Adiabatic quenches and characterization of amplitude excitations in a continuous quantum phase transition

Significance

Symmetry-breaking phase transitions play important roles in many areas of physics, including cosmology, particle physics, and condensed matter. The freezing of water provides a familiar example: The translational and rotational symmetries of water are reduced upon crystallization. In this work, we investigate symmetry-breaking phase transitions of the magnetic properties of an ultracold atomic gas in the quantum regime. We measure the excitations of the quantum magnets in different phases and show that the excitation energy (gap) remains finite at the phase transition. We exploit the nonzero gap to demonstrate an adiabatic (reversible) quench across the phase transition. Adiabatic quantum quenches underlie proposals for generating massively entangled spin states and are fundamental to the ideas of adiabatic quantum computation.

Abstract

Spontaneous symmetry breaking occurs in a physical system whenever the ground state does not share the symmetry of the underlying theory, e.g., the Hamiltonian. This mechanism gives rise to massless Nambu–Goldstone modes and massive Anderson–Higgs modes. These modes provide a fundamental understanding of matter in the Universe and appear as collective phase or amplitude excitations of an order parameter in a many-body system. The amplitude excitation plays a crucial role in determining the critical exponents governing universal nonequilibrium dynamics in the Kibble–Zurek mechanism (KZM). Here, we characterize the amplitude excitations in a spin-1 condensate and measure the energy gap for different phases of the quantum phase transition. At the quantum critical point of the transition, finite-size effects lead to a nonzero gap. Our measurements are consistent with this prediction, and furthermore, we demonstrate an adiabatic quench through the phase transition, which is forbidden at the mean field level. This work paves the way toward generating entanglement through an adiabatic phase transition.

The amplitude mode and phase mode describe two distinct excitation degrees of freedom of a complex order parameter $\psi =A{e}^{i\varphi }$ appearing in many quantum systems such as the order parameter of the Ginzburg–Laudau superconducting phase transition (1) and the two-component quantum field of the Nambu–Goldstone–Anderson–Higgs matter field model (2–5). In a zero-dimensional system of an interacting spin-1 condensate, the transverse spin, ${\mathbf{S}}_{\perp },$ plays the role of an order parameter in the quantum phase transition (QPT) with ${\mathbf{S}}_{\perp }$ being zero in the polar (P) phase and nonzero in the broken axisymmetry (BA) phase (Fig. 1A). Representing the transverse spin vector as a complex number, ${\mathbf{S}}_{\perp }=S_x+i{S}_y,$ with the real and imaginary parts being expectation values of spin-1 operators, the amplitude mode corresponds to the amplitude oscillation of ${\mathbf{S}}_{\perp }.$

Fig. 1.

(A) Effective spinor potential energy V in the BA phase ($q/\left|c\right|<2$), at the QCP ($q/\left|c\right|=2$), and in the P phase ($q/\left|c\right|>2$). In the P phase, there are two gapped modes (blue lines) along the radial direction about the GS. In the BA phase, the GS occupies a minimum energy ring (gray circle) with one gapped mode along the radial direction (blue line) and one NG mode (red line) in the azimuthal direction. (B) The GS on the $\left\{S_{\perp },Q_{\perp },Q_z\right\}$ unit spheres is represented by a red-shaded region. Coherent orbiting (phase winding) dynamics are represented by red (green) curves and the blue curve is the separatrix. The magenta (black) arrow represents the radio-frequency (microwave) pulse used for the initial state preparation (Supporting Information). (C) The energy gap for 40,000 atoms (cyan curve) is calculated from the eigenvalues of the quantum Hamiltonian (Supporting Information). (D) The energy gap at the QCP (blue solid line) calculated from the eigenvalues of the quantum Hamiltonian matches the GS oscillation frequencies from simulations (red circles).

The amplitude mode can be studied in different spinor phases by tuning the relative strengths of the quadratic Zeeman energy per particle $q\propto B^2$ and spin interaction energy c of the condensate (6) by varying the magnetic field strength B (Fig. 1). In the P phase, both the effective spinor potential energy V and the ground state (GS) spin vector have SO(2) rotational symmetry about the vertical axis (Fig. 1A), and there are two degenerate collective amplitude modes along the radial directions about the GS located at the bottom of the parabolic bowl. These amplitude excitations are gapped modes, which vary both the amplitude of ${\mathbf{S}}_{\perp }$ and the energy.

In the BA phase, the effective spinor potential energy V acquires a Mexican-hat shape with the GS occupying the minimal energy ring of radius $\sqrt{4c^2-q^2}/\left(2\left|c\right|\right).$ The GS spin vector, ${\mathbf{S}}_{\perp }$ (orange arrow in Fig. 1A), spontaneously breaks the SO(2) symmetry and acquires a definite direction (7, 8). This broken symmetry induces a massless Nambu–Goldstone (NG) mode in which it costs no energy for the spin vector to rotate about the vertical axis. Recently, the magnetic dipolar interaction was used to open a gap in the NG mode by breaking the rotational symmetry of the spin interaction (9). In our condensate, the magnetic dipolar interaction can be ignored due to spatial isotropy, and therefore, the NG mode in the BA phase remains gapless. The other excitation, the amplitude mode, manifests itself as an amplitude oscillation of the transverse spin in the radial direction. This amplitude mode is similar to the massive mode in the Goldstone model (3).

In this work, we measure the amplitude modes in a spin-1 Bose–Einstein condensate (BEC) through measurements of very low amplitude excitations from the GS. The results show a quantitative agreement with gapped excitation theory (10–12) and provide a platform to probe the amplitude excitation, which plays a crucial role in the KZM in spinor condensates (11, 13–15). Although in the thermodynamic limit the amplitude mode energy gap goes to zero at the quantum critical point (QCP), a small size-dependent gap persists for finite-size systems (12). Measurements of the energy gap near the QCP are challenging; however, our results are consistent with a small nonzero gap. Furthermore, by using a very slow, optimized magnetic field ramp, we demonstrate an adiabatic quench across the QCP. Such adiabatic quenches in finite-sized systems underlie proposals for generating massively entangled spin states including Dicke states (12) and are fundamental to the ideas of adiabatic quantum computation (16).

The experiments use a tightly confined ⁸⁷Rb BEC with $N=\mathrm{40,000}$ atoms in optical traps such that spin domain formation is energetically suppressed. The Hamiltonian describing this spin system in a bias magnetic field B along the z axis is (17–21)

$$\widehat{H}=\tilde{c}{\widehat{S}}^2-q\left({\widehat{Q}}_z-N\right)/2,$$ [1]

where ${\widehat{S}}^2$ is the total collective spin-1 operator and ${\widehat{Q}}_z$ is proportional to the spin-1 quadrupole moment, ${\widehat{Q}}_{zz}.$ The coefficient $\tilde{c}$ is the collisional spin interaction energy per particle integrated over the condensate and quadratic Zeeman energy per particle $q=q_z{B}^2$ with $q_z=72$ Hz/${\text{G}}^2$ (hereafter, $h=1$). The longitudinal magnetization $\langle {\widehat{S}}_z\rangle$ is a constant of the motion ($\langle {\widehat{S}}_z\rangle =0$ for these experiments); hence the first-order linear Zeeman energy $p{\widehat{S}}_z$ with $p\propto B$ can be ignored. The spin-1 coherent states can be represented on the surface of a unit sphere shown in Fig. 1B with axes $\left\{S_{\perp },Q_{\perp },Q_z\right\},$ where the expectation value of transverse spin is $S_{\perp }^2=S_x^2+S_y^2,$ $Q_{\perp }$ is the transverse off-diagonal nematic moment $Q_{\perp }^2=Q_{xz}^2+Q_{yz}^2,$ and $Q_z=2{\rho }_0-1,$ where ${\rho }_0$ is the fractional population in the $|F=1,m_F=0\rangle$ state. In this representation, the coherent dynamics evolve along the constant energy contours of $\mathrm{\mathscr{H}}=\left(1/2\right)c{S}_{\perp }^2-\left(1/2\right)q\left(Q_z-1\right),$ where $c=2N\tilde{c}$ (22–24) (red and green orbits in Fig. 1B). The phase space for the single-mode spin-1 condensate is similar to that for the Lipkin–Meshkov–Glick (LMG) model (25), which in turn describes the infinite coordination number limit of the XY model or quantum Ising model (26). The dynamics of the QPT in these zero-dimensional quantum systems have been explored theoretically and experimental realizations include the double-well Bose–Hubbard (27, 28), pseudospin-1/2 BEC (29, 30), and many-atom cavity quantum electrodynamics systems (31).

In the mean-field (large atom number) limit, quantum fluctuations can be ignored and the wavefunction for each spin state, $m_F=0,\pm 1,$ can be represented as a complex vector with components, ${\psi }_{0,\pm 1}=\sqrt{{\rho }_{0,\pm 1}}\exp \left(i{\theta }_{0,\pm 1}\right).$ Using Bogoliubov analysis (10) and mean-field theory (24, 32), the energy gap of the amplitude mode in the P phase and the BA phase in the long-wavelength limit corresponds to the oscillation frequency of small excitations in ${\rho }_0$ from the GS

$${\mathrm{\Delta }}_{\text{P}}\equiv f_{\text{P}}=2\sqrt{q\left(q+2c\right)},\hspace{1em}{\mathrm{\Delta }}_{\text{BA}}\equiv f_{\text{BA}}=2\sqrt{c^2-q^2/4}.$$ [2]

Here the energy gap is ${\mathrm{\Delta }}_E$ ($\equiv {\mathrm{\Delta }}_{\text{P}}$ and ${\mathrm{\Delta }}_{\text{BA}}$) and coherent oscillation frequency is f ($\equiv f_{\text{P}}$ and $f_{\text{BA}}$). Although these relations show a vanishing gap at the QCP, quantum fluctuations due to finite atom number result in a nonzero gap. In the quantum theory, the energy gap can be exactly calculated from the eigenenergy values of the Hamiltonian in Eq. 1 (Supporting Information). Fig. 1C shows the energy gap between the GS and the first excited state with a small nonzero gap at the QCP as a result of a finite atom number. Fig. 1D shows the relation of energy gap at the QCP to the atom number in condensates ranging from ${10}^1\hspace{0.5em}\text{to}\hspace{0.5em}{10}^5$ atoms, which scales as ${\mathrm{\Delta }}_E\propto N^{-1/3}$ (12). The energy gap curve compares well to the oscillation frequencies of GS spinor population ${\rho }_0$ obtained from quantum simulations (Supporting Information) for a broad range of atom numbers (red circles in Fig. 1D). The equivalence relation between the energy gap and the coherent oscillation (32) frequency in Eq. 2 is a general statement connecting the amplitude modes to the observable dynamics and is key to this study.

Energy Gap Measurement

To characterize the energy gap ${\mathrm{\Delta }}_E,$ we measure coherent dynamics for states initialized close to the GS (Fig. 1B) for different values of $q/\left|c\right|$ ranging from 0.1 to 3 and fit the measurements to sinusoidal functions to determine the oscillation frequencies (Supporting Information). For each $q/\left|c\right|$ value, several measurements of the population ${\rho }_0$ are made for a series of initial states approaching the GS as illustrated in Fig. 2A. The GS population ${\rho }_{0,\text{GS}}$ can be obtained by minimizing the spinor energy (Supporting Information) (24)

$${\rho }_{0,\text{GS}}=1\hspace{0.17em}\left(P\right),\hspace{1em}{\rho }_{0,\text{GS}}=1/2+q/\left(4\left|c\right|\right)\left(\text{BA}\right).$$ [3]

The oscillation amplitude of ${\rho }_0$ has a lower limit given by the Heisenberg standard quantum limit ($\text{SQL}=N^{-1/2}$) projected onto the ${\rho }_0$ axis ($\propto Q_z$ axis in Fig. 1B) (21); hence the best estimate of the energy gap is obtained from the measurement with the lowest observable oscillation amplitude. An alternate method to determine the energy gap for states centered on the pole is to measure the oscillations of the transverse spin fluctuations, $\mathrm{\Delta }{S}_{\perp }.$ Although this method requires many more data because the signal is in the fluctuations instead of the mean value, it provides higher contrast for states localized at the pole. Measurements obtained with this technique at the QCP are shown in Fig. 2B for a state prepared in the polar GS (Supporting Information).

Fig. 2.

Energy gap measurements. (A) Coherent oscillation data (red circles) obtained at $q/\left|c\right|=2.03.$ In clockwise order, the oscillation amplitude decreases as the initial state is prepared closer to the GS. Each data point is an average of 10 measurements and the data are fitted to a sinusoidal function with a varying frequency (solid line). (B) The time evolution of $\mathrm{\Delta }{S}_{\perp }$ data (blue squares) at the QCP ($q/\left|c\right|=2$) is fitted to a sinusoidal function (solid line). Each data point is the noise of 45 measurements. The corresponding simulation is represented by an orange curve with the shaded region being $q/\left|c\right|=2\pm 0.005.$ (C) The energy gap ${\mathrm{\Delta }}_E$ for different $q/\left|c\right|$ values is obtained from the frequency fits of coherent oscillation data. Circles (triangles) are obtained from an average of 10 (or 3) measurements of ${\rho }_0$ coherent oscillation, and the blue square is the frequency fit of $\mathrm{\Delta }{S}_{\perp }$ dynamics. The theoretical energy gap is represented by the purple curve. Inset shows the region around the QCP with the shaded orange region being the energy gap for an imperfect initialization of the GS (main text). The dashed line shows the theoretical energy gap when the initial population ${\rho }_0$ is about 0.05 away from the actual GS, reflecting an oscillation amplitude of 0.05.

The results of the energy gap measurements are shown in Fig. 2C for both methods. Overall, the measurements capture the characteristics of energy gap predicted by gapped excitation theory for a spin-1 BEC (10–12). In the P phase, the energy gap data show a good agreement with the theoretical prediction within the uncertainty of the measurements. In the BA phase, the measured gap data are also in reasonable agreement with the theory; however, the measured values are 20% lower than the theory for the smallest values of $q/\left|c\right|<1.$ This finding is possibly a result of small violations of the single-mode approximation or the presence of a small thermal fraction, both of which would be more significant in this spin interaction-dominated regime. In a study of an antiferromagnetic condensate, using an initial state (${\rho }_0=0.5$) prepared far away from the antiferromagnetic GS (${\rho }_{0,\text{AGS}}=1$), slightly lower oscillation frequencies than those in the theory were also observed (33); it was suggested that these resulted from excess magnetization noise from the radio frequency (RF) pulse of the initial state preparation; however, this noise is not large enough to explain the difference in our measurements.

In the neighborhood of the QCP, the energy gap decreases dramatically. As shown in Fig. 2C, Inset, the measurements are in good agreement with the theoretical prediction in this region. For measurements at $q=2\left|c\right|,$ the minimum measured gap is ${\mathrm{\Delta }}_E=0.15\left(1\right)\left|c\right|,$ which is consistent with the nonzero gap predicted by the quantum theory, ${\mathrm{\Delta }}_{E,\text{th}}=0.165\left|c\right|;$ here $c\approx -7.5\left(1\right)$ Hz (Supporting Information). We point out, however, that there are experimental challenges to these measurements. The initial state is prepared in the high magnetic field GS ($q/\left|c\right|=38$). This state has symmetric $\sqrt{N}$ fluctuations in the $S_{\perp },Q_{\perp }$ plane. When the condensate is rapidly quenched to a lower $q/\left|c\right|$ for the energy gap measurement, this projects the condensate to slightly excited states of the final $q/\left|c\right|$ Hamiltonian. The subsequent evolution of this state will have an oscillation frequency higher than the calculated gap frequency, particularly in the region $1.95\le q/\left|c\right|\le 2.05.$ We can accurately calculate this effect, and the results are indicated by the orange-shaded region in Fig. 2C.

A further complication in the measurement at the QCP is that the value $q/\left|c\right|$ is not truly constant during the measurement of the gap, but drifts to slightly higher values because of a reduction of density due to the finite lifetime of the condensate. The spin interaction energy depends on the density and atom number as $c\left(t\right)\propto n\left(t\right)\propto N{\left(t\right)}^{2/5}.$ For these measurements, the condensate lifetime was $1.6\left(1\right)$ s, which results in a drift of $\mathrm{\Delta }q/\left|c\right|=0.05$ in 100 ms in the neighborhood of the QCP. The atom loss is taken into account in the simulations, an example of which is shown in Fig. 2C, and the energy gap is determined by the frequency at $t=0.$ Despite these challenges to the measurements near the QCP, the data indicate the presence of a nonzero gap that is of the same size as predicted by theory.

In the BA phase of a $\langle {\widehat{S}}_z\rangle =0$ spin-1 BEC, two of the three excitation modes (Supporting Information) are massless NG modes that appear due to broken global symmetries. The third mode is a massive amplitude mode with a dispersion relation: $E^2\left(k_0\right)={\mathrm{\Delta }}_E^2+{\left(2c\hslash /m\right)}^2{k}_0^2$ with $k_0$ being the wavenumber and m being the atomic mass (10, 11). The energy gap ${\mathrm{\Delta }}_E$ is equivalent to the rest mass energy of the quasiparticle corresponding to the excitation mode. Our experiments are in the long-wavelength limit in which the wave vector approaches zero, ${\mathbf{k}}_0\to 0.$ The massive amplitude mode that appears when a global symmetry is broken has properties analogous to those of the Higgs mode, which relates to the amplitude fluctuation of the order parameter of the phase transition. Such a Higgs-like mode has been observed as a collective excitation in the superfluid/Mott insulator transition as an amplitude fluctuation of a complex order parameter (34), in the XY model of antiferromagnetic materials as an amplitude fluctuation of the spin vector (35), in superconducting systems (36–39), and here as the amplitude mode of the spin-1 BEC in the BA phase.

Adiabatic QPT

In the thermodynamic ($N\to \infty$) limit, the vanishing gap at the QPT prohibits adiabatic crossing between phases and gives rise to excitations characterized by the Kibble–Zurek mechanism (KZM). However, the opening of the gap at the QCP due to finite-size effects makes it possible, in principle, to cross the QCP adiabatically using a carefully tailored ramp from $q\gg 2\left|c\right|$ to $q<2\left|c\right|,$ while remaining in the GS of the Hamiltonian. Recently, adiabaticity in sodium spin-1 condensates has been studied (40); however, these experiments were performed using condensates with nonzero longitudinal magnetization ($\langle {\widehat{S}}_z\rangle \ne 0$) that do not have a QCP. Here, we focus on the very challenging case of the small energy gap at the QCP.

Due to the small size of the gap, the ramp in q needs to be very slow in the region of $2\left|c\right|$ to maintain adiabaticity. To allow longer ramps, we used a single-focus dipole trap in which the condensate lifetime is 15–19 s. To determine the optimal ramp, we performed simulations using measured values of the trap lifetime, the atom number, and the spin interaction energy. The ramp is determined from a piecewise optimization of the Landau–Zener adiabaticity parameter $\left(\text{d}{\mathrm{\Delta }}_E/\text{d}t\right)\left(1/{\mathrm{\Delta }}_E^2\right)$ (41–44) and includes the effects of atom loss on c (Supporting Information). The simulations (45, 46) show that it is possible to adiabatically cross the phase transition in ∼35 s, starting with a condensate initially containing 40,000 atoms; here we use an adiabatic invariant to determine the condition for adiabaticity (44, 47).

The experiment starts with atoms at the GS in the polar phase at a high magnetic field, $q/\left|c\right|=140.$ Then, the magnetic field is ramped through the QCP to $q/\left|c\right|=1.33$ in 35 s along the trajectory represented by the green line in Fig. 3A. The measured evolution of the population ${\rho }_0$ is shown in the same graph and compared with that predicted by the simulation. The data show excellent agreement with the theoretical values for the evolving GS population ${\rho }_{0,\text{GS}}$ (Eq. 3), which provides a strong indication of adiabaticity.

Fig. 3.

Adiabatic and nonadiabatic dynamics. (A–C) Adiabatic dynamics of population ${\rho }_0$ and the uncertainty $\mathrm{\Delta }{\rho }_0$ (main text). (D and E) Nonadiabatic dynamics of population ${\rho }_0$ of 1-s and 28-s linear ramp from $q/\left|c\right|=140\to 1.33.$ Red circles (blue squares) are the adiabatic (nonadiabatic) measurements, gray-shaded regions are the theoretical ${\rho }_{0,\text{GS}}$ values, green arrow lines represent the ramp of $q/\left|c\right|$ (vertical axis on the right), and cyan curves represent the dynamics simulation of ${\rho }_0$ with the corresponding $q/\left|c\right|$ ramp. Each adiabatic data point is an average of 3 measurements and each nonadiabatic data point is an average of 15 measurements.

There are about 9,000 atoms remaining after the adiabatic ramp. The theoretical value of the GS population and uncertainty is ${\rho }_{0,\text{GS}}=0.833\pm 0.0047,$ where the uncertainty is the SQL for 9,000 atoms, projected onto the ${\rho }_0$ axis ($\propto Q_z$ axis in Fig. 1B) (Supporting Information). Immediately after the adiabatic ramp ($t\approx 36$ s), the measured mean population and fluctuations are ${\rho }_0=0.830\pm 0.007,$ which are very close to the theoretical values and further indicate adiabiticity. Following the adiabatic ramp, the ratio $q/\left|c\right|=1.33$ is held constant for 2 s to verify that the system remains in the GS. As shown in Fig. 3B, the mean value of ${\rho }_0$ stays close to the theoretical value ${\rho }_{0,\text{GS}}.$ In Fig. 3C, the variance $\mathrm{\Delta }{\rho }_0^2$ is plotted. Although the measurements of $\mathrm{\Delta }{\rho }_0^2$ (red circles) tend above the theoretical SQL squared (dashed line) after holding, atom loss increases fluctuations in the spin populations (assuming uncorrelated losses) to the level shown in the green-shaded region (Supporting Information).

For comparison, in Fig. 3 D and E we show data from nonadiabatic ramps from $q/\left|c\right|=140\to 1.33.$ In Fig. 3D, a 1-s linear ramp is used, whereas in Fig. 3E, a 28-s ramp is used. In both cases, the spin population ${\rho }_0$ does not follow the theoretical GS population during the ramp and the variance $\mathrm{\Delta }{\rho }_0^2$ grows dramatically. The fluctuations at the end of the 28-s ramp are compared with those from the adiabatic ramp in Fig. 3C (blue squares), and it is clear that the nonadiabaticity gives rise to increased fluctuations.

Adiabatically crossing the QPT in a spin-1 zero magnetization condensate is predicted to generate massively entangled spin states (12). Broadly speaking, this is an example of the fundamental principle underlying adiabatic quantum computing, in which the initial, simple GS is transformed into a highly entangled final GS by tuning the Hamiltonian adiabatically through a QCP. This final GS of the Hamiltonian is a solution to a computation problem (16). In our case, the final state for a ramp to $q=0$ is predicted to be the Dicke state $|S=N,S_z=0\rangle .$ In this study, we stop the adiabatic ramp at $q/\left|c\right|=1.33.$ The entanglement of the GS at this $q/\left|c\right|$ can be calculated as in ref. 12, $\xi =\left(\langle {\widehat{S}}_x^2\rangle +\langle {\widehat{S}}_y^2\rangle \right)N/\left(1+4\langle {\left(\mathrm{\Delta }{\widehat{S}}_z\right)}^2\rangle N^2\right).$ The uncertainty in transverse magnetization is $\langle {\widehat{S}}_x^2\rangle +\langle {\widehat{S}}_y^2\rangle \approx 1-{\left(2{\rho }_0-1\right)}^2.$ In the ideal case, the longitudinal magnetization is zero and conserved $\langle {\left(\mathrm{\Delta }{\widehat{S}}_z\right)}^2\rangle \to 0,$ and the expected entanglement is $\xi =0.56N$ or roughly $\mathrm{5,000}$ atoms are entangled out of $\mathrm{9,000}$ atoms at the end of the adiabatic ramp. However, atom loss induces noise in the magnetization, $\mathrm{\Delta }{S}_z\approx 0.5\%,$ in our experiment. This small magnetization noise reduces the entanglement to $\xi <1$ atom.

In summary, we have explored the amplitude mode in small spin-1 condensates. The energy gap measurements show evidence of a nonzero gap at the QCP arising from finite-size effects, and using a carefully tailored slow ramp of the Hamiltonian parameters, we have adiabatically crossed the QCP with no apparent excitation of the system. We hope that this work stimulates similar investigations in related many-body systems, and in particular, we anticipate that the results of this study could directly inform investigations in double-well Bose–Josephson junction systems, (pseudo)spin-1/2 interacting systems (48, 49), and the Lipkin–Meshkov–Glick (LMG) model (43, 50), which share similar Hamiltonians.

Materials and Methods

The experiment is carried out using small condensates of $\mathrm{40,000}$ atoms in the $F=1$ hyperfine GS of ⁸⁷Rb. In the energy gap experiment, atoms are confined in a spherical optical dipole force trap with trap frequencies $\sim 2\pi \times 160\hspace{0.17em}\text{Hz},$ formed by crossing the focus of a $10.6$-μm wavelength laser with an 850-nm wavelength laser. This tight confinement ensures that the condensate is well described by the single-mode approximation (SMA), such that the spin dynamics can be considered separately from the spatial dynamics (17–19). The spin interaction energy $c\approx -7.5\left(1\right)$ Hz and trap lifetime is $\approx 1.6\left(1\right)$ s. The spin populations of the condensate are measured by releasing the trap and allowing the atoms to expand in a Stern–Gerlach magnetic field gradient to separate the $m_F$ spin components. The atoms are probed for 200 μs with three pairs of counter-propagating orthogonal laser beams, and the fluorescence signal collected by a CCD camera is used to determine the number of atoms in each spin component.

Initial State Heisenberg Uncertainty

All atoms are prepared in the $m_F=0$ spin state (i.e., the high-field GS), which can be expressed in the Fock representation as $|N_1=0,N_0=N,N_{-1}=0\rangle ,$ where $N_{0,\pm 1}$ is the total number of atoms in $m_F=0,\pm 1$ spin states. With the total atom number $N=N_1+N_0+N_{-1}$ and conservation of the magnetization $M=N_1-N_{-1}=0,$ the Fock basis can be represented as $|N,k,M=0\rangle$ or more concisely as $|k\rangle ,$ where k is the number of pairs of atoms in the $m_F=\pm 1$ states. This initial state yields nonzero expectation values of the following commutator operators for the spin and quadrupole operator (21)

$$\langle 0,N,0|\left[{\widehat{S}}_x,{\widehat{Q}}_{yz}\right]|0,N,0\rangle =-2iN$$ [S1]

$$\langle 0,N,0|\left[{\widehat{S}}_y,{\widehat{Q}}_{xz}\right]|0,N,0\rangle =2iN,$$ [S2]

giving the uncertainty relations

$$\mathrm{\Delta }{S}_x\mathrm{\Delta }{Q}_{yz}\ge N$$ [S3]

$$\mathrm{\Delta }{S}_y\mathrm{\Delta }{Q}_{xz}\ge N.$$ [S4]

The initial state, $|k=0\rangle$ can be approximated by a Gaussian distribution with

$$\mathrm{\Delta }{S}_x=\mathrm{\Delta }{S}_y=\mathrm{\Delta }{Q}_{yz}=\mathrm{\Delta }{Q}_{xz}=\sqrt{N}.$$ [S5]

Note that $S_{\perp }^2=S_x^2+S_y^2$ and $Q_{\perp }^2=Q_{xz}^2+Q_{yz}^2.$ Projecting the 2D Gaussian distribution of the initial state in $S_x{S}_y$ and $Q_{xy}{Q}_{yz}$ phase spaces onto the $S_{\perp },Q_{\perp }$ phase space, the uncertainty of the initial state normalized to the atom number is

$$\mathrm{\Delta }{S}_{\perp }=\mathrm{\Delta }{Q}_{\perp }=N^{-1/2}.$$ [S6]

Energy Gap Calculation

The gap ${\mathrm{\Delta }}_E$ can be obtained by computing the eigenvalues of the quantum Hamiltonian in the Fock basis, using the number of pairs of atoms in the $m_F=\pm 1$ states, $|k\rangle$ (12, 19, 20, 45),

$$\begin{array}{c}{\mathrm{\mathscr{H}}}_{k,k\prime }=2{\delta }_{k\prime ,k}\left[\tilde{c}\left(2k\left(N-2k\right)+N-k\right)+2qk\right]\\ +\hspace{0.17em}2{\delta }_{k\prime ,k-1}\left[\tilde{c}k\sqrt{\left(N-2k+1\right)\left(N-2k+2\right)}\right]\\ +\hspace{0.17em}2{\delta }_{k\prime ,k+1}\left[\tilde{c}\left(k+1\right)\sqrt{\left(N-2k\right)\left(N-2k-1\right)}\right],\end{array}$$ [S7]

where $\tilde{c}=c/2N,$ and ${\delta }_{k,k\prime }$ is the Kronecker delta function. The energy gap ${\mathrm{\Delta }}_E$ is the excitation energy from the lowest to the first excited eigenstate. The energy gap as a function of q is illustrated for different atom numbers in Fig. S1.

Mean-Field Energy Gap.

In the mean-field (large atom number) limit, the energy gap in the polar (P) phase ($q\ge 2\left|c\right|$) and the BA phase ($q<2\left|c\right|$) can be obtained using a Bogoliubov approximation, as described in ref. 10:

$${\mathrm{\Delta }}_{\text{P}}=2\sqrt{q\left(q+2c\right)}$$ [S8]

$${\mathrm{\Delta }}_{\text{BA}}=2\sqrt{c^2-q^2/4}.$$ [S9]

There are three excitation modes. In the BA phase, there are two gapless phase modes and one gapped amplitude mode with the dispersion relations (10)

$$E_{\gamma }^2\approx q\hslash {\mathbf{k}}_0^2/\left(2m\right)$$ [S10]

$$E_{\beta }^2\approx 2g_{2}n\hslash {\mathbf{k}}_0^2/\left(2m\right)$$ [S11]

$$E_{\alpha }^2\approx {\mathrm{\Delta }}_{BA}^2-4c\hslash {\mathbf{k}}_0^2/\left(2m\right),$$ [S12]

where ${\mathbf{k}}_0$ is the wave vector, $g_2=4\pi \hslash a_2/m,$ $a_2$ is the scattering length of the $F=2$ channel, n is the atom density, and m is the atomic mass.

Mean-Field Spinor Energy.

From Eq. 1 in the main text, the mean-field spinor energy can be written as

$$\mathrm{\mathscr{H}}=\frac{c}{2}{S}_{\perp }^2-\frac{q}{2}\left(Q_z-1\right).$$ [S13]

Transforming into the variables ${\rho }_0$ (population fraction in $m_F=0$) and ${\theta }_s$ (spinor phase), using the relations $Q_z=2{\rho }_0-1$ and $S_{\perp }=2\sqrt{{\rho }_0\left(1-{\rho }_0\right)}\cos \left({\theta }_s/2\right),$ we obtain

$$\mathrm{\mathscr{H}}=2c{\rho }_0\left(1-{\rho }_0\right)\left(1+\text{cos}\left({\theta }_s\right)\right)+2q\left(1-{\rho }_0\right).$$ [S14]

${\rho }_0$ and ${\theta }_s$ are constrained by the spherical geometry of the phase space (main text Fig. 1B) (46). In particular, $0\le {\rho }_0\le 1$ and $-\pi \le {\theta }_s\le \pi .$ Minimizing Eq. S14 under these constraints, we obtain the P-phase and BA-phase GSs (Eq. 3 of the main text):

$${\rho }_{0,\text{GS}}=\{\begin{array}{l}1,q>2\left|c\right|\left(P\hspace{0.17em}\text{phase}\right)\hfill \\ \frac{1}{2}+\frac{q}{4\left|c\right|},q<2\left|c\right|\left(\text{BA}\hspace{0.17em}\text{phase}\right).\hfill \end{array}$$ [S15]

Coherent Oscillations in the Amplitude Mode.

From Eq. S15 the GS value of $S_{\perp }$ is

$$S_{\perp ,\text{GS}}=\{\begin{array}{l}0,q>2\left|c\right|\left(P\hspace{0.17em}\text{phase}\right)\hfill \\ \sqrt{1-\frac{q^2}{4|c{|}^2}},q<2\left|c\right|\left(\text{BA}\hspace{0.17em}\text{phase}\right).\hfill \end{array}$$ [S16]

In the P phase, the set of GSs is represented by a point in the $S_x,S_y$ plane. In the BA phase, the set of GSs forms a circle of radius $\sqrt{1-q^2/4{\left|c\right|}^2}.$ Therefore, a QPT from the P phase to the BA phase is accompanied by a spontaneous breaking of an $\text{SO}\left(2\right)$ symmetry.

The mean-field equation for $S_{\perp }$ reads ${\dot{S}}_{\perp }=-q\sqrt{1-S_{\perp }^2-Q_z^2}$ (46). Eliminating $Q_z$ between this equation and Eq. S13, we obtain the following dynamical equation for $S_{\perp }:$

$$-{\dot{S}}_{\perp }^2=q^2\left[S_{\perp }^2-1+{\left(\frac{c{S}_{\perp }^2-2\mathrm{\mathscr{H}}}{q}+1\right)}^2\right].$$ [S17]

$\mathrm{\mathscr{H}}$ is the conserved mean-field energy. The right-hand side of the above equation can be considered as an effective potential, $V\left(S_{\perp }\right)$ governing the dynamics of $S_{\perp }$ in the $S_x,S_y$ plane. In effect, obtaining the above equation is a Legendre transformation of the Hamiltonian. The potential $V\left(S_{\perp }\right)$ takes the shape of a Mexican hat (Fig. 1 of main text). By Taylor expanding this potential around the GS, we obtain the following dynamical equation for small oscillations, $\delta S_{\perp }$ about $S_{\perp ,\text{GS}}:$

$$\delta {{\dot{S}}_{\perp }}^2+\left(4{\left|c\right|}^2-q^2\right)\delta S_{\perp }^2=\frac{{\mathrm{\Delta }}_{\mathrm{\mathscr{H}}}{q}^2}{\left|c\right|}.$$ [S18]

${\mathrm{\Delta }}_{\mathrm{\mathscr{H}}}$ is the corresponding excitation above the GS. Note that the frequency of this amplitude oscillation is formally identical to the Bogoliubov excitation energy, Eq. S8. The amplitude mode oscillation corresponds to an excitation in the massive Bogoliubov mode (10), described in Eq. S12. The remaining two excitation modes are the massless NG modes arising due to the spontaneous breaking of $\text{U}\left(1\right)\times \text{SO}\left(2\right)$ symmetry. The massive mode is the analog of the Higgs mode, which in general can be regarded as an amplitude excitation of an order parameter in a spontaneous symmetry breaking (34, 35, 37, 39). This Higgs mode is different from the Anderson–Higgs mechanism, in which a massless gauge field in combination with the spontaneous symmetry breaking can generate a massive boson (4, 5).

Measuring Spin Interaction Energy

The spin interaction energy is defined as $c=2N\tilde{c}$ with $\tilde{c}\propto N^{-3/5}$ (22), so that $c\propto N^{2/5}.$ If $q/\left|c\right|>2,$ a state initialized with ${\rho }_0=1$ remains at this value. To search for the critical point $q/\left|c\right|=2,$ we measure the population ${\rho }_0$ after a short evolution time (150 ms) at different magnetic field values with $q=q_z{B}^2$ (similar to the experiment carried out in ref. 23). At some critical value, the population ${\rho }_0$ will begin to evolve away from 1. This critical value, $B_{\text{c}},$ is used to calculate the spin interaction energy as $c=-q_z{B}_{\text{c}}^2/2.$ We can compare the ${\rho }_0$ data with simulations of different c values and determine c with $\sim 1$% uncertainty as shown in Fig. S3.

Energy Gap Experiment

In our experiment, the condensate is prepared at a high magnetic field ($q/\left|c\right|\sim 38$). This state has symmetric fluctuations in the $S_{\perp },Q_{\perp }$ plane set by the uncertainty limit as in Eq. S6. The system is subsequently quenched to a lower field in 2 ms. A radio frequency (RF) pulse (transition between $|m_F=0\rangle \leftrightarrow |m_F=\pm 1\rangle$) is applied to prepare the system at the ${\rho }_{0,\text{GS}}$ value and a subsequent microwave pulse (transition between $|F=1,m_F=0\rangle \leftrightarrow |F=2,m_F=0\rangle$) (21) is applied to rotate the spinor phase ${\theta }_s={\theta }_{+}+{\theta }_{-}-2{\theta }_0$ to the GS value. After this initial state preparation, the condensate is allowed to evolve, following the energy contours seen on the spheres in Fig. 1B. Coherent dynamics are observed through time evolution of the population ${\rho }_0$ and transverse spin component $S_{\perp }.$ The measurements of coherent oscillations are shown in Fig. S4 for different $q/\left|c\right|$ values.

Measuring $\mathrm{\Delta }{S}_{\perp }$.

Note that a $\pi /2$-RF pulse can be used to rotate $S_x$ into the $S_z$ measurement axis. Details of this protocol are described in ref. 21. As quantum states evolve along their respective energy contours, the projection onto the $S_{\perp }$ axis oscillates with a frequency equivalent to the energy gap ${\mathrm{\Delta }}_E.$ Because we are unable to track the Larmor phase of the spin vector due to its fast dynamics, the Larmor phase is considered to be uniformly distributed in the $S_x,S_y$ plane of the spin space. Therefore, measuring the standard deviation $\mathrm{\Delta }{S}_x$ is equal to measuring $\mathrm{\Delta }{S}_{\perp }.$

In the main text Fig. 2B, the condensate is prepared at the polar GS at a high magnetic field ($q/\left|c\right|\sim 38$). Whereas the mean-field GS ($|m_F=0\rangle$ state) remains unchanged as the system is quenched to the QCP for $S_{\perp }$ measurements, the initial quantum fluctuation projects the condensate to slightly excited states of the Hamiltonian at the QCP. Therefore, the state used in the $S_{\perp }$ measurements is a close approximation of the actual GS, because the QCP is limited by the quantum noise.

Sinusoidal Fitting.

To extract the oscillation frequency, the data are fitted to a sinusoidal function of the form ${\rho }_0\left(t\right)={\rho }_{00}+A\cos \left(2\pi f\left(t\right)t+\varphi \right)+a\times t$ as shown in Fig. S4. Fitting parameters include the initial population value (${\rho }_{00}$), the oscillation amplitude (A), the initial phase(ϕ), and the drift of the GS population due to atom loss (a). Due to atom loss, the frequency is a function of time, $f\left(t\right).$ Because $f_0\equiv {\mathrm{\Delta }}_E\left(0\right)$ and $f\left(t\right)\equiv {\mathrm{\Delta }}_E\left(t\right),$ the frequency $f\left(t\right)=f_0\left({\mathrm{\Delta }}_E\left(t\right)/{\mathrm{\Delta }}_E\left(0\right)\right),$ where $f_0$ is the oscillation frequency at $t=0.$ The energy gap $\mathrm{\Delta }\left(t\right)$ is calculated using the Hamiltonian in Eq. S7 with spin interaction energy depending on the number of atoms $c\left(t\right)=c\left(0\right)\times {\left(N\left(t\right)/N\right)}^{2/5}\approx c\left(0\right)\text{exp}\left(-\left(2/5\right)\left(t/\tau \right)\right).$ The trap lifetime τ is determined using the coherent oscillation data. The oscillation frequency of the $\mathrm{\Delta }{S}_{\perp }$ dynamics is extracted using a similar fit function.

Energy Gap Raw Data.

The energy gap is obtained by fitting the data to the sinusoidal functions described above. Fig. S5 shows the summary of all of the frequency fits. Of the several measurements at each $q/\left|c\right|$ value, the one with the smallest-amplitude oscillation that could still provide a reliable fit corresponds to the closest measurement of the energy gap. The resulting points are shown in Fig. 2C.

QPT Dynamics

We investigate the dynamics of the QPT through simulations by numerically integrating the quantum Hamiltonian (Eq. S7). In the simulations, the condensate is prepared at the P-phase ground state just above the QCP. The quadratic Zeeman coefficient is ramped linearly from an initial value $q_{\text{i}}=3\left|c\right|$ to a final value $q_{\text{f}}=0,$ with quench rate $v_q$ so that $q\left(t\right)=3\left|c\right|-v_{q}t.$ We investigate the adiabaticity with different ramp speeds for a broad range of atom numbers. To quantify the adiabaticity of the phase transition crossing, we calculate the probability of excitation out of the GS ($P_{\text{E}}$), the fractional population ${\rho }_0,$ and the variance $\mathrm{\Delta }{\rho }_0^2$ after the quench (Fig. S2). The excitation probability is calculated from the final wavefunction ${\psi }_{\text{f}}$ and the GS wavefunction ${\psi }_{\text{GS}}$ at $q_{\text{f}}=0$ as $P_{\text{E}}=1-{\left|\langle {\psi }_{\text{f}}|{\psi }_{\text{GS}}\rangle \right|}^2.$ When the system crosses the phase transition adiabatically, the excitation probability is close to zero; i.e., the final state is the ground state.

Adiabaticity can be achieved with a fast ramp in a small atom number condensate whereas a slower ramp is required for a large atom number condensate (Fig. S2A). This can be understood by considering the energy gap for different atom numbers (Fig. S1). In the large condensate, the minimum energy gap is very small, whereas the minimum gap is larger for the smaller atom number in the condensate. The larger minimum gap represents an easier adiabatic passage.

A meaningful connection to Landau–Zener (LZ) theory can be drawn by considering only the GS and first excited state. This is justified by the exactness of the equivalence between the Bogoliubov approximation of the energy gap and coherent oscillation frequency, where the latter is obtained assuming possible excitations beyond the first excited state. The contribution of excitations beyond the first excited state is thereby expected to be small. The two-level energy gap is taken to be the same as is in Fig. S1. Then the probability to excite the system out of the GS can be approximated using LZ theory as (41, 43, 44)

$$P_{\text{LZ}}\approx e^{-\pi {\mathrm{\Gamma }}_{\text{LZ}}/4}.$$ [S19]

Here ${\mathrm{\Gamma }}_{\text{LZ}}$ is LZ parameter

$${\mathrm{\Gamma }}_{\text{LZ}}\equiv \frac{{\mathrm{\Delta }}_E^2}{\left(d{\mathrm{\Delta }}_E/dt\right)}=\frac{{\mathrm{\Delta }}_E^2}{v_q\left(d{\mathrm{\Delta }}_E/dq\right)},$$ [S20]

where ${\mathrm{\Delta }}_E$ is a function of N, c, and $q\left(t\right).$ It is seen from Eq. S19 that a large value of ${\mathrm{\Gamma }}_{\text{LZ}}$ corresponds to low excitation, i.e., adiabaticity. From Eq. S20, this is accomplished with a small $v_q$ or a slow quench. The most pivotal point in evolution is near the minimum of ${\mathrm{\Delta }}_E,$ where ${\mathrm{\Gamma }}_{\text{LZ}}$ reaches a minimum for a constant $v_q$ (linear) quench. This is where adiabaticity is most difficult to achieve. We therefore specify this minimum value ${\mathrm{\Gamma }}_{\text{LZ,min}}$ and study the excitation probability as a function of this minimum LZ parameter. We observe that the adiabatic QPT occurs when ${\mathrm{\Gamma }}_{\text{LZ,min}}\mathrm{\gtrsim }10$ for a broad range of atom numbers (Fig. S2B). The probability $P_{\text{LZ}}$ (Eq. S19) indicated by a dashed curve sets a lower bound for the excitation probability of the many-body system (41, 43, 44).

Adiabatic QPT Experiment

In the adiabatic QPT experiment, the atoms are confined in the focus of a $10.6$-μm wavelength laser with a trap lifetime of $15-19$ s and spin interaction energy $c\approx -2.1$ Hz. Although the Thomas–Fermi radii of the condensate in the longitudinal direction in this trap are larger than the spin healing length, the spin domains are unlikely to be formed because the adiabaticity maintains the condensate at the GS, leaving no extra energy for domain formation. The single-mode approximation is still valid in describing the adiabatic process.

The LZ parameter depends sharply on q. Therefore, a linear adiabatic ramp is not optimal; it will be much slower than necessary at values of q away from the critical point. An optimal adiabatic ramp would be nonlinear, obtained by adjusting the ramp speed according to the local value of the energy gap. Such a ramp is calculated by a piecewise linear approximation, through an iterative procedure using a semiclassical simulation (45, 46). The ramp is divided into 100 linear sections. For each section, the optimal slope is determined by maintaining adiabaticity within the section. This is done using the following condition for adiabaticity.

An adiabatic invariant is used to determine the condition for adiabaticity (44). The mean-field variables ${\rho }_0$ and ${\theta }_s$ are canonically conjugate with energy function given by Eq. S14. Therefore, the obvious choice of the adiabatic invariant is the action (47) $I=\left(1/2\pi \right){\displaystyle \oint {\rho }_{0}d{\theta }_s},$ where the integral is evaluated along the closed orbit of the state on the phase space. This is equal to the area enclosed inside an orbit on the nematic sphere.

At all noncritical points ($q\ne 2\left|c\right|$), the orbits around the GS are elliptical for small excitations. This can be seen by Taylor expanding the energy Eq. S14 about the GS. It is straightforward to evaluate the action for such orbits:

$$I=\{\begin{array}{l}\frac{2{\mathrm{\Delta }}_{\mathrm{\mathscr{H}}}}{\sqrt{q\left(q-2\left|c\right|\right)}},q>2\left|c\right|\hfill \\ \frac{2{\mathrm{\Delta }}_{\mathrm{\mathscr{H}}}}{\sqrt{4{\left|c\right|}^2-q^2}},q<2\left|c\right|.\hfill \end{array}$$ [S21]

${\mathrm{\Delta }}_{\mathrm{\mathscr{H}}}$ is the difference between the energy of the orbit and the GS energy. We start with an orbit around the pole in P phase with ${\rho }_0=1-1/\left(2N\right).$ The initial action is $I_{\text{initial}}=1/N.$ In the adiabatic limit, i.e., $v_q\to 0,$ the final action is also expected to be $1/N.$ $\mathrm{\Delta }{\rho }_0^2$ is a measure of the action; more precisely, ${\mathrm{\Delta }}_{\mathrm{\mathscr{H}}}=4\mathrm{\Delta }{\rho }_0^2\left|c\right|$ in BA phase. This is obtained by a Taylor expansion of Eq. S14 around the broken symmetry GS, followed by averaging over samples and one oscillation time. Thus, we obtain a condition for adiabaticity:

$$\mathrm{\Delta }{\rho }_0^2\approx \frac{\sqrt{1-q^2/\left(4{\left|c\right|}^2\right)}}{4N}.$$ [S22]

The slope of each section of the ramp is chosen to be the largest one that maintains this condition. Thus, our adiabatic ramp is optimal.

In a linear ramp used within each section, the dynamics are frozen for a short period, depending on the ramp speed as shown by the KZM (11, 13, 14). The value $\mathrm{\Delta }{\rho }_0$ of a fast (nonadiabatic) ramp will not have enough time to increase right after the ramp, and it may satisfy the adiabatic condition (Eq. S22) even though the dynamics are not adiabatic. To avoid this problem, while determining the optimal slope for each section, the system is allowed to evolve for some time before computing $\mathrm{\Delta }{\rho }_0.$

In our experiment, the state preparation is done at a very high magnetic field ($q=140\left|c\right|$). Therefore, before the adiabatic ramp, the system is quenched from the high value of q to a lower value within the polar phase. The adiabatic ramp continues from this lower value of q. It is necessary to verify adiabaticity of the initial quench as well. In the polar phase, it is more convenient to verify adiabaticity using $\mathrm{\Delta }{S}_{\perp }.$ The SQL Eq. S6 serves as a condition for adiabaticity.

Quantifying Nonadiabaticity

In this section, we quantitatively determine how adiabatically the phase transition was carried out experimentally. The action remains invariant in the adiabatic limit: $v_q\to 0.$ However, practically we have a small but nonzero value for $v_q.$ Therefore, the final action, $I_{\text{final}}$ deviates from $I_{\text{initial}}$ by a small amount. For ramps with constant $v_q,$ this deviation has been shown to be proportional to $v_q\log \left(N\right)$ for a system similar to our system (44). Adapting this work to our system, we get

$$I_{\text{final}}=I_{\text{initial}}+\alpha v_q\log \left(N\right).$$ [S23]

Here, α is a numerical constant. This expression is derived by integrating the action over the one single open orbit at the critical point. The deviation from adiabaticity is predominantly accumulated through this orbit. This equation, when expressed in terms of the LZ parameter, reads $\mathrm{\Delta }I\propto 1/{\mathrm{\Gamma }}_{\text{LZ}}.$ This can be compared with the excitation probability Eq. S19, which is an equivalent measure of nonadiabaticity.

In our experiment, we do not use a constant $v_q$ through the evolution. However, most of the nonadiabaticity is accumulated in the vicinity of the critical point. Therefore, we assume that our experiment is equivalent to a linear ramp with ramp speed ${\left\{v_q\right\}}_{\max },$ where the maximum is evaluated over the region $q\in \left[1.95\left|c\right|,2.05\left|c\right|\right].$ This turns out to be ${\left\{v_q\right\}}_{\text{max}}=0.0003{\left|c\right|}^2$ (Fig. 3A in the main text).

Using the correction terms from Eq. S23, we obtain correction terms for Eq. S22:

$$\mathrm{\Delta }{\rho }_0^2=\frac{\sqrt{1-q_{\text{f}}^2/\left(4{\left|c\right|}^2\right)}}{4N}+\alpha v_q\log \left(N\right).$$ [S24]

Here, $q_{\text{f}}$ is the final value of q, at the end of the adiabatic ramp. Fig. S2C shows numerical estimates of $\mathrm{\Delta }{\rho }_0^2,$ averaged over samples as a function of $v_q\log \left(N\right)$ for linear ramps. This simulation was done for $q_{\text{f}}=0$ and in the range $v_q\in \left[0.0004,0.1\right].$ In this range of $v_q,$ the data do not fit well with the above expression; instead, they fit with a different power law scaling of the deviation with $v_q:$

$$\mathrm{\Delta }{\rho }_0^2=\frac{1}{4N}+\beta \left(N\right)v_q^{3/2}.$$ [S25]

β is a constant depending on N. This curve for $N=\mathrm{10,000}$ is shown in Fig. S2C.

We assume that the atom fluctuations are identical for all spin components, $m_F=0,\pm 1.$ If the atom number fluctuation is $\mathrm{\Delta }N,$ the uncertainty in the number of atoms in the spin state $|m_F=0\rangle$ is $\mathrm{\Delta }N/\sqrt{3}$ atoms, which translates into the uncertainty $\mathrm{\Delta }{\rho }_{0,\text{atoms}}=\mathrm{\Delta }N/\left(\sqrt{3}N\right).$ Combining this result with Eq. S22 gives

$$\mathrm{\Delta }{\rho }_{0,\text{total}}^2=\mathrm{\Delta }{\rho }_{0,\text{atoms}}^2+\mathrm{\Delta }{\rho }_0^2.$$

During a 2-s period, the fluctuation is about 300 atoms with the mean number 9,000 atoms, which yields $\mathrm{\Delta }{\rho }_0\approx 0.0047$ and $\mathrm{\Delta }{\rho }_{0,\text{atoms}}\approx 0.0227$ as seen in the main text Fig. 3C. These calculations are done using the remaining number of atoms in the condensate.

Simulation Tools

The details of the simulation method are described in detail in refs. 45 and 46.