Kinetic phase transition with global coupling in the resonantly driven atomic trap

The last few decades have witnessed an immense amount of research activity, both in experimental and theoretical work, focused on understanding the detailed dynamics of quantum [1] and classical systems [2–5] that are under time-dependent external perturbations. In general, such systems exhibit an interplay of three characteristic components: (i) nonlinearity, (ii) nonequilibrium behavior and (iii) quantum tunneling (for quantum systems) or large rare fluctuation (for classical systems). In particular, the large rare fluctuations provide a variety of phase transitions far from equilibrium in classical nonlinear dynamical systems [6–10].

The study of large rare fluctuations has mainly been focused on prominent qualitative changes of an equilibrium system, such as the nucleation at phase transitions, chemical reactions, mutations in DNA sequences and protein transport in biological cells [11–14]. On the other hand, the fluctuating systems of interest are far from thermal equilibrium; for example, lasers, pattern-forming systems, radio-frequency-driven Josephson junctions [15], micromechanical and nanomechanical oscillators [16–18] and periodically driven trapped electrons [19–22]. In particular, in the periodically driven nonlinear dynamic system out of equilibrium, which exhibits intrinsic bistable or multistable states, the fluctuation triggers noise-induced switching between stable states [5, 8, 9], where the fluctuation-induced switching dynamics shows characteristics similar to the equilibrium phase transition [5, 8].

It is well known that, in thermodynamic systems, the increase of fluctuations at the critical point is a universal characteristic of phase transition [23]. In particular, even in the single-particle-driven nonlinear systems [5], such as the Duffing oscillator, significant enhancement of fluctuations that arise from noise-induced switching between coexisting states was also observed in the measured spectral density of fluctuations (SDFs) near the specific driving frequency, which is indicative of a phase transition in nonequilibrium systems—the so-called kinetic phase transition (KPT) [8]. Nonetheless, in such a driven mechanical system, it is experimentally difficult to go beyond the single-particle behavior and investigate many-body effects associated with the inter-particle interactions between bistable states, although the coupled oscillator arrays have been explored theoretically [24–27]. In the parameterically driven cold atom trap [9, 10], on the other hand, the globally coupled atom–atom attractive interaction between attractors was shown to produce numerous qualitative changes in the nonequilibrium phase transition, while exhibiting the ideal mean-field transition. Therefore, it is essential to make an experimental study of KPT in a weakly interacting ensemble system [28], which still lacks despite many KPT works performed so far.

In this paper, we experimentally realize the globally coupled Duffing oscillators in an optically modulated magneto-optical trap of ⁸⁵Rb in the presence of long-range interatomic interactions associated with the shadow-induced light force [10] and advance a theory. This paper is organized as follows. In section 2, the experimental results are presented and compared to the theoretical results. In section 3, we demonstrate the theoretical approach of global coupling long-range interaction between dynamical bistable states consisting of many particles. Section 4 summarizes the results.

The resonantly driven nonlinear oscillator, Duffing oscillator, exhibits two coexisting dynamical states (CS) as well as the large-amplitude state (LS) and small-amplitude state (SS), which have differing vibrational amplitude and phase at a specific range of driving frequency (figure 1(a)). In the presence of noise, such as thermal fluctuations, there occurs noise-induced switching between CS, and the transition probability as well as the occupied population between them are determined by the activation energy and noise intensity, given by

$$\begin{equation} \mathcal{P}_{n}=\frac{W_{mn}}{W_{nm}+W_{mn}},\quad W_{nm}=C\exp(-R_n/D), \end{equation} \tag{ 1 }$$

where W_nm and $\mathcal {P}_{n}$ are the transition probabilities from the nth state to the mth state (n,m = 1,2,n ≠ m) and the occupied population in the nth state, respectively. Here $\mathcal {P}_{1(2)}$ is the normalized atomic populations of LS (SS), so that $\mathcal {P}_{1}+\mathcal {P}_{2}=1$, and C is a constant. The subscript 1 and 2 denote LS and SS, respectively. R_n stands for the activation energy of the nth state and D is the noise intensity. Notice that W_nm and $\mathcal {P}_{n}$ have been calculated previously for a single Duffing oscillator, which is expected to exhibit differing behavior for an interacting many-oscillator system.

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The many-body system of the Duffing oscillators is experimentally realized in the magneto-optical trap of ⁸⁵Rb atoms by modulating the intensity of trap lasers counterpropagating along the anti-Helmholtz coil axis with the phase difference between the two lasers maintained at π (figure 1(b)). We measure the atomic populations in each state to investigate the noise-induced switching between two attractors, and compare experimental results with simulation. Near the KPT point where the populations of two attractors are comparable, we also observe the enhanced intensity of the SDFs. Notice that, in our ensemble system, 10⁶–10⁷ atoms are distributed over dynamical states, and thus one can readily measure the steady-state populations occupied in each state without any statistical analysis. Figure 1(a) shows the atomic cloud (T ≈ 0.4 mK) in the LS, CS and SS within a specific frequency range. Notice that because the distance between the two attractors is not far enough, there exists a region where the two atomic clouds overlap, manifested by the long cloud shape.

Figure 1(c) shows the normalized atomic populations versus ω_F/2π near the KPT point at 35 Hz. As shown, the two attractors are equally occupied at the KPT point because the activation energies of the two attractors are equal. Here the important feature of a phase transition is to observe large fluctuations associated with transitions between the two states.

To confirm this feature, we have obtained the fluctuations near the KPT point. In our system consisting of many particles, it is hard to obtain the time trace of single-particle trajectory. Instead of analyzing the time correlation of single-particle trajectory [2, 5], therefore, in our ensemble system of many atoms, we obtain the intensity I of the SDF on the distance from KPT point, which represents the integrated spectral density, given by [8]

$$\begin{equation} I \equiv\int^\infty_{-\infty}{\rm d}\omega Q_{\rm tr}(\omega)=\frac{(z^{\rm l}_{\rm max}-z^{\rm s}_{\rm max})^2}{4}\mathcal{P}_1\mathcal{P}_2, \end{equation} \tag{ 2 }$$

where z^l(s)_max is the maximum amplitude of LS (SS) and Q_tr(ω) is the fluctuational noise-induced spectral peak that arises due to noise-induced transition between two dynamical states, which $Q_{\mathrm { tr}}(\omega )\approx \frac {2\omega _{\mathrm { F}}|\delta \omega |(W_{12}+W_{21})}{3\pi B_0[(W_{12}+W_{21})^2+(\omega -\omega _{\mathrm { F}})^2]}(z_{\mathrm { max}}^{\mathrm {l}}-z_{\mathrm { max}}^{\mathrm { s}})^2\mathcal {P}_1\mathcal {P}_2$ where δω = ω_F − ω₀. Figure 1(d) clearly shows the maximum SDF intensity near the driving frequency 35 Hz, as expected.

The atom–atom interaction induces change of the activation energy, which is measurable because the cumulative effect of the atomic interactions between clouds becomes significant [10, 29]. Figure 2 presents the simulation results on the effects of atomic interactions on the activation energy and the associated atomic populations. To control the cumulative atomic interaction, one can change N_tot; the long-range attractive shadow force increases with N_tot because the atomic clouds in each state, while absorbing the incoming laser beams, attract each other due to the resulting imbalance of laser intensity.

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As shown in figure 2(a), as N_tot increases, the atom–atom interaction induces the change of R₁ and R₂; the activation energy of LS (SS) increases (decreases) with N_tot. Figure 2(b) shows that the change of R₁₍₂₎ then shifts the KPT point, where ΔR (= R₂ − R₁) = 0, toward the higher ω_F at the higher N_tot. In addition, at a fixed modulation frequency ω_F, ΔR decreases with the increase of N_tot. Figure 2(c) presents the resulting population change; $\mathcal {P}_1$ of LS increases with N_tot (solid line) where $\mathcal {P}_2$ of SS decreases (dot-dashed line), shifting the KPT point toward the higher ω_F.

We have experimentally observed the effects of atom–atom interactions on the single-particle behavior of KPT (i.e. KPT without interaction). As shown in figure 3(a), as N_tot increases, atoms in SS transfer to LS in a unidirectional way, so that the atom–atom interaction plays the role as the one-way bias field (notice that the effect of the interaction in the parametric resonance system is to induce the time-translational symmetry, breaking which results in the ideal mean-field transition [10]). To detail the characteristics of the interaction as a one-way bias field, we have measured the difference of atomic populations of each state, $\mathcal {P}_1-\mathcal {P}_2$. Figure 3(b) shows that the population difference grows larger at the higher N_tot. The experimental results are in qualitatively good agreement with the theoretical results (figure 3(c)).

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The characteristic interaction of the one-way bias field originates from the difference of the optimal path and equilibrium position of each stable state because the interatomic force depends on the position of all individual atoms. The coefficient α_nm in equation (17) is related to the interatomic force, where the terms with n = m and n ≠ m indicate the force from the atoms in the same and in the different cloud, respectively. In our system, we find α₁₁ ≠ α₂₂ and α₁₂ ≠ α₂₁, which means the interatomic force that an atom in the cloud m experiences by atoms in the same as well as in different clouds is not equal to that for an atom in the cloud n. In particular, notice that the effective switching activation energy R⁽¹⁾_n (equation (16)) linearly depends on the number of atoms in the clouds, and also numerical calculation shows that α₁₁ > α₂₂,α₁₂,α₂₁. For example, when α₁₁ > 0 while α₂₂, α₁₂ and α₂₁ are negative, we observe that the effective switching activation energy of LS (SS) becomes positive (negative), so that the total activation energy of LS (SS) increases (decreases) with the total number of atoms, as shown in figure 2(a).

Figure 4 presents the phase-transition map in the parameter space of driving frequency ω_F and modulation amplitude (∝F₀). In figure 4(a), the calculated KPT lines (black and green dashed lines) are shown, where the changes of the nonequilibrium first-order phase transition boundary are expected to depend on the total number of trapped atoms. The experimental results in figure 4(b) are in qualitatively good agreement (the differing absolute values of N_tot may be attributed to the fact that the experiment is three-dimensional whereas the theory is one-dimensional).

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Interestingly, a similar phenomenon is observed in the first-order gas–liquid transition in equilibrium states [6, 23]. The gas–liquid transition described by the van der Waals equation takes into account the nonzero radius of atoms as well as the attractive interactions between atoms. In the pressure–temperature (P–T) map, the phase transition can be plotted where there is a coexistence region and a critical point. The atomic radius and interaction depend on the species of atom, and thus the coexistence line in the P–T plane changes with respect to the atomic species. Therefore, the attractive interaction between cold Duffing-oscillator atoms in the coexisting periodic attractors out of equilibrium exhibits close similarity with the gas–liquid transition in equilibrium. These results provide another piece of evidence for the similarity of phase transitions between equilibrium and nonequilibrium systems.

3.1. Model

The vibrations in the periodically driven cold ⁸⁵Rb atom system are subject to the combined influence of a periodic field, a weak random force and a long-range attractive interaction. The equation of motion for this many-body system is derived from the atom–photon interaction theory for a (1 + 3) atomic energy structure model [30, 31], which is given by

$$\begin{eqnarray} &&\fl \ddot{z}_i+\gamma\dot{z}_i+\omega_{0}^{2}z_i+B_{0}(z_i+\eta \dot{z}_i)^3 =F_{0}\cos\omega_{\rm F} t+F_{\mathrm{sh}}+f(t)~~(i=1,\ldots,N_{\rm tot}), \end{eqnarray} \tag{ 3 }$$

where z_i and $\dot {z}_i$ are the position and velocity of the ith atom, and N_tot is the total number of trapped atoms. We ignore the term $\eta \dot {z}_i$ as $\eta \dot {z}_i\ll 1$. ω₀ is the natural frequency, γ is the damping coefficient, B₀ is the nonlinear coefficient and F₀ is the magnitude of the force. They are given by

$$\begin{equation} \omega_{0}^{2}=C_{\mathrm{f}}\omega_{t}^{2},~~\gamma=C_{\mathrm{d}}\frac{\hbar k}{\mu_{\rm B}b}\omega_{t}^{2}, \end{equation} \tag{ 4 }$$

$$\begin{equation} B_{0}=C_{\mathrm{n}}\frac{8\mu_{\rm B}^{2}b^2(4\delta^{2}/\Gamma^2-1)}{\hbar^{2}\Gamma^{2}(4\delta^{2}/\Gamma^2+1)^{2}}\omega_{t}^{2}, \end{equation} \tag{ 5 }$$

$$\begin{equation} F_{0}=C_{\mathrm{d}}\frac{\hbar k\Gamma s_{0}\epsilon}{m_{\rm a}(4\delta^{2}/\Gamma^2+1)}, \end{equation} \tag{ 6 }$$

where ω²_t = 8kμ_Bbs₀(− δ/Γ)/[m_a(1 + 4δ²/Γ²)²], and correction factors C_f, C_d and C_n, considering an atomic multilevel energy structure, are 0.54, 0.41 and 0.19, respectively. s₀ is the on-resonance saturation parameter. stands for the modulation amplitude of the laser intensity. k is the wave vector, μ_B is the Bohr magneton, m_a is the mass of an atom and Γ is the decay rate of the excited state (= 2π × 6.07 MHz). δ and b are the detuning of the laser frequency relative to the transition line F_g = 3 → F_e = 4 and the magnetic gradient field, respectively.

f(t) is the white noise, which originates from the atomic random motion due to spontaneous emission, and $\left \langle f(t)f(t')\right \rangle =2\gamma (k_{\mathrm { B}} T/m_{\mathrm { a}})\delta (t-t')$. k_B and T are the Boltzmann constant and the temperature of the atom. The long-range shadow force F_sh results from 'shielding' of atoms from the laser light by other atoms, which produces all-to-all attractive coupling between atoms in the two CS [10]

$$\begin{equation} F_{\mathrm{sh}}=-f_{\mathrm{sh}}\sum_{j=1}^{N_{\rm tot}}\mathrm{sgn}(z_i-z_j), \end{equation} \tag{ 7 }$$

where the interaction strength f_sh is a very small constant value (9.19 × 10⁻³² N kg⁻¹). The sgn(z) indicates the sign function and z_j represents the position of the jth atom of a stable state.

The nonlinear oscillator in the periodically perturbed magneto-optical trap consists of a number of atoms, and there is a long-range attractive interaction between atoms, unlike in the single nonlinear mechanical oscillator [32]. The atom–atom interaction modifies the activation energy R_n in equation (1), including the interaction effect, and thus the switching probability W_nm is changed. The influence of the variation of the switching probability owing to the atom–atom interaction remarkably displays the qualitative and quantitative discrepancies between a single particle and many particles in nonlinear dynamic systems, as reported in [10]. For understanding the interaction effect in the Duffing oscillator clearly, it is useful to transform equation (3) into the rotating frame, $(z_i,\dot {z}_i)\rightarrow (Q_i,P_i)$ [33].

3.2. Rotating-wave approximation

The atomic dynamics in equation (3) can be described by changing to the rotating frame using a standard transformation

$$\begin{eqnarray} &&\begin{array}{l} z_i=C_{\mathrm{RWA}}\left[P_i\cos(\omega_{\rm F} t)-Q_i\sin(\omega_{\rm F} t)\right],\\ \dot{z}_i=-\omega_{\rm F} C_{\mathrm{RWA}}\left[P_i\sin(\omega_{\rm F} t)+Q_i\cos(\omega_{\rm F} t)\right] \end{array} \end{eqnarray} \tag{ 8 }$$

with $C_{\mathrm {RWA}}=\left (2\omega _{\mathrm { F}}|\delta \omega |/3|B_0|\right )^{1/2}$ and δω = ω_F − ω₀. In the rotating-wave approximation (RWA) [10, 33], the equation of motions for slow variables q_i ≡ (Q_i,P_i) in slow time τ = |δω|t are

$$\begin{eqnarray} &&\frac{\mathrm{d}\mathbf{q}_i}{\mathrm{d}\tau}=\mathbf{K}_i(\mathbf{q}_i)+\mathbf{f'}(\tau),~\mathbf{K}_i(\mathbf{q}_i)=\mathbf{K}^{(0)}_i(\mathbf{q}_i)+\hat{\epsilon}\partial_{\mathbf{q}_i}H_{\mathrm{sh}} \end{eqnarray} \tag{ 9 }$$

and

$$\begin{equation} \mathbf{K}^{(0)}_i(\mathbf{q}_i)=-\frac{\gamma}{2|\delta\omega|}\mathbf{q}_i+\hat{\epsilon}\partial_{\mathbf{q}_i}H(\mathbf{q}_i), \end{equation} \tag{ 10 }$$

$$\begin{equation} H(\mathbf{q}_i)=\frac{1}{16}\left(Q_i^2+P_i^2\right)^2-\frac{1}{2}\left(Q_i^2+P_i^2\right)-\frac{F_0}{2C_{\mathrm{RWA}}\omega_{\rm F}|\delta\omega|}P_i, \end{equation} \tag{ 11 }$$

where i = 1,...,N_tot, ∂_qi ≡ (∂_Qi,∂_Pi) and f'(τ) is white Gaussian noise with two asymptotically independent components

$$\begin{eqnarray} &&\left\langle f'_\kappa(\tau)f'_{\kappa'}(\tau')\right\rangle=2D_{\tau}k_{\rm B}T\delta_{\kappa\kappa'} \delta(\tau-\tau'), \end{eqnarray} \tag{ 12 }$$

where D_τ = 2D/k_BT and D = 3B₀k_BT/(4m_aω³_Fγ).

The generalized force K_i(q_i) on the ith atoms depends on the dynamical variables q_i of all particles. The term K⁽⁰⁾_i(q_i) in equation (9) is the force in the absence of interaction. The tensor $\hat {\epsilon }$ is the permutation tensor: _QiQi = _PiPi = 0, _QiPi = −_PiQi = 1. The Hamiltonian H_sh that describes the shadow-effect-induced interaction has the form

$$\begin{equation} H_{\mathrm{sh}}=\frac{2f_{\mathrm{sh}}}{\pi \omega_{\rm F} |\delta\omega| m_{\rm a} C_{\mathrm{RWA}}}\sum_{i,j=1}^{N_{\rm tot}}\sqrt{Q_{ij}^2+P_{ij}^2}, \end{equation} \tag{ 13 }$$

where Q_ij = Q_i − Q_j and P_ij = P_i − P_j. Thus the interaction from the shadow effect is described by a simple Hamiltonian in the slow variables.

3.3. Switching rate modification by the atom–atom interaction

The interaction between atoms leads to an extra force F_sh in equation (3). This force is weak in the sense that it weakly affects the intracloud atomic dynamics. As reported in [10, 33], the modification of activation energy by the interatomic force on an atom in cloud n, to first order in H_sh, can be written as

$$\begin{equation} R_{n}\approx R_n^{(0)}-\frac{1}{2}\int^\infty_{-\infty}\,{\rm d}t\left[\dot{\mathbf{q}}_{\mathrm{opt}}-\mathbf{K}^{(0)}_{\mathrm{opt}}\right]\hat{\epsilon}\partial_{\mathbf{q}}H_{\mathrm{sh}}, \end{equation} \tag{ 14 }$$

where q_opt is the optimal path for the ith atom in the absence of the interatomic interaction, and K⁽⁰⁾_opt ≡ K⁽⁰⁾(q_opt). Here, $\dot {\mathbf {q}}_{\mathrm {opt}}-\mathbf {K}^{(0)}_{\mathrm {opt}}$ are the solutions of the variational problem of minimizing the functional $\mathcal {R}^{(0)}$ [34]

$$\begin{equation} \mathcal{R}^{(0)}=(4D_{\tau})^{-1}\int {\rm d}\tau \,\mathbf{f'}(\tau)^2+\int {\rm d}\tau\,\boldsymbol{\lambda}(\tau)\left[\frac{\mathrm{d}\mathbf{q}}{\mathrm{d}\tau}-\mathbf{K}^{(0)}(\mathbf{q})-\mathbf{f'}(\tau)\right], \end{equation} \tag{ 15 }$$

where λ(τ) is the Lagrange multiplier that relates the dynamics of the atom and the force to each other. In fact, q_opt determines the most probable path followed in switching.

By assuming that all the other atoms stay close to either of the attractors, the total activation energy R_n(n = 1,2) considering the interaction between atoms can be approximately written as

$$\begin{equation} R_{n}=R_n^{(0)}+R^{(1)}_n,\quad R^{(1)}_n=\sum_{m=1,2}\alpha_{nm}N_m, \end{equation} \tag{ 16 }$$

where N_n is the number of atoms in cloud n, and α_nm is given by the explicit expressions

$$\begin{eqnarray} &&\fl \alpha_{nn}=\frac{f_{\mathrm{sh}}}{\pi \omega_{\rm F} |\delta\omega| m_{\rm a} C_{\mathrm{RWA}}}\int^{\infty}_{-\infty}{\rm d}t\!\left[\!\left(\dot{P}^{(n)}_{\mathrm{opt}}-K^{(0)}(P)\right)\aleph_{Q}^{(n)}(t)-\left(\dot{Q}^{(n)}_{\mathrm{opt}}-K^{(0)}(Q)\right)\aleph_{P}^{(n)}(t)\!\right],\nonumber\\\\ &&\fl \alpha_{n3-n}=\frac{f_{\mathrm{sh}}}{\pi \omega_{\rm F} |\delta\omega| m_{\rm a} C_{\mathrm{RWA}}}\int^{\infty}_{-\infty}{\rm d}t\!\left[\!\left(\dot{P}^{(n)}_{\mathrm{opt}}-K^{(0)}(P)\right)\tilde{\aleph}_{Q}^{(n)}(t)-\left(\dot{Q}^{(n)}_{\mathrm{opt}}-K^{(0)}(Q)\right)\tilde{\aleph}_{P}^{(n)}(t)\!\right].\nonumber \end{eqnarray} \tag{ 17 }$$

Here

$$\begin{equation} \aleph_{Q}^{(n)}(t)=\frac{Q^{(n)}_{\mathrm{opt}}(t)-Q^{(n)}_{\mathrm{eq}}}{\sqrt{\xi^{(n)}_1(t)}},\quad \tilde{\aleph}_{Q}^{(n)}(t)=\frac{Q^{(n)}_{\mathrm{opt}}(t)-Q^{(3-n)}_{\mathrm{eq}}}{\sqrt{\xi^{(n)}_2(t)}}, \end{equation} \tag{ 18 }$$

where ℵ⁽ⁿ⁾_P and $\tilde {\aleph }_{P}^{(n)}$ are obtained by changing P and Q with each other in ℵ⁽ⁿ⁾_Q and $\tilde {\aleph }_{Q}^{(n)}$.

The denominators in equation (18) are given by

$$\begin{eqnarray*} &&\xi^{(n)}_1(t)=\left(P^{(n)}_{\mathrm{opt}}(t)-P^{(n)}_{\mathrm{eq}}\right)^2+\left(Q^{(n)}_{\mathrm{opt}}(t)-Q^{(n)}_{\mathrm{eq}}\right)^2,\\ &&\xi^{(n)}_2(t)=\left(P^{(n)}_{\mathrm{opt}}(t)-P^{(3-n)}_{\mathrm{eq}}\right)^2+\left(Q^{(n)}_{\mathrm{opt}}(t)-Q^{(3-n)}_{\mathrm{eq}}\right)^2. \end{eqnarray*}$$

R⁽¹⁾_n in equation (16) shows that the effective switching activation energy linearly depends on the number of atoms in the clouds and α_nm, for weak interatomic coupling. α_nm depends on the systemic parameters, optimal path (Q_opt,P_opt) and equilibrium position (Q_eq,P_eq) as shown in equation (17), and according to equation (16), the activation energy of each stable state is changed by the growth of the total number of atoms. From equations (1), (16) and (17) we can obtain the switching probability as follows:

$$\begin{eqnarray} &&W_{nm}(N_n;N_{\rm tot})=W^{(0)}_{nm}\exp[(\alpha^{nm}+\beta^{nm})N_{\rm tot}-2\alpha^{nm} N_n] \end{eqnarray} \tag{ 19 }$$

and

$$\begin{eqnarray} &&\begin{array}{l} W^{(0)}_{nm}=C_W\exp(-R^{(0)}_{n}/D),\\ \alpha^{nm}=(\alpha_{nn}-\alpha_{nm})/2D,\\ \beta^{nm}=-(\alpha_{nn}+\alpha_{nm})/2D. \end{array} \end{eqnarray} \tag{ 20 }$$

3.4. Master equation

To describe noise-induced switching between the bistable states, consisting of many particles, we need to solve the master equation describing the discrete jump process. Through the steady-state solution of the master equation, we can calculate the occupied population of each state as follows. Figure 5 depicts a schematic representation of the discrete jump process describing the master equation [35] for the time-dependent probability P₁(N₁,t) of state 1, where N_n and N_tot mean the number of particles in state n (= 1,2) and the total number of atoms, respectively. The master equation for the time-dependent probability P₁(N₁,t) ≡ P₁(N₁) of state 1, which means the probability of having N₁ atoms in cloud (state) 1 at time t, is as follows:

$$\begin{eqnarray} &&\fl \partial_tP_1(N_1)=-[\mu(N_1)+\nu(N_1)]P_1(N_1) +\nu(N_1-1)P_1(N_1-1)+\mu(N_1+1)P_1(N_1+1),~ \end{eqnarray} \tag{ 21 }$$

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where μ(N₁) = N₁W₁₂(N₁;N_tot) and ν(N₁) = (N_tot − N₁)W₂₁(N_tot − N₁;N_tot). W_nm(N_n;N_tot) is the transition probability per unit time for one atom in state n, as a result of an n → m transition, and then for N_n atoms in state n, the total transition probability per unit time becomes N_nW_nm(N_n;N_tot). Therefore, in the case in which atoms transit from state 1 to state 2, the total transition probability is expressed by μ(N₁) = N₁W₁₂(N₁;N_tot). Conversely, in the case in which atoms transit from state 2 to state 1, ν(N₁) = N₂W₂₁(N₂;N_tot). Because the total number of atoms is conserved, it becomes ν(N₁) = (N_tot − N₁)W₂₁(N_tot − N₁;N_tot). Let us calculate the time derivative of the average of the population in state 1, $\partial _t\left \langle N_1\right \rangle$. It is given by

$$\begin{equation} \partial_t\left\langle N_1\right\rangle=\partial_t\left[\sum^{N_{\rm tot}}_{N_1=0}N_1P_1(N_1)\right], \end{equation} \tag{ 22 }$$

where $\sum ^{N_{\mathrm { tot}}}_{N_1=0}P_1(N_1)=1$, and then

$$\begin{eqnarray} &&\fl \partial_t\left\langle N_1\right\rangle=\left[\sum^{N_{\rm tot}}_{N_1=0}N_1\partial_tP_1(N_1)\right]\nonumber\\ &&=\sum^{N_{\rm tot}}_{N_1=0}N_1\left[-\{\mu(N_1)+\nu(N_1)\}P_1(N_1)+\nu(N_1-1)P_1(N_1-1)\right.\nonumber\\ &&\quad \left.+\,\mu(N_1+1)P_1(N_1+1)\right]. \end{eqnarray} \tag{ 23 }$$

Because N_tot ≫ 1, $\partial _t\left \langle N_1\right \rangle$ is simplified as follows:

$$\begin{equation} \partial_t\left\langle N_1\right\rangle=\sum^{\infty}_{N_1=0}\left[\nu(N_1)-\mu(N_1)\right]P_1(N_1). \end{equation} \tag{ 24 }$$

In the absence of fluctuations, we can obtain the deterministic equation from equation (24), which becomes ∂_tN₁ = ν(N₁) − μ(N₁) [35]. The stationary state occurs when ν(N₁) = μ(N₁), which provides the following result:

$$\begin{equation} N_{2}W_{21}(N_{2};N_{\rm tot})=N_1W_{12}(N_1;N_{\rm tot}). \end{equation} \tag{ 25 }$$

From the above formula, we can obtain the ratio of N₁ to N₂ as follows:

$$\begin{eqnarray} \frac{N_1}{N_2}&=&\frac{W_{21}(N_{2};N_{\rm tot})}{W_{12}(N_1;N_{\rm tot})}\nonumber\\& =&\frac{W^{(0)}_{21}\exp[(\alpha^{21}+\beta^{21})N_{\rm tot}-2\alpha^{21} N_2]}{W^{(0)}_{12}\exp[(\alpha^{12}+\beta^{12})N_{\rm tot}-2\alpha^{12} N_1]}\nonumber\\& =& \exp\{-(\Delta R^{(0)}+\Delta R^{(1)})/D\}, \end{eqnarray} \tag{ 26 }$$

where

$$\begin{equation} \Delta R^{(0)}=R^{(0)}_2-R^{(0)}_1, \end{equation} \tag{ 27 }$$

$$\begin{equation} \Delta R^{(1)}=D\{2(\alpha^{21}N_2-\alpha^{12}N_1)-(\alpha^{21}-\alpha^{12}+\beta^{21}-\beta^{12})N_{\rm tot}\}. \end{equation} \tag{ 28 }$$

ΔR⁽⁰⁾ is the activation energy difference between two states in the non-interacting case, and ΔR⁽¹⁾ is the activation energy difference between them, induced by the atom–atom interaction, depending on the number of atoms. Equation (26) can be simplified further in terms of $\mathcal {P}_1\,({\equiv }\frac {N_1}{N_{\mathrm { tot}}})$ as follows:

$$\begin{equation} \mathcal{P}_1=\frac{1}{1+\exp[\Delta R(\mathcal{P}_1)/D]}, \end{equation} \tag{ 29 }$$

where $\Delta R(\mathcal {P}_1)\equiv \Delta R^{(0)}+\Delta R^{(1)}$ and $\mathcal {P}_1+\mathcal {P}_2=1$. According to the above equation, we can numerically calculate the density of the population, $\mathcal {P}_{1(2)}$.

In conclusion, two CS have been realized in a resonantly driven trapped cold atoms, interacting weakly and globally with each other. In contrast to the single-particle system, we directly measure the change of atomic population with respect to the variation of activation energy, and thereby confirm the fluctuation enhancement near the KPT point, which is characteristic of KPT far from equilibrium. Moreover, we observe that the attractive atom–atom interaction shifts the phase-transition boundaries of KPT, similar to the effect of interaction on the system governed by the van der Waals equation, which provides evidence of similarity between equilibrium and nonequilibrium phase transition. Our system serves as a unique simple platform suitable for investigating the nonlinear dynamics of many-body cold atoms far from equilibrium and relating this dynamics to other domains of physics (dynamical systems, statistical physics).

This work was supported by National Research Foundation of Korea (NRF) grant no. 2012-047677 funded by the Korean government (MEST). GM was supported by Korea Student Aid Foundation (KOSAF) grant no. S2-2009-000-01627-1 funded by the Korean government (MEST).