Observation of heat scaling across a first-order quantum phase transition in a spinor condensate

In quantum mechanics, at zero temperature, when we drive an isolated system by tuning a system parameter, if the driving rate is so slow such that the process is adiabatic, the transition between energy levels cannot occur, and heat cannot be created. Yet, when the system undergoes a second-order QPT during the process, the relaxation time diverges and thus adiabaticity cannot be maintained. As a result, transition between energy levels does occur, producing the heat Polkovnikov2008PRL ; Polkovnikov2008NP . In fact, across the transition point, the physics can be described by the quantum Kibble-Zurek mechanism (KZM) and universal scaling laws for various quantities, such as the temporal onset of excitations, the density of defects and the heat, are predicted polkovnikov2011colloquium ; de2010adiabatic . While important aspects of the quantum KZM have been experimentally observed chen2011quantum ; Chapman2016PRL ; clark2016universal ; zhang2017defect ; Nature2019Lukin ; Qiueaba7292 , the experimental measurement of the heat still remains a significant challenge.

Such non-equilibrium dynamics is of crucial importance ranging from cosmology to condensed matter kibble1980some ; zurek1985cosmological ; PhysRevLett.95.035701 ; zurek2005dynamics ; PhysRevB.72.161201 . Yet the existence of scaling laws is not limited to non-equilibrium dynamics across a second-order QPT. It has been predicted that the scaling of some quantities can also occur across a first-order QPT where multiple phases coexist panagopoulos2015off ; Zhong2017PRE ; coulamy2017dynamics ; pelissetto2017dynamic ; Shimizu_2018 . In particular, very recently, the KZM has been generalized to account for a power-law scaling of the temporal onset of spin excitations present in a spinor condensate across the first-order QPT Qiueaba7292 . The generalized KZM (GKZM) has also been experimentally observed in a spinor condensate Qiueaba7292 . Similar to a second-order QPT, it is natural to ask whether the heat can still exhibit a power-law scaling across the first-order QPT.

A spinor Bose-Einstein condensate (BEC), described by a vector order parameter, provides a controllable platform to explore non-equilibrium dynamics, and various interesting relevant phenomena have been experimentally observed sadler2006spontaneous ; PhysRevLett.103.250401 ; PhysRevLett.108.035301 ; PhysRevLett.105.090402 ; PhysRevA.84.063625 ; parker2013direct ; Yang2019PRA ; Tian2020PRL . In some parameter regime for the condensate, the spin and spatial degrees of freedom are decoupled because all spin states have the same spatial wave function under the single-mode approximation Ueda2012 . As a consequence, the physics is significantly simplified so that the spin degrees of freedom can be separately studied. For an antiferromagnetic (AFM) sodium condensate, there is a first-order QPT between a polar phase with atoms all occupying the $m_{F}=0$ level and an AFM phase with atoms equally occupying the $m_{F}=\pm 1$ levels, where $m_{F}$ is the magnetic quantum number. The system thus provides an ideal platform to study the non-equilibrium physics across a first-order QPT.

Here we theoretically and experimentally demonstrate the existence of a power-law scaling of the heat in a sodium spinor condensate for two dynamical processes: a one-way process where a system is driven from a polar phase to an AFM phase and a cyclic process where a system ends up at the initial polar phase. For the one-way process, the power-law scaling is well characterized by the GKZM. In experiments, we prepare an initial condensate in the polar phase and then slowly vary the quadratic Zeeman energy $q$ by controlling magnetic and microwave fields to realize the two non-equilibrium processes. Since the energy gap vanishes at the transition point, adiabaticity cannot be maintained no matter how the system is driven, leading to the appearance of excitations as well as heat, which can be used as a measure of how strongly adiabaticity is broken. Based on Refs. Polkovnikov2008PRL ; Polkovnikov2008NP , the heat density can be defined as an energy increase per atom relative to the ground state at the final quadratic Zeeman energy $q_{f}$ over the entire process. In experiments, it can be evaluated by measuring the atom number occupying the $m_{F}=0$ level owing to a simple approximate relation between the heat density and the particle number when $|q_{f}|$ is large.

We start by considering the following Hamiltonian describing a spin-1 BEC under single-mode approximation

where $\hat{a}_{m_{F}}^{\dagger}$ ($\hat{a}_{m_{F}}$) is the Boson creation (annihilation) operator for an atom in the hyperfine level $|F=1,m_{F}\rangle$, $\hat{\bf L}$ is the total spin operator with $\hat{L}_{\mu}=\sum_{i,j}\hat{a}^{\dagger}_{i}(f_{\mu})_{ij}\hat{a}_{j}$ ($\mu=x,y,z$) and $f_{\mu}$ being the spin-1 angular momentum matrix, $c_{2}$ is the spin-dependent interaction ($c_{2}>0$ for sodium atoms) and $N$ is the total number of atoms. Here $p$ and $q$ describe linear and quadratic Zeeman energies, respectively. In our experiments, we initialize our condensates in the polar phase so that the dynamics is restricted to the eigenspace of $L_{z}=0$, since the magnetization along $z$ is conserved, i.e., $[\hat{L}_{z},\hat{H}]=0$. The linear Zeeman term $p$ thus becomes irrelevant in the dynamics even though its value is not equal to zero in the experiment. So the ground states of the spinor condensate correspond to the polar and AFM phases when $q>0$ and $q<0$, respectively. The phase diagram is shown in Fig. 1(a) where the mean value $\langle\rho_{0}\rangle$ with $\rho_{0}=\hat{a}_{0}^{\dagger}\hat{a}_{0}/N$ taken as an order parameter drops to zero from one at $q=0$, indicating the occurrence of the first-order QPT.

We investigate the heat production in two types of quench processes: a one-way process for linearly ramping $q$ from $q_{i}$ ($q_{i}>0$) to $q_{f}$ ($q_{f}<0$) and a cyclic process for linearly ramping $q$ from $q_{i}$ to $q_{m}$ ($q_{m}<0$) and then back to $q_{f}$, forming a cyclic process when $q_{f}=q_{i}$ [see Fig. 1(a)]. In both scenarios, we calculate the energy increase at the end of the quench for different ramp rates $v$. To numerically determine the energy of a spinor condensate, we diagonalize the Hamiltonian under the Fock state basis in the subspace of zero magnetization, $\{|{N}/{2},0,{N}/{2}\rangle,|{N}/{2}-1,2,{N}/{2}-1\rangle,...,|0,N,0\rangle\}$, yielding instantaneous eigenstates $|\phi_{n}(q)\rangle$ $(n=1,2,...,{N}/{2}+1)$ of $\hat{H}(q)$ satisfying $\hat{H}(q)|\phi_{n}(q)\rangle=E_{n}(q)|\phi_{n}(q)\rangle$. We also solve the Schrödinger equation $i\hbar\partial|\Psi(t)\rangle/\partial t=\hat{H}(t)|\Psi(t)\rangle$ (we take $h=1$ as a natural unit) to determine the evolving state of the spinor condensate. The energy per atom at the end of the quench is given by $\langle\Psi(q_{f})|\hat{H}(q_{f})|\Psi(q_{f})\rangle=\langle H_{I}\rangle/N+q_{f}-q_{f}\langle\rho_{0}\rangle$ with $H_{I}=c_{2}\hat{\bf L}^{2}/(2N)$ characterizing the interactions. Since the corresponding ground state energy per atom in the AFM phase is $q_{f}$, the produced heat per atom over the one-way process is given by $Q_{o}=\langle H_{I}\rangle/N-q_{f}\langle\rho_{0}\rangle$. For the cyclic process, the produced heat per atom is $Q_{c}=\langle H_{I}\rangle/N+q_{f}(1-\langle\rho_{0}\rangle)$ since the ground state energy of the polar phase is zero.

In Fig. 1(b), we plot our numerical simulation results of the heat density $Q_{o}$ for a one-way process, remarkably showing the existence of a power-law scaling, i.e., $Q_{o}\propto v^{\eta}$ with $\eta=0.319$. This power-law scaling persists even when $q_{f}$ is far away from the transition point, but the exponents change as a function of $q_{f}$ and increase very slowly when $|q_{f}|$ is large [see Fig. 1(c)]. The exponents are independent of $q_{i}$ given that two distinct $q_{i}^{\prime}s$ are connected by an adiabatic process. For a cyclic process, we also observe the power-law scaling of the heat density $Q_{c}$ as shown in Fig. 1(e). While the scaling does not depend on $q_{i}$ for the same reason, Fig. 1(f) shows that the scaling exponents increase slightly with increasing $q_{f}$ (but they are irrelevant of $q_{m}$).

It is a well-known fact that the universal scaling laws across a second-order QPT are accounted for by the quantum KZM PhysRevLett.95.035701 ; zurek2005dynamics ; PhysRevB.72.161201 . Its essential basis is the existence of impulse and adiabatic regions. Specifically, suppose at $t=0$, $q=q_{c}=0$ and the system is in the polar phase. When we linearly drive the system into the AFM phase, the system cannot respond (impulse region) until the response time $\tau(t_{a})=1/\Delta(t_{a})=t_{a}$, where $\Delta$ is the relevant energy gap. When $t>t_{a}$, adiabaticity is restored (adiabatic region). For a second-order QPT, the relevant energy gap refers to the gap between the ground state and the first excited state. Based on the KZM, the heat induced by a slow quench across the critical point is shown to exhibit a power-law dependence on the ramp rate with the scaling exponent determined by the equilibrium critical exponents polkovnikov2011colloquium ; de2010adiabatic ; You2018PRA .

For a first-order QPT, we have demonstrated the existence of impulse and adiabatic regions when a spinor condensate is linearly driven across the transition point Qiueaba7292 . Yet, in stark contrast to the KZM, the relevant energy gap is the gap between the maximally occupied state (the metastable state) and its corresponding first excited state in the first-order case. For example, when $q<0$, the metastable state refers to the many-body metastable polar phase Qiueaba7292 . We now apply the GKZM to determine the heat scaling. To be more precise, we use the equation $|1/\Delta(t)|=|\Delta(t)/\dot{\Delta}(t)|$ to calculate the critical time $t_{a}$ and the corresponding $q_{a}=-vt_{a}$. Since the evolving state is frozen to the initial state when $t<t_{a}$, the heat density can be evaluated by $Q_{o}(q_{f})=[\sum_{n}P_{n}E_{n}(q_{f})-E_{g}(q_{f})]/N$, where $P_{n}=|\langle\phi_{n}(q_{a})|\Psi(q_{i})\rangle|^{2}$ is the probability that the initial state $|\Psi(q_{i})\rangle$ occupies the eigenstate $|\phi_{n}(q_{a})\rangle$ of $\hat{H}(q_{a})$ corresponding to the eigenenergy $E_{n}(q_{a})$.

In Fig. 1(b), we show the power-law scaling of the heat density $Q_{o}(q_{f})$ with respect to $v$ calculated by the GKZM, which agrees very well with the numerical simulation results. Figure 1(c) also displays their comparison of scaling exponents versus $q_{f}$, showing very good agreement with less than $1.5\%$ discrepancy. In comparison, we further compute the heat scaling based on the KZM, which exhibits conspicuous discrepancy especially for large $|q_{f}|$ as shown in Fig. 1(c). For example, when $q_{f}=-20c_{2}$, the scaling exponent obtained by the KZM has about $8.6\%$ difference from the numerical simulation results, while the GKZM only exhibits about $1.0\%$ difference. This indicates that the GKZM gives a better account of the heat scaling law at a first-order QPT.

In fact, based on the GKZM, the heat per atom produced during a quench process ending at $q=q_{a}$ is given by $-q_{a}$, implying that the heat scaling is determined by the scaling of $q_{a}$. In Fig. 1(d), we display the scaling obtained by the numerical simulation, the GKZM and the KZM, demonstrating that the GKZM gives a closer prediction of the power-law exponent to the numerical simulation result than the KZM.

To experimentally probe the heat density is a formidable task due to the complexity of the spin interactions, which is hard to measure. Fortunately, for the one-way process, when $|q_{f}|$ is large, the heat density $Q_{o}$ is dominated by the second part, $\widetilde{Q}_{o}=-q_{f}\langle\rho_{0}\rangle$, which can be experimentally evaluated by measuring $\langle\rho_{0}\rangle$ and $q_{f}$. We call $\widetilde{Q}_{o}$ the quasi-heat density to distinguish it from the heat density $Q_{o}$. Similarly, for the cyclic process, we define the corresponding quasi-heat density as $\widetilde{Q}_{c}=q_{f}(1-\langle\rho_{0}\rangle)$. For both processes, the heat density and the quasi-heat density are related by the following equation

with $s=o,c$ referring to the one-way and cyclic processes, respectively.

Figure 2(a) shows the decline of the relative difference between the heat density $Q_{o}$ and the quasi-heat density $\widetilde{Q}_{o}$ when $|q_{f}|$ is increased; the relative difference decreases to less than $20\%$ when $q_{f}=-3c_{2}$ when $v=0.02c_{2}^{2}$. In fact, the scalings determined by these two energy increases agree much better than their energy differences even for not very large $|q_{f}|$, which is experimentally achievable. For instance, when $q_{f}=-3c_{2}$, while there exists $18\%$ difference of $\widetilde{Q}_{o}$ compared to $Q_{o}$, their scaling exponents are in excellent agreement with only less than $3\%$ discrepancy for the one-way process [see Fig. 2(b)]. For the cyclic process when $q_{f}=6c_{2}$, the scaling exponent difference is also smaller than $3\%$ [see Fig. 2(c) and (d)]. This allows us to obtain the scaling of the heat density by experimentally measuring $\langle\rho_{0}\rangle$ for relatively large $|q_{f}|$.

In experiments, we produce a sodium BEC in an all-optical trap by evaporation of atoms Yang2019PRA . At the evaporation cooling stage, we apply a strong magnetic field gradient to remove the atoms on the hyperfine levels $|F=1,m_{F}=\pm 1\rangle$ out of the trap, leaving all atoms on the $|F=1,m_{F}=0\rangle$ level. After that, a weak and nearly resonant microwave pulse is applied to excite the atoms from $|F=1,m_{F}=0\rangle$ to $|F=2,m_{F}=0\rangle$ (the corresponding detuning is $\delta\simeq-6\,\textrm{kHz}$). Since the atoms on the latter level suffer a significant loss due to three body decay, this process kicks many atoms out of the BEC cloud, resulting in less than $3000$ atoms remaining in the trap. The reduction of the atom number allows us to avoid the unwanted relaxation to the AFM ground state when $q$ is tuned to negative values. The atoms are then immersed in a uniform magnetic field with $q_{i}\approx 10\,\textrm{Hz}$ for $2\,\textrm{s}$ to equilibrate to the polar phase. Afterwards, we slowly decrease the magnetic field strength so as to linearly vary $q$ according to the relation $q\approx B^{2}\cdot 277\,\textrm{Hz}/\textrm{G}^{2}$. When $q$ is changed to around $5\,\textrm{Hz}$, we immediately switch on a microwave field with a frequency of $1.7701264\,\textrm{GHz}$ (the detuning is $\delta=-1.5\,\textrm{MHz}$ relative to the transition from $|F=1,m_{F}=0\rangle$ to $|F=2,m_{F}=0\rangle$) and then gradually raise its amplitude so that $q$ is linearly driven to the negative regime YouLi2017Science (Appendix A). During the process, the amplitude of the microwave field is precisely controlled by a proportional-integral-derivative (PID) feedback system according to a careful calibration of $q$’s values. For each quench rate $v$, $\rho_{0}$ is measured by a standard Stern-Gerlach fluorescence imaging at the end of the linear quench in each experiment and $\langle\rho_{0}\rangle$ is evaluated by averaging over 40 repeated measurements (Appendix C).

In Fig. 3, we show the experimental results of the quasi-heat density $\widetilde{Q}_{o}$ with respect to the ramp rate $v$ for the one-way quench with four different sets of quench parameters. In (a-c), we drive a BEC with about $1100$ atoms and $c_{2}=8.1\pm 0.9\,\textrm{Hz}$ to three distinct $q_{f}$. The experimental data clearly demonstrate the existence of a power-law scaling for these different conditions. We fit the data by a power-law function, i.e., $\widetilde{Q}_{o}\propto v^{\widetilde{\eta}}$, giving the fitting exponents of $\widetilde{\eta}=0.35\pm 0.04$, $\widetilde{\eta}=0.35\pm 0.03$ and $\widetilde{\eta}=0.35\pm 0.06$, respectively. The results agree well with the numerical simulation results with the power-law fitting exponents of $0.340$, $0.340$ and $0.342$, respectively [see the insets of Fig. 3]. Here the numerically calculated exponents for $\widetilde{Q}_{o}$ exhibit about $10\%$ difference from the exponents of $Q_{o}$, which is larger than the result shown in Fig. 2(b) due to larger ramp rates considered here to reduce the relaxation to the ground states in experiments (Appendix D). In Fig. 3(d), we further plot the experimental results for a BEC with roughly $3000$ atoms and $c_{2}=11.8\pm 0.8\,\textrm{Hz}$, showing the existence of a power-law scaling with the fitting exponent of $0.33\pm 0.06$, which is in good agreement with the numerically obtained exponent of $0.337$.

For a cyclic quench, similar to the one-way one, we initially prepare the condensate in the polar phase with the quadratic Zeeman energy $q_{i}$ (e.g., $q_{i}\approx 15\,\textrm{Hz}$) provided by a uniform magnetic field and then linearly decrease $q$ to roughly $5\,\textrm{Hz}$ by decreasing the magnetic field strength. After that, we shine a microwave field to the BEC to linearly vary $q$ from about $5\,\textrm{Hz}$ to $q_{m}$ ($q_{m}<0$, e.g., $q_{m}=-22\,\textrm{Hz}$) and then back to $5\,\textrm{Hz}$ by controlling the field amplitude. We then turn off the microwave field and raise the magnetic field strength until $q$ slowly rises to $q_{f}$ (e.g., $q_{f}=60.1\,\textrm{Hz}$). Since the results do not depend on the value of $q_{i}$ when it is sufficiently large so that the dynamics is adiabatic under the quench rate at $q=q_{i}$ [see Fig. 1(e)], in experiments, we use $q_{i}$ and $q_{f}$ with $q_{f}>q_{i}$ for experimental convenience. The entire ramping process is precisely controlled to be linear according to the calibration of $q$. Similarly, at the end of each quench, the quasi-heat density $\widetilde{Q}$ is measured by probing $\langle\rho_{0}\rangle$ through the Stern-Gerlach fluorescence imaging.

For a cyclic quench, in Fig. 4, we show the experimental measurement of the quasi-heat density $\widetilde{Q}_{c}$ as a function of the quench rate $v$ under different quench parameters. For a BEC cloud with about $1100$ atoms and $c_{2}=8.1\pm 0.9\,\textrm{Hz}$, the results shown in Fig. 4(a-c) evidently illustrate a power-law scaling of the quasi-heat density with fitting exponents of $0.33\pm 0.04$, $0.32\pm 0.03$ and $0.33\pm 0.06$, respectively. The exponents agree well with the exponents of $0.348$, $0.352$ and $0.352$ numerically obtained for $\widetilde{Q}_{c}$, which are larger than $0.330$, $0.330$ and $0.335$ (numerically calculated scaling exponents for $Q_{c}$) by about $5.4\%$, $6.7\%$ and $5.1\%$, respectively. This also shows that our experimental measurements cannot differentiate the slight difference between $\widetilde{Q}_{c}$ and $Q_{c}$. Additionally, we raise the atom number to around $3000$ corresponding to $c_{2}=11.8\pm 0.8\,\textrm{Hz}$ and perform the experiments under the quench parameters of $q_{i}=17.55\,\textrm{Hz}$, $q_{m}=-18.10\,\textrm{Hz}$ and $q_{f}=66.73\,\textrm{Hz}$. Figure 4(d) reveals the existence of a power-law scaling with a fitting exponent of $0.34\pm 0.05$, in good agreement with the numerical result of $0.345$.

Our work demonstrates the first experimental observation of a power-law scaling of heat with respect to a ramp rate for non-equilibrium dynamics. Two types of quench processes including one-way and cyclic processes are studied across a first-order QPT in a spinor condensate. The experimentally measured scaling exponents for both non-equilibrium processes agree well with our numerical simulation results.