Many-Body Dephasing in a Trapped-Ion Quantum Simulator

How a closed interacting quantum many-body system relaxes and dephases as a function of time is a fundamental question in thermodynamic and statistical physics. In this work, we observe and analyse the persistent temporal fluctuations after a quantum quench of a tunable long-range interacting transverse-field Ising Hamiltonian realized with a trapped-ion quantum simulator. We measure the temporal fluctuations in the average magnetization of a finite-size system of spin-$1/2$ particles and observe the experimental evidence for the theoretically predicted regime of many-body dephasing. We experiment in a regime where the properties of the system are closely related to the integrable Hamiltonian with global spin-spin coupling, which enables analytical predictions even for the long-time non-integrable dynamics. We find that the measured fluctuations are exponentially suppressed with increasing system size, consistent with theoretical predictions.

Introduction.-Investigating the relaxation and dephasing dynamics of a closed many-body quantum system is of paramount importance to the study of thermodynamics and statistical physics. Most commonly, this problem is investigated by studying the time evolution of the expectation value of a local observable, e.g., particle density or magnetization, after quenching the system from an initial out-of-equilibrium state [1,2]. For a generic nonintegrable system, the expectation value tends to relax to a constant in the thermodynamic limit which can be described by a thermal state at some temperature depending on the initial state [3][4][5][6][7][8][9]. However, if the system size is finite, there exist persistent temporal fluctuations around the constant average value, as sketched in Fig. 1(a). Importantly, these persistent temporal fluctuations in the expectation value after a quench are distinct from the usual fluctuations of observables in equilibrium (where expectation values are constant). Studying these temporal fluctuations represents the next level of the description of quench dynamics going beyond merely looking at long-time observable averages.
A crucial question for statistical physics is how the temporal fluctuations are suppressed with increasing system size N . In the case of integrable systems mappable to free quasiparticles, it has been found that the variance of temporal fluctuations scales as 1/N [10][11][12]. In the case of generic nonintergrable systems [13][14][15][16][17], or the integrable systems solvable with the Bethe ansatz (not mappable to noninteracting ones) [18], the temporal fluctuations are exponentially suppressed by the system size due to the highly nondegenerate spectrum. This was first found only numerically. However, in Ref. [17], the authors were able, for the first time, to provide an exact analytical result for the exponential scaling of fluctuations with N spins in a weakly nonintegrable system. In this setting, they identified a general dynamical regime which they termed "many-body dephasing" [19]. In the thermalization process, the dephasing mechanism comes from the relaxation of the quasiparticle distribution to thermal equilibrium by quasiparticle scattering described by the Boltzmann equation. In contrast, many-body dephasing results from lifting of the exponentially large degeneracies of transition energies in integrable systems while the quasiparticle distribution remains practically unchanged [17].
Nevertheless, the exponential size scaling due to manybody dephasing in nonintegrable systems has not yet been verified in experiments. Here, we give the first experimental observation of persistent temporal fluctuations after a quantum quench characterized as a function of system size, employing a trapped-ion quantum simulator. We present a direct measurement of relaxation dynamics in the nonintegrable system by measuring the temporal fluctuations in the average magnetization of a finite-size system of spin-1/2 particles. After including the experimental noise in the data analysis, the temporal fluctuations from experimental data are consistent with our numerical simulations and theoretical analysis based on the concept of many-body dephasing.
Model Hamiltonian.-The Hamiltonian implemented in this experiment is the long-range transverse-field Ising model, where J ij ≈ J 0 /|i − j| α > 0, is a long-range coupling that falls off approximately as a tunable power-law. The Hamiltonian (1) is implemented using an applied laser field which creates spin-spin interactions through spindependent optical dipole forces. The spin chain is initial-arXiv:2001.02477v1 [cond-mat.stat-mech] 8 Jan 2020   ized to the |↓↓ ... ↓ z state, then a quench is performed using Hamiltonian (1), and the magnetization along thê z axis is measured as a function of time. The cases of α −1 = 0 and α = 0 correspond to two integrable limits, i.e., the nearest neighbour coupling and global coupling models respectively. For a finite α > 0, Hamiltonian (1) is in general nonintegrable.
Temporal fluctuations.-In the present experiment, the observable is the magnetization, i.e., where the temporal averaging is restricted within the time window between t i and t i + T . The variance of temporal fluctua- with σ A the standard deviation. We use |Φ n (n = 1, 2, · · · , 2 N ) to represent the many-body eigenstates of Hamiltonian (1) with eigenenergy E n . Given the initial state |ψ(0) , the exact time evolution of the observable is where ∆ mn ≡ E m − E n is the transition energy between the two energy levels |Φ m and |Φ n ( = 1). It is not difficult to prove that, in the long time window limit (T → +∞), we have the temporal average and the variance of temporal fluctuation with ∆ denoting the set of all the possible values of ∆ mn . For the integrable models (α −1 = 0 and α = 0), there are exponentially many degeneracies with the number of spins for a given transition energy ∆ mn , since each many-body eigenstate can be labelled by many independent conserved quantities. However, for a generic nonintegrable model with finite α > 0, there are no conserved quantities except the Hamiltonian itself. Thus, it is reasonable to assume that all the degeneracies of transition energies are lifted, making ∆ mn = 0 only possible for m = n in the nonintegrable model, so Eq. (2) simplifies to Upon closer analysis, this is the basic reasoning that leads to the exponential suppression of fluctuations with system size [13]. However, in general cases, it is impossible to evaluate this expression analytically.
Theoretical results.-We investigated numerically the temporal fluctuation σ A as a function of α for fixed dimensionless parameter λ ≡ 2J 0 /B. We also extract from our numerical simulations the size scaling exponent κ from the fit σ A ∝ e −κN by calculating σ A for N = 3 − 10 spins (see Fig. 1(b)). We find two distinct regimes, at small and large α, separated by the crossover value of α * = ln(2|λ|)/ ln 2 [23]. The crossover between those regimes can be understood from the competition between the two terms in Hamiltonian (1), i.e., the magnetic field energy −B i s z i (where s z i ≡ 1 2 σ z i is the spin operator) and the next-nearest-neighbor (NNN) spin-spin coupling 2 −α 4J 0 i s x i s x i+2 , which, for α > 0, is the leading term responsible for breaking integrability [23]. In the regime of α α * , by neglecting the NNN coupling (and other long-range coupling terms), the Hamiltonian is reduced into an integrable model with NN coupling. Adding the NNN coupling terms weakly breaks the integrability and results in many-body dephasing [17]. We cannot reach this regime in the experiment since the power-law exponent is α ≈ 0.7. Therefore this work lies in the opposite regime of α α * , where the long-range coupling terms are dominant over the magnetic field energy. The general concept of many-body dephasing still applies in this regime, and an analytical prediction can be obtained, as we will show below.
We start our analysis from the global coupling limit α = 0: The Hamiltonian (4) is also called Lipkin-Meshkov-Glick (LMG) model [28], which is integrable [29,30] since there exist N conserved quantities. For example, S 2 n ≡ S x2 n + S y2 n + S z2 n (n = 2, . . . , N ) and the Hamiltonian (4) itself satisfy [ S 2 n , H α=0 ] = 0, where S β n ≡ n i=1 1 2 σ β i with β = x, y, z. In the special case of λ → ∞ (B → 0), we can label each energy level by |S 1 , S 2 , · · · , S N −1 , S N , S x N and group all the eigenstates into N +1 subspaces according to S x N . In each subspace of S x N , we can prove that there are We define the notation |Φ λ=∞ where P λ=∞ . Based on this assumption and the eigenstate thermalization hypothesis (ETH) [6,7,[20][21][22], we are able to obtain an approximate formula for Eq. (3) For large N , we have the asymptotic expression that , predicting the size scaling exponent κ = ln √ 2 ≈ 0.35. Considering both λ and α finite, the formula (6) holds as long as α α * . However, the eigenstate |Φ λ  [31]. However, Eq. (6) reduces the calculation of σ A to an N × N eigenvalue problem which can easily be solved on a computer. As we will show further below, the analytical predictions compare well with the experiment (see Fig. 4(a)). Experimental results.-To perform this experiment, each effective spin 1/2 particle is encoded in the hyperfine ground state of one 171 Yb + with | ↑ ≡ 2 S 1/2 |F = 1, m F = 0 and |↓ ≡ 2 S 1/2 |F = 0, m F = 0 . The Hamiltonian of Eq. (1) is realized by global spin-dependent optical dipole forces from laser beams, which modulate the Coulomb interaction to create an effective Ising coupling between spins [32]. The field term is implemented by asymmetrically detuning the two laser beatnotes generating the optical dipole forces.
The magnetization fluctuations σ A are characterized by measuring the standard deviation of the average magnetization of the sum of all ions in the chain, i.e., A = N −1 j σ z j . This is measured with B-fields ranging from ± 2π × 0.5 kHz to 2π × 2.0 kHz. The two plots in Fig. 2 show the magnetization data measured as a function of time with a 4-ion chain and B = ±2π × 0.5 kHz. Although the decoherence time in our trapped-ion simulator is long enough to consider J 0 and B unchanged within a single time evolution up to t = 2 ms, the values of J 0 and B may vary between different time evolutions. We assume the coupling strength and magnetic field in the experiments to be independent and normally distributed. Then, the averaged observable A at a fixed time t also needs to be averaged over the experimental values of J 0 and B, resulting in: In Fig. 2, the red curves are the theory fits by setting σ J0 and σ B both to approximately 2π × 0.1 kHz. To fit the experimental data, we use the gradient descent method to search for the optimal values of σ J0 and σ B , which happen to be roughly equal. Therefore, we set σ J0 and σ B to be the same values for simplicity.  In general, with a positive B-field, we observe more significant oscillations than when using a negative B-field. This can be understood by analyzing the overlap between the pre-quench state and the post-quench energy eigenstates (obtained for the post-quench J 0 and B values). For the system parameters given in Fig. 2, the structure of the post-quench spectrum is such that at high energies there is a non-vanishing energy gap in the thermodynamic limit. Conversely, in the low energy sector of the spectrum the level spacing decreases with system size and the gap vanishes in the thermodynamic limit. For the positive B-field, the pre-quench state is the superposition of several of the highest excited states of the spectrum and the energy gap leads to more persistent oscillations. For the negative B-field, the pre-quench state is very close to the ground state of the spectrum [33], suppressing the oscillations.
We plot the standard deviation of the average magnetization σ A as a function of λ = 2J 0 /B for fixed N in Fig. 3. The data for N = 3 to N = 6 agree with the theoretical prediction. The N = 7 data largely agrees with theory excluding the two outlying points at negative λ values. For N = 8, the data points tend to gather around the 0.07 level indicating that the measurement noise in this case obscures the measured fluctuations. In these plots, the values near λ = 0 were not taken because when B J 0 the ions are predominantly acting paramagnetically. In this regime, fluctuations are expected to be very small and well below the noise floor of this experiment. The shape of the data is asymmetric with a pronounced slope at 2J 0 /B = 1/2. This point marks the ferromagnetic (FM) to paramagnetic (PM) phase transi-  tion of the ion chain. The fluctuations are enhanced here as this is an unstable point for the system. In contrast, the antiferromagnetic (AFM) to PM transition [34] for λ < 0 is not as pronounced.

Number of spins
System size scaling.-The temporal fluctuation variance σ 2 A given by Eq. (3) is obtained by averaging over an infinite time window J 0 t ∈ [0, +∞]. However, in the experiment, we can only average over a finite time win-dow up to t ∼ 2.0 ms ( i.e., 3 or 4 oscillations depending on the value of λ), as the long-time fluctuations are suppressed by the noise in the Hamiltonian parameters J 0 and B. Fig. 4(a) shows that short-time-window averaging only makes sense for small system size, e.g., the time window J 0 t ∈ [0, 2π] works up to N = 6 spins. Larger system sizes result in smaller level splittings and makes the period of temporal fluctuations longer, thus necessitating a longer time window to calculate the temporal fluctuations. We compare the numerical results from Eq. (3) (red empty circles) and with the analytical result from Eq. (6) (solid blue dots), showing good agreement. The fit to the infinite-time-window averaging (red dashed line) shows that the system size scaling exponent is κ ≈ 0.35 consistent with the theoretical prediction κ = ln √ 2. In Fig. 4(b), we compare the experimental data (black dots with error bars) with the numerical results (red circle dots) by averaging over the finite time window t ∈ [0.08 ms, 2.0 ms]. The finite variances σ J0 and σ B reduce the temporal fluctuations and also set a lower limit for the measurement accuracy, as indicated by the blue region in the figure. As a result, only the first three data points (N = 3 − 5) provide the information on system size scaling. The fits to the experimental and numerical values of σ A for N = 3 − 5 (dashed lines) both show a clear exponential size suppression with the scaling exponent κ ≈ 0.24, smaller than the ideal scaling exponent κ ≈ 0.35 shown in Fig. 4(a). This is caused by the experimental drifts in the Hamiltonian parameters J 0 and B. Indeed with larger σ J0 and σ B , the measured fluctuations σ A versus number of spins N would be completely flat as shown in the interval N = 6 − 8 in Fig.  4(b) and the extracted scaling exponent κ would be zero.
Summary.-Using our trapped-ion quantum simulator, we present the first experimental observation of persistent temporal fluctuations after a quantum quench with a long-range interacting transverse-field Ising model. We characterized how the fluctuations in the average magnetization of the spin chain depend on the transverse field and the spin-spin interactions. This experiment was performed in the near-integrable regime where analytical solutions are available, though the system is non-integrable. Numerical simulations compared with experiment show that, as a function of system size N , the exponential suppression of temporal fluctuations matches well with the theoretical value.