Glimmers of a Quantum KAM Theorem: Insights from Quantum Quenches in One Dimensional Bose Gases

Real-time dynamics in a quantum many-body system are inherently complicated and hence difficult to predict. There are, however, a special set of systems where these dynamics are theoretically tractable: integrable models. Such models possess non-trivial conserved quantities beyond energy and momentum. These quantities are believed to control dynamics and thermalization in low dimensional atomic gases as well as in quantum spin chains. But what happens when the special symmetries leading to the existence of the extra conserved quantities are broken? Is there any memory of the quantities if the breaking is weak? Here, in the presence of weak integrability breaking, we show that it is possible to construct residual quasi-conserved quantities, so providing a quantum analog to the KAM theorem and its attendant Nekhoreshev estimates. We demonstrate this construction explicitly in the context of quantum quenches in one-dimensional Bose gases and argue that these quasi-conserved quantities can be probed experimentally.

A milestone in the dynamics of classical many-body systems is the Kolmogorov-Arnold-Moser (KAM) theory [1].Generically, classical many-body systems exhibit chaotic behaviour -that is to say, giving the bodies of such systems slightly different initial positions and velocities results in the bodies following radically different trajectories.An exception to this rule is made for a special set of systems termed integrable which possess conserved quantities beyond energy and momentum.The existence of these conserved quantities promises the availability of a set of action-angle {p i , q i } whose action variables are constants of motion.In such variables the Hamiltonian, H, is solely a function of {p i }, and Hamilton's equations of motion become particularly simple: Trajectories of bodies in integrable systems are not sensitive to initial conditions, but instead lie on invariant tori in phase space described by frequencies {ω i } parameterizing solutions to the equations of motion: qi = ω i .However integrable systems and their attendant simple behaviour are comparatively rare.And so the question arises what can one expect with a system which is merely close to being integrable.Is the motion of bodies in this system chaotic?Or is there some influence on the system's dynamic from being close to an integrable point?One answer to this question is given by the KAM theorem.It tells us that if we weakly perturb a classical integrable system, we do not immediately transit to completely chaotic dynamics, but rather see a smooth crossover.Specifically, the KAM theorem promises that a subset of the solutions {ω i } survive under a sufficiently small perturbation, H pert (p i , q i ), provided their frequencies are sufficiently irrational.What of quantum analogs to the KAM theorem?There is tremendous interest  in the role exotic con-served quantities play in the dynamics of low dimensional systems.This interest [6-13, 21, 22] arises in the context of one dimensional (1D) Bose gases from the ability to manipulate isolated gases and observe their relaxation in closed surroundings, both when the gases are near integrable points [27][28][29] as well as far away [30].In the context of quantum spin chains [14][15][16][17][18][19][20], it comes about from the wish to understand related thermalization questions as well as whether integrable systems can sustain ballistic transport.It also appears in the burgeoning field of many-body localization [31,32] of disordered interacting systems and associated attempts to construct sequences of conserved charges in what one would traditionally consider a non-integrable setting [33,34].
To understand crossover behavior arising from integrability breaking, both indirect measures such as level spacing statistics [35][36][37][38] as well as studies of systems in their quasi-classical limit using such tools as the semi-classical eigenfunction hypothesis [39][40][41] are oft employed.Such behavior is often phrased in terms of pre-thermalization plateaus [42][43][44][45][46], where a system's observables, in relaxing from some non-equilibrium initial state, remain nearly constant over some finite time interval before decaying to their final equilibrium value.Such plateaus have been argued to be controlled by the remnants of the conserved quantities of the nearby integrable system [45,46].
In this work we go beyond this and show that in finite systems it is possible to construct an infinite sequence of nearly conserved local operators, i=1 , in the presence of a perturbation that weakly breaks integrability, We will show that this near-conservation is good for all times.The Q Q Q i are conserved in the sense that if we consider a (non-eigen)state, |s , with average energy per particle E = s|H|s /N less than some bound Λ(N Q ), then for all times where δ( , N Q ) can be made arbitrarily small.These conserved operators are constructed as finite linear combinations (length The quality of this conservation can be controlled (i.e.Λ can be made larger and δ smaller) by adjusting how many, N Q , of the charges, Qi , appear in the linear combinations.
Our construction is akin not so much to the KAM theorem, but to what are known as Nekhoroshev estimates [2] inasmuch as the charges Q Q Q we construct are nearly conserved on the entirety of the low energy Hilbert space.While the KAM theorem promises that some subset of solutions of the equations of motion survive a perturbation and remain "close" to their integrable counterparts for all time, the Nekhoroshev estimates tell us that all solutions remain close to their integrable counterparts in the sense that for exponentially long times: , where here P * , T * , and a are constants and N is the number of degrees of freedom the system has [2].
While general, we perform this construction in the context of quantum quenches in one dimensional (1D) Bose gases.This setting is particularly appropriate as it is the experimental study of quantum quenches in these gases [27] that has led to tremendous interest in the role of exotic conserved quantities in quantum dynamics.Quenches are moreover directly relevant to understanding these experiments.Because of the one-body potentials that trap the gases, they can be at most approximately integrable.Thus the construction of a quantum version of the KAM theorem and its variants can only help yield insights into the dynamics of these gases in their experimental settings.

QUANTUM QUENCH DYNAMICS IN 1D BOSE GASES
To set the scene, we first describe the quantum quench in a 1D Bose gas as described by the Lieb-Liniger model [47].The Lieb-Liniger model is believed to provide an excellent description of a 1D Bose gas [48].In the absence of external (trapping) one-body terms, it is integrable with an infinite number of conserved operators, { Qi }.It's Hamiltonian with the addition of a one body potential, V (x), is given by The type of quantum quench we will study is found in preparing the gas on a ring of length L in the ground state of a parabolic trap [10,21,22], i.e.V (x) = 1 2 mω 2 x 2 , then at time t = 0, releasing the gas from the parabolic trap into a one-body cosine potential, V (x) = A cos(2πn cos x/L), and observing the subsequent dynamics of the gas.This quench protocol is illustrated in Fig. 1.
This form of the Hamiltonian, an integrable model together with an integrability breaking perturbation, allows us to determine the ground and excited states of the model pre-and post-quench through a numerical renormalization group (NRG) designed precisely to attack such problems [10,36,49,50] together with a set of routines known as ABACUS that allow numerically exact computation of matrix elements of operators in the Lieb-Liniger model [51].In turn, this gives us access to the post-quench dynamics of the gas.In particular we employ an NRG able to study perturbations of integrable and conformal continuum field theories.This approach, as it is an extension of a methodology known as the truncated conformal spectrum approach [3,4], has been primarily used to study perturbations of relativistic field theories [36,49,50], but has recently been applied to the Lieb-Liniger model perturbed by a one-body potential [10], the problem at hand.The NRG uses the eigenstates of the Lieb-Liniger model as a computational basis.Because this basis accounts for the interactions of the Bose gas particles with one another, this numerical method builds in the strong correlations present in the problem right at the start.We discuss details of this method in Appendix A1.
In Fig. 2 we show the time evolution of the gas after the quench.At time t = 0 we see the density profile of the gas in the ground state of the parabolic potential.After quenching the potential to a cosine, the gas moves away from the center, oscillates a number of times before settling into the minima of the cosine.This occurs at times of the order of t = 50t F -we are able to run the simulation out to times of t = 80t F .
While we are able to compute the dynamics of such observables as the density and the momentum distribution function, the key to the work in this paper will be our ability to compute the dynamics of the (formerly) conserved Lieb-Liniger charges, Qi .Our numerical approach makes this extremely simple because of our use t>0 gas in a parabolic poten/al, t<0 gas in a cosine poten/al, t>0 V confining V confining t=0 FIG.1: Quench protocol: We prepare the one dimensional Bose gas in its ground state in a harmonic trap.At time t=0 we release the gas into a cosine potential and track the subsequent dynamics.The shaded green regions are illustrations of the equilibrium density profiles of the gas in the presence of the confining potentials.
of the eigenstates of the integrable Lieb-Liniger model as a basis.Each Lieb-Liniger state of an N -particle gas |ψ LL is characterized by N -rapidities, λ i , i = 1, . . ., N , which should be thought of as, more or less, the momenta of the gas's particles.These rapidities determine the ac- FIG.2: The density profile of the gas at selected times postquench as computed with the NRG.Here this time dependence is computed after releasing a N = L = 14, c = 7200 gas prepared in a parabolic potential with mω 2 0 L 2 /2EF = 10.36 (shown with a green dashed line in the t = 0tF frame, tF = 1/EF , EF = k 2 F /(2m), and kF = π(N − 1)/L) into a cosine potential Vcosine(x)/EF = 0.35(cos( 4π L x) + 1) (plotted with a dashed line in the t = 43tF frame).In the t = 0 frame, we show the density profile as computed analytically in the hardcore limit (see Appendix A12).Using the NRG we can run the time evolution as far out as t = 85tF before dephasing exceeds 1%.We see however by t = 43tF the gas' density profile has already come close to its long time average (black dashed line in the final panel).
tion of the conserved operators on the Lieb-Liniger states.For example both the energy, E = Q2 and momentum, P = Q1 , operators act on |ψ LL via, The action of all of the higher non-trivial charges, Qn , n = 3, 4, 5, • • • in the Lieb-Liniger model are simply higher power sums of the same rapidities: While the actual expression of the charges in terms of the Bose field operators is complicated and unwieldy [54], the action of the charges on the Lieb-Liniger eigenstates turns out to be extremely simple.This will be crucial in facilitating our construction of effective Q Q Q's.

CONSTRUCTION OF CONSERVED QUANTITIES IN THE BOSE GAS POST-QUENCH
We now turn to the core of the paper.We have shown in the previous section that we can describe the temporal dynamics of various quantities post-quench.In that section we specifically considered the density profile of the gas after release into the cosine potential.We now consider the time evolution of the Lieb-Liniger charges.They are of course not conserved and so their evolution will be non-trivial.We however show that one can construct linear combinations of the Lieb-Liniger charges whose expectation values are nearly time invariant under unitary evolution by the post-quench Hamiltonian.The quality of this time invariance can be controlled by allowing more charges in the linear combination.Moreover we show that these linear combinations of charge are not merely time invariant with respect to the particular initial condition created in the quench protocol, but as operators acting on the low energy Hilbert space.
The post-quench time evolution of the Lieb-Liniger charges normalized by their mean value as described in the text.
Here the time dependence is computed after releasing a N = L = 8, c = 10 gas prepared in a parabolic potential of strength mω 2 0 L 2 /2EF = 3.24 into a cosine potential cos( 4π L x).We show this behavior at late times (for details of how long we can run the simulation, see Appendix A11).b) The post-quench time evolution of a sequence of effective charges, We begin by first considering the time evolution of the individual Lieb-Liniger charges themselves.We plot this evolution for the first four Lieb-Liniger charges in Fig. 3 for a gas with N = L = 8 and c = 10.In plotting the time evolution we have normalized each charge to its mean value post-quench so that all of the charges fluctuate about 1.The mean value of the unnormalized n-th charge, given by, where T is the time out to which we can track the evolution, grows rapidly with n as the charge's action on a Lieb-Liniger eigenstate We see from Fig. 3 that even after normalization, the size of the oscillations increases with n.
We now consider linear combinations of the Lieb-Liniger charges of the form: where we choose the constant a 0 such that the mean value of Q Q Q(N Q ) is about 0 and the remaining constants a i such that the fluctuations in We plot the time evolution for a c = 10 gas of these effective charges in panel b) of Fig. 3 for three different values of N Q , the number of charges in the linear combination.In panel c) we plot the fluctuations of this charge as a function of N Q .We see that these fluctuations drop exponential with N Q .In the bottom part of panel c) we do the same for a quench involving a c = 1 gas.In order to be sure that we are not simply reconstructing the post-quench Hamiltonian as some linear combination of the Lieb-Liniger charges, in both cases (c = 10, 1), we demonstrate we can construct simultaneous multiple effective charges.In panel c) we show that the fluctuations of a second effective charge built as a linear combination of charges drawn from { Q2n } 16 n=9 also die off exponentially.
This exponential dependence in N Q is possible to understand at large c.To do so, we write the initial condition of the gas in terms of post-quench cosine eigenstates: With the initial condition as above, the time dependence of the charge takes the form: We demonstrate in Appendix A21 that each Lieb-Liniger with Λ(N Q ) = (2π(N Q − 2)/L) 2 .We then see the weight that is not zeroed out and so can contribute to Q Q Q's temporal fluctuations goes as e −(Λ(N Q )/mω0) 2 .For large amplitude A cosine potentials, the temporal fluctuations die off much more slowly with N Q : In this latter case, essentially the number of non-zero matrix elements of Q Q Q(t) proliferate, making a construction where it is nearly time invariant much more difficult.So far we have only demonstrated that we can construct charges Q Q Q as linear combinations of the original Lieb-Liniger charges, Qn , whose time fluctuations can be made arbitrarily small supposing we start the system in a specific initial condition, |ψ GS,para .However we now demonstrate that these charges are quasi-conserved not just relative to a specific initial state, but as operators, at least when projected onto the low energy post-quench Hilbert space.
To do so we compute the off-diagonal matrix elements in Fig. 4 of one of the two Q Q Q's we have constructed (the one constructed with Lieb-Liniger charges, Q2 , • • • , Q16 ) relative to the basis of the low-lying energy eigenstates of the post-quench Hamiltonian.These matrix elements are plotted in Fig. 4. In the rightmost panel we display the off-diagonal matrix elements of Q2 (normalized as described previously) to set the scale of how large these matrix elements are for the individual Lieb-Liniger charges.
In the middle panel we then plot the matrix elements of Q Q Q (8).We see that most of the previous non-zero matrix elements of Q2 are now dramatically reduced.We quantify this disappearance in panel c) of Fig. 4.There we present the average magnitude of the off-diagonal matrix elements as a function of N Q .We present data for both effective charges considered in Fig. 3 for both values of c = 1, 10.We see in all cases the size of these matrix elements drops exponentially in N Q .Roughly speaking, if the average energy per particle of two distinct states, |s , |s , is less than Λ(N Q ), then s|Q Q Q|s will be exponentially small.We conclude that the Q Q Q's are then nearly conserved as operators.This conclusion is supported by an analytic construction of the Q Q Q's that we present in Appendix A2.
this minimization being done for a particular quench protocol, the conservation of the charge occurs at the operator level.Specifically, off-diagonal matrix elements of the charges are small.We demonstrated that both postquench temporal fluctuations and the off-diagonal matrix elements can be made exponentially small in the number of charges, N Q , in the linear combination.We have supported this construction by demonstrating an equivalent analytic construction of these charges (Appendix A2).
In the introduction to this paper, we have billed these constructions as being quantum equivalents to the quantum KAM theorem and its counterparts such as the Nekhoreshev estimates.There are some similarities in the consequences of our constructions as well as some dissimilarities.Nekhoreshev estimates tell us that the values of the classical action variables in the face of a small nonintegrable perturbation change only very slowly in time, as controlled by both the size of the perturbation and the number of degrees of freedom (see Eqns. 5 and 6).
For the quantum case, we see something analogous but with certain differences.These differences arise both because we are forming linear combinations of the originally conserved charges, and because of how in our construction we segregate portions of the quantum phase (Hilbert) space.Nekhoreshev estimates apply to the entire phase space of the weakly perturbed model (i.e.Eqn. 5 is good for any p i (t = 0)).In contrast, in our constructions, the approximate time invariance of the charge is restricted to a portion of the low energy Hilbert space as marked by the integer N max .While we can make N max as large as we want (provided we are willing to make N Q correspondingly large), we cannot take it to be infinite.
Another difference between the two constructions is the role played by the strength of the integrability breaking perturbation.Here the Nekhoreshev estimates provide a bound on the temporal variation of the original action variable going as a fractional power (a function of the system's degrees of freedom) of the strength of the perturbation.We, in contrast, can construct effective charges, Q Q Q, whose temporal variation is controlled not directly by A, but N Q the number of Lieb-Liniger charges forming Q Q Q.To be sure if A is large, N Q will need to be correspondingly larger in order to produce the same minimum of temporal variation (see Eqn. 14).
In constructing these charges the nature of the potential here is important.Our potential mixes the momenta of different (unperturbed) eigenstates solely through the wavevector of the cosine potential.This is then considerably different than the integrability breaking considered in Refs.[52,53] where they considered integrability that respected no selection rules and correspondingly saw an extremely rapid crossover from quantum integrable to quantum chaoticity.8) in the post-quench eigenbasis as derived for the c = 1 quench discussed previously in Figs. 3 and 4.

Experimental consequences
Having constructed these charges, we can ask what are the consequences of their existence.That they take non-zero values on the eigenstates means that the long time dynamics of the gas post-quench is going to be constrained.In this light, we have one way to understand the "quantum Newton's cradle" experiment presented in Ref. [27].As we discussed in the introduction, it was argued there that the post-quench dynamics of a gas were very slow to achieve equilibration and that this slowness was indicative of the underlying integrability of the Lieb-Liniger model.However, strictly speaking, the gas in this experiment was not integrable.The gas was confined in a one-body parabolic potential, a potential that breaks integrability [55].Our construction of effective quasi-conserved charges in the presence of an integrability breaking one-body potential thus provides a means to understand the slow thermalization of the gas postquench in this experiment despite the presence of integrability breaking.More generally, our construction helps explain the finding of [56,57] where weak integrability breaking does not lead to immediate thermalization in finite systems.
In constructing these operators, it should be stressed that the operators we construct are local (in the sense that they are spatial integrals over operators that are defined at a single point in space).This follows as the effective charges, Q Q Q, are constructed as linear combinations of the Lieb-Liniger charges, which are all local quantities.Thus we are not constructing, in effect, projection operators corresponding to eigenstates of the post-quench gas.Such projection operators are necessarily always present in a model regardless of its integrability.To demonstrate this we plot the diagonal matrix elements of the charges, Q Q Q, which are linear combinations of eight Lieb-Liniger charges and whose average off-diagonal matrix elements are presented in Fig. 5.We see that these matrix elements are all O(1).
If the nearly conserved quantities are governing the long time dynamics of 1D Bose gases as in Ref. [27], a second question that must be asked is whether this influence is merely confined to a pre-thermalization plateau or whether it influences the dynamics of the gas at all times.There have been at least two constructions [45,46] of quasi-conserved quantities that are thought to govern pre-thermalization plateaus.Our construction is fundamentally different inasmuch as the quasi-conserved operators are such for all times.This, in particular, implies that a modified form of Mazur's inequality [58] holds.This inequality relates the long time average of a correlation function lim t→∞ O(t)O(0) with the projection OQ of the operators O onto conserved charges, Q.This inequality continues to hold with quasi-conserved charges Q Q Q but with the addition of an error term that is proportional to the size of Q Q Q's off-diagonal matrix elements (which, in our construction, can be made arbitrarily small), something immediately clear from the proof of Mazur's inequality found in Ref. [59].This implies that Q Q Q will control the long time limit of a host experimental observables in systems with weak integrability breaking.
Our approach to describing the dynamics associated to the quantum quench of the gas is to employ a numerical renormalization group [A1] that employs the eigenstates of the Lieb-Liniger model as a computational basis to determine the relatively low lying eigenstates of the Bose gas in a one-body potential.This numerical renormalization group is built upon both ideas taken from K. Wilson's development of a numerical renormalization group used to study quantum impurity problems [A2] as well as Al.B. Zamolodchikov's numerical treatment of perturbed conformal field theories [A3, A4].The use of the Lieb-Liniger basis as such a basis trades on our ability to be able to efficiently compute matrix elements of relevant operators such as the density operator exactly.While there are compact determinental expressions for such matrix elements [A5, A6], their evaluation is still a non-trivial numerical task and to this end we use a set of computerized routines named ABACUS [A7-A9].We have already demonstrated that we are able to perform the first step in our quench protocol: we have shown in Fig. 2 that we can accurately compute the ground state of the gas in the parabolic trap.In this figure we plotted our numerical determination (black) of the density profile of a gas with N = 14 particles in a system of length L = 14 with an interaction parameter of c = 7200 in a trap of strength V para = 1 m ω 2 0 x 2 with mω 2 0 L 2 /2E F = 10.36 against the density profile determined analytically (red) by mapping these (nearly) hardcore bosons onto free fermions.The details of the analytic description of the gas in its hardcore limit are found in Appendix A12.
In the second step of the quench protocol, we released the gas into a one-body cosine potential, In order to compute the post-quench dynamics, we need to be able to describe not only the ground state in the cosine potential, but some large number of excited states.
In our quench protocol, we take as our initial t = 0 state the ground state of the gas in the parabolic potential, |ψ GS,para .If we can compute a wide range of eigenstates in the cosine potential, both ground and excited states, |ψ α,cos , we can expand this initial state in terms of the post-quench basis: Of course for this expansion to be exact, we would need to know all of the eigenstates of the gas in the cosine potential.We will instead settle for a determination of the post-quench eigenbasis that allows us to include enough states so that α |c α | 2 > 0.99.In computing the spectrum of states in the cosine potential, we employ the variant of the NRG discussed in Ref. [A10].The NRG in its plain vanilla formulation [A1] can compute the spectrum of the low lying states of the gas in the one-body potential [A11].But to capture accurately an appreciable fraction of the spectrum, we need to employ a sweeping routine [A10] analogous to that used in the finite volume routine of the density matrix renormalization group [A12, A13].
In Fig. A1 we present results for the spectra of an N = L = 14 gas in the hardcore limit c = 7200.Here we plot in black (r.h.s.) the numerical determination of the first 365 energy levels of the gas in a cosine potential.In red (l.h.s.) we plot the corresponding analytic determination of the levels.This analytic determination is possible by mapping the bosons to nearly free fermions who interact with a four-body term of strength 1/c.Again the details of the analytics is found in Appendix A12.The difference between the numerics and the analytics here is less than 10 −3 (in absolute units).
Once we have this expansion of our initial condition |ψ GS,para in terms of the eigenstates in the cosine potential, we can readily determine the time evolution of the state post-quench: We can track time evolution of the state to a point in time determined by the accuracy by which we can determine E α .If the accuracy to which we determine E α is δE α , we can only track time evolution while tδE α 2π before we can no longer trust the numerics.Concretely, we call a state |ψ α,cos dephased at time t if δE α t > 0.01 × 2π and we conservatively will not track the time evolution beyond a point where the sum of states that are dephased have a weight exceeding 0.01, i.e.
run out to times ∼ 80t F , while for the N = 8 data in Figs.3,4 and 5 of the main text, we can run considerably longer, to t ∼ 6000t F .
With the time evolved state in hand we are able to compute the time evolution of a number of observables and operators.Because we use the eigenstates of the Lieb-Liniger model absent a one-body potential, |ψ α,LL , as the computational basis of the NRG, the NRG gives any eigenstate in a one-body potential as a linear combination of such states: Thus the dynamics of any operator whose matrix elements are known in the Lieb-Liniger basis can be determined.As one example, we plotted in Fig. 2 of the main text the time evolution post-quench of the density profile of the gas.
A12: Description of the gas in the cosine potential in the large c limit In this appendix we provide a description of the hardcore limit (c → ∞) of the Lieb-Liniger model defined on a ring of length L in the presence of a cosine potential: The ability to do analytics in the hardcore limit will then serve as a check on our numerical results.
For c 1 the system can be mapped onto a system of fermions with Hamiltonian [A14-A16] where in the dual picture we have an ultra-local interaction term of strength 1/c.For c → ∞ the fermions are noninteracting and the physics becomes effectively one-body [A18].We then must only solve the following single-body Schrödinger equation: This equation can be put in the standard form of the Mathieu equation, What we now will argue is that we can systematically zero out all low energy matrix elements (below some designated cutoff) of the first term involving the commutator of the one-body potential with Q Q Q(t).This results in a charge Q Q Q which has a t 2 (and higher) time dependence on the low energy Hilbert space.However this higher order dependence is only nominal.What we observe is that for a cosine one-body potential, zeroing out first-order matrix elements also zeros out a large number of matrix elements from higher order commutators that arise from one particle-hole processes.To keep things tractable in this construction we will only explicitly consider the c = ∞ limit where there are no more than one particle-hole processes.
To understand why higher orders remain zeroed out, we first need to describe the Hilbert space as spanned by the Lieb-Liniger eigenstates in a bit more detail.An eigenstate of the Lieb-Liniger model is described by Nrapidities, λ i , which in turn are determined by N -integers (or halfintegers) via the Bethe ansatz equations: We use the notion of these quantum numbers both to delineate the zeroed-out portion of the Hilbert space as well as to describe how it changes under higher order processes.
Let us now construct the effective charge by defining it to have the following property: if the integers characterizing |s and |s are all such that then the following matrix element vanishes: This condition amounts to insisting that Here for c = ∞ we illustrate how the zeroed matrix elements of C1 on a low-energy block of the Hilbert space become successively non-zero with increasing order of the higher order commutators, Cn>1. where Provided we are willing to include enough Lieb-Liniger charges in Q Q Q (i.e.choose N Q large enough) we can always find a Q Q Q satisfying Eqn.(A26) as the collection of constraints in Eqn.(A26) form a set of homogenous linear equations: The number of charges, N Q we need to include to be able to find a non-trivial solution behaves as N Q = N max + 2, a number that is effectively proportional to the log of the size of Hilbert space.
So we now suppose we have constructed a Q Q Q where a block of states of its commutator with V cosine have been zeroed out -see the top square in Fig. A2 for a graphical representation of this.But now how does this zero block fare when we consider higher order commutators,  In this section we estimate the quality of the conservation of the charges Q Q Q(t) that we have constructed in the previous section.We do so for both weak and strong amplitudes of the post-quench cosine potential.
To determine the magnitude of the time variation in Q Q Q(t) following the quench, we first express the initial condition, |ψ para , in terms of the post-quench eigenbasis |ψ α,cos : and then in turn express Q Q Q(t) in terms of matrix elements of Q Q Q in this basis: We have argued in Appendix A21 that the construction of Q Q Q(t) is such that the time dependence (at least up to some order in time) of the low energy off-diagonal matrix elements of Q Q Q(t) are zeroed out.This implies that some of the terms in the above expansion will be either zero (or at least small).But which ones and what weight do they carry?The matrix elements we have zeroed are not in the post-quench basis but in the Lieb-Liniger eigenbasis, the eigenbasis of the gas without a one-body potential.
To see the effects of this zeroing out, we expand |ψ α,cos in terms of this basis: where the state |I 1 , • • • , I N is constructed according to Eqns.(A22) and (A23).We then in turn rewrite Q Q Q(t) in terms of matrix elements involving this Lieb-Liniger basis: From our construction of Q Q Q, we see that the matrix elements involving states |I 〈ρ(x)〉 av /ρ 0 4, and 8. c) The standard deviation of the fluctuations of two sequences of effective charges Q Q Q.We build the first sequence (in black) using linear combinations of the charges { Q2m} m=8 m=1 , while the second sequence (in red) is formed with the next eight Lieb-Liniger charges, i.e. { Q2m} m=16 m=9 .d) We show the fluctuations of the two effective charges built following the quench of a c = 1 gas prepared in a parabolic trap of strength, mω 2 0 L 2 /2EF = 0.13, and released into the same cosine potential, cos( 4π L x).
FIG.4: a) We plot the intensity of the off-diagonal matrix elements of Q2, comparing it to b) the off diagonal m.e.'s of Q Q Q(8) for the quench of the c = 1 gas discussed in Fig.3c.c) We plot the average size of the off-diagonal matrix elements of two sequences of effective charges Q Q Q(NQ), in black is the sequence constructed from Q2m, m = 1, • • • , 8, while in red is the sequence constructed from Q2m, m = 9, • • • , 16.We show this for both the c = 1 (same quench as in a) and b)) and the c = 10 case (same quench as described in Fig.3a)-c).

FIG. 5 :
FIG. 5:We plot the values of the diagonal matrix elements of Q Q Q(8) in the post-quench eigenbasis as derived for the c = 1 quench discussed previously in Figs.3 and 4.
FIG.A1:A plot of the energy spectra for an N=14 gas with c=7200 in a cosine potential of amplitude A/EF = 0.35 (as in Fig.2of the main text).The analytic results are given in red while in black are the corresponding numerics.On the r.h.s.we expand a range of energy with a dense number of states so as to better exhibit agreement between the numerics and the analytics.We can determine the first 365 states (up to energies of E = 65) with accuracy of 10 −3 .

C 2 =C 3 =C 4 =C 5 =C 6 C 7 =
s [V cosine ,C 1 ] s' = 0 s [V cosine ,C 2 ] s' s [V cosine ,C 3 ] s' = 0 s [V cosine ,C 4 ] s' = s [V cosine ,C 5 ] s' = 0 s [V cosine ,C 6 ] s' FIG. A3: The magnitude of the off-diagonal matrix elements of an effective charge constructed from a) four Lieb-Liniger charges, Q Q Q(4), and b) from eight Lieb-Liniger charges, Q Q Q(8).c) The size of the post-quench temporal fluctuations of the effective charges Q Q Q(NQ) as a function of the number of charges in the linear combination.Here the quench is performed by preparing an N = 8, c = 7200 gas in a parabolic potential of strength mω 2 0 L 2 /2EF = 6.48 and released into a cosine potential of strength A = 1.We show the size of the temporal fluctuations for two sequences of effective charges, the first (in black) constructed from Lieb-Liniger charges, Q2m, m = 1, • • • , 8 and the second (in red) constructed from Q2m, m = 9, • • • , 16. d) The size of the off-diagonal matrix elements of these same two sequences of Q Q Q as a function of the number, NQ, of Lieb-Liniger charges in the linear combination.

TABLE A1 :
Degree to which numerical Q Q Q lies in null space of analytic Q Q Q's for a c = 7200, N = L = 8 gas:Nmax NQ Dim.null space the Temporal Variation of Q Q Q(t) 1