for symmetric ground states of the Heisenberg chain and for maximally mixed states.

Spin squeezing: transforming one-axis twisting into two-axis twisting

Squeezed spin states possess unique quantum correlation or entanglement and are significantly promising for advancing quantum information processing and quantum metrology. In recent back-to-back publications [C. Gross et al., Nature (London) 464, 1165 (2010) and Max F. Riedel et al., Nature (London) 464, 1170 (2010)], reduced spin fluctuations are observed leading to spin squeezing at -8.2 and -2.5 dB, respectively, in two-component atomic condensates exhibiting one-axis-twisting interactions. The noise reduction limit for the one-axis twisting scales as ∝1/N(2/3), which for a condensate with N∼10(3) atoms is about 100 times below the standard quantum limit. We present a scheme using repeated Rabi pulses capable of transforming the one-axis-twisting spin squeezing into the two-axis-twisting type, leading to Heisenberg limited noise reduction ∝1/N or an extra tenfold improvement for N∼10(3).

Reduced-density-matrix spectrum and block entropy of permutationally invariant many-body systems

Spectral properties of the reduced density matrix (RDM) of permutational invariant quantum many-body systems are investigated. The RDM block diagonalization which accounts for all symmetries of the Hamiltonian is achieved. The analytical expression of the RDM spectrum is provided for arbitrary parameters and rigorously proved in the thermodynamical limit. The existence of several sum rules and recurrence relations among RDM eigenvalues is also demonstrated and the distribution function of RDM eigenvalues (including degeneracies) characterized. In particular, we prove that the distribution function approaches a two-dimensional Gaussian in the limit of large subsystem sizes n>>1. As a physical application we discuss the von Neumann entropy (VNE) of a block of size n for a system of hard-core bosons on a complete graph, as a function of n and of the temperature T. The occurrence of a crossover of VNE from purely logarithmic behavior at T=0 to a purely linear behavior in n for T≥Tc, is demonstrated.

Storage of spin squeezing in a two-component Bose-Einstein condensate

A simple scheme for storage of spin squeezing in a two-component Bose-Einstein condensate is investigated by considering rapidly turning-off the external field at a time that maximal spin squeezing occurs. We show that strong reduction of spin fluctuation can be maintained in a nearly fixed direction. We explain the underlying physics using the phase model and present analytical expressions of the maximal-squeezing time and the corresponding squeezing parameter.

Bose-Einstein condensation with an entangled order parameter

We propose a practically accessible non-mean-field ground state of Bose-Einstein condensation, which occurs in an interspecies two-particle entangled state, and is thus described by an entangled order parameter. A suitably defined entanglement entropy is used as the characterization of the non-mean-field nature, and is found to persist in a wide parameter regime. The interspecies entanglement leads to novel interference terms in the dynamical equations governing the single-particle orbital wave function. Experimental feasibility and several methods of probe are discussed. We urge the study of multichannel scattering between different species of atoms.

Spin-dependent forces on trapped ions for phase-stable quantum gates and entangled states of spin and motion

Favored schemes for trapped-ion quantum logic gates use bichromatic laser fields to couple internal qubit states with external motion through a "spin-dependent force." We introduce a new degree of freedom in this coupling that reduces its sensitivity to phase decoherence. We demonstrate bichromatic spin-dependent forces on a single trapped 111Cd+ ion, and show that phase coherence of the resulting entangled states of spin and motion depends critically upon the spectral arrangement of the optical fields. This applies directly to the operation of entangling gates on multiple ions.

Spin squeezing via one-axis twisting with coherent light

We propose a new method of spin squeezing of atomic spin, based on the interactions between atoms and off-resonant light which are known as paramagnetic Faraday rotation and the fictitious magnetic field of light. Since the projection process, squeezed light, or special interactions among the atoms are not required in this method, it can be widely applied to many systems. The attainable range of the squeezing parameter is zeta greater, similarS(-2/5), where S is the total spin, which is limited by additional fluctuations imposed by coherent light and the spherical nature of the spin distribution.

Entanglement interferometry for precision measurement of atomic scattering properties

We report on a matter wave interferometer realized with entangled pairs of trapped 87Rb atoms. Each pair of atoms is confined at a single site of an optical lattice potential. The interferometer is realized by first creating a coherent spin superposition of the two atoms and then tuning the interstate scattering length via a Feshbach resonance. The selective change of the interstate scattering length leads to an entanglement dynamics of the two-particle state that can be detected in a Ramsey interference experiment. This entanglement dynamics is employed for a precision measurement of atomic interaction parameters. Furthermore, the interferometer allows us to separate lattice sites with one or two atoms in a nondestructive way.

Quantum Zeno subspace and entangled Bose-Einstein condensates

We discuss a proposal for the efficient generation of the maximally entangled atomic N-GHZ state in a spinor-1 condensate by driving internal state atomic Raman transitions using (classical) laser fields. We illustrate the dynamics in terms of a quantum Zeno subspace and identify the resultant atomic elastic collision in facilitating the deterministic entanglement creation. Our proposal can be readily implemented in several laboratories where ferromagnetic spinor condensates (of 87Rb atoms) are investigated.

Creating maximally entangled atomic states in a Bose-Einstein condensate

We propose a protocol to create maximally entangled pairs, triplets, quartiles, and other clusters of Bose-condensed atoms starting from a condensate in the Mott insulator state. The essential element is to drive single atom Raman transitions using laser pulses. Our scheme is simple, efficient, and can be readily applied to the recent experimental system as reported by M. Greiner 413, 44 (2002)].

Directional "superradiant" collisions: bosonic amplification of atom pairs emitted from an elongated Bose-Einstein condensate

We study spontaneous directionality in the bosonic amplification of atom pairs emitted from an elongated Bose-Einstein condensate, an effect analogous to superradiant emission of atom-photon pairs. Using a simplified model, we make analytic predictions regarding directional effects for both atom-atom and atom-photon emission. These are confirmed by numerical mean-field simulations, demonstrating the feasibility of nearly perfect directional emission along the condensate axis. The dependence of the emission angle on the pump strength for atom-atom pairs is significantly different than for atom-photon pairs.

Photoassociation of sodium in a Bose-Einstein condensate

We form ultracold Na2 molecules by single-photon photoassociation of a Bose-Einstein condensate, measuring the photoassociation rate, linewidth, and light shift of the J = 1, v = 135 vibrational level of the A1 Sigma (+)(u) molecular state. The photoassociation rate constant increases linearly with intensity, even where it is predicted that many-body effects might limit the rate. Our observations are in good agreement with a two-body theory having no free parameters.

Nonlinear and quantum atom optics

Coherent matter waves in the form of Bose-Einstein condensates have led to the development of nonlinear and quantum atom optics - the de Broglie wave analogues of nonlinear and quantum optics with light. In nonlinear atom optics, four-wave mixing of matter waves and mixing of combinations of light and matter waves have been observed; such progress culminated in the demonstration of phase-coherent matter-wave amplification. Solitons represent another active area in nonlinear atom optics: these non-dispersing propagating modes of the equation that governs Bose-Einstein condensates have been created experimentally, and observed subsequently to break up into vortices. Quantum atom optics is concerned with the statistical properties and correlations of matter-wave fields. A first step in this area is the measurement of reduced number fluctuations in a Bose-Einstein condensate partitioned into a series of optical potential wells.

Cold atoms and quantum control

This overview prefaces a collection of Insight review articles on the physics and applications of laser-cooled atoms. I will cast this work into a historical perspective in which laser cooling and trapping is seen as one of several research directions aimed at controlling the internal and external degrees of freedom of atoms and molecules.