Dynamical instabilities due to spin-mixing collisions in a $^{87}$Rb F=1 spinor Bose-Einstein condensate are used as an amplifier of quantum spin fluctuations. We demonstrate the spectrum of this amplifier to be tunable, in quantitative agreement with mean-field calculations. We quantify the microscopic spin fluctuations of the initially paramagnetic condensate by applying this amplifier and measuring the resulting macroscopic magnetization. The magnitude of these fluctuations is consistent with predictions of a beyond-mean-field theory. The spinor-condensate-based spin amplifier is thus shown to be nearly quantum-limited at a gain as high as 30 dB.
We propose a theoretical scheme to realize a sensitive amplification of quantum discord (QD) between two atomic qubits via a cavity-Bose-Einstein condensate (BEC) system which was used to firstly realize the Dicke quantum phase transition (QPT) [Nature 464, 1301 (2010)]. It is shown that the influence of the cavity-BEC system upon the two qubits is equivalent to a phase decoherence environment. It is found that QPT in the cavity-BEC system is the physical mechanism of the sensitive QD amplification.
We analyze a scheme for generating entanglement between two spinor Bose-Einstein condensates (BECs). The BECs are off-resonantly coupled to a common photon mode and are controlled by external lasers to induce a $ S^z S^z $ interaction, where $ S^z $ is the total spin of the BEC. We directly simulate the scheme numerically for small systems and show the performance of the scheme. The scaling of the entanglement to large scale systems under realistic conditions of spontaneous emission and cavity photon loss is analyzed. It is shown that both entanglement in the beyond-continuous variables regime…
We study the time-evolution of an optically trapped spinor Bose-Einstein condensate under the influence of a dominating magnetic bias field in the z-direction, and a perpendicular smaller field that couples the spinor states. We show that if the bias field depends quadratically on time, the relative phases of the spinor components affect the populations of the final state. This allows one to measure the differences in the time-evolution of the relative phases, thereby realizing a multi-arm interferometer in a spinor Bose-Einstein condensate.
We examine the internal structure of the ground states of a trapped Bose-Einstein condensate in which atoms have three internal hyperfine spins. We determine a set of collective spin states which minimize the interaction energy between condensate atoms. We also examine the internal dynamics of an initially spin polarized condensate. The time scale of spin-mixing is predicted.
In a spinor Bose-Einstein gas, the non-zero hyperfine spin of the gas becomes an accessible degree of freedom. At low temperature, such a gas shows both magnetic and superfluid order, and undergoes both density and spin dynamics. These lecture notes present a general overview of the properties of spinor Bose-Einstein gases. The notes are divided in five sections. In the first, we summarize basic properties of multi-component quantum fluids, focusing on the specific case of spinor Bose-Einstein gases and the role of rotational symmetry in defining their properties. Second, we consider the magne…
Magnon excitation in a spinor Bose-Einstein condensate by a driven magnetic field is shown to have a close analogy with the dynamical Casimir effect. A time-dependent external magnetic field amplifies quantum fluctuations in the magnetic ground state of the condensate, leading to magnetization of the system. The magnetization occurs in a direction perpendicular to the magnetic field breaking the rotation symmetry. This phenomenon is numerically demonstrated and the excited quantum field is shown to be squeezed.
We investigate a radial spin hedgehog, analogous to the Dirac monopole, in an optically trapped atomic spin-1 Bose-Einstein condensate. By joining together a monopole-antimonopole pair, we may form a vortex line with free ends. We numerically simulate the three-dimensional dynamics and imaginary time relaxation of these structures to nonsingular textures and show they can be observable for realistic experimental parameters.
We introduce a Floquet spinor Bose-Einstein condensate induced by a periodically driven quadratic Zeeman coupling whose frequency is larger than any other energy scales. By examining a spin-1 system available in ultracold atomic gases, we demonstrate that such an external driving field has great effect on the condensate through emergence of a unique spin-exchange interaction. We uncover that the ferromagnetic condensate has several unconventional stationary states and thus exhibits rich continuous phase transitions. On the other hand, the antiferromagnetic condensate is found to possess a nont…
We calculate, in the standard Bogoliubov approximation, the ground state energy of the spinor BEC with hyperfine spin $f=1$ where the two-body repulsive hard-core and spin exchange interactions are both included. The coupling constants characterized these two competing interactions are expressed in terms of the corresponding s-wave scattering lengths using second-order perturbation methods. We show that the ultraviolet divergence arising in the ground state energy corrections can be exactly eliminated.
We propose a scheme to generate and detect various kinds of quantum entanglement in a spin-1 Bose-Einstein condensate. It is shown that substantial many-particle entanglement can be generated directly in the spin-1 condensate by free dynamical evolution with a properly prepared initial state. The scheme also provides a simple method to generate three-mode entanglement in the second-quantization picture and to detect the continuous variable type of entanglement between two effective modes in the spin-1 condensate.
Josephson junctions are essential ingredients in the superconducting circuits used in many existing quantum technologies. Additionally, ultracold atomic quantum gases have also become essential platforms to study superfluidity. Here, we explore the analogy between superconductivity and superfluidity to present an intriguing effect caused by a thin finite barrier in a quasi-one-dimensional toroidal spinor Bose--Einstein condensate (BEC). In this system, the atomic current density flowing through the edges of the barrier oscillates, such as the electrical current through a Josephson junction in …
Manipulation of the quantum state by the Majorana transition in spinor BEC system has been realized by altering the rotation frequency of the magnetic field's direction. This kind of manipulation method has no limitation on the transition speed in principle and the system is well closed, which provides a new and superior tool to manipulate quantum states. Using this methord on pulsed atom laser, multicomponent spinor atom laser is generated. We demonstrate that the experiment results are agreed with the theoretical predication.
Excited-state quantum phase transitions (ESQPTs) extend the notion of quantum phase transitions beyond the ground state. They are characterized by closing energy gaps amid the spectrum. Identifying order parameters for ESQPTs poses however a major challenge. We introduce spinor Bose-Einstein condensates as a versatile platform for studies of ESQPTs. Based on the mean-field dynamics, we define a topological order parameter that distinguishes between excited-state phases, and discuss how to interferometrically access the order parameter in current experiments. Our work opens the way for the expe…
We propose to generate macroscopic superposition states of a large number of atoms in the ground state of a three-mode spinor Bose-Einstein condensate. The ground state is protected by a finite energy gap, is immune to phase noise, and the measurement of the number of particles in one mode heralds the coherent preparation of the two other modes simultaneously empty and filled. Highly entangled macroscopic superposition states are generated with large probability also taking into account quasi-adiabatic preparation of the ground state. All the key ingredients necessary to realize our proposal a…
Helical spin textures in a $^{87}$Rb F=1 spinor Bose-Einstein condensate are found to decay spontaneously toward a spatially modulated structure of spin domains. This evolution is ascribed to magnetic dipolar interactions that energetically favor the short-wavelength domains over the long-wavelength spin helix. This is confirmed by eliminating the dipolar interactions by a sequence of rf pulses and observing a suppression of the formation of the short-range domains. This study confirms the significance of magnetic dipole interactions in degenerate $^{87}$Rb F=1 spinor gases.
We propose the use of stimulated Raman adiabatic passage (STIRAP) to offer a fast high fidelity method of performing SU(2) rotations on spinor Bose Einstein condensates (BEC). Past demonstrations of BEC optical control suffer from difficulties arising from collective enhancement of spontaneous emission and inefficient two-photon transitions originating from selection rules. We present here a novel scheme which allows for arbitrary coherent rotations of two-component BECs while overcoming these issues. Numerical tests of the method show that for BECs of \ce{^{87}Rb} with up to $ 10^4 $ atoms an…
We analyze the spinor dynamics of a Rb-87 F=2 condensate initially prepared in the m_F=0 Zeeman sublevel. We show that this dynamics, characterized by the creation of correlated atomic pairs in m_F=+/-1, presents an intriguing multi-resonant magnetic field dependence induced by the trap inhomogeneity. This dependence is directly linked to the most unstable Bogoliubov spin excitations of the initial m_F = 0 condensate, showing that, in general, even a qualitative understanding of the pair creation efficiency in a spinor condensate requires a careful consideration of the confinement.
Parametric amplification of quantum fluctuations constitutes a fundamental mechanism for spontaneous symmetry breaking. In our experiments, a spinor condensate acts as a parametric amplifier of spin modes, resulting in a twofold spontaneous breaking of spatial and spin symmetry in the amplified clouds. Our experiments permit a precise analysis of the amplification in specific spatial Bessel-like modes, allowing for the detailed understanding of the double symmetry breaking. On resonances that create vortex-antivortex superpositions, we show that the cylindrical spatial symmetry is spontaneousl…
We study the richer structures of quasi-one-dimensional Bogoliubov-de Genes collective excitations of F = 1 spinor Bose-Einstein condensate in a harmonic trap potential loaded in an optical lattice. Employing a perturbative method we report general analytical expressions for the confined collective polar and ferromagnetic Goldstone modes. In both cases the excited eigenfrequencies are given as function of the 1D effective coupling constants, trap frequency and optical lattice parameters. It is shown that the main contribution of the optical lattice laser intensity is to shift the confined phon…
We present an analytical model for the theoretical analysis of spin dynamics and spontaneous symmetry breaking in a spinor Bose-Einstein condensate (BEC). This allows for an excellent intuitive understanding of the processes and provides good quantitative agreement with experimental results in Phys. Rev. Lett. 105, 135302 (2010). It is shown that the dynamics of a spinor BEC initially prepared in an unstable Zeeman state mF=0 (|0>) can be understood by approximating the effective trapping potential for the state |+-1> with a cylindrical box potential. The resonances in the creation efficiency …
We consider the possibility of parametric amplification of a mechanical vibration mode of a nanowire due to its interaction with a Bose-Einstein condensate (BEC) of ultracold atoms. The magneto-mechanical coupling is mediated by the vibrationally modulated magnetic field around the current-carrying nanowire, which can induce atomic transitions between different hyperfine sublevels. We theoretically analyze the limitations arising from the fact that the spin inverted atomic medium which feeds the mechanical oscillation has a finite bandwidth in the range of the chemical potential of the condens…
We have studied computationally the collision dynamics of spin-2 Bose--Einstein condensates initially confined in a triple-well trap. Depending on the intra- and inter-component relative phases of the initial state spinor wave function, the collision of the three condensate segments produces one of many possible vortex-antivortex lattices after which the system transitions to quantum turbulence. We find that the emerging vortex lattice structures can be described in terms of multi-wave interference. We show that the three-segment collisions can be used to systematically produce staggered vorte…
Spinor Bose gases form a family of quantum fluids manifesting both magnetic order and superfluidity. This article reviews experimental and theoretical progress in understanding the static and dynamic properties of these fluids. The connection between system properties and the rotational symmetry properties of the atomic states and their interactions are investigated. Following a review of the experimental techniques used for characterizing spinor gases, their mean-field and many-body ground states, both in isolation and under the application of symmetry-breaking external fields, are discussed.…
We theoretically investigate the creation of squeezed states of a Bose-Einstein Condensate (BEC) trapped in a magnetic double well potential. The number or phase squeezed states are created by modulating the tunnel coupling between the two wells periodically with twice the Josephson frequency, i.e., through parametric amplification. Simulations are performed with the multi configurational Hartree method for bosons (MCTDHB). We employ optimal control theory to bring the condensate to a complete halt at a final time, thus creating a highly squeezed state (squeezing factor of 0.12, $ξ_S^2=-18$ dB…
We investigate, both experimentally and theoretically, the quench dynamics of antiferromagnetic spinor Bose-Einstein condensates in the vicinity of a zero temperature quantum phase transition at zero quadratic Zeeman shift q. Both the rate of instability and the associated finite wavevector of the unstable modes - show good agreement with predictions based upon numerical solutions to the Bogoliubov de-Gennes equations. A key feature of this work is inclusion of magnetic field inhomogeneities that smooth the phase transition. Once these were removed, we observed a dramatic sharpening of the tra…
The spinor condensate with spin states degenerated in the ground spin-space provides a unique platform for investigating the edge of quantum mechanics and statistical physics. We study the evolution of the condensate under the scattering of a coherent atom beam. The time-dependent magnetization, entanglement entropy, thermal entropy and the entropy production rate are calculated. A novel spontaneous symmetry breaking is found during the evolution, It is shown that the stationary spin distribution can be controlled by the incoming coherent spin state of the incident atom beam, therefore the ato…
We have analytically solved one-dimensional interacting two-component bosonic gases with spin-orbit (SO) coupling by the Bethe-ansatz method. Through a gauge transformation, the effect of SO coupling is incorporated into a spin-dependent twisted boundary condition. Our result shows that the SO coupling can influence the eigenenergy in a periodical pattern. The interplay between interaction and SO coupling may induce the energy level crossing for the ground state, which leads to a transition from the ferromagnetic to antiferromagnetic state.
We study the zero temperature phase diagram of the disordered spin-1 Bose-Hubbard model in a 2-dimensional square lattice. To this aim, we use a mean field Gutzwiller ansatz and a probabilistic mean field perturbation theory. The spin interaction induces two different regimes corresponding to a ferromagnetic and antiferromagnetic order. In the ferromagnetic case, the introduction of disorder reproduces analogous features of the disordered scalar Bose-Hubbard model, consisting in the formation of a Bose glass phase between Mott insulator lobes. In the antiferromagnetic regime the phase diagram …
We investigate the dynamics of a spinor Bose-Einstein condensate inside an optical cavity, driven transversely by a laser with a controllable polarization angle. We focus on a two-component Dicke model with complex light-matter couplings, in the presence of photon losses. We calculate the steady-state phase diagram and find dynamical instabilities in the form of limit cycles, heralded by the presence of exceptional points and level attraction. We show that the instabilities are induced by dissipative processes which generate non-reciprocal couplings between the two collective spins. Our predic…
We show how the length scale hierarchy, resulting from different interaction strengths in an optically-trapped spin-1 sodium-23 Bose-Einstein condensate, can lead to intriguing core deformations in singular topological defects. In particular, a point defect can be unstable with respect to the formation of a stable half-quantum vortex ring (an Alice ring), providing a realistic scheme to use dissipation as a sophisticated state engineering tool. We compute the threshold for stability of the point monopole, which is beyond the current experimental regime.
We propose to use spatial control of the Zeeman energy shifts in an ultracold atomic gas to engineer an interface between topologically distinct regions. This provides an experimentally accessible means for studying the interface physics of topological defects and textures. Using the spin-1 Bose-Einstein condensate as an example, we find spinor wave functions that represent defects and textures continuously connecting across the interface between polar and ferromagnetic regions induced by spatially varying Zeeman shifts. By numerical energy minimization we characterize the defect core structur…
We measure the mass, gap, and magnetic moment of a magnon in the ferromagnetic $F=1$ spinor Bose-Einstein condensate of $^{87}$Rb. We find an unusually heavy magnon mass of $1.038(2)_\mathrm{stat}(8)_\mathrm{sys}$ times the atomic mass, as determined by interfering standing and running coherent magnon waves within the dense and trapped condensed gas. This measurement is shifted significantly from theoretical estimates. The magnon energy gap of $h\times 2.5(1)_\mathrm{stat}(2)_\mathrm{sys}\;\mathrm{Hz}$ and the effective magnetic moment of $-1.04(2)_\mathrm{stat}(8)\,μ_\textrm{bare}$ times the …
We demonstrate detection of a weak alternate-current magnetic field by application of the spin echo technique to F = 2 Bose-Einstein condensates. A magnetic field sensitivity of 12 pT/Hz^1/2 is attained with the atom number of 5*10^3 at spatial resolution of 99 μm^2. Our observations indicate magnetic field fluctuations synchronous with the power supply line frequency. We show that this noise is greatly suppressed by application of a reverse phase magnetic field. Our technique is useful in order to create a stable magnetic field environment, which is an important requirement for atomic experim…
We report on the spin textures produced by cooling unmagnetized $^{87}$Rb F=1 spinor gases into the regime of quantum degeneracy. At low temperatures, magnetized textures form that break translational symmetry and display short-range periodic magnetic order characterized by one- or two-dimensional spatial modulations with wavelengths much smaller than the extent of the quasi-two-dimensional degenerate gas. Spin textures produced upon cooling spin mixtures with a non-zero initial magnetic quadrupole moment also show ferromagnetic order that, at low temperature, coexists with the spatially modul…
