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

Edited by Subir Sachdev, Harvard University, Cambridge, MA, and approved June 28, 2016 (received for review January 7, 2016)

## 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 *P*) phase and nonzero in the broken axisymmetry (BA) phase (Fig. 1*A*). Representing the transverse spin vector as a complex number,

The amplitude mode can be studied in different spinor phases by tuning the relative strengths of the quadratic Zeeman energy per particle *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. 1*A*), 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

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 *A*), 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 ^{87}Rb BEC with *B* along the *z* axis is (17⇓⇓⇓–21)*B* with axes *B*). 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, *P* phase and the BA phase in the long-wavelength limit corresponds to the oscillation frequency of small excitations in *f* (**1** (*Supporting Information*). Fig. 1*C* 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. 1*D* shows the relation of energy gap at the QCP to the atom number in condensates ranging from *Supporting Information*) for a broad range of atom numbers (red circles in Fig. 1*D*). 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 *B*) for different values of *Supporting Information*). For each *A*. The GS population *Supporting Information*) (24)*B*) (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, *B* for a state prepared in the polar GS (*Supporting Information*).

The results of the energy gap measurements are shown in Fig. 2*C* 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

In the neighborhood of the QCP, the energy gap decreases dramatically. As shown in Fig. 2*C*, *Inset*, the measurements are in good agreement with the theoretical prediction in this region. For measurements at *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 (*C*.

A further complication in the measurement at the QCP is that the value *C*, and the energy gap is determined by the frequency at

In the BA phase of a *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: *m* being the atomic mass (10, 11). The energy gap

## Adiabatic QPT

In the thermodynamic (

Due to the small size of the gap, the ramp in *q* needs to be very slow in the region of *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, *A*. The measured evolution of the population **3**), which provides a strong indication of adiabaticity.

There are about 9,000 atoms remaining after the adiabatic ramp. The theoretical value of the GS population and uncertainty is *B*) (*Supporting Information*). Immediately after the adiabatic ramp (*B*, the mean value of *C*, the variance *Supporting Information*).

For comparison, in Fig. 3 *D* and *E* we show data from nonadiabatic ramps from *D*, a 1-s linear ramp is used, whereas in Fig. 3*E*, a 28-s ramp is used. In both cases, the spin population *C* (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

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 ^{87}Rb. In the energy gap experiment, atoms are confined in a spherical optical dipole force trap with trap frequencies

## Initial State Heisenberg Uncertainty

All atoms are prepared in the *k* is the number of pairs of atoms in the

## Energy Gap Calculation

The gap *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 (*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*B*) (46). In particular, **S14** under these constraints, we obtain the *P*-phase and BA-phase GSs (Eq. **3** of the main text):

### Coherent Oscillations in the Amplitude Mode.

From Eq. **S15** the GS value of *P* phase, the set of GSs is represented by a point in the *P* phase to the BA phase is accompanied by a spontaneous breaking of an

The mean-field equation for **S13**, we obtain the following dynamical equation for **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

## Measuring Spin Interaction Energy

The spin interaction energy is defined as *c* values and determine *c* with

## Energy Gap Experiment

In our experiment, the condensate is prepared at a high magnetic field (**S6**. The system is subsequently quenched to a lower field in 2 ms. A radio frequency (RF) pulse (transition between *B*. Coherent dynamics are observed through time evolution of the population

### Measuring Δ S ⊥ .

Note that a

In the main text Fig. 2*B*, the condensate is prepared at the polar GS at a high magnetic field (

### Sinusoidal Fitting.

To extract the oscillation frequency, the data are fitted to a sinusoidal function of the form *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, **S7** with spin interaction energy depending on the number of atoms *τ* is determined using the coherent oscillation data. The oscillation frequency of the

### 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 *C*.

## 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

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. S2*A*). 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)*N*, *c*, and **S19** that a large value of **S20**, this is accomplished with a small *B*). The probability **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

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 **S14**. Therefore, the obvious choice of the adiabatic invariant is the action (47)

At all noncritical points (**S14** about the GS. It is straightforward to evaluate the action for such orbits:*P* phase with **S14** around the broken symmetry GS, followed by averaging over samples and one oscillation time. Thus, we obtain a condition for adiabaticity:

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 **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

In our experiment, the state preparation is done at a very high magnetic field (*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 **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: *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 **S19**, which is an equivalent measure of nonadiabaticity.

In our experiment, we do not use a constant *A* in the main text).

Using the correction terms from Eq. **S23**, we obtain correction terms for Eq. **S22**:*q*, at the end of the adiabatic ramp. Fig. S2*C* shows numerical estimates of *β* is a constant depending on *N*. This curve for *C*.

We assume that the atom fluctuations are identical for all spin components, **S22** gives*C*. These calculations are done using the remaining number of atoms in the condensate.

## Acknowledgments

We thank K. Wiesenfeld, F. Robicheaux, T. Li, and T.A.B. Kennedy for useful discussions. The authors acknowledge support from National Science Foundation Grant PHY-1506294.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: mchapman{at}gatech.edu.

Author contributions: T.M.H. and M.S.C. designed research; T.M.H. performed research; T.M.H. analyzed data; T.M.H., H.M.B., M.J.B., M.A., B.A.R., and M.S.C. wrote the paper; M.S.C. conceived the study and supervised the research; T.M.H. conceived the study, developed essential theory, and carried out the simulations.; M.A. and B.A.R. assisted data taking; and H.M.B. and M.J.B. developed essential theory and carried out the simulations.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1600267113/-/DCSupplemental.

Freely available online through the PNAS open access option.

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