Many-body effects on atom−molecule conversion in a Floquet spin-boson model

Abstract

We investigate the quantum many-body dynamics of ultracold atom−molecule conversion using a Floquet spin-boson model, where the periodic energy detuning between molecules and atomic pairs is utilized to explore various dynamical regimes. We find that the upper bound of the adiabatic driving frequency increases continuously with the strength of molecule−molecule interactions, indicating that many-body interactions are beneficial in meeting the requirement of the adiabatic condition, thereby facilitating the realization of adiabatic atom−molecule conversion. This enhancement of the fulfillment of the adiabatic condition is further evidenced by the stabilization of periodic oscillations in the mean molecule number over time, protected by these interactions, even when the frequency lies within the localized regime. Interestingly, in the diffusive regime, while the many-body interaction has little effect on the dynamical equilibrium of atom−molecule conversion, it significantly expands the diffusive regime. In the high-frequency limit, many-body interactions are found to completely suppress atom−molecule conversion. Our results shed light on how molecule−molecule interactions influence the boundaries between different dynamical regimes.

Graphical abstract

Keywords

many-body effects / atom−molecule conversion / spin-boson model

1 Introduction

The ultracold atom−molecule conversion has received significant attention in various fields of physics, such as condensed matter and atomic−molecular physics [1-6]. Ultracold molecules, including both diatomic and polyatomic species, exhibit rich physics, such as ultracold chemical reactions [7], molecular decoherence [8], and quantum computing [9], offering insights into fundamental physics and potential applications in quantum technology. Various methods have been developed for producing ultracold molecules, both directly and indirectly [10-15]. Among these, a widely used technique involves converting ultracold atoms into ultracold molecules by adiabatically sweeping a magnetic field in the vicinity of a Feshbach resonance [16-19]. Notably, the formation of ultracold molecules from Fermi atoms via Feshbach resonance has attracted extensive investigations both experimentally and theoretically. The ability to create and manipulate ultracold fermionic molecules also opens new avenues for quantum precision measurement [20] and quantum information processing [21], facilitating studies in controlled dynamics of many-body systems [22-25].

A paradigm model that effectively describes the dynamics of ultracold atom−molecule conversion is the spin-boson model [26, 27], which has been widely exploited in investigations of fundamental problems such as quantum simulation [28, 29], quantum phase transitions [30-33], and quantum chaos [34-37]. In particular, a high-dimensional generalization of the spin-boson model has several equivalent representations [38], including the conversion of atomic pairs into molecules, the binding of spinful fermion pairs into bosonic molecules, and the time-dependent Dicke model in quantum optics. Recently, the dynamics in an ultracold atom−molecule conversion system described by the spin-boson model has received considerable attention. It has been found that four dynamically distinct regimes, namely, adiabatic, localized, diffusive, and high-frequency, exist for atom−molecule conversion, depending on the driving frequencies in a periodically driven spin-boson model [39].

It has been found, both experimentally and theoretically, that many-body interactions and periodic driving have significant implications in atom molecule conversion. Our previous investigation uncovered that the adiabatic conditions can be modified within a modified spin-boson model by involving molecule−molecule interactions [40]. In this context, we explore the interplay of molecule-molecule interactions and periodic driving on conversion dynamics. In this work, we find that the minimum value of the adiabatic frequency, which is closely linked to both the particle number and the driving amplitude, increases contineously with the strength of the molecule−molecule interaction. This indicates that the adiabatic regime can be expanded by adjusting the many-body interactions. We numerically investigate the many-body effects on atom−molecule conversion, quantified by both the mean molecule number and its variance, across four distinct regimes: adiabatic, localized, diffusive, and high-frequency. In the localized regime, periodic oscillations of the mean molecule number with time appear, especially with strong interactions, similar to the adiabatic regime. The corresponding variance fluctuates around a saturated value with time, which decreases as the interaction strength increases. This behavior indicates that many-body interactions protect the adiabatic atom−molecule conversion. In the diffusive regime, classical chaos causes both the mean molecule number and its variance to saturate, unaffected by many-body interactions. In the high-frequency region, both the mean molecule number and its variance saturate rapdily with time evolution under the effects of many-body interaction. And their saturated value decreases with the increase of the frequency.

The paper is organized as follows. In Section 2 we introduce the periodically driven spin-boson model, involving the molecule−molecule interaction. In Section 3 we investigate the effects of many-body interaction on ultracold atom−molecule conversion. The summary and discussion are presented in Section 4.

2 Model

The Hamiltonian of the spin-boson model reads (\hslash =1)

(1)\begin{array}{rl}\hat{H}=& \frac{\xi (t)}{2}{\hat{b}}^{†}\hat{b}-\frac{\xi (t)}{4}\sum _{i=1}^N({\hat{a}}_{i\uparrow }^{†}{\hat{a}}_{i\uparrow }+{\hat{a}}_{i\downarrow }^{†}{\hat{a}}_{i\downarrow })\\ & +\frac{\upsilon }{\sqrt{N}}\sum _{i=1}^N(\hat{b}{\hat{a}}_{i\uparrow }^{†}{\hat{a}}_{i\downarrow }^{†}+{\hat{b}}^{†}{\hat{a}}_{i\downarrow }{\hat{a}}_{i\uparrow })+\frac{c}{N}{\hat{b}}^{†}{\hat{b}}^{†}\hat{b}\hat{b},\end{array}

where {\hat{a}}_{i\uparrow }^{†}({\hat{a}}_{i\uparrow }) and {\hat{a}}_{i\downarrow }^{†}({\hat{a}}_{i\downarrow }) are creation (annihilation) operators of a fermionic atom, {\hat{b}}^{†}(\hat{b}) is creation (annihilation) operator of a bosonic molecule [40]. The parameter c indicates the strength of molecule−molecule interaction. The parameter \upsilon measures the Feshbach coupling strength between atoms and molecules, and N is the total number of particles. Without loss of generality, we assume that the time-dependent detuning takes the form \xi (t)={\xi }_0\cos (\omega t), with both amplitude {\xi }_0 and frequency \omega being scaled by \upsilon, i.e., \frac{{\xi }_0}{\upsilon }\to {\xi }_0 and \frac{\omega }{\upsilon }\to \omega [39]. Other parameters in Eq. (1) are also scaled by \upsilon, hence \upsilon = 1. By defining the collective pseudo-spin operators, {\hat{S}}_{+}={\sum }_{i=1}^N {\hat{a}}_{i\uparrow }^{†}{\hat{a}}_{i\downarrow }^{†}, {\hat{S}}_{-}={\sum }_{i=1}^N{\hat{a}}_{i\downarrow }{\hat{a}}_{i\uparrow }, and {\hat{S}}_z=\frac{1}{2} {\sum }_{i=1}^N( {\hat{a}}_{i\uparrow }^{†}{\hat{a}}_{i\uparrow }+{\hat{a}}_{i\downarrow }^{†} {\hat{a}}_{i\downarrow }-1), we obtain the Hamiltonian

(2)\widehat{H^{{}^{\prime }}}= [\xi (t)- {\displaystyle \frac{c}{N}}]{\hat{b}}^{†} \hat{b}+{\displaystyle \frac{c}{N}}{\hat{b}}^{†}\hat{b}{\hat{b}}^{†} \hat{b}+{\displaystyle \frac{1}{\sqrt{N}}}({\hat{b}}^{†}{\hat{S}}_{-}+\hat{b}{\hat{S}}_{+}) ,

where we have used the conserved relation of the particles, {\hat{a}}_{i\uparrow }^{†} {\hat{a}}_{i\uparrow }+ {\hat{a}}_{i\downarrow }^{†}{\hat{a}}_{i\downarrow } +2{\hat{b}}^{†} \hat{b}=2 N [40-42].

3 Results and analysis

It is known that there are four dynamical regimes, i.e., adiabatic, localized, diffusive, and high-frequency regimes [39]. A fundamental problem is how the repulsive molecule−molecule interaction (i.e., c>0) affect the borders of these four regimes. The adiabatic condition is expressed as \omega \ll {\omega }_{ad}=min(\frac{{\mathrm{\Delta }}_{01}^2}{|⟨{\psi }_1(\mu )|\frac{\mathrm{\partial }{\hat{H}}^{{}^{\prime }}}{\mathrm{\partial }\mu }|{\psi }_0(\mu )⟩|}), where the symbol \text{min} (\cdots ) indicates the operation of taking the minimum value, {\mathrm{\Delta }}_{01}=E_1(\mu )-E_0(\mu ) denotes the gap between the ground state energy E_0(\mu ) and the excited state energy E_1(\mu ), | {\psi }_0(\mu )⟩ and |{\psi }_1(\mu )⟩ are the corresponding ground state and excited state, and \mu =\omega t is the adiabatic phase. For c=0, the frequency {\omega }_{ad} =1/(N{\xi }_0) provides the border between the adiabatic and localized regime [39]. We numerically investigate the {\omega }_{ad} for different c. Fig.1 shows that {\omega }_{ad} monotonically increases with the rise of c, and the fitting function of our numerical results is in the form {\omega }_{ad}\approx \alpha (1-{\mathrm{e}}^{-c/\beta })+\gamma, where \alpha, \beta, and \gamma are positive constantly and the values are determined by N and {\xi }_0. Therefore, the many-body interactions are beneficial to meet the requirement of adiabatic condition and thus facilitate the realization of adiabatic atom−molecule transformation. More importantly, the values of the adiabatic frequency {\omega }_{ad} is increased by an order of magnitude (e.g., from 0.01 to 0.1) because of particle interactions.

For convenience, we define the dimensionless time as m=\omega t/(2\pi ). We further investigate the time evolution of the average particle number ⟨n⟩ =⟨{\mathrm{\Psi }}^{{}^{\prime }} |{\hat{b}}^{†}\hat{b}|\mathrm{\Psi }⟩/N for different c. The expansion of a quantum state |\mathrm{\Psi } ⟩ in the basis of Fock state takes the form |\mathrm{\Psi }⟩= \sum _{k=0}^N{\mathrm{\Psi }}_k|k⟩, where {\mathrm{\Psi }}_k is the probability amplitude for the Fock state |k⟩ and the normalization condition is \sum _k|{\mathrm{\Psi }}_k{|}^2=1. The state |k⟩ stands for the eigenstates of the particle number operator {\hat{b}}^{†}\hat{b}. In the adiabatic regime, we find that for interacting case (e.g., c=1), ⟨n⟩ varies periodically with m [see Fig.2(a), \omega =2\times {10}^{-3}], similar to the behavior observed for c=0 [39]. The value of the variance, i.e., \mathrm{\Delta } n^2=⟨n^2⟩ -⟨n{⟩}^2 [39] is close to 0 for different c [see Fig.2(d)], which owing to the adiabatic dynamics. In the localized regime, with an moderate value of \omega [e.g., \omega =2\times {10}^{-2} in Fig.2(b)], for c=0, the amplitude of the periodic oscillation in atom−molecule conversion gradually decreases over time. For c\ne 0, the periodical oscillation emerges especially for large c [e.g., c=3 in Fig.2(b)], the phenomenon exhibits similarities to those observed within the adiabatic regime, which demonstrates that many-body interaction among molecules assists the atom−molecule conversion. The \mathrm{\Delta }{n}^2 fluctuates significantly around a saturated value as time evolves, which decreases clearly with the increase of c [see Fig.2(e)]. This a signature of the protection of the adiabatic atom−molecule conversion by many-body interaction. In the diffusive regime with a large value of \omega [e.g., \omega = 2\times {10}^{-1} in Fig.2(c)], the ⟨n⟩ saturates rapidly to 0.5 over time, exhibiting some fluctuations regardless of whether c is zero or not, indicating the dynamical equilibrium of the atom−molecule conversion. The saturated value is in good agreement with the classical ergodic limit, i.e., \mathrm{\Delta } n_{cel}^2=1/12, for different c [see Fig.2(f)]. The wavepackets’ dynamics follows classically chaotic behavior, wherein the ergodicity of trajectories in phase space results in the saturation of the variance [39].

We use the dynamic of ⟨n⟩ and \mathrm{\Delta } n^2 above-mentioned for the measurement. The following we will intuitively observe the impact of c on them in the state space. We numerically calculate the distribution of final state and Husimi distributions, i.e., quasi-probability distribution functions in phase space. The quasi-probability reads P=| ⟨n, \phi |\mathrm{\Psi }⟩{|}^2 where the coherent spin state is | n,\phi ⟩ =\sum _{k=0}^N\sqrt{\frac{N!}{k!(N-k)!}}{(1-n)}^{\frac{N-k}{2}} n^{\frac{k}{2}}\times {\mathrm{e}}^{-\mathrm{i}(2k-N)(\phi +\pi )/2}|k⟩ with n being number of molecules and \phi the phase [39]. The quantum dynamics of the system is governed by the time-dependent Schrödinger equation \mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}t}|\mathrm{\Psi }⟩={\hat{H}}^{\prime }|\mathrm{\Psi }⟩. In the Fock basis, it can be rewritten as (N+1) ordinary differential equations (\hslash =1) as follows:

(3)\mathrm{i}{\displaystyle \frac{\mathrm{d}{\mathrm{\Psi }}_k}{\mathrm{d}t}}={\displaystyle \sum _k{H}_{k,k^{\prime }}{\mathrm{\Psi }}_{k^{\prime }},}

where the Hamiltonian matrix elements are H_{k,k^{\prime }}= [\frac{\xi (t)}{2}+\frac{c}{N}(k -1)]k{\delta }_{k,k^{\prime }}+\frac{1}{\sqrt{N}}(k^{\prime }+1)\sqrt{N-k^{\prime }}{\delta }_{k,k^{\prime }+1}+\frac{1}{\sqrt{N}}{k}^{\prime }\sqrt{N-k^{\prime }+1}{\delta }_{k,k^{\prime }-1}. In numerical simulations, we use the 4−5th Runge−Kutta step-adaptive algorithm to solve the Schrödinger equation (3).

We numerically calculate both |{\mathrm{\Psi }}_k{|}^2 and P for different c in the localized regime. Fig.3(a) and (c) show that both |{\mathrm{\Psi }}_k{|}^2 and P resemble the distribution in classically chaotic phase space for c=0. They all dramatically suppressed when c\ne 0 [see Fig.3(b) and (d)], which demonstrate that the interaction induces a transition from the localized regime to the adiabatic regime.

We further investigate the effects of many-body interaction on the atom−molecule conversion in the high-frequency region, i.e., \omega \gg 1. As shown in Fig.4(a), for \omega = 1.92, the average particle number ⟨n⟩ with c=0 exhibits a clear revival phenomenon, distinct from the periodic oscillations of ⟨n⟩ observed in the adiabatic regime. For interacting case (e.g., c=1), however, the time-dependent of ⟨n⟩ fluctuates around the saturated value, indicating the disappearance of the periodical atom−molecule conversion induced by many-body interactions. We find that for \omega = 1.92, the variance \mathrm{\Delta } n^2 rapidly saturates over time with large fluctuations when c=0, while these fluctuations are dramatically suppressed as c increases [see Fig.4(c)]. In this situation, the \mathrm{\Delta } n^2 fluctuates around a saturated value, i.e., \mathrm{\Delta } n^2\approx 1/12, which is the same as in the diffusive regime. Although we have observed the similar phenomena, the mechanisms underlying them are distinct. The phenomenon in the diffusive regime is induced by periodically driven resonance, whereas the phenomenon here arises from nonlinearity. When \omega increases further [e.g., \omega =4 in Fig.4(b)], we observe the emergence of atom−molecule conversion with no perfect periodical revival for c=0. The saturation level of ⟨n⟩ clearly decreases as c increases, demonstrating the suppression of atom−molecule conversion by the many-body interactions. The \mathrm{\Delta } n^2 rapidly saturates to the classical ergodic limit with some fluctuations when c=0, while it approaches almost zero for nonzero values of c [see Fig.4(d)]. We also calculated the evolution for higher frequencies, such as \omega =10 and \omega = 100. Our findings reveal that the temporal evolution behaviors of ⟨n⟩ and \mathrm{\Delta } n^2 are similar to the case with \omega = 1.92 and \omega =4.

We use the time-averaged value of the variance, \mathrm{\Delta } n_{sat}^2={\sum }_{j=1}^m\mathrm{\Delta } n^2(t_j)/m, to quantify the saturation level. We numerically investigate the dependence of \mathrm{\Delta }{n}_{sat}^2 on \omega for different values of c. Our results show that, for a specific c, the \mathrm{\Delta } n_{sat}^2 remains nearly zero when \omega is smaller than a critical value {\omega }_c. Beyond this point, it gradually increases and saturates at approximately \mathrm{\Delta }{n}_{sat}^2\approx 1/12 as \omega increases [See Fig.5]. Interestingly, the {\omega }_c increases with the increase of c, indicating that many-body interactions enhance the fulfillment of the adiabatic condition. In fact, we have calculated a wide range of modulation frequencies within four distinct regimes. However, because the \mathrm{\Delta }{n}_{sat}^2 ultimately converges to 1/ 12 for diffusive and high-frequency regime. Therefore, in Fig.5, we only show the boundary of the adiabatic and local regimes.

4 Conclusion

In this work, we numerically investigated the many-body effects on the ultracold atom−molecule conversion in the Floquet spin-boson model. We numerically find that the frequency {\omega }_{ad} monotonically increase with the increase of c, demonstrating the enlargement of the adiabatic regime due to many-body interactions. In the adiabatic regime, the periodic oscillation of ⟨n⟩ is almost unaffected by many-body interactions. However, in the localized regime, the interaction enhances the oscillation. Meanwhile, \mathrm{\Delta } n^2 fluctuates around a saturated value, decreasing as c increases. Therefore, many-body interactions can facilitate atom−molecule conversion. In the diffusive regime, the ⟨n⟩ stabilizes at 0.5 over time for different c, indicating dynamical equilibrium in atom−molecule conversion. The \mathrm{\Delta } n^2 fluctuates around 1 /12, reflecting the ergodicity of chaotic trajectories. In the high-frequency region, for moderate \omega, the time evolution of both ⟨n⟩ and \mathrm{\Delta }{n}^2 resembles that of the diffusive regime, suggesting that many-body interactions expand the diffusive regime. For sufficiently large \omega, the interaction causes both ⟨n⟩ and \mathrm{\Delta }{n}^2 saturated, with the saturation decreasing as c increases. For a specific c, the \mathrm{\Delta }{n}_{sat}^2 is nearly zero for and gradually saturates to approximately 1/12 as \omega exceeds {\omega }_c. Notably, the {\omega }_c increases with increasing c, highlighting the influence of many-body interactions on the boundary between the adiabatic and localized regimes.

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