We propose a generalization of the Feynman path integral using squeezed coherent states. We apply this approach to the dynamics of Bose-Einstein condensates, which gives an effective low energy description that contains both a coherent field and a squeezing field. We derive the classical trajectory of this action, which constitutes a generalization of the Gross Pitaevskii equation, at linear order. We derive the low energy excitations, which provides a description of second sound in weakly interacting condensates as a squeezing oscillation of the order parameter. This interpretation is also supported by a comparison to a numerical c-field method.
Dynamical generation of phase-squeezed states in a two-component Bose-Einstein condensates
作为线性（Mach-Zehnder或Ramsey）干涉仪的“输入”状态，Berry和Wiseman提出的相位压缩状态表现出接近Heisenberg极限的最佳灵敏度[Phys. Rev. Lett. 85，5098（2000）]。在本文中，我们发现，它可以动态生成原子玻色-爱因斯坦凝聚体限制在一个对称的双阱。与Berry态和Wiseman态类似，制备态沿自旋算符S_y方向存在压缩，沿自旋算符S_z方向存在反压缩，导致相位灵敏度的亚散粒噪声。
Within the framework of the Gaussian-state theory, we show that the quantum many-body ground state of a trapped condensate with weakly attractive interaction is a single-mode squeezed vacuum state, as oppose to the coherent state under repulsive interaction. The spatial mode of the squeezed-state condensates satisfies a Gross-Pitaevskii like equation in which the interaction strength is augmented by a factor $3$ due to the large particle fluctuation of the squeezed state. We also study the collective excitations of the condensates by the tangential space projection, which leads to new two-part…
Light with a chemical potential and no mass is shown to possess a general phase-transition curve to Bose-Einstein condensation. This limiting density and temperature range is found by the diverging in-medium potential range of effective interaction. The inverse expansion series of the effective interaction from Bethe-Salpeter equation is employed exceeding the ladder approximation. While usually the absorption and emission with Dye molecules is considered, here it is proposed that squeezing can create also such a mean interaction leading to a chemical potential. The equivalence of squeezed lig…
We employ the Gross-Pitaevskii equation to study acoustic emission generated in a uniform Bose gas by a static impurity. The impurity excites a sound-wave packet, which propagates through the gas. We calculate the shape of this wave packet in the limit of long wave lengths, and argue that it is possible to extract properties of the impurity by observing this shape. We illustrate here this possibility for a Bose gas with a trapped impurity atom -- an example of a relevant experimental setup. Presented results are general for all one-dimensional systems described by the nonlinear Schrödinger equ…
We study the propagation of the two sound modes in two-dimensional Bose gases across the Berezinksii-Kosterlitz-Thouless (BKT) transition using classical-field dynamics, which is motivated by recent measurements of Christodoulou et al. Nature 594, 191 (2021). Based on the dynamic structure factor, we identify the two sound modes as the Bogoliubov (B) and the non-Bogoliubov (NB) sound mode below the transition, and as the diffusive and the normal sound mode above the transition. The NB sound mode velocity is higher than the B sound mode velocity, which we refer to as the weak-coupling regime of…
We investigate the out-of-equilibrium dynamics following a sudden quench of the interaction strength, in a one-dimensional quasi-condensate trapped at the surface of an atom chip. Within a linearized approximation, the system is described by independent collective modes and the quench squeezes the phase space distribution of each mode, leading to a subsequent breathing of each quadrature. We show that the collective modes are resolved by the power spectrum of density ripples which appear after a short time of flight. This allows us to experimentally probe the expected breathing phenomenon. Our…
We present an alternative method for determining the sound velocity in atomic Bose-Einstein condensates, based on thermodynamic global variables. The total number of trapped atoms was as a function of temperature carefully studied across the phase transition, at constant volume. It allowed us to evaluate the sound velocity resulting in consistent values from the quantum to classical regime, in good agreement with previous results found in literature. We also provide some insight about the dominant sound mode (thermal or superfluid) across a wide temperature range.
Motivated by recent experiments in Phys. Rev. Lett. 121, 145301 (2018), we study sound propagation in a two-dimensional (2D) Bose gas across the superfluid-thermal transition using classical field dynamics. Below the transition temperature we find a Bogoliubov and a non-Bogoliubov mode, above it we find the normal sound mode and the diffusive mode, as we determine from the dynamical structure factor. Our simulations of the experimental procedure agree with the measured velocities, and show that below the transition temperature the measurements detect the Bogoliubov mode. Above the transition, …
A unified model of a dilute Bose-Einstein condensate is proposed, combining of the logarithmic and Gross-Pitaevskii nonlinear terms in a wave equation, where the Gross-Pitaevskii term describes two-body interactions, as suggested by standard perturbation theory; while the logarithmic term is essentially non-perturbative, and takes into account quantum vacuum effects. The model is shown to have excellent agreement with sound propagation data in the condensate of cold sodium atoms known since the now classic works by Andrews and collaborators. The data also allowed us to place constraints on two…
In superfluid systems several sound modes can be excited, as for example first and second sound in liquid helium. Here, we excite propagating and standing waves in a uniform two-dimensional Bose gas and we characterize the propagation of sound in both the superfluid and normal regime. In the superfluid phase, the measured speed of sound is well described by a two-fluid hydrodynamic model, and the weak damping rate is well explained by the scattering with thermal excitations. In the normal phase the sound becomes strongly damped due to a departure from hydrodynamic behavior.
We theoretically investigate sound modes in a weakly-interacting collisional Bose gas in $D$ dimensions. Using the Landau's two-fluid hydrodynamics and working within the Bogoliubov theory, we observe the hybridization of the first and second sound modes for $D\ge 2$. To model the recent measurements of the sound velocities in 2D, obtained in the weakly-interacting regime and around the Berezinskii-Kosterlitz-Thouless transition temperature, we derive a refined calculation of the superfluid density, finding a fair agreement with the experiment. In the 1D case, for which experimental results ar…
Many-particle entanglement is a fundamental concept of quantum physics that still presents conceptual challenges. While spin-squeezed and other nonclassical states of atomic ensembles were used to enhance measurement precision in quantum metrology, the notion of entanglement in these systems remained controversial because the correlations between the indistinguishable atoms were witnessed by collective measurements only. Here we use highresolution imaging to directly measure the spin correlations between spatially separated parts of a spin-squeezed Bose-Einstein condensate. We observe entangle…
The collective modes of a quantum liquid shape and impact its properties profoundly, including its emergent phenomena such as superfluidity. Here we present how a two-dimensional Bose gas responds to a moving lattice potential. In particular we discuss how the induced heating rate depends on the interaction strength and the temperature. This study is motivated by the recent measurements of Sobirey {\it et al.} arXiv:2005.07607 (2020), for which we provide a quantitative understanding. Going beyond the existing measurements, we demonstrate that this probing method allows to identify first and s…
Using Landau's theory of two-fluid hydrodynamics, we investigate first and second sound propagating in a two-dimensional Bose gas. We study the temperature and interaction dependence of both sound modes and show that their behaviour exhibits a deep qualitative change as the gas evolves from the weakly interacting to the strongly interacting regime. Special emphasis is given to the jump of both sounds at the Berezinskii-Kosterlitz-Thouless transition, caused by the discontinuity of the superfluid density. We find that the excitation of second sound through a density perturbation becomes weaker …
Using linear response theory within the Random Phase Approximation, we investigate the propagation of sound in a uniform two dimensional (2D) Bose gas in the collisionless regime. We show that the sudden removal of a static density perturbation produces a damped oscillatory behavior revealing that sound can propagate also in the absence of collisions, due to mean-field interaction effects. Our analysis points out the crucial role played by Landau damping. We support our predictions by performing numerical simulations with the stochastic (projected) Gross-Pitaevskii equation. The results are co…
We apply quantum optimal control to shape the phase-space distribution of Bose-Einstein condensates in a one-dimensional optical lattice. By a time-dependent modulation of the lattice position, determined from optimal control theory, we prepare, in the phase space of each lattice site, translated and squeezed Gaussian states, and superpositions of Gaussian states. Complete reconstruction of these non-trivial states is performed through a maximum likelihood state tomography. As a practical application of our method to quantum simulations, we initialize the atomic wavefunction in an optimal Floq…
We theoretically investigate the use of quantum non-demolition measurement to enhance the sensitivity of atom interferometry with Bose-condensed atoms. In particular, we are concerned with enhancing existing high-precision atom interferometry apparatuses, so restrict ourselves to dilute atomic samples, and the use of free-propagating light, or optical cavities in the weak-coupling regime. We find the optimum parameter regime that balances between spin squeezing and atomic loss, and find that significant improvements in sensitivity are possible. Finally, we consider the use of squeezed light, a…
A comparative study is done of the propagation of sound pulses in elongated Bose liquids and Bose-Einstein condensates in Gross-Pitaevskii and logarithmic models, by means of the Thomas-Fermi approximation. It is shown that in the linear regime the propagation of small density fluctuations is essentially one-dimensional in both models, in the direction perpendicular to the cross section of a liquid's lump. Under these approximations, it is shown that the speed of sound scales as a square root of particle density in the case of the Gross-Pitaevskii liquid/condensate, but it is constant in a cas…
The two-fluid model is fundamental for the description of superfluidity. In the nearly-incompressible-liquid regime, it successfully describes first and second sound, corresponding, respectively, to density and entropy waves, in both liquid helium and unitary Fermi gases. Here, we study the two sounds in the opposite regime of a highly compressible fluid, using an ultracold $^{39}$K Bose gas in a three-dimensional box trap. We excite the longest-wavelength mode of our homogeneous gas, and observe two distinct resonant oscillations below the critical temperature, of which only one persists abov…
Using parametric conversion induced by a Shapiro-type resonance, we produce and characterize a two-mode squeezed vacuum state in a sodium spin 1 Bose-Einstein condensate. Spin-changing collisions generate correlated pairs of atoms in the $m=\pm 1$ Zeeman states out of a condensate with initially all atoms in $m=0$. A novel fluorescence imaging technique with sensitivity $ΔN \sim 1.6$ atom enables us to demonstrate the role of quantum fluctuations in the initial dynamics and to characterize the full distribution of the final state. Assuming that all atoms share the same spatial wave function, w…
A protocol for the creation of time-stationary squeezed states in a spin-1 Bose condensate is proposed. The method consists of a pair of controlled quenches of an external magnetic field, which allows tuning of the system Hamiltonian in the vicinity of a phase transition. The quantum fluctuations of the system are well described by quantum harmonic oscillator dynamics in the limit of large system size, and the method can be applied to a spin-1 gas prepared in the low or high energy polar states.
Scalable Spin Squeezing for Quantum-Enhanced Magnetometry with Bose-Einstein Condensates
量子精密测量的一个主要挑战是产生具有宏观原子数的纠缠态。在这里，我们实验证明，通过非线性动力学产生的原子压缩在玻色-爱因斯坦凝聚体，结合适当的陷阱几何形状，允许缩放到大的合奏大小。对12300个粒子，我们实现了5.3（5）dB的涨落抑制，这意味着对于10个以上的$^7$原子也可以实现类似的压缩。有了这个资源，我们展示了量子增强磁强计交换压缩状态的磁敏感的超精细水平，具有可忽略的非线性。我们发现一个量子增强的单次激发的灵敏度为310（47）pT的静态磁场中的探头体积小到90 $μ$ m $^3$。
Resolving the problem of complex sound velocity in binary Bose mixtures with attractive intercomponent interactions
Center for Theoretical Physics, Khazar University（哈扎尔大学理论物理中心）；Physics Department, National University of Uzbekistan（乌兹别克斯坦国立大学物理系）；Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research（俄罗斯联合核研究所鲍里索夫理论物理实验室）；Instituto de Fisica de São Carlos, Universidade de São Paulo（圣保罗大学圣卡洛斯物理研究所）
We experimentally and theoretically investigate the lowest-lying axial excitation of an atomic Bose-Einstein condensate in a cylindrical box trap. By tuning the atomic density, we observe how the nature of the mode changes from a single-particle excitation (in the low-density limit) to a sound wave (in the high-density limit). Throughout this crossover the measured mode frequency agrees with Bogoliubov theory. Using approximate low-energy models we show that the evolution of the mode frequency is directly related to the interaction-induced shape changes of the condensate and the excitation. Fi…
We investigate collective excitations of density fluctuations and a dynamic density structure factor in a mixture of Bose and Fermi gases in a normal phase. With decreasing temperature, we find that the frequency of the collective excitation deviates from that of the hydrodynamic sound mode. Even at temperature much lower than the Fermi temperature, the collective mode frequency does not reach the collisionless limit analogous to zero sound in a Fermi gas, because of collisions between bosons and fermions.
