Observation of Many-Body Quantum Phase Transitions beyond the Kibble-Zurek Mechanism

Quantum critical behavior of many-body phase transitions is one of the most fascinating yet challenging questions in quantum physics. Here, we improved the band-mapping method to investigate the quantum phase transition from superfluid to Mott insulators, and we observed the critical behaviors of quantum phase transitions in both the dynamical steady-state-relaxation region and the phase-oscillation region. Based on various observables, two different values for the same quantum critical parameter are observed. This result is beyond a universal-scaling-law description of quantum phase transitions known as the Kibble-Zurek mechanism, and suggests that multiple quantum critical mechanisms are competing in many-body quantum phase transition experiments in inhomogeneous systems.

Figure 1

Experimental setup and improved band-mapping method. (a) Three orthogonal standing waves form 3D optical lattices for ${}^{87}\mathrm{Rb}$, and absorption images are taken along the $z$ direction. The inset: the $z$ lattice is turned off instantaneously while the $x$ and $y$ lattices are ramped down in 2 ms for the band mapping. (b) The incoherent fraction ${\gamma }_{\mathrm{inc}}$ versus the trap depth $V$ for the SF-MI phase transitions. The red circles correspond to the measurement with adiabatic ramping, while the blue squares correspond to the measurement with linear ramping at rates $k=2.5E_r/\mathrm{ms}$. The shadow area locates quantum phase transition at $V_c=13E_r$ for $\overline{n}=1$’s lobe. Gray arrows indicate data in the panel (c) and (d). The inset shows the decomposition of quasimomentum profiles and analysis of ${\gamma }_{\mathrm{inc}}$. (c) and (d) The band mapping profiles performed at trap depth $V=7$, 13, 19, and $35E_r$. Error bars (1 standard deviation) are smaller than the marker size.

Figure 2

Dynamical response of SF-MI phase transitions. (a) The time sequence of the trap depth ramping. The first stage of 80 ms and the second stage of 20 ms prepare superfluid samples from the condensates. The third stage is the linear ramp with a ramping rate $k$ when the atoms experience phase transitions. The final stage is the band mapping to distinguish the coherent component from the incoherent one. (b) The incoherent fraction ${\gamma }_{\mathrm{inc}}$ versus trap depth $V$ for different $k$. The scattered markers are the experimental data and the solid lines are the polynomial fits (see SM-V [37]). A furcation appears at critical point $V_c=13E_r$. And ${\gamma }_{\mathrm{inc}}$ approaches 1 for different ramping rate $k$ for small $k$. When $k$ gets larger, ${\gamma }_{\mathrm{inc}}$ starts to oscillate with retard thermalization. The pink triangles denote the dynamics at $k=14E_r/\mathrm{ms}$, which oscillates drastically and fail to relax. Here we label the definition of SF response time ${\tau }_{\mathrm{SF}}$ (time to reach ${\gamma }_{\mathrm{inc}}=0.6$) and the MI dynamical relaxation time ${\tau }_{\mathrm{MI}}$ (time between ${\gamma }_{\mathrm{inc}}=0.6$ and ${\gamma }_{\mathrm{inc}}=0.9$) on the plot. (c) ${\tau }_{\mathrm{SF}}$ versus $k$. The blue circles are the experimental data and the blue solid line is the fit based on power laws with ${\tau }_{\mathrm{SF}}\sim k^{-0.91(3)}$. The red diamonds are the GMFT simulation with a fit ${\tau }_{\mathrm{SF}}\propto k^{-0.95(2)}$. (d) ${\tau }_{\mathrm{MI}}$ versus $k$. The red filled circles are the extracted data. The blue dashed line is ${\tau }_{\mathrm{MI}}\sim k^{-1}$ for the adiabatic response (see SM-VI [37]). The black solid line is the fit with ${\tau }_{\mathrm{MI}}\sim k^{-0.50(5)}$ for $k\ge 0.7E_r/\mathrm{ms}$, which indicates $\nu z=1.0(2)$. (e) The universal response of ${\gamma }_{\mathrm{inc}}$ versus the rescaled trap depth $V_{\mathrm{eff}}$, where $V_{\mathrm{eff}}=(V-V_c)k^{-(1-0.50)}+V_c$ and $k$ is in the unit of $E_r/\mathrm{ms}$. Error bars (see SM-V [37]) correspond to 1 standard deviation.

Figure 3

Excitation fraction $n_{\mathrm{ex}}$ in the MI region. (a) $n_{\mathrm{ex}}(k)$ measured at $V\sim 18E_r$ (SM-VI [37]). Blue circles are experimental results with the blue line fit of $n_{\mathrm{ex}}\sim k^{\alpha }$ at $\alpha =0.97(12)$. Orange squares are GMFT simulation results with the orange line fit of $n_{\mathrm{ex}}\sim k^{\alpha }$ at $\alpha =0.97(5)$. (b) The fitted scaling coefficient $\alpha$ is robust against the chosen trap depth $V$, as long as $V>13E_r$ (goes through phase transition) and $V<19E_r$ (far from deep MI region). Error bars correspond to 1 standard deviation.

Figure 4

Phase oscillations under a fast ramp. (a) Time sequences for oscillation measurements. The atoms are hold at $V=25E_r$ for a varying time $t$. (b) ${\gamma }_{\mathrm{inc}}$ oscillates versus time $t$ for different $k$. The solid lines are cubic splines fit. (c) The fitted oscillation amplitude $A$ versus $k$. Blue circles are experimental data and the red squares are GMFT simulations. Two dashed lines are the fits based on power laws. For experimental data we obtain $A(k)\propto k^{1.49(17)}$, and the GMFT gives $A(k)\propto k^{1.41(15)}$. (d) An illustration of the dynamics in the SF-MI phase diagram. The blue lines represents our actual system and the on- or off-tip locations are labeled by arrows with $\nu z=1/2$ or 1. The on-tip location has square-root gap opening and the off-tip one has linear gap opening.