Floquet spinor Bose gases

Abstract

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 nontrivial metastable region, which supports unusual elementary excitations and hysteresis phenomena.Introduction.-Quantum degenerate systems with multiple order parameters emerge in diverse fields of physics such as unconventional superconductors [1, 2] and superfluid 3 He [3,4] in condensed matter, p-wave superfluids in neutron stars [5,6], and color superconductors in quark matter [7,8]. Due to presence of nontrivial orderparameter manifolds, such systems are known to exhibit a variety of phase structures, low-energy excitations, and topological defects absent in single order-parameter systems including conventional s-wave superconductors.Currently, spinor Bose-Einstein condensates (BECs) realized in ultracold atomic gases offer testing grounds for examining fundamental properties of multiple orderparameter systems [9][10][11][12][13]. In fact, cold-atom experiments have successfully observed rich phase structures [14][15][16], exotic topological excitations such as solitons [17][18][19][20][21], skyrmions [22][23][24], knots [25,26], and vortices [27][28][29], and universal non-equilibrium dynamics [30][31][32].One of the strengths in ultracold atomic gases is high controllability of experimental parameters, e.g. atomphoton interactions [33]. Of particular interest using this controllability is Floquet engineering [34][35][36][37], where a periodically oscillating field is applied to a system and thereby generates unconventional states absent in equilibrium [38,39]. In ultracold atomic gases, Floquet engineering has successfully been implemented [40][41][42][43][44], and one of the remarkable realizations is an artificial gauge field [45][46][47][48]. Despite the surge of great interest in Floquet engineering, such a technology in BECs has mainly been limited to engineering of kinetic energy terms.In this Letter, we introduce a Floquet spinor BEC induced by a high-frequency modulation of an external field (see Fig. 1), and uncover emergence of an unconventional spin-exchange interaction. As a possible external field, we consider a microwave, which is known to cause an effective quadratic Zeeman shift in spinor systems [15,16,[49][50][51][52]. By employing high-frequency expansion in the Floquet formalism [46,53,54], we obtain an effective Hamiltonian having the unconventional interaction, which is in sharp contrast to the case of an artificial gauge field (note however Ref. [55]). Applying the the-

Dynamic renormalization group theory for open Floquet systems

“…By this, we refer to the following: Common static renormalization methods account for the rapid periodic drive through a renormalization of the static, time-independent description, such as the Floquet Hamiltonian (see e.g. [50,[118][119][120][121]). These powerful methods are able to capture fundamental phenomena, like the stabilization of the Kapitza pendulum [122,123].…”

“…Although we work at large drive frequencies, we go beyond the common approaches based on the Floquet Hamiltonian and/or variants of the Magnus expansion (see e.g. [50,[118][119][120][121]) in two distinct ways: First, we include loop integrals (as opposed to tree-level processes) to account for fluctuations, and second, we include the renormalization of the dynamic sector, i.e. higher Fourier modes n = 0, without integrating them out.…”

“…( 49) to compute the RG flow of the drive coefficients [Eq. (50)] and show how these flow equations can be simplified in the presence of a monochromatic drive. This ultimately leads to Eq.…”

“…They have recently gained a lot of interest, because of their fundamentally new properties. or example, periodic drives can be used to generate artificial gauge fields for ultracold atoms [22][23][24][25][26], time crystals in atomic [27], ionic [28] or dissipative [29][30][31][32][33] systems, novel topological states without equilibrium counterparts [26,[34][35][36][37][38][39][40][41][42][43][44][45], driven analogs of many-body localization [46][47][48][49] or Floquet spinor Bose gases [50][51][52].…”

Solvable Model of a Generic Driven Mixture of Trapped Bose–Einstein Condensates and Properties of a Many-Boson Floquet State at the Limit of an Infinite Number of Particles

“…A particularly exciting arena for quantum dynamics across many disciplines is that in which the system’s Hamiltonian is periodic in time [ 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 ]. The time-periodic Hamiltonian defines the Floquet Hamiltonian and governs the Floquet eigenvalue equation in an extended Hilbert space [ 11 , 12 ], in analogy to the static Hamiltonian, which governs the Schrödinger eigenvalue equation in the standard Hilbert space of a time-independent system.…”