Quantum Simulation of Massive Relativistic Fields in 2 + 1 Dimensions

Abstract

Quantum field theories provide fundamental models of complex interacting systems, from high-energy physics and cosmology to condensed matter. However, solving these models in non-perturbative and dynamical regimes is often extremely challenging, particularly in more than one spatial dimension. Analog simulation using tunable synthetic quantum systems can both verify existing theoretical predictions and lead to new physical insights. Here, we realize quantum simulation of massive relativistic fields in $2+1$ dimensions (two spatial dimensions and time), using two coherently coupled spin components in a uniform two-dimensional Bose-Einstein condensate. Specifically, we encode the paradigmatic sine-Gordon model in the field describing the relative phase, $φ$, of the two components. We show that, in the perturbative regime, collective field excitations exhibit a relativistic dispersion with a tuneable mass gap. We also observe explicitly non-perturbative phenomena, including the existence of topological domain walls across which $φ$ rapidly winds by $2π$. Our work opens possibilities for studies of cosmologically relevant phenomena including preheating, dynamics of topological defects, and relativistic false-vacuum decay.

Figures (5)

• Figure 1: Two-dimensional quantum simulator for massive relativistic fields.a, The concept. In our two-dimensional BEC, the density $n$ is essentially uniform, and we use a spatially varying spin state, shown by the Bloch vector, to encode a massive relativistic field. The conjugate variables $Z(x,y)$ and $\phi(x,y)$ are, respectively, the local population imbalance and relative phase of spin components $\ket{\uparrow}$ and $\ket{\downarrow}$, which are coherently coupled by a radio-frequency (RF) field, indicated by the Rabi vector. b, Sine-Gordon model. In the Josephson regime, $\mu_\text{s} \gg \hbar \Omega$, where $\mu_\text{s}$ is the spin chemical potential and $\Omega$ the RF Rabi frequency, $\phi$ evolves under a sinusoidal potential $U$. c, The experiment. Our $\ket{\uparrow}$ and $\ket{\downarrow}$ are two hyperfine states of $^{39}$K. We image both the local populations $n_{\uparrow,\downarrow} = |\psi_{\uparrow,\downarrow}|^2$ (where $\psi_{\uparrow,\downarrow}$ are the wavefunctions) and the transverse spin projections $n_{+,-}=|\psi_\uparrow\pm \psi_\downarrow|^2/2$ and $n_{i,-i}= |{ i}\psi_\uparrow\pm \psi_\downarrow|^2/2$. Here, we show example images of $n_+$, $n_{ -i}$, and $n_\uparrow$, for a cloud with a magnetically imprinted $\phi$ gradient. The scale bar corresponds to $20~\upmu$m; OD is optical density.
• Figure 2: Robust simulator initialization.a, Experimental protocols. Spin and total-density dynamics decouple for $Z=Z_0$ (see text). In protocol ①, we prepare the system at the predicted $Z_0$ using strong RF coupling, and then reduce this coupling to reach the Josephson regime, $\mu_{\rm s} \gg\hbar \Omega$. Alternatively, in ②, we start with a strongly coupled symmetric mixture ($Z=0$) and, as we reduce $\Omega$, a nonzero $Z$ spontaneously emerges and tends towards $Z_0$. b, Experimental data for protocols ① (open symbols) and ② (solid symbols), for two sets of interaction parameters corresponding to different $Z_0$ values (Methods). The dashed lines show the predicted $Z_0$. The solid lines show predictions based on Eq. (\ref{['eq:susceptibility']}): for the same $\Omega$, the RF field in protocol ② is detuned from that in ① by $-Z_0\Omega$, so $Z \simeq Z_0 (1 - \chi \Omega)$.
• Figure 3: Plasma oscillations.a, A jump in the RF-field phase, $\phi_{\rm RF}$, initiates global spin oscillations. Interactions squash the oscillation trajectory (blue ellipse) in the $Z$ direction. b, An example of $Y$ and $Z$ oscillations, for $\mu_{\rm s}/(\hbar\Omega)\simeq11$; note that $\phi \simeq Y$. Damped-sine fits (solid lines) give the plasma frequency $\omega_\text{p}$ and the oscillation amplitudes $A_Y$ and $A_Z < A_Y$. c, Dependence of $\omega_\text{p}/\Omega$ and $A_Z/A_Y$ on $\mu_\text{s}/(\hbar\Omega)$. The Rabi and Josephson regimes, respectively, correspond to $\mu_\text{s}/(\hbar\Omega) \ll 1$ and $\gg 1$. The solid lines show predictions from Eqs. (\ref{['eq:aspect-ratio']}) and (\ref{['eq:plamsa-freq']}). For measurements here, $Z_0 = 0.31$.
• Figure 4: Massive relativistic dispersion relation.a, Experimental protocol. We parametrically excite the system by modulating $\Omega$ at frequency $\omega_\text{m}$. The injected energy, in quanta of $\hbar\omega_\text{m}$, is converted to pairs of spatial excitations with opposite momenta, each carrying energy $\hbar\omega_\text{m}/2$. b, Example measurements. Top: After the parametric modulation, images of $n_{-i}$ (top) show fluctuations with a characteristic $\omega_\text{m}$-dependent length scale; the scale bar corresponds to $20\,\upmu$m. Bottom: Normalizing the spatial power spectrum of the fluctuations observed with modulation to that without modulation, we reveal the dominant $k$ excited at each $\omega_{\rm m}$, which gives $\omega_{\rm s}(k) = \omega_{\rm m}/2$. Solid lines show Gaussian fits. c, Dispersion relations for different field mass $m^*$, tuned by varying the value of $\Omega$ (around which we modulate). The solid lines show theoretical predictions from Eq. (\ref{['eq:dispersion']}). The error bars in $k$ denote the Gaussian widths of the spectra such as shown in b. We also include $k=0$ plasma-oscillation measurements. The dotted line shows the inverse spin healing length $1/\xi_{\rm s} = \sqrt{2m\mu_\text{s}/\hbar^2}$, and the dashed ones the inverse magnetic healing lengths, $1/\xi_\text{M} =\sqrt{2m\Omega/\hbar}$; in our system, $\xi_\text{M}$ corresponds to the reduced Compton wavelength $\hbar/(m^*c)$. For these measurements $Z_0=0.31$.
• Figure 5: Non-perturbative sine-Gordon dynamics.a, Experimental concepts. We study non-perturbative dynamics following a large initial phase displacement $\phi_0$. Left: For large $|\phi_0|<\pi$, the concavity of the sine-Gordon potential $U(\phi)$ leads to an amplitude-dependent oscillation period. Right: $\phi_0=\pi$ is an unstable equilibrium point, and fluctuations cause different regions of the system to settle to different potential minima. This leads to the formation of domain walls across which $\phi$ rapidly winds by $2\pi$. b, Large-amplitude plasma oscillations. Left: We measure $\phi(t)$ for different $|\phi_0| < \pi$ and extract the time, $\tau$, for the system to return to $\phi =0$. Solid lines show sinusoidal fits. Note that for the largest $\phi_0$ here the oscillation is visibly anharmonic. Right: $\tau/\tau_0$ versus $\phi_0$; here $\tau_0 = \pi/(2\omega_{\rm p})$ is the $\phi_0$-independent result in the perturbative (harmonic) regime. The solid line is the theoretical prediction based on the mechanical-pendulum model. For these measurements, $Z_0 = 0.31$, $\Omega/(2\pi) \simeq 74\,$Hz, $\mu_\text{s}/(\hbar\Omega) \simeq11$, and $m^*/m\simeq0.41$. c, Domain walls. We show examples of domain walls seen in the $n_+(x,y)$ and $n_-(x,y)$ images, where they appear as, respectively, dark and white lines; each image shows an independent experimental realization. The system is prepared at $\phi\simeq\pi$ at time $t=0$, and relaxes towards $\phi=0$ or $2\pi$ within $10\,$ms (here corresponding to $\approx 7\tau_0$). The small red scale bar corresponds to $\lambdabar = \hbar/(m^* c) =\sqrt{\hbar/(2m\Omega)} \simeq 2.4\,\upmu$m, the theoretically predicted characteristic width of a domain wall. The black scale bar corresponds to $20\,\upmu$m. For these measurements, $Z_0 = 0.53$, $\Omega/(2\pi) \simeq 22\,$Hz, $\mu_{\rm s}/(\hbar\Omega)\simeq39$, and $m^*/m\simeq0.22$. d, Effect of $\Omega$ on the domain-wall width. Here the protocol is as in c, except that for the last $2\,$ms of relaxation, we jump $\Omega/(2\pi)$ from $22\,$Hz to either $45\,$Hz or $10\,$Hz. As predicted, the walls are narrower in the former case and wider in the latter.