Quantum Simulation of Landau-Zener Model Dynamics Supporting the Kibble-Zurek Mechanism

The Kibble-Zurek mechanism (KZM) captures the key physics of nonequilibrium dynamics in second order phase transitions, and accurately predicts the density of topological defects formed in such processes. However, the central prediction of KZM—i.e., the scaling of the density of defects with the quench rate—still needs further experimental confirmation, particularly for quantum transitions. Here, we perform a quantum simulation of the nonequilibrium dynamics of the Landau-Zener model based on a nine-stage optical interferometer with an overall visibility of $0.975\pm 0.008$. The results support the adiabatic-impulse approximation, which is the core of Kibble-Zurek theory. Moreover, the developed high-fidelity multistage optical interferometer can support more complex linear optical quantum simulations.

Figure 1

The relationship between the Kibble-Zurek mechanism and the Landau-Zener model. (a) Avoided level crossing LZM [see Eq. (1)]. (b) The relaxation time $\tau$ in the second order phase transition, where the relaxation time is divergent near the critical point. The whole dynamical process can be divided into three regions according to KZM (adiabatic, impulse, and adiabatic), signified by the graduated color background and separated by the freeze out time $\widehat{t}$ determined by Zurek’s Eq. [2]. (c) The inverse of the energy gap in LZM, which plays a similar role to the relaxation time in second order transitions. The inverse of the gap in LZM is not divergent at the crossing point. The quench velocity in this case cannot be too small, which guarantees that the dynamics are not adiabatic throughout. Under this condition, the dynamics can also be divided into the same three regions under adiabatic-impulse approximation, and a similar freeze out time can be introduced and estimated [15, 20].

Figure 2

Experimental setup. (a) A standard Mach-Zender interferometer (MZI), whose logic diagram is shown in (b) (detailed information is given in the Supplemental Material [33]), (c) provides the experimental details. The initial polarization state is prepared with a polarized beam splitter (PBS) followed by a half wave plate (HWP). Each step of the evolution process is realized by a module composed of a polarization-dependent Sagnac interferometer. By using two quarter-wave plates (QWPs) independently in each arm, and two HWPs (one in front of and the other after the interferometer), we can adjust the evolution parameters. A phase compensator (PC) is inserted in one arm to correct the phase error and there is an adjustable slit in the other arm for calibration. A 1 m long single-mode fiber (SMF) connects two cascaded interferometers and also acts as a spatial filter. After a nine-step evolution, the photons are guided to the polarization analysis module, which consists of an HWP, a Wollaston prism, and two single photon detectors (SPDs). A 1 nm interference filter (IF) is introduced before the Wollaston prism to increase the coherence length. Two QWPs, one after the state preparation and the other before the polarization analysis module, are inserted to perform the process tomography.

Figure 3

Experimental demonstration of the dynamics of LZM. We show here the dynamics of LZM with different parameters. In order to study the AI approximation of KZM in LZM, we need to observe the probability of H polarization with different parameters ${\tau }_Q/{\tau }_0={\omega }_0^2/\mathrm{\Delta }$. The upper, middle, and lower lines are typical dynamics corresponding to ${\tau }_Q/{\tau }_0=4.5\times {10}^{-3}$, 0.2, and 1.25, respectively, where ${\omega }_0=0.03$, 0.2, and $0.50{\tau }_s^{-1}$, respectively, and $\mathrm{\Delta }=0.2{\tau }_s^{-2}$ for all cases. The points are experimental results and the solid lines are theoretical predictions by numerical calculation with the same parameters as in the experiment. The horizontal axis is the evolution step. Due to current physical limitations, our complete experiment consists of 9 steps. In the experiment, the evolution time for each step equals ${\tau }_s$. From these dynamics, we can roughly find the freeze out time $\widehat{t}$: approximately $2{\tau }_s$, $1{\tau }_s$, and less than $1{\tau }_s$, respectively, from upper to lower. Error bars (not shown) have a width of 0.011 and are dominated by the inaccuracies in wave plate rotation.

Figure 4

Experimental results for demonstrating the AI approximation of KZM in LZM. The blue dashed line is the theory prediction from the AI approximation of KZM with $\alpha =0.8$ (the free parameter in the theory and found through numerical fitting) following the solution put forth by Damski [15, 20]. The pink dot-dash line is the direct analytical solution of the dynamical LZM equation with the same parameters. These two methods give almost the same results, showing the validity of the AI approximation of the Kibble-Zurek theory in the non-equilibrium dynamics of LZM. Our experimental results are shown by the points, and the green solid line shows the numerical solution of our experimental system. Our experimental results agree quite well with the numerical solution. Error bars (not shown) are the same as in Fig. 3.