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Experimentally Testing Quantum Critical Dynamics Beyond The ARTICLE https://doi.org/10.1038/s42005-020-0306-6 OPEN Experimentally testing quantum critical dynamics beyond the Kibble–Zurek mechanism ✉ ✉ Jin-Ming Cui1,2, Fernando Javier Gómez-Ruiz 3,4,5, Yun-Feng Huang1,2 , Chuan-Feng Li1,2 , ✉ Guang-Can Guo1,2 & Adolfo del Campo3,5,6,7 1234567890():,; The Kibble–Zurek mechanism (KZM) describes the dynamics across a phase transition leading to the formation of topological defects, such as vortices in superfluids and domain walls in spin systems. Here, we experimentally probe the distribution of kink pairs in a one- dimensional quantum Ising chain driven across the paramagnet-ferromagnet quantum phase transition, using a single trapped ion as a quantum simulator in momentum space. The number of kink pairs after the transition follows a Poisson binomial distribution, in which all cumulants scale with a universal power law as a function of the quench time in which the transition is crossed. We experimentally verified this scaling for the first cumulants and report deviations due to noise-induced dephasing of the trapped ion. Our results establish that the universal character of the critical dynamics can be extended beyond KZM, which accounts for the mean kink number, to characterize the full probability distribution of topological defects. 1 CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China. 2 CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China. 3 Donostia International Physics Center, E-20018 San Sebastián, Spain. 4 Departamento de Física, Universidad de los Andes, A.A. 4976 Bogotá D. C., Colombia. 5 Department of Physics, University of Massachusetts Boston, 100 Morrissey Boulevard, Boston, MA 02125, USA. 6 IKERBASQUE, Basque Foundation for Science, E-48013 ✉ Bilbao, Spain. 7 Theoretical Division, Los Alamos National Laboratory, MS-B213, Los Alamos, NM 87545, USA. email: [email protected]; cfl[email protected]; [email protected] COMMUNICATIONS PHYSICS | (2020) 3:44 | https://doi.org/10.1038/s42005-020-0306-6 | www.nature.com/commsphys 1 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0306-6 he understanding of nonequilibrium quantum matter systems. This result forms the basis of much of the progress on stands out as a fascinating open problem at the frontiers of the study of quantum phase transitions and can be established via T – 38 physics. Few theoretical tools account for the behavior the Jordan Wigner transformation , Fourier transform, and away from equilibrium in terms of equilibrium properties. One Bogoliubov transformation39. We detail these steps in the Sup- such paradigm is the so-called Kibble–Zurek mechanism (KZM) plementary Note 1, where it is shown that the TFQIM Hamil- that accounts for the universal critical dynamics of a system tonian can be alternatively written in terms of independent driven across a phase transition. Pioneering insights on the KZM modes as37,40 1 X X were conceived in a cosmological setting and applied to describe y y – H¼^ ^ ¼ ϵ ðÞ^γ ^γ þ ^γ ^γ À ; thermal continuous phase transitions2 4. The resulting KZM was Hk k g k k Àk Àk 1 ð2Þ later extended to quantum phase transitions5–8, see ref. 9 for a k>0 k>0 ^γ review. Its central prediction is that the average total number of where k are quasiparticle operators, with k labeling each mode topological defects 〈nT〉, formed when a system is driven through ¼ π ðÞÀ ¼ÀN þ ; ¼ ; N and taking values k N 2m 1 with mqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 2 . The a critical point in a time scale τQ, is given by a universal power Àβ ϵ ϵ ðÞ¼ ðÞÀ 2 þ 2 h Tiτ β = ν∕ + ν fi energy k of the k-th mode is k g 2J g cos k sin k. law n Q . The power-law exponent D (1 z )is xed by the dimensionality of the system D and a combination of the Other physical observables can as well be expressed in both the equilibrium correlation length and dynamic critical exponents, spin and momentum representations. We are interested in denoted by ν and z, respectively. Essentially, the KZM is a characterizing the number of kinks. As conservation of statement about the breakdown of the adiabatic dynamics across momentum dictates that kinks are formed in pairs, we focus on ^ P ÀÁ a critical point. As such, it provides useful heuristics for the N N ^ À σ^z σ^z = the kink-pair number operator m¼1P1 m mþ1 4. preparation of ground-state phases of matter in the laboratory, N¼^ ^γy^γ The later can be equivalently written as k>0 k k, where e.g., in quantum simulation and adiabatic quantum computa- ^γy^γ tion10. It has spurred a wide variety of experimental efforts in k k is a Fermion number operator, with eigenvalues {0, 1}. As superfluid helium11–13, liquid crystals14,15, convective fluids16,17, different k-modes are decoupled, this representation paves the superconducting rings18–20, trapped ions21–23, colloids24, and way to simulate the dynamics of the phase transition in the ” ultracold atoms25–29, to name some relevant examples. This TFQIM in "momentum space : the dynamics of each mode can be simulated with an ion-trap qubit, in which the expectation activity has advanced our understanding of critical dynamics, e.g., y 30,31 ^γ ^γ by extending the KZM to inhomogeneous systems . In the value of k k can be measured. To this end, we consider the quantum domain, experimental progress has been more limited quantum critical dynamics induced by a ramp of the magnetic and led by the use of quantum simulators in a variety of plat- field – forms32 34. t ð Þ¼ þ ð Þ; ð Þ Beyond the focus of the KZM, the full counting statistics enco- g t τ g 0 3 ded in the probability distribution can be expected to shed further Q τ light. The number distribution of topological defects has become in a time scale Q that we shall refer to as the quench time. We available in recent experiments35. In addition, the distribution of further choose g(0) < −1 in the paramagnetic phase. In momen- kinks has recently been explored theoretically in the one- tum space, driving the phase transition is equivalently described dimensional transverse-field quantum Ising model (TFQIM)36,a by an ensemble of Landau–Zener crossings. This observation paradigmatic testbed of quantum phase transitions37. Indeed, it has proved key in establishing the validity of the KZM in the quan- ^ been argued that the distribution of topological defects is generally tum domain7,32,33: the average number of kink pairs hNi¼hni determined by the scaling theory of phase transitions and thus after the quench scales as exhibits signatures of universality beyond the KZM36. sffiffiffiffiffiffiffiffiffiffi We aim at validating this prediction experimentally using a N _ h i ¼ ; ð4Þ n KZM π τ single-trapped ion as a quantum simulator and probe the quan- 4 2J Q tum critical dynamics of the TFQIM in momentum space. Spe- À ν cifically, we show that the measured distribution of kink pairs is h i / τ 1þzν in agreement with the universal power law n KZM Q , with Poisson binomial and that low-order cumulants scale as a uni- critical exponents ν = z = 1 and one spatial dimension (D = 1). versal power law with the quench time. The full counting statistics of topological defects is encoded in the probability P(n) that a given number of kink pairs n is Results ^ obtained as a measurement outcome of the observable N .To Quantum critical dynamics. The TFQIM is described by the explore the implications of the scaling theory of phase transitions, Hamiltonian we focus on the characterization of the probability distribution P XN ÀÁ (n) of the kink-pair number in the final nonequilibrium state H¼À^ σ^z σ^z þ σ^x ; ð Þ J m mþ1 g m 1 upon completion of the crossing of the critical point induced by m¼1 Eq. (3). Exploiting the equivalence between the spin and σ^z ¼ that we shall consider with periodic boundary conditions 1 momentum representation, the dynamics in each mode leads to σ^z σ^z σ^x two possible outcomes, corresponding to the mode being found Nþ1 and N even. Additionally, m and m are Pauli matrices at the site m and g plays the role of a (dimensionless) magnetic field in the excited state (e) or the ground state (g), with probabilities = = − that favors the alignment of the spins along the x-axis. This pe pk and pg 1 pk, respectively. Thus, one can associate with system exhibits a quantum phase transition between a para- each mode k > 0 a discrete random variable, with excitation ∣ ∣≫ ¼h^γy^γ i magnetic phase ( g 1) and a doubly degenerated phase with probability pk k k . The excitation probability of each mode ∣ ∣≪ ferromagnetic order ( g 1). There are two critical points at gc is that of Bernoulli type. As the kink-pair number n is associated = ±1. We shall consider the nonadiabatic crossing of the critical with the number of modes excited, P(n) is a Poisson binomial = − distribution, with the characteristic function36 point gc 1 after initializing the system in the paramagnetic Z phase and ending in a ferromagnetic phase. Our quantum YÂÃÀÁ eðθÞ¼Àππ θ ð Þ inθ ¼ þ iθ À ; simulation approach relies on the equivalence of in the TFQIM in P d P n e 1 e 1 pk ð5Þ one spatial dimension and an ensemble of independent two-level k>0 2 COMMUNICATIONS PHYSICS | (2020) 3:44 | https://doi.org/10.1038/s42005-020-0306-6 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0306-6 ARTICLE Fig.
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