Equilibration of quantum cat states

In this paper, we study the equilibration properties of isolated ergodic quantum systems initially prepared in a cat state, i.e a macroscopic quantum superposition of states. Our main result consists in showing that, even though decoherence is at work in the mean, there exists a remnant of the initial quantum coherences visible in the strength of the fluctuations of the steady state. We back-up our analysis with numerical results obtained on the XXX spin chain with a random field along the z-axis in the ergodic regime and find good qualitative and quantitative agreement with the theory. We also present and discuss a framework where equilibrium quantities can be computed from general statistical ensembles without relying on microscopic details about the initial state, akin to the eigenstate thermalization hypothesis.


I. INTRODUCTION
Upon encountering the quantum statistical ensembles for the first time, one is often struck by the strong similitude they share with their classical counterpart. Indeed, quantum ensembles such as e.g the Gibbs ensemble, appear like a mere transcription of classical ones where one would have replaced the possible classical configurations by the eigenstates of the Hamiltonian. An explanation dating back to the early days of quantum mechanics [1,2] is that, assuming ergodicity, the off-diagonal elements undergo a dephasing that time-averages to zero given that the different frequencies of the Hamiltonian are incommensurate. Thus, in the mean steady-state, purely quantum mechanical features such as state superposition and entanglement are lost : this is a decoherence effect. Furthermore, the previous years have seen the development of a general framework known as the eigenstate thermalization hypothesis (ETH) which explains the emergence of statistical ensembles from a given set of assumptions on the spectral properties of the observables of the system [3][4][5][6] . The validity or invalidity of the ETH has been tested numerically in a certain number of studies [6][7][8][9][10].
Therefore, one could legitimately ask what is the consequences of having purely quantum features such as superposition of states and entanglement in the initial state of the system on the final equilibrium properties, if there are any at all? In this letter, we intend to prove that, even if on average information about the quantum coherence of the initial state is lost at equilibrium, there is a remnant of the latter visible in the fluctuations around the stationary state. This phenomenon was already seen in a model of stochastic fermionic chain on a discrete lattice [11,12] and we provide here the generalization of these results to any ergodic quantum system.
In the context of ETH, one important assumption is that the initial states considered must have an energy comprised in a narrow energy shell. This assumption is tightly bound with having initial states which fulfill a cluster decomposition [13] constraint, i.e that the typical coherence length is small compared to the size of the system. Within this hypothesis, the fluctuations of the state around its average value scale like the inverse of the dimension of the Hilbert space and are thus exponentially suppressed as one increase the system size [14].
We are interested in situations where these hypothesis are relaxed, i.e for which the initial state of the system can be a superposition of states which have energies largely spread across the spectrum or equivalently that entangle large part of the system together. This typically happens for cat states which are quantum superposition of macroscopically distinct states -see fig.1. Cat states have attracted humongous interest from the physics community in the recent years [15][16][17][18][19][20]  as their creation, stabilization and manipulation constitute key steps towards quantum computing and simulation. We will see that such states present non trivial, possibly non-local fluctuations of the off-diagonal components in the steady-state that are fixed by the initial quantum coherences.
This paper is organized as follows : First, we introduce our definition for quantum ergodicity and following it, compute first and second order correlation functions for elements of the density matrix. In a second part, we show that these results are in qualitative and quantitative agreement with numerical results obtained in a quantum ergodic spin chain. We then discuss a more general framework where the equilibrium ensembles don't depend on the fine structure of the initial states and discuss connection with ETH. We finally end by some concluding remarks and perspectives.

II. THEORY
In classical physics, ergodicity is the hypothesis that at long-times, when the system reaches equilibrium, there is an equivalence between the time average of quantities and an ensemble average over a microcanonical distribution. The microcanonical distribution stipulates that, at fixed energy for an isolated system, the probability of all microscopic configurations are equal. Physically, the equivalence between the two averages comes from the assumption that at long-time the system explores isotropically all the degrees of freedom available under the constraint of fixed energy.
This work is devoted to the formulation and study of a similar quantum ergodic hypothesis : Let ρ 0 be the density matrix containing information about the initial conditions of the system. Given a set of conserved observables,Ĥ 1 ,Ĥ 2 , · · ·Ĥ n , an accessible state is defined as a density matrix ρ such that there exists a unitary U commuting with all theĤ i and fulfilling U ρ 0 U † = ρ. The quantum ergodic hypothesis asserts that in the long-time equilibrium state, all accessible density matrices have same probability weight or equivalently, that time average of elements of ρ t is equivalent to ensemble average over all possible unitary evolution U that commutes with the conserved quantities.
For simplification, in this paper, we will consider a unique conserved quantityĤ but generalization to a set of mutually commuting observables is straightforward.
Let us introduce some notations. We call G the group formed by all the unitaries such that the eigenvector corresponding to energy E i forĤ where ν i is an index accounting for possible degeneracies. We will call d i the dimension of the subspace associated to energy E i . Because of the commutativity of the U 's with H, the group G can be decomposed as a Alternatively, this means that in the eigenbasis of the Hamitonian, U can be written in blocks indexed by i with an element This constitutes a fundamental representation of G. We also introduce the decomposition of ρ 0 into different sectors ρ We will now illustrate how our quantum ergodic hypothesis allow to compute equilibrium quantities. We begin by considering the average of ρ denoted by E[ρ 0 ] with respect to the ensemble average we just introduced. By definition : have : By the left invariance of the Haar measure we have that for i = j, proportional to the identity. The proportionality coefficient is determined by taking the trace so that we get : Thus, in average, as one expects from decoherence, information about «off-diagonal» correlations between different energy sectors is lost. However, we will see in what follows that there is actually a remnant of the latter when one goes to higher order correlations.
Before going on, let's notice two extreme cases of interest : the case where all the energy level are degenerate, in which case E[ρ 0 ] is just the diagonal ensemble and the case where there is only one energy sector that is the whole Hilbert space itself in which case we just have E[ρ 0 ] = 1 d I, i.e the usual microcanonical ensemble. We will discuss this point further in the last section.
The second momenta of elements of the density matrix is by definition : . This quantity can be computed by generalizing arguments used for the mean. Again, it relies on the decomposition of ρ ⊗2 j 2 | and identifying the invariant objects under the action of U ⊗ U . We simply state the result and present the full derivation in app.A: The subscripts (2) and (1, 1) refers respectively to the symmetric and antisymmetric irreducible The important point is that contrary to (2), (3) contains information about quantum superposition of states both in the case where they belong to the same sector (line 1 of (3)) but also when the superposition involves states belonging to different sectors (line 3 of (3)). Thus, two initial states having the same diagonal elements may relax to the same density matrix in average but present differences at the level of fluctuating quantities, providing a signature of the presence or the absence of initial quantum coherences.
One can carry this procedure to get access to higher momenta but their explicit expression becomes more and more involved. We present the general formula in the app.B.
We will now illustrate these ideas on a concrete numerical example.

III. NUMERICAL RESULTS
We test the predictive power of our model on the XXX model with random local fields : with L the lattice size and σ a j the usual Pauli matrices.
The boundaries are open. The h j are independent random variables distributed uniformly in an interval [−h, h]. The transition from an ergodic to a localized regime of this model has been studied in [21] and characterized by the spectral properties ofĤ. A quantity of particular interest is the mean eigenvalue spacing known to be close to the one of the Wigner distribution (≈ 0.53) in the ergodic regime and to the one of the Poisson distribution (≈ 0.38) in the localized regime. For J = 1 and lattice size ranging from 11 to 22, it has bee shown in [21] that the transition between the two regimes occurred for h ≈ 2.5. Since we are interested in the ergodic regime we will fix the value of h to 1. We work in the minimal magnetization sector, i.e 0 for L even and 1 for L odd.
LetÔ be an observable. We will compute the time-evolution of O(t) ≡ tr(ρ tÔ ) by using exact diagonalization methods [22][23][24]. We denote the time-average by E t [•] ≡ lim T →∞ 1 T T t=0 •dt and will be interested in first and second order correlation We will study a quench situation in which the initial state is expressed in terms of eigenvalues of the HamiltonianĤ from which we remove the XXX coupling between sites L/2 − 1 and L/2 The initial states chosen this way have well-defined energies E R and E L and will be denoted |E R , E L . At time t = 0, we switch the Hamiltonian from H 0 toĤ, so that the system is now in an out-of-equilibrium situation.
To illustrate the importance of the presence or absence of initial quantum coherences in the steady state of the system, we propose to study two different set of initial conditions. They will be both indistinguishable from the point of view of their mean energy but they will encode for different off-diagonal quantum coherences which effect will be visible in the equilibrium fluctuations of the system. Let E min , E max be the minimum and maximum energy of the spectrum and |Φ 1 ≡ The decomposition of these states in the eigenbasis ofĤ are shown in the app.C.
In the protocol I, corresponding to a cat state made of a quantum superposition of two states with «macroscopically» distinct energies E 1 R + E 1 L and E 2 R + E 2 L , the initial state is chosen to be : In the protocol II, corresponding to a classicaly mixed state, the initial state is described by the density matrix : In both protocols, the reduced density matrices on R and L are the same.
We compute the time-evolution of two observables : H R (t) ≡ tr(ρ tĤR ) and Q(t) = tr(ρ tQ ), From formula (2,3) we can deduce the predictions for first and second momenta of these quantities given by ensemble averages in both protocols. Importantly we have that : We show in fig.2, the time-evolution of H R (t) and Q(t) in both protocols for a given realization of the disorder. We can clearly see that H R (t) is independent of the protocol, contrarily to Q(t). The predicted value for the different quantities is also in quantitative arguments with the simulations (see tab.I). Theory Cat state −7.13 * 10 −1 1.51 * 10 −1 2.09 * 10 −2 Mixed state −7.14 * 10 −1 1.51 Mixed state −7.14 * 10 −1 ± 7.8 * 10 −4 1.52 * 10 −1 ± 0.3 * 10 −2 5.83 * 10 −5 ± 1.0 * 10 −6 Details on how these values are obtained and the meaning of the error bars are given in app.C.

IV. DISCUSSION
So far, we were only concerned about the equilibrium state of the system and haven't gone into the thermalization properties. Thermalization is stronger as it implies that the steady-state properties of the system can be described by one of the canonical ensemble of thermodynamics. In this section, we informally discuss possible links between the theory presented and the Eigenstate Thermalization Hypothesis (ETH). The ETH conjectures that for any initial state prepared as mixture of eigenstates of the total Hamiltonian with energies in a narrow window [E − δE, E + δE], the matrix elements of observables in the energy eigenbasis is given by are smooth function of their arguments and that R mn is a random variable with zero mean and unit variance.
The important remark is that for the situations we looked at in the paper, this has no reason to be true anymore, essentially because we don't expect the off-diagonal correlations to be exponentially suppressed as there is no notion of «narrow window» around a given energy for the cat states considered.
However the great advantage of the ETH is to explain why one can forget about all microscopic details contained in the diagonal ensemble and instead work with the microcanonical ensemble ρ m ≡ 1 d I. It would be great to have an equivalent statement here. A possible way for obtaining such simplification in our case already discussed in [25] would be the following : We can suppose that the «actual» group whose action leaves invariant the stationary state is not given by the set of all unitaries that commutes with H but rather with an Hamiltonian H = H + δH with δH a small perturbation which «mixes» the different energy sectors separated by energy ≈ δH. The microcanonical ensemble is recovered in the case where the spectrum of H is fully degenerate in the energy window of interest [E − δE, E + δE]. Indeed, in that case, from (2) we see that the average density matrix is the microcanonical one : E[ρ 0 ] = ρ m and the second momenta is for simplicity, we suppose that the initial state is a pure state). For the case where the initial state has two peaks E 1 and E 2 in its energy spectrum as shown in fig.1, we can conjecture that H is such that it mixes energies in the interval I 1 = [E 1 − δE, E 1 + δE] and I 2 = [E 2 − δE, E 2 + δE] around E 1 and E 2 but not altogether so that the average density matrix is given by : ). This has the simple interpretation that, on average at equilibrium, the mean density matrix is a statistical mixture of a state at energy E 1 and a state at energy E 2 . The only information retained from the initial state is the weights corresponding to each energy sector. Similarly, a direct application of (3) shows that at the second order, the information about the connected correlations between the two energy sectors is contained in a compact way in tr(ρ 0 ⊗ ρ 0 I (1,2,2,1) ) with I (1,2,2,1) = E∈I 1 ,E ∈I 2 |E, E E , E|. Thus, one would not need fine information about the initial state to describe the equilibrium properties of the system. Of course, as with ETH, the range of applicability of these hypothesis is for now rather elusive and needs to be determined via careful numerical or experimental studies. We wish to report more on that in future studies.

V. CONCLUSION AND OUTLOOKS
We presented a theoretical framework enabling one to compute equilibrium properties of isolated quantum systems upon an assumption of quantum ergodicity which postulates that time-averages are equivalent to unitary ensemble averages in the stationary state. We brought specific attention to the relaxation of cat states, i.e quantum states which are a superposition of two macroscopically distinct states. We showed both analytically and numerically that a remnant of the initial quantum coherence was visible in the fluctuations around average quantities in the steady state whose amplitudes can be computed exactly. In the last part, we sketched a possible framework describing the equilibrium fluctuations in terms of statistical ensembles that do not require full knowledge of the microscopic details of the initial state.
In non-integrable or integrable systems a subject of debate of the previous decades has been to determine which conserved quantities were relevant to describe the thermal ensembles determining the local equilibrium properties. The question is of particular relevance for integrable systems since they comprise in principle a macroscopic number of conserved quantities [26]. It is now commonly accepted that one should only consider local (or quasilocal [27]) quantities to characterize such ensembles. However, our study stipulates that these ensembles no longer suffice when one looks at the equilibrium fluctuations of the system. There, additional information about possibly non-local conserved quantities are required.
Another important affirmation of ETH concerns the notion of typicality [28,29]. Typicality states that for all eigenstates of the energy window, few-body operators have thermal distributions in the thermodynamic limit. It would be very interesting to test whether some notion of typicality remains in our case, i.e that the fluctuations of -possibly non-local-few-body observables are described by typical distributions in the thermodynamic limit starting from any superposition of states belonging to the macroscopically different energy sectors.
Another interesting point is that the equilibrium formulae (2,3) can in principle be applied to non chaotic or integrable system. Of course, there is no reason for the ergodic property to be fulfilled anymore so there is no guarantee that they provide the right predictions. Typically, for finite-size integrable systems for example, one expects the existence of long-lived oscillations that prevents the system from equilibrating [30,31]. However one should remark that the various symmetries of the system that lead to ergodicity breaking are accounted for in the structure of the group.

Supplementary material Appendix A: Second order fluctuations
In this section we show how to compute formula (3) from the main text , i.e we want to compute : In essence, the calculations will rely on the same mechanics than for order 1 with some twists. Once again, we define the decomposition of ρ ⊗2 0 into different sectors (ρ ⊗2 0 ) (i 1 ,j 1 ,i 2 ,j 2 ) as follows : The average of a block is given by : Indeed, suppose it's not the case : then, there exists a k such that ∀k , i k = j k . For definiteness, say k = 1. From the left invariance of the Haar measure, we then have ∀V (i 1 ) ∈ SU (d i 1 ) that : which implies E[(ρ ⊗2 0 ) (i 1 ,j 1 ,i 2 ,j 2 ) ] = 0. We thus learn the important fact that {j 1 , j 2 } must be a permutation of {i 1 , i 2 } for the average of the block to be non zero.
There are three possible cases that we will examine separately : Case I : Let D (i) be the fundamental representation of SU (d i ). The tensor product representation D (i) ⊗D (i) admits a decomposition onto irreducible representations indexed by Young diagrams. We denote by {2} the possible partitions of 2, i.e (2) and (1, 1). Following the usual convention for indexation of irreducible representations of the unitary group by Young tableaux, D (2)(i) corresponds to and the tensor representation that is symmetric under permutation of two indices while D (1,1)(i) corresponds to and denotes the antisymmetric representation. We have [32] : The representation D (2)(i) preserves the symmetric eigenbasis made of One can further block-decompose (ρ 0 ⊗ ρ 0 ) (i 1 ,i 1 ,i 1 ,i 1 ) according to these basis. Introducing : By Schur lemma, we then have than the only non-zero block components of E[(ρ ⊗2 0 ) (i 1 i 1 i 1 i 1 ) ] are the diagonal ones, i.e E[(ρ ⊗2 0 ) (i 1 i 1 i 1 i 1 ),((2),(2)) ] and E[(ρ ⊗2 0 ) (i 1 i 1 i 1 i 1 ),((1,1),(1,1)) ] and these blocks are proportional to the identity : is the identity matrix associated to the representation D {2}(i 1 ) . The proportionality coefficient is determined by taking the trace. Explicitly, we have : )I ((1,1),(1,1) Case II : We look at : Since D (i 1 ) and D (i 2 ) are two irreducible representations of SU (d i 1 ) and SU (d i 2 ), the tensor The proof comes from Schur orthogonality relation : Let D 1 and D 2 be two irreducible representations over vector spaces V 1 and V 2 : If G1 and G2 are finite, we show that the tensor product representation is again irreducible. Indeed the Schur orthogonality relation for an irreducible representation states (with normalized measure with respect to the group volume) that : where χ(g) is the character of the representation. Then : dη(g 1 × g 2 )|χ ⊗ (g 1 × g 2 )| 2 = dη(g 1 )|χ 1 (g 1 )| 2 dη(g 2 )|χ 2 (g 2 )| 2 = 1 and the representation D ⊗ is again irreducible.
By Schur lemma, we then have that As before, we take the trace to determine the proportionality coefficient, we get : Case III : We look at : Let M be an element of M ∈ L(H i , H j ) ⊗ L(H k , H l ) and σ ∈ S 2 . We define the right action of σ on M , M · σ as : As it will be useful later, we define in the same way, the left action σ · M acting on M as : We then have that : where σ is here the permutation {1, 2} → {2, 1}.
Applying Schur lemma as before then leads to : Regrouping the results for all three cases proves (3) of the main text : Appendix B: General formula at any order In this section, we wish to compute the generalization of the formula for the mean and second order correlation of density matrix elements to higher order, i.e : This is equivalent to knowing the generating function Z(A) defined as which is reminiscent of the Harish-Chandra Itzykson Zuber integral [33] except that the group G upon which the integration is performed is not an unitary group so we can't directly use that result.
To compute (B1), we will rely on the same approach than for the mean and the second order correlations, i.e we will first decompose ρ ⊗n 0 into different sectors transforming according to different combination of U (i k ) and identify the different invariants under such transformations. In spirit, it will be close to the proof of the invariant theory presented in the appendix of [11]. We introduce once again the block decomposition of ρ ⊗n 0 into (ρ ⊗n 0 ) (ij) ≡ (ρ ⊗n 0 ) (i 1 i 2 ···in,j 1 j 2 ···jn) defined by : The average of a block is given by : To each term U (i 1 ) ⊗U (i 2 ) · · ·⊗U (in) , we associate a standard ordering defined as the tensor product ≡ U (m 1 )⊗n 1 ⊗ · · · ⊗ U (mn)⊗n k .
We can use the same argument as before to show that the average is null unless the j k s are a permutation of the i k 's. We call this permutation γ : j γ(k) = i k . γ is related to σ, σ * by γ = σ * −1 σ.
Appendix C: More details on the case study In this appendix, we provide additional details on the numerics presented in the main text. As a reminder, the Hamiltonian we study is : h j σ z j with L = 12, J = 1 and h j picked at random between −1 and 1 with the uniform distribution.
We work in the 0 magnetization sector which has a dimension of 924. We choose a seed such that the mean level spacing of the spectrum is close to the Wigner distribution one : 0.53069. The  The theoretical values for the mean and variances of our different quantities are computed from equations (2) and (3) of the main text. Recall that for the protocol I, the initial state was chosen as the pure state 1 √ 2 (|Φ 1 + |Φ 2 ) and for the protocol II, it was the classical mixture described by the density matrix ρ 0 = 1 2 (|Φ 1 Φ 1 | + |Φ 2 Φ 2 |). For a fully degenerate spectrum, (2,3) gives that the average of a given observableM and its second order connected correlations are given by : Below, we explain why there is no difference in the first and second order correlation ofĤ R between protocol I and II while there are for the second order correlation ofQ.
Mean quantities : Since the overlap between |Φ 1 and |Φ 2 is small, we can approximate the previous expression by : Similarly : tr(E[ρ We see that as far as the mean quantities are concerned, the cat state and the classical mixture provide the same results. Second order fluctuations : From (3), we have that : Where in the last line we used the fact that i 1 |Φ 1 Φ 2 |i 2 is non zero only for E i 1 close to E 1 and E i 2 close to E 2 . But this in turns imply that i 1 |Ĥ R |i 2 ≈ 0 .
ForQ : Where to go to the last line we used the fact that if | i 1 |Φ 1 Φ 2 |i 2 + i 1 |Φ 2 Φ 1 |i 2 | 2 is non zero, the E i 1 is close to either E 1 or E 2 and E i 2 the other way around. This in turn implies that This also tells us that tr(E[ρ (II)⊗2 0Q ⊗Q] ≈ 0 We thus see a clear difference in the fluctuations ofQ for the two protocols.