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arXiv:2312.11910 · Physical Review A

Quantum echo in two-component Bose-Einstein condensates

Chang-Yan Wang

Published Jun 24, 2024• 6 citations
arXivHTMLPDF

Abstract

The development of ultracold atom technology has enabled precise investigations on the quantum dynamics of quantum gases. Recently, inspired by experimental advancement, the $\text{SU}(1,1)$ echo, akin to the well-known $\text{SU}(2)$ spin echo, was proposed for the single-component Bose-Einstein condensate (BEC). In this paper, we investigate the possibility of quantum echo in the more intricate two-component BEC by fully exploiting its underlying symmetry, which is the Lie group $\text{Sp}(4,R)$. We demonstrate that quantum echo can occur for the two-component BEC by applying a driving protocol consisting of two steps in each period. The first step can be any Bogoliubov Hamiltonian, while the second step is a Hamiltonian with interactions turned off, which plays a similar role as the $\ensuremath{\pi}$ pulse in spin echo. We confirm our theoretical results with numerical calculations for different examples of the two-component BEC. We further consider the effect of interactions between the excited boson modes on the quantum echo process and discuss the possible experimental implementation of this quantum echo.

DOI
10.1103/physreva.109.063327

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The development of ultracold atom technology has enabled precise investigations on the quantum dynamics of quantum gases. Recently, inspired by experimental advancement, the SU(1,1)\text{SU}(1,1)SU(1,1) echo, akin to the well-known SU(2)\text{SU}(2)SU(2) spin echo, was proposed for the single-component Bose-Einstein condensate (BEC). In this paper, we investigate the possibility of quantum echo in the more intricate two-component BEC by fully exploiting its underlying symmetry, which is the Lie group Sp(4,R)\text{Sp}(4,R)Sp(4,R). We demonstrate that quantum echo can occur for the two-component BEC by applying a driving protocol consisting of two steps in each period. The first step can be any Bogoliubov Hamiltonian, while the second step is a Hamiltonian with interactions turned off, which plays a similar role as the \ensuremathπ\ensuremath{\pi}\ensuremathπ pulse in spin echo. We confirm our theoretical results with numerical calculations for different examples of the two-component BEC. We further consider the effect of interactions between the excited boson modes on the quantum echo process and discuss the possible experimental implementation of this quantum echo.
Bose-Einstein condensation is a unique phase transition in that it is not driven by interparticle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question, ‘ How is this ideal Bose gas condensation modified in the presence of interactions between the particles?’ This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers. Introduction Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally. The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.
Implications of the internal symmetries on the dynamics of the trapped two-component atomic vapors are discussed. In the cases of 87^{87}87Rb (bosons) as well as of 40^{40}40K (fermions) trapped in the two hyperfine states, the intrinsic su(2)su(2)su(2) symmetry can be realized with a very good precision. Such a symmetry protects the global operators, which are the generators of the symmetry, from any decoherence. The case of boson-fermion mixture is discussed as well. The role of external factors, breaking the symmetry, in inducing the decoherence of the global operators is considered. It is shown that, the loss of the correlations is not faster than the rates of the induced heating or losses, provided the noise is characterized by short correlation length. The case of the extrinsic long-ranged fluctuations is also considered. Intrinsic mechanisms of decoherence of the correlators of the condensate operators of the two-component condensate are analyzed. The atomic echo is discussed as a test for the reversibility of the phase diffusion effect. The intensity and the profile of the multiple echo are calculated numerically as well as analytically.
We theoretically study the properties of Nambu-Goldstone bosons in an interacting single-component Bose-Einstein condensate (BEC). We first point out that the proofs of Goldstone's theorem by Goldstone, et al. [Phys. Rev. {\bf 127} (1962) 965] may be relevant to distinct massless modes of the BEC: whereas the first proof deals with the poles of the single-particle Green's function G^\hat{G}G^, the second one concerns those of the two-particle Green's function. Thus, there may be multiple Nambu-Goldstone bosons even in the single-component BEC with broken U(1) symmetry. The second mode turns out to have an infinite lifetime in the long-wavelength limit in agreement with the conventional viewpoint. In contrast, the first mode from G^\hat{G}G^, i.e., the Bogoliubov mode in the weak-coupling regime, is shown to be a "bubbling" mode fluctuating temporally out of and back into the condensate. The substantial lifetime originates from an "improper" structure of the self-energy inherent in the BEC, which has been overlooked so far and will be elucidated here, and removes various infrared divergences pointed out previously.
Periodic waves in a two-component Bose-Einstein condensate are considered with allowance for mixing of components admitted by the interaction between atoms. The possibility of relative motion of the components leads to significant variability of the system dynamics as compared to the case of a single-component condensate. In particular, it was experimentally established that, if one component flows relative to another, waves with counterphase oscillations of the components can be excited in the two-component condensate. A theory of one-phase periodic waves is developed for the case of weak mixing of the components, in which the condensate dynamics is described with sufficiently good precision by a system of Gross-Pitaevskii equations with equal constants of nonlinear interaction between atoms both of the same and different kinds. Both condensates with repulsive and attractive interactions between atoms are considered. The limiting case of waves with constant amplitude is studied in detail. The structure of soliton analogs in these systems is considered. The possibility of applying the proposed theory to description of experimentally observed wavy periodic structures in two-component Bose-Einstein condensate is discussed.
We study Bose-Einstein condensation (BEC) in three-dimensional two-component bosonic gases, characterizing the universal behaviors of the critical modes arising at the BEC transitions. For this purpose, we use field-theoretical (FT) renormalization-group (RG) methods and perform mean-field and numerical calculations. The FT RG analysis is based on the Landau-Ginzburg-Wilson ${\mathrm{\ensuremath{\Phi}}}^{4}$ theory with two complex scalar fields which has the same symmetry as the bosonic system. In particular, for identical bosons with exchange Z2{\mathbb{Z}}_{2}Z2​ symmetry, coupled by effective density-density interactions, the global symmetry is ${\mathbb{Z}}_{2,e}\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\bigotimes}\mathrm{U}(1)$. At the BEC transition, it may break into ${\mathbb{Z}}_{2,e}\ensuremath{\bigotimes}{\mathbb{Z}}_{2}\ensuremath{\bigotimes}{\mathbb{Z}}_{2}$ when both components condense simultaneously, or to $\mathrm{U}(1)\ensuremath{\bigotimes}{\mathbb{Z}}_{2}$ when only one component condenses. This implies different universality classes for the corresponding critical behaviors. Numerical simulations of the two-component Bose-Hubbard model in the hard-core limit support the RG prediction: when both components condense simultaneously, the critical behavior is controlled by a decoupled XY fixed point, with unusual slowly decaying scaling corrections arising from the onsite interspecies interaction.
We study the spin squeezing in a spin-1/2 Bose-Einstein condensates (BEC) with Raman-induced spin-orbit coupling (SOC). Under the condition of two-photon resonance and weak Raman coupling strength, the system possesses two degenerate ground states, using which we construct an effective two-mode model. The Hamiltonian of the two-mode model takes the form of the one-axis-twisting Hamiltonian, which is known to generate spin squeezing. More importantly, we show that the SOC provides a convenient control knob to adjust the spin nonlinearity responsible for spin squeezing. Specifically, the spin nonlinearity strength can be tuned to be comparable to the two-body density-density interaction, and hence is much larger than the intrinsic spin-dependent interaction strength in conventional two-component BEC systems such as 87Rb^{87}\mathrm{Rb}87Rb and 23Na^{23}\mathrm{Na}23Na in the absence of the SOC. We confirm the spin squeezing by carrying out a fully beyond-mean-field numerical calculation using the truncated Wigner method. Additionally, the experimental implementation is also discussed.