Bogoliubov Modes Research Articles

Many-body dynamics with time-dependent interaction

Recent advances in optical Feshbach resonance technique have enabled the experimental investigation of atomic gases with time-dependent interaction. In this work, we study the many-body dynamics of weakly interacting bosons subject with an arbitrary time varying scattering length. By employing a variational ansatz, we derive an effective Hamiltonian that governs the dynamics of thermal particles. Crucially, we show that there exists a hidden symmetry in this Hamiltonian that can map the many-body dynamics to the precession of an SU(1,1) "spin". As a demonstration, we calculate the situation where the scattering length is sinusoidally modulated. We show that the non-compactness of the SU(1,1) group naturally leads to solutions with exponentially growth of Bogoliubov modes and causes instabilities.

Spin-orbit-coupled fluids of light in bulk nonlinear media

We show that nonparaxial polarized light beams propagating in a bulk nonlinear Kerr medium naturally exhibit a coupling between the motional and the polarization degrees of freedom, realizing a spin-orbit-coupled mixture of fluids of light. We investigate the impact of this mechanism on the Bogoliubov modes of the fluid, using a suitable density-phase formalism built upon a linearization of the exact Helmholtz equation. The Bogoliubov spectrum is found to be anisotropic, and features both low-frequency gapless branches and high-frequency gapped ones. We compute the amplitudes of these modes and propose a couple of experimental protocols to study their excitation mechanisms. This allows us to highlight a phenomenon of hybridization between density and spin modes, which is absent in the paraxial description and represents a typical fingerprint of spin-orbit coupling.

Perfect higher-order squeezing via strong nonlinearity in microwave-modified electromagnetically induced transparency

We show that it is possible to obtain perfect higher-order squeezing via strong nonlinearities in the microwave-modified electromagnetically induced transparency (EIT). In a typical $\mathrm{\ensuremath{\Lambda}}$ three-level system coupled to a control field and a microwave field, the strong nonlinearity is existent under the full-resonant conditions and can be effectively controlled by the relative intensity of the optical and microwave fields. Via dressed-state and Bogoliubov mode transformation, we explore that the internal nonlinearity is closely related to the squeezing parameter and dissipative rate for two cavity modes. As a result, it is found that the two-mode higher-order squeezing is nearly close to $100%$ under ideal conditions, which is verified by our numerical and analytical results. In addition, we reveal that the higher-order squeezing in the present scheme is robust against the dephasing rate between two lower levels. This may find potential applications in high-precision measurement and provide a convenient way for experimental implementation.

Manipulation of the asymmetric Einstein–Podolsky–Rosen steering via coherent population transfer

We investigate the Einstein–Podolsky–Rosen (EPR) steering of two cavity modes generated via four-wave mixing processes. For a three-level V-type configuration, a strong coherent field as a dressing field couples the atoms, and a weak coherent field as a control filed resonantly couples one dressed state to the auxiliary state. By using the approach of dressed states and Bogoliubov modes, we find that only one Bogoliubov mode is mediated into the interaction with the dressed atoms, and the other one is decoupled. Due to the presence of the coherent control filed, the channel for dressed state population transfer is established, which plays a key role in the generation of the asymmetric EPR steering of two cavity fields. It is shown that the one-way EPR steering can be manipulated by the Rabi frequencies of the control field and the cooperativity parameters.

Stationary Gaussian entanglement between levitated nanoparticles

Coherent scattering of photons is a novel mechanism of optomechanical coupling for optically levitated nanoparticles promising strong, versatile interactions with light and between nanoparticles. We show that it allows efficient deterministic generation of Gaussian entanglement between two particles in separate tweezers. A combination of red- and blue-detuned tweezers brings a mechanical Bogoliubov mode to its ground state. An additional, dispersively coupled cavity mode can reduce noise in the orthogonal mode, resulting in strong entanglement as quantified by the logarithmic negativity and verifiable with the Duan criterion for realistic experimental parameters. Such an important resource for quantum sensing and quantum simulations is pivotal for current experiments and presents an important step towards optomechanics with multiple particles in the quantum regime.

Dispersion relations and self-localization of quasiparticles in coupled elongated Bose-Einstein condensates

We present a detailed study of the spectrum and dispersion of Bogoliubov quasiparticles in two coupled elongated Bose-Einstein condensates. We develop an analytically solvable model that approximates two infinite homogeneous condensates and compare its predictions to a numerical simulation of a realistic trapped system. While the comparisons show a reasonable agreement between the two models, they also manifest the existence of several anomalous Bogoliubov modes in the spectrum. These modes show degeneracy in both energy and momentum together with self-localization in the coordinate space.

Generation of optical-photon-and-magnon entanglement in an optomagnonics-mechanical system

We present a scheme to implement a steady optical-photon-and-magnon entanglement in a hybrid optomagnonics-mechanical system by adiabatical elimination of the auxiliary microwave (MW) cavity and effective laser cooling of a delocalized Bogoliubov mode. The system consists of a magnon, an optical and a MW cavities, and a mechanical vibrator. To achieve a direct entangling interaction between the magnon and the optical photon, we drive the optical cavity and magnon at the red and blue sideband associated with the mechanical resonator, respectively. In particular, by eliminating the MW cavity and optimizing the relative ratio of effect couplings, rather than merely increasing their magnitudes, we achieve a strong entanglement between optical photons and magnons.

Sound propagation in a two-dimensional Bose gas across the superfluid transition

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, they either detect the normal sound mode for low densities or weak interactions, or the diffusive mode for high densities or strong interactions. As a key observation, we discuss the weak coupling regime in which the non-Bogoliubov mode has a higher velocity than the Bogoliubov mode, in contrast to a hydrodynamic scenario. We propose to detect this regime via step-pulse density perturbation, which simultaneously detects both sound modes

Bogoliubov excitations in the quasiperiodic kicked rotor: Stability of a kicked condensate and the quasi–insulator-to-metal transition

We study the dynamics of a Bose-Einstein condensate in the quasiperiodic kicked rotor described by a Gross-Pitaevskii equation with periodic boundary conditions. As the interactions are increased, Bogoliubov excitations appear and deplete the condensate; we characterize this instability by considering the population of the first Bogoliubov mode, and show that it does not prevent, for small enough interaction strengths, the observation of the transition. However, the predicted subdiffusive behavior is not observed in the stable region. For higher interaction strengths, the condensate may be strongly depleted before this dynamical regimes set in.

Strong mechanical squeezing in a standard optomechanical system by pump modulation

Being beneficial for the amplitude modulation of the pump laser, we propose a simple yet surprisingly effective mechanical squeezing scheme in a standard optomechanical system. By merely introducing a specific kind of periodic modulation into the single-tone driving field to cool down the mechanical Bogoliubov mode, the far beyond 3-dB strong mechanical squeezing can be engineered without requiring any additional techniques. Specifically, we find that the amount of squeezing is not simply dependent on the order of magnitude of the effective optomechanical coupling but strongly on the ratio of sideband strengths for it. To maximize the mechanical squeezing, we numerically and analytically optimize this ratio in the steady-state regime, respectively. The mechanical squeezing engineered in our scheme also has strong robustness and can survive at a high bath temperature. Compared with previous schemes based on the two-tone pump technique, our scheme involves fewer external control laser source and can be extended to other quantum systems to achieve strong squeezing effect.

FACt: FORTRAN toolbox for calculating fluctuations in atomic condensates

We develop a FORTRAN code to compute fluctuations in atomic condensates (FACt) by solving the Bogoliubov–de Gennes (BdG) equations for two component Bose–Einstein condensate (TBEC) in quasi-two dimensions. The BdG equations are recast as matrix equations and solved self consistently. The code is suitable for handling quantum fluctuations as well as thermal fluctuations at temperatures below the critical point of Bose–Einstein condensation. The code is versatile, and the ground state density profile and low energy excitation modes obtained from the code can be easily adapted to compute different properties of TBECs — ground state energy, overlap integral, quasi particle amplitudes of BdG spectrum, dispersion relation and structure factor and other related experimental observables. Program summaryProgram Title: FACtProgram Files doi:http://dx.doi.org/10.17632/5h348ndydg.1Licensing provisions: MITProgramming language: FORTRAN 90External routines/libraries: ARPACKNature of problem: Compute the ground state density profile, ground state energy and chemical potential for individual species, evaluate the quasiparticle mode energies and corresponding amplitudes which can capture the transformation of the modes against the change of the parameters (intraspecies interaction, interspecies interaction, anisotropy parameter etc.) using Hartree–Fock–Bogoliubov theory with the Popov approximation. Calculate the overlap integral, dispersion relation and structure factor.Solution method: In the first step, the pair of coupled Gross–Pitaevskii equations (CGPEs) is solved using split time-step Fourier pseudospectral method to compute the condensate density. To solve the BdG equations, as a basic input the first Nb harmonic oscillator eigenstates are chosen as a basis to generate the BdG 011 matrix with dimension of 4(Nb+1)×4(Nb+1). Since the matrix size rapidly increases with Nb, Arpack routines are used to diagonalize the BdG matrix efficiently. To compute the fluctuation and non-condensate density, a set of the low energy quasiparticle amplitudes above a threshold value of the Bose factor are considered. The equations are then solved iteratively till the condensate, and non-condensate densities converge to predefined accuracies. To accelerate the convergence we use the method of successive under-relaxation (SUR).Additional comments including restrictions and unusual features: For a large system size, if the harmonic oscillator basis size is also taken to be large, the dimension of the BdG matrix becomes huge. It may take several days to compute the low energy modes at finite temperature and this package may be computationally expensive.After successful computation of this package, one should obtain the equilibrium density profiles for TBEC, low energy Bogoliubov modes and the corresponding quasiparticle amplitudes. In addition, one can calculate the dispersion relation, structure factor, overlap integral, correlation function, etc. using this package with minimal modifications. In the theory section of the manuscript, we have provided the expressions to compute the above quantities numerically.

Ground-state and Excitation Spectra of Bose–Fermi Mixtures in a Three-Dimensional Optical Lattice

Motivated by recent developments in the experimental study of Bose-Fermi mixtures, we investigate ground-state phase diagrams and excitation spectra for Bose-Fermi mixtures in a three-dimensional (3D) optical lattice. The Gutzwiller approximation is used to identify a new phase in which both superfluid bosons and metal fermions coexist. As a useful probe to identify the quantum phases, we also calculate the excitation spectra. In Mott insulator phase, two excitation features appear in the spectra that correspond to particle and hole excitations. In superfluid phase, there are Bogoliubov modes and amplitude modes. In coexisting phase, two gapless dispersive modes are identified, which shift due to interaction between bosons and fermions.

Control of electromagnetically induced transparency and Fano resonances in a hybrid optomechanical system

We study the control of electromagnetically induced transparency and Fano resonances in a hybrid optomechanical system based on Bose-Einstein condensate trapped inside a high finesse Fabry Perot cavity coupled with two charged nanomechanical resonators. The system is driven by a pump field and probed by a probe field. The nanomechanical resonators are connected to an external bias voltage V1(−V2) and generated electrostatic Coulomb coupling gc. In this study, we show that the electromagnetically induced transparency can be controlled by the s-wave scattering frequency ωsw as well as by the coupling frequencies of the system. Further, studying the effect of Bogoliubov mode, the transparency window splits from double to triple windows. At strong atom-atom interaction, we explain the asymmetric windows’ and explain their physical understanding.

Coherent population trapping based atomic reservoir for almost perfect higher-order squeezing.

Due to either coherent or dissipative interactions with the coherent population trapping (CPT)-based atoms, the evolutions of the Bogoliubov modes towards the vacuum states have been shown to lead to second-order squeezing of the involved optical fields. Here we push the CPT-based dissipative interactions towards higher-order squeezing, which is not simply determined by second-order squeezing but instead by different criteria involving higher-order moments. It is shown that the CPT-based atomic reservoir supports the dissipative evolution of the Bogoliubov modes almost completely to the vacuum states and then yields almost perfect fourth-order squeezing (90%∼100%). The present mechanism is robust against spontaneous emission since the atoms stay largely in the ground states. As a by-product, a comparison is given with two-level atoms, in which the excitation of a large fraction reduces the degree of higher-order squeezing.

A Renormalization-Group Study of Interacting Bose–Einstein Condensates: Absence of the Bogoliubov Mode below Four (T > 0) and Three (T = 0) Dimensions

We derive exact renormalization-group equations for the $n$-point vertices ($n=0,1,2,\cdots$) of interacting single-component Bose-Einstein condensates based on the vertex expansion of the effective action. They have a notable feature of automatically satisfying Goldstone's theorem (I), which yields the Hugenholtz-Pines relation $\Sigma(0)-\mu=\Delta(0)$ as the lowest-order identity. Using them, it is found that the anomalous self-energy $\Delta(0)$ vanishes below $d_{\rm c}=4$ ($d_{\rm c}=3$) dimensions at finite temperatures (zero temperature), contrary to the Bogoliubov theory predicting a finite "sound-wave" velocity $v_{\rm s}\propto [\Delta(0)]^{1/2}>0$. It is also argued that the one-particle density matrix $\rho({\bf r})\equiv\langle\hat\psi^\dagger({\bf r}_1)\hat\psi({\bf r}_1+{\bf r})\rangle$ for $d<d_{\rm c}$ dimensions approaches the off-diagonal-long-range-order value $N_{\bf 0}/V$ asymptotically as $r^{-d+2-\eta}$ with an exponent $\eta>0$. The anomalous dimension $\eta$ at finite temperatures is predicted to behave for $d=4-\epsilon$ dimensions ($0<\epsilon\ll 1$) as $\eta\propto\epsilon^2$. Thus, the interacting Bose-Einstein condensates are subject to long-range fluctuations similar to those at the second-order transition point, and their excitations in the one-particle channel are distinct from the Nambu-Goldstone mode with a sound-wave dispersion in the two-particle channel.

Strong quadrature squeezing and quantum amplification in a coupled Bose–Einstein condensate—optomechanical cavity based on parametric modulation

We consider an optomechanical cavity with a movable end-mirror as a quantum mechanical oscillator (MO) containing an interacting cigar-shaped Bose–Eisenstein condensate (BEC). It is assumed that both the MO and the BEC interact with the radiation pressure of the cavity field in the red-detuned and weak coupling regimes while the two-body atomic collisions frequency of the BEC and the mechanical spring coefficient of the MO are coherently modulated. By analyzing the scattering matrix, we show that in the largely different cooperativities regime together with strong modulations, the mechanical mode of the MO and the Bogoliubov mode of the BEC exhibit quadrature squeezing which can surpass the so-called 3dB limit (up to 75 dB) with high robustness to the thermal noises. Surprisingly, in this regime by controlling the system and modulation parameters, a very high degree of squeezing (up to 16 dB) together with high purity of quantum state for the output cavity field is achievable. Furthermore, one can attain simultaneous strong quantum amplification, added-noise suppression, and controllable gain-bandwidth for the complementary quadratures of squeezed ones in the subsystems.

Parametric heating in a 2D periodically-driven bosonic system: Beyond the weakly-interacting regime.

We experimentally investigate the effects of parametric instabilities on the short-time heating process of periodically-driven bosons in 2D optical lattices with a continuous transverse (tube) degree of freedom. We analyze three types of periodic drives: (i) linear along the x-lattice direction only, (ii) linear along the lattice diagonal, and (iii) circular in the lattice plane. In all cases, we demonstrate that the BEC decay is dominated by the emergence of unstable Bogoliubov modes, rather than scattering in higher Floquet bands, in agreement with recent theoretical predictions. The observed BEC depletion rates are much higher when shaking both along x and y directions, as opposed to only x or only y. We also report an explosion of the decay rates at large drive amplitudes, and suggest a phenomenological description beyond Bogoliubov theory. In this strongly-coupled regime, circular drives heat faster than diagonal drives, which illustrates the non-trivial dependence of the heating on the choice of drive.

Atomic quadrature squeezing and quantum state transfer in a hybrid atom–optomechanical cavity with two Duffing mechanical oscillators

In this paper, we investigate theoretically the quantum state transfer in a laser driven hybrid optomechanical cavity with two Duffing-like anharmonic movable end mirrors containing an ensemble of identical two-level trapped atoms. The quantum state transfer from the Bogoliubov modes of the two anharmonic oscillators to the atomic mode results in the atomic quadrature squeezing beyond the standard quantum limit of 3 dB which can be controlled by both the optomechanical and atom-field coupling strengths. Interestingly, the generated atomic squeezing can be made robust against the noise sources by means of the Duffing anharmonicity. Moreover, the results reveal that the presence of the Duffing anharmonicity provides the possibility of transferring strongly squeezed states between the two mechanical oscillators in a short operating time and with a high fidelity.

Parametric resonance of a Bose-Einstein condensate in a ring trap with periodically driven interactions

We study the instability of a ring Bose-Einstein condensate under a periodic modulation of inter-atomic interactions. The condensate exhibits temporal and spatial patterns induced by the parametric resonance, which can be characterized by Bogoliubov quasi-particle excitations in the Floquet basis. As the ring geometry significantly limits the number of excitable Bogoliubov modes, we are able to capture the non-linear dynamics of the condensate using a three-mode model. We further demonstrate the robustness of the temporal and spatial patterns against disorder, which are attributed to the mode-locking mechanism under the ring geometry. Our results can be observed in cold atomic systems and are also relevant to physical systems described by the non-linear Schrodinger's equation.

Momentum distribution and coherence of a weakly interacting Bose gas after a quench

We consider a weakly interacting uniform atomic Bose gas with a time-dependent nonlinear coupling constant. By developing a suitable Bogoliubov treatment we investigate the time evolution of several observables, including the momentum distribution, the degree of coherence in the system, and their dependence on dimensionality and temperature. We rigorously prove that the low-momentum Bogoliubov modes remain frozen during the whole evolution, while the high-momentum ones adiabatically follow the change in time of the interaction strength. At intermediate momenta we point out the occurrence of oscillations, which are analogous to Sakharov oscillations. We identify two wide classes of time-dependent behaviors of the coupling for which an exact solution of the problem can be found, allowing for an analytic computation of all the relevant observables. A special emphasis is put on the study of the coherence property of the system in one spatial dimension. We show that the system exhibits a smooth "light-cone effect," with typically no prethermalization.