Experimental verification of generalized eigenstate thermalization hypothesis in an integrable system

Identifying the general mechanics behind the equilibration of a complex isolated quantum system towards a state described by only a few parameters has been the focus of attention in non-equilibrium thermodynamics. And several experimentally unproven conjectures are proposed for the statistical description of quantum (non-)integrable models. The plausible eigenstate thermalization hypothesis (ETH), which suggests that each energy eigenstate itself is thermal, plays a crucial role in understanding the quantum thermalization in non-integrable systems; it is commonly believed that it does not exist in integrable systems. Nevertheless, integrable systems can still relax to the generalized Gibbs ensemble. From a microscopic perspective, understanding the origin of this generalized thermalization that occurs in an isolated integrable system is a fundamental open question lacking experimental investigations. Herein, we experimentally investigated the spin subsystem relaxation in an isolated spin–orbit coupling quantum system. By applying the quantum state engineering technique, we initialized the system with various distribution widths in the mutual eigenbasis of the conserved quantities. Then, we compared the steady state of the spin subsystem reached in a long-time coherent dynamics to the prediction of a generalized version of ETH and the underlying mechanism of the generalized thermalization is experimentally verified for the first time. Our results facilitate understanding the origin of quantum statistical mechanics.

Without loss of generality, we can set the initial state as where a k and b k are the normalized complex amplitudes satisfying |a k | 2 +|b k | 2 = 1, P(k) is the initial probability distribution in momentum space, |n u(d) k denotes the spinor eigenstates in the upper (lower) energy band with the momentum k, and |k is eigenstate of the momentum operator. At any given time t, the time-evolving density matrix of the spin subsystem can be written as [7] ρ(t) = 1 2 {I + where n i (k) denotes the Bloch vector of the initial spin state at each momentum k and satisfies n i (k) · n H (k) = |a k | 2 − |b k | 2 .
Moreover, the steady state for any initial state can be obtained by using its "diagonal ensemble" [8,9] as ρ DE ≡ α |c α | 2 |α α|, where |α is the eigenstate of the Hamiltonian, and c α = α|φ 0 is the initial coefficient. Thus, we can predict the steady state of the spin subsystem as Then the dynamics of the reduced density matrix of the spin subsystem can be investigated by comparing the time-evolving state with the steady state ρ st . Herein, we use the trace distance D to measure the difference between two states as where || · || denotes Schatten 1-norm of a matrix, and | · | denotes the norm of a vector.
If P(k) is dense enough in momentum space, such that we can approximately replace the sum by the integral as k → dk, which can be satisfied for the infinite lattice. Moreover, by applying the Riemann-Lebesgue Lemma, D can vanish in the long-time limit. The SOC system thus relaxes to a steady state that can be predicted by the "diagonal ensemble". It is noteworthy that ρ st can also be directly obtained based on Eq. (S3) through the long-time average of the spin states [2,5].

B. EQUILIBRATION TIME SCALE OF THE ISOLATED QUANTUM SYSTEM.
In this part, we will determine the the upper bound on the equilibration time scale T eq for the isolated quantum system [10]. Here, we use a normalized trace distance, comparing the time-evolved expected value of an observable A to its equilibrium value, to give a measurement for the distinguishability between the evolved state ρ t and the long-time average where {G 1 , G 2 , ...} is all energy gaps of the Hamiltonian, v α ≡ ρ j,k A k,j . For convenience of the proof, we define a probability distribution where the normalization factor N = | α vα| α |vα| . Therefore, we get Moreover, based on the result in [10], we have where can be any positive real number, P max (∆) ≡ max x 0 ∈R δ:G δ ∈[x 0 ,x 0 +∆] p δ , d( ) ≡ Pmax( ) and δ( ) ≡ P max ( ).
According to the inequality Eq. (S11), the system will eventually be equilibrated with respect to A when the second term on the right-hand side is samll N 2 δ( ) << 1. And the system wll equilibrate for the time T >> T eq , where the equilibration time scale T eq , in which the decay of distinguishability happens, is then given by (S12)

C. FAILURE OF THE EIGENSTATE THERMALIZATION HYPOTHESIS
It is generally believed that thermalization can occur in the chaotic system; it is broken down in the integrable system. This can be explained by utilizing the validity (failure) of the eigenstate thermalization hypothesis in the chaotic (integrable) system. Eigenstate thermalization hypothesis (ETH) suggests that all the energy eigenstates can be locally equal to the thermal state which is obtained by averaging on an uniformly distributed microcanonical ensemble (ME) [8,11], i.e., |E α is the eigenstate with energy E α , H δE ME is the Hilbert subspace spanned by the eigenstates whose energies are belong to the energy window [E α − δE ME , E α + δE ME ], dim(·) denotes the dimension of a space and I H δE ME is the identity matrix in this subspace. Moreover, the energy window should be macroscopically small but microscopically large to cover enough energy eigenstates [12]. Based on the ETH, all the energy eigenstates with the similar energy should be locally close to each other. As a result, any superposition of the eigenstates whose energies lie within a small energy window can locally relax to an equivalent state.
However, for the case of SSQW, the energy eigenstate has two-fold degeneracy, i.e., . Considering the scenario that two individual eigenstates with the opposite momentum ±k α (k α > 0) have a same energy E α (see Fig. S1), the reduced density matrices of the two eigenstates are totally different

Thus, for a general energy eigenstate
its reduced density matrix highly depends on the complex coefficient a ± , which indicates the breakdown of the ETH. That is to say, only the information of energy is not enough to determine the steady state of the spin subsystem, and the other nontrivial conserved quantity (i.e., the momentum) is necessary to be introduced to the SSQW.
Consider the initial state in a small energy window The ω l(r) (k) is the phase, and the expected energy of the initial state is E 0 . Moreover, the positive momentum window |k − k E 0 | < δk E is directly gotten by the chosen energy window (see Fig. S1). According to Eq. (S4), the steady state of the spin system can be written as where "±" represents that the chosen energy eigenstate lies in the upper (lower) energy band, andP R(L) ≡ δk E P(±k). On the other hand, based on the ETH, the steady state of the spin subsystem can also be predicted by the micro-canonical ensemble as where N E 0 ,δE ME is the number of the eigenstates whose energies located in the energy window [E 0 −δE ME , E 0 +δE ME ], and δk E ME is the half-width of the corresponding momentum window. Obviously, ρ st = ρ th , where ρ th dependents on the energy E 0 , and ρ st remains the details of and δk E is the half-width of the momentum window, which is determined by δE.
the initial conditionP R(L) . Thus, due to the breakdown of the ETH, the steady state of the spin subsystem cannot be characterized by a thermal state.

HYPOTHESIS.
As demonstrated in the previous section, in integrable system such as SSQW, the ETH is broken down; thus, the thermal state fails to characterize the steady state. However, the steady state in integrable system can still be characterized by the generalized microcanonical ensemble (GME) state [9], and this generalized thermalization is explicable by using the generalized ETH (GETH) [13]. According to the GETH, the mutual eigenstates of the conserved quantities should be locally equal to the GME state [13,14], i.e., where |i α denotes the eigenstate with the values of the conserved quantities i α ≡ {i 0 α , i 1 α , ..., i N α } (in the SSQW, the conserved quantities are the energy and momentum), H {δi GME } is the Hilbert subspace spanned by the eigenstates whose values of the conserved quantities i µ (µ = 0, 1, 2, ..., N ) lie within the window [i µ α − δi µ GME , i µ α + δi µ GME ] (i 0 is defined as the energy), and I H {δi GME } is the identity matrix in this subspace. Moreover, the GETH predicts that the mutual eigenstates with the similar values of the conserved quantities should be locally close to each other. As a result, the steady state of any superposition of these mutual eigenstates within the windows all gives the same result, which shows the "universality" of the generalized thermalization. Then, we will demonstrate that the SOC system satisfies the GETH and exhibits the generalized thermalization, i.e., the steady state can be predicted by the GME state.
For a mutual eigenstate with momentum k 0 and energy E u(d) 0 = ±|E(k 0 )|, its reduced density matrix reads where "±" denotes that the mutual eigenstate lies in the upper (lower) band. On the other hand, the reduced state can also be obtained by the ensemble of the right-hand side in Eq. (S19) as where δk GME denots the half-width of the momentum window of the GME state (at the same time, the energy window is determined based on the function E(k)), which should be macroscopically small but can cover enough mutual eigenstates, ..
A denotes the second derivative of A with respect to k, and N δk GME is the number of the mutual eigenstates whose momentum located in the momentum window. Since the GME state should be independent of the width of the chosen window [8], it satisfies the condition where ∆ 2 k GME = 1 N δk GME |k−k 0 |<δk GME (k −k 0 ) 2 , and ∆ k GME is the momentum standard deviation of the GME state. Obviously, the state in Eq. (S20) and Eq. (S21) are the same, indicating the validity of the GETH.
where ω(k) denotes the phase.
According to Eq. (S4), the initial state finally relaxes to a steady state as Clearly, the steady state is only determined by the conserved quantities k 0 and E(k 0 ) (H(k) is determined by k and E(k)). Moreover, the momentum standard deviation of the initial state ∆ k (for the Gaussian initial state adopted in the QW experiment, δk = 2ln(2)∆ k ) should satisfy Eq. (S22) to make sure the small conserved quantities fluctuation. Thus, the maximal value of can be estimated by the maximum value of ∆ k GME satisfying Eq. (S22). If the ∆ k (and the associated δk) of the initial state satisfies this condition, then its steady state is identical to the prediction of the GME. , the orange region is the small momentum window [k 0 − δk, k 0 + δk], and the blue region is its corresponding energy window where δE k is determined by δk. We choose herein that the energy window is in upper energy band, so E 0 > 0. Moreover, there is only one part of the mutual eigenstates within this energy-momentum window with respect to the case in Fig. S1.

E. THE GENERALIZED THERMALIZATION IN EXTENDED SITUATIONS
As discussed in section C, due to the validity of the GETH, any superposition state of the mutual eigenstate within a small connected window exhibits the generalized thermalization.
Further, we consider a superposition state of the mutual eigenstates within two separated windows and show the occurrence of the generalized thermalization in this extended situation can also be understood by the GETH.
Firstly, we consider the situation that both the two separated energy-momentum windows are within the same momentum window but different energy windows (see Fig. S3). The initial state then takes the form where δk denotes the half-width of the small momentum window. The initial state's expected energy is E 0 = |k−k 0 |<δk E(k)P(k)(|a k | 2 − |b k | 2 ) and expected momentum is where δE k is determined by δk. k 0 = |k−k 0 |<δk k P(k). In addition, when a k → 0 or b k → 0, it reduces to the scenario discussed in the section C. According to Eq. (S4), we have whereP up ≡ |k−k 0 |<δk P(k) |a k | 2 andP down ≡ |k−k 0 |<δk P(k) |b k | 2 are the upper and lower band occupancy of the initial state, respectively. The approximation of the last line can be gotten by applying E 0 ≈ E(k 0 ) |k−k 0 |<δk P(k)(|a k | 2 − |b k | 2 ). Moreover, the GME state in this extended situation is the weighted average of the GME states in the individual connected window, and takes the form Furthermore, we consider the situation that both the two separated energy-momentum windows are within the same energy window but different momentum windows (see Fig. S4), which has been previously used in section B to indicate the failure of the ETH (and thermalization) in the SSQW system. The initial state then reads where ω l(r) (k) denotes the phase, and |k − k E 0 | < δk E (k E 0 > 0) is the positive momentum window, which is determined by the chosen energy window. The initial expected , and expected momentum is According to the Eq. (S4), we have are the probabilities to be found in the right and left connected windows, respectively. The last approximation is obtained by applying k 0 ≈ k E 0 k>0; |k−k E 0 |<δk E (P(k) − P(−k)). Moreover, the GME state in this extended situation can also be the weighted average of the GME states in the right and left connected windows (the weights areP R andP L ), and takes the form is the Hilbert subspace spanned by the eigenstates located in the right (left) window (see Fig. S4), and I H {δi GME1 } (I H {δi GME2 } ) is the identity matrix in this subspace. As discussed in the section B, the SSQW system in this extended situation cannot be thermalized, and only with the energy E 0 is not enough to characterize its steady state; however, with another information of the conserved quantity-momentum k 0 , the steady state can be predicated by the GME state in Eq. (S29), which is explicable by using the GETH.

F. RELATION WITH SU-SCHRIEFFER-HEEGER MODEL
In this part, we will show that the validity of GETH in SOC system will lead to a special case of the generalized thermaliztion in an equivalent many-body system like the Su-Schrieffer-Heeger (SSH) model [15]. For the SSH model, its Hamiltonian reads [16] with c k = (c s 1 ,k , c s 2 ,k ). c i,k (i = s 1 , s 2 ) are the annihilation operators in momentum space for two different sublattice in SSH model, and is what we defined in the SOC system. By performing an unitary transformation for each we can rewrite the Hamiltonian as where c u,k = ξ * 1,k c s 1 ,k − ξ 2,k c s 2 ,k , and c d,k = ξ * 2,k c s 1 ,k + ξ 1,k c s 2 ,k . The energy eigenstate thus reads with |0 is vacuum state, which means there is no quasi-particle in the system. M ≡ {M k |M k = c † u,k or c † d,k or I k , k ∈ [−π, π]} where I k means that the eigenstate has no quasiparticle with momentum k, and P represents a permutation of M k . As an example, considering a two quasi-particles state, where the momentum of the quasi-particles are k 1 and k 2 , the permutation of M k is two, the first one reads |e , and the second one reads |e . Obviously, these two states are not independent. Actually, it is enough to only choose one permutation to generate a complete set of eigenbasis. In the following, for the energy eigenbasis we use, we choose the momentum sequential permutation.
As we know, the Hermitian operator for a single spinor reads is a vector. Now, considering the one-body observable with a specific momentum in this system, there are four kinds (In the following, we ignore the trivial parts of the observable, i.e. the terms proportional to the identity matrix). The first kind of the one-body observable , and σ k is the Pauli matrices in a new coordinate whose z-axis is parallel with the n H (k). In other word, and o 1,k = ( o 1,k · σ x,k , o 1,k · σ y,k , o 1,k · σ z,k ). Besides, the eigenstate expectation value of the first kind of one-body observable is The second kind of one-body observable reads O 2,k = c † 2,k ( O 2,k · σ k )c 2,k where c 2,k = (c † u,k , c d,k ), whose eigenstate expectation value is The third kind of one-body observable reads O 3,k = c † 3,k ( O 3,k · σ k )c 3,k where c 3,k = (c u,k , c † d,k ), whose eigenstate expectation value is Furthermore, if the initial state |φ 0 = j l M j |e M j with mean energy E 0 and mean momentum k 0 only occupies the energy eigenstates whose quasi-particle momentum is within a small momentum window W k = [ k 0 − δk, k 0 + δk], which is similar with the case in the SOC system we have showed, and a one-body observable reads the approximation. Namely, we can replace all n H (k α ), where k α is within the momentum window, by n H ( k 0 ). This is due to the validity of the GETH in the SOC system.
Moreover, considering the conserved charges of the initial state, i.e. the energy and the momentum, we have where both two approximation is valid when W k is small. We thus can get Then, the expectation of the one-body operator we defined above is only dependent on three initial factors, i.e., the initial energy E 0 , the initial momentum k 0 , and the mean momentum of the momentum window k 0 . Thus, the generalized thermalization can occur in this special situation.
All above, it is the validity of GETH in SOC system that causes the satisfaction of the generalized thermalization in SSH model.