Necessary and sufficient condition for quantum adiabaticity in a driven one-dimensional impurity-fluid system
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Necessary and sufficient condition for quantum adiabaticityin a driven one-dimensional impurity-fluid system
Oleg Lychkovskiy , , Oleksandr Gamayun , and Vadim Cheianov Skolkovo Institute of Science and Technology, Skolkovo Innovation Center 3, Moscow 143026, Russia Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia, Institute for Theoretical Physics, Universiteit van Amsterdam,Science Park 904, Postbus 94485, 1098 XH Amsterdam, The Netherlands and Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands (Dated: October 18, 2018)We study under what conditions the quantum adiabaticity is maintained in a closed many-bodysystem consisting of a one-dimensional fluid and an impurity particle dragged through the latter byan external force. We employ an effective theory describing the low-energy sector of the system toderive the time dependence of the adiabaticity figure of merit – the adiabatic fidelity. We find thatin order to maintain adiabaticity in a large system the external force, F N , should vanish with thesystem size, N , as 1 /N or faster. This improves the necessary adiabatic condition F N = O (1 / log N )obtained for this system earlier [1]. Experimental implications of this result and its relation to thequasi-Bloch oscillations of the impurity are discussed. I. INTRODUCTION
Quantum adiabatic theorem is a fundamentally im-portant result in the theory of quantum systems withtime-dependant Hamiltonians. In essence, it states thata system initially prepared in an instantaneous eigenstateof a Hamiltonian remains arbitrarily close to the (time-evolving) instantaneous eigenstate provided the ramprate (i.e. the rate of change of the Hamiltonian) is slowenough [2, 3]. When it comes to applying the adiabatictheorem in practice, the key question to be addressedis how slow ”slow enough” is. While this question canbe exhaustively answered for a simple two-level system[4, 5], it becomes complicated for many-body systemsand/or for continuous quantum systems with infinite-dimensional Hilbert spaces. Although numerous suffi-cient conditions for adiabaticity are known (see the pi-oneering work [6] and the review [7]), they often proveto be inapplicable for continuous quantum systems dueto the divergence of operator norms entering these con-ditions. Recently a necessary condition for adiabaticityhave been proven [8] which is free from this shortcomingand is well-suited for applying to many-body systems.Anyway, any adiabatic condition, whether sufficient ornecessary, provides only a bound on the driving rate,without an indication how tight this bound is. In the present paper we derive a necessary and suffi-cient condition for quantum adiabaticity in a many-bodysystem consisting of a one-dimensional fluid and an im-purity particle dragged through the latter by a constantexternal force. This system exhibits a spectacular phe-nomenon predicted in refs. [9–11] and experimentallyobserved in ref. [12] – quasi-Bloch oscillations of the One could imagine that both sufficient and necessary conditionare available and provide bounds which are close to each other.In practice, however, such a fortunate occurrences are rare if notextinct, at least in the many-body context. impurity’s velocity and position. These oscillations aresomewhat reminiscent to the Bloch oscillations of a singleparticle in a periodic potential. Their root cause is thenontrivial spectral edge of the impurity-fluid system inone dimension, which is periodic in the thermodynamiclimit as a function of the total momentum, with the pe-riod 2 πρ determined by the density of the fluid ρ ≡ N/L (here N is the number of particles of the fluid, L is the lin-ear dimension of the system, and we set ~ = 1 throughoutthe paper). There are two important features, however,distinguishing quasi-Bloch oscillations from the conven-tional Bloch oscillations. First, intriguingly, quasi-Blochoscillations occur in the translation-invariant system, inthe absence of any external periodic potential. Second,quasi-Bloch oscillations are a genuinely many-body phe-nomenon.The exact conditions for the occurrence of the quasi-Bloch oscillations is a matter of a controversy [9–11, 13–18]. However, it is undisputable that the many-bodyadiabaticity is a sufficient (though, in general, not neces-sary) condition for the quasi-Bloch oscillations [10]. Thisis the reason for our interest in conditions for adiabaticityin the one-dimensional impurity-fluid system.Recently we have applied a necessary adiabatic con-dition of ref. [8] to the impurity-fluid system, with theresult that in order to maintain adiabaticity the drivingforce F N should vanish with the system size (the densityof the fluid being fixed) at least as fast as O (1 / log N ) [1].This result demonstrates that the adiabaticity does notsurvive in the thermodynamic limit, as is expected ongeneral grounds for a gapless many-body system [19].However, if the O (1 / log N ) scaling were a true scaling ofthe maximal force tolerated by adiabaticity, it would bewell possible to observe the adiabatic evolution in state-of-the-art cold atom experimental settings with moder-ately large N , e.g. with N ∼
100 like in the experimentof ref. [12]. This observation motivated us to searchfor a necessary and sufficient adiabatic condition in theimpurity-fluid system.Here we report such a condition obtained in the frame-work of an effective theory describing an impurity slowlymoving in a 1D quantum fluid [20–22]. The conditionhas the form F N < O (1 /N ), which is a dramatic quan-titative difference from the logarithmic scaling obtainedpreviously. This result implies that maintaining many-body adiabaticity in the impurity-fluid system is a verychallenging experimental task.The paper is organized as follows. After a generaldiscussion of the notion of adiabaticity in Sec. II, weintroduce the impurity-fluid system and its effective de-scription in Sec. III. The diagonalization of the effectiveHamiltonian at a given moment of time is reviewed inSec. IV. The solution of the full dynamical problem ispresented in Sec. V. In Sec. VI the results are presentedand their immediate experimental implications are dis-cussed. In Sec. VII we summarize our results and makea couple of concluding remarks. Most of the technicalitiesare reserved to the appendices. II. ADIABATICITY: FIGURE OF MERIT
We start from introducing the notion of adiabaticityin quantitative terms. Consider a parameter-dependentHamiltonian H Q , Q being for a moment an abstract pa-rameter. We introduce time dependence in this Hamil-tonian by assuming that Q linearly varies with time t , Q = F N t. At the moment F N is treated merely as another abstractparameter quantifying the dirivng rate. The subscript N in F N indicates that the driving rate may, in general,scale with the system size. For each Q one can definean instantaneous ground state, Φ Q , which is the lowesteigenvalue solution to the Schr¨odinger’s stationary equa-tion, ˆ H Q Φ Q = E Q Φ Q . (1)Here E Q is the instantaneous ground state energy. We as-sume that the ground state is non-degenerate for any Q .The dynamics of the system is governed by theSchr¨odinger equation, which can be written in a rescaledform as (cid:16) ˆ H Q − i F N ∂ Q (cid:17) Ψ Q = 0 . (2)Here Ψ Q is the state vector of the system which dependson time through the time-dependent parameter Q . Ini-tially, at t = 0 (or, equivalently, at Q = 0), the system isprepared in the instantaneous ground state:Ψ = Φ . (3)The evolution is called adiabatic as long as the state ofthe system, Ψ Q , stays close to the instantaneous groundstate, Φ Q . To what extent adiabaticity is preserved dur-ing the evolution is quantified by the adiabatic fidelity F Q , which is the probability that the evolved state coin-cides with the instantaneous ground state, F Q ≡ |h Φ Q | Ψ Q i| . (4)Perfect adiabaticity would imply F Q = 1. Calculating F Q for the driven impurity-fluid system is the main goalof our study.Another useful quantity is the adiabatic mean free path Q ∗ which quantifies how far can the system travel in theparameter space for a given driving rate before the adi-abaticity breaks down. We define this quantity as thesmallest positive solution of the equationlog F Q ∗ = − . (5) III. IMPURITY-FLUID SYSTEMA. Preliminary considerations
An object of our study is a one dimensional many-bodysystem, consisting of a quantum fluid and an impurityparticle. A constant external force F N is exerted uponthe impurity. The fluid consists of N identical particles,either fermions or bosons. The particles of the fluid in-teract with the impurity and, in general, with each other.As a preliminary step, we discuss how to describe aone-body problem of a noninteracting impurity particlepulled by an external force. This can be done conve-niently by introducing a time-dependent Hamiltonianˆ H imp Q = ( ˆ P + Q ) m , (6)where m is the mass of the impurity, ˆ P ≡ − i∂/∂X isthe canonical momentum of the impurity and X is thecoordinate of the impurity. Periodic boundary conditionswith some period L are implied. In this context Q = F N t is the impulse of the force.The interacting impurity-fluid system is described bythe microscopic Hamiltonianˆ H micro Q = ˆ H imp Q + ˆ H f + ˆ H if , (7)where ˆ H f is the Hamiltonian of the fluid which includesthe kinetic term and the pairwise interactions betweenthe particles of the fluid, and ˆ H if is the impurity-fluidcoupling. We do not specify microscopic Hamiltoniansˆ H f and ˆ H if explicitely since our analysis will be basedon an effective low-energy model described in the nextsection.A general feature of translation-invariant one-dimensional systems described by the Hamiltonian (7)is that eigenenergy as a function of Q is periodic in ther-modynamic limit [23]. The period is determined by thenumber density of particles and, in our case, is given by2 k F up to finite size corrections, where k F ≡ πN/L = πρ .The latter quantity sets the typical momentum scale ofthe problem. For the fluid consisting of noninteract-ing fermions, k F coincides with the Fermi momentum.It should be emphasized, however, that in general casewe do not ascribe any “fermionic” meaning to k F . Inparticular, we consider bosonic and fermionic fluids onequal footing. Note that Fermi statistics plays no role inthe above-mentioned periodicity of eigenenergies, whichis present for bosons as well. B. Effective Hamiltonian
Under fairly general conditions the low-lying excita-tions of a one-dimensional quantum fluid can be treatedby means of an effective Luttinger liquid theory [24]. Thistheory can be extended to describe the low-energy sec-tor of the one-dimensional impurity-fluid system (7) [20–22, 25]. This extension is valid for sufficiently small ab-solute value of the velocity of the impurity, v Q (belowwe will discuss this condition in more detail). The corre-sponding effective Hamiltonian reads [20–22, 25]ˆ H = v Q ˆ P + v s X q | q | ˆ a † q ˆ a q − √ πL X q √ q δ qQ ∆ v qQ (ˆ a † q e − iqX + ˆ a q e iqX ) . (8)Here v s is the sound velocity of the fluid, a † q , ˆ a q are cre-ation and annihilation operators of bosonic excitations ofthe fluid carrying momentum q ,∆ v qQ ≡ v s − v Q sign q (9)and δ qQ = δ + Q , q > , , q = 0 δ − Q , q < In eq. (8) and throughout the paper the increment inthe sums over q equals the momentum quantum δk ≡ π/L . For definitness, we employ the ultraviolet cutoff Observe that the definition of v Q is self-consistent in the frame-work of the model (8): ˆ V ≡ i [ ˆ H, X ] = v Q , i.e. the operator of im-purity’s velocity, ˆ V , is equal to the time-dependent c -number v Q . Strictly speaking, the boson operators a q , a † q might also dependon Q . This subtle issue is a particular instance of a generalproblem of ambiguity of a connection on the bundle of Hilbertspaces over a space of external parameters which vary in time(see e.g. ref. [26] for a discussion). Following an establishedpractice [20–22, 25], we ignore this possible dependence. This isto say, we assume that this dependence is either absent or pro-duces corrections which are subleading for considered ranges of Q (see below). This assumption is supported by an independentanalysis within a microscopic integrable model, see Appendix E. equal to k F , although the exact cutoff value does notenter the final results.The total canonical momentumˆ P tot ≡ ˆ P + X q q ˆ a † q ˆ a q (11)commutes with the Hamiltonian H and therefore is con-served. Its eigenvalue is quantized in units of δk . Thecanonical momentum should not be confused with thetotal kinetic momentum, ˆ P kin = ˆ P tot + Q , which is notquantized and grows linearly with time due to the actionof the external force.In fact, the effective model (8) is well-defined only ina subspace of the Hilbert space corresponding to someeigenvalue P tot of the total canonical momentum. In thissubspace the kinetic momentum also has a well-defined,though varying with time, eigenvalue P kin . Effective“constants” v Q , δ + Q and δ − Q are in fact functions of thekinetic momentum, P kin = P tot + Q (and, therefore, no-tations v P kin , δ + P kin and δ − P kin would be more consistent).We, however, choose to fix P tot and refrain from referringto it explicitly throughout the paper, except the presentsection and Appendix A. The only importance of the pre-cise value of P tot is to fix v Q , δ + Q and δ − Q at Q = 0.The range of validity of the effective Hamiltonian (8)is a somewhat subtle issue. This Hamiltonian is designedto describe a low-energy sector of the Hilbert space, i.e.an energy shell of a width ∆ E P kin above the ground state.The subscript in ∆ E P kin indicates that the range of va-lidity of the effective model varies with P kin . In cer-tain cases ∆ E P kin is nonzero in the whole Brillouin zone, − k F < P kin < k F , and vanishes only at its ends [20–22].In particular, this is the case for an integrable modelsolved by McGuire [27] which is discussed in AppendixE. In other cases, however, the effective model (8) breaksdown in a finite portion of the Brillouin zone. In partic-ular, this happens for a sufficiently light impurity weaklyinteracting with a one-dimensional fluid [13, 15]. In gen-eral, one expects that ∆ E P kin is nonzero as long as theimpurity moves with the velocity below the generalizedcritical velocity v c ≤ v s which ensures absence of theCherenkov-like radiation [15]. The latter critical velocityis typically on the order of v s . To summarize, to be onthe safe side, one can assume v Q ≪ v s , (12)although the actual range of validity of the effectiveHamiltonian (8) can be much wider. IV. INSTANTANEOUS GROUND STATE
The first ingredient required for calculating the adia-batic fidelity F Q is the instantaneous ground state Φ Q of the Hamiltonian (8). The latter can be diagonalizedexactly [20–22]. We describe and discuss the diagonal-ization procedure and identification of the ground statein the Appendix A. Here we give the final result whichreads e ˆ W Q ˆ H Q e − ˆ W Q = ˆ H d Q + C, (13)where ˆ H d Q = v Q ˆ P + v s X q | q | ˆ a † q a q , (14)and C is a c -number which is omitted in what follows. The anti-Hermitian operator W readsˆ W Q = X q ( α qQ ˆ a † q e − iqX − α qQ ˆ a q e iqX ) , (15)where α qQ = − δ qQ p πL | q | (16)and the overbar in α qQ and elswhere refers to the complexconjugation.The ground state of ˆ H Q for a fixed total canonicalmomentum P tot readsΦ Q = e − ˆ W Q | vac , P tot i , (17)where | vac , P tot i ≡ | vac i ⊗ | P tot i is a product state with | vac i being a Fock vacuum with respect to bosonic oper-ators ˆ a q and | P tot i being the state of impurity with themomentum P tot . V. DYNAMICSA. Dynamical diagonalization
The second ingredient for calculating F Q is the dy-namical state vector Ψ Q evolving according to theSchr¨odinger equation (2). Remarkably, the dynamicsdescribed by this equation is integrable, in the sensethat the operator (cid:16) ˆ H Q − i F ∂ Q (cid:17) can be diagonalized bya unitary transformation analogous to the transforma-tion (13): e ˆ Y Q (cid:16) ˆ H Q − i F ∂ Q (cid:17) e − ˆ Y Q = ˆ H d Q − i F ∂ Q + C ′ . (18)Here C ′ is a c -number which will be omitted in whatfollows, ˆ Y Q = X q ( β qQ ˆ a † q e − iqX − β qQ ˆ a q e iqX ) (19) In fact, a Q -dependent c -number responsible for reproducing thecorrect ground state energy is already omitted in the definition(8) of H . Predicting the ground state energy is beyong the scopeof the effective low-energy model. and the coefficients β qQ satisfy the differential equation iF ∂ Q β qQ − | q | ∆ v qQ β qQ − δ qQ √ πL p | q | ∆ v qQ = 0 . (20)The key insight behind the dynamical diagonalization isthat [ Y Q , ∂ Q Y Q ] is a c -number, and therefore − i F e ˆ Y Q (cid:16) ∂ Q e − ˆ Y Q (cid:17) = i F ( ∂ Q Y Q + 12 [ Y Q , ∂ Q Y Q ])(21)is linear in boson operators and has the same structureas Y Q and interaction term of the Hamiltonian (8).In order to satisfy the initial condition of theSchr¨odinger equation (3) we supplement the differentialequation (20) with the initial condition β q = α q . (22)Eq. (20) can be solved in quadratures, with the result β qQ = α qQ − Z Q ∂ Q ′ α qQ ′ exp − i | q | F N Z QQ ′ d e Q ∆ v q e Q ! dQ ′ (23)Eq. (18) entails that the dynamical state of the systemevolves according toΨ Q = e − ˆ Y Q | vac , P tot i . (24)where an irrelevant c -number phase factor is omitted. B. Dynamics of adiabatic fidelity
With Φ Q and Ψ Q in hand, we are prepared to proceedto calculating the adiabatic fidelity. Substituting eqs.(17) and (24) into the definition (4) and applying theBaker-Campbell-Hausdorff formula, we obtain F Q = exp − X q | β qQ − α qQ | ! . (25) VI. RESULTSA. Adiabatic fidelity and adiabatic mean free path
In principle, eqs. (25) along with eqs. (16) and (23)allows one to calculate F Q for any values of parameterswithin the validity range of the model (8). However, afurther asymptotic analysis is required to reveal the scal-ing properties of F ( Q ). Here we present the main resultsof such an analysis, referring the reader to Appendix Bfor details.The asymptotic behavior of the adiabatic fidelity in thelimit of large system size depends on how the force F N scales with N . Before turning to a general case, we con-sider an important special case of the force independentfrom the system size. In this case we obtainlog F Q = − ξ (cid:18) Qk F (cid:19) log N (26)with ξ = k ∂ Q δ + Q π ! + ∂ Q δ − Q π ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q =0 (27)This is the leading term of the double asymptotic expan-sion of log F Q in the limit of N ≫ , N/L fixed , (28) Q ≪ k F , (29)where the limit 28 is taken first.Quite remarkably, the force does not enter eq. (26).In fact, the adiabatic fidelity follows the orthogonalitycatastrophe overlap, F Q ≃ C Q ≡ |h Φ | Φ Q i| , in ac-cordance with the scenario described in ref. [8]. Onecan obtain eq. (26) without solving the Schr¨odingerequation (2) by calculating the orthogonality overlap |h Φ | Φ Q i| and applying the general result of ref. [8], asdetailed in Appendix C. However, this method is appli-cable in a superficially narrow range of Q which shrinkswhen N grows, Q = o (1). Solving the dynamical prob-lem shows that the validity range of the approximation F Q ≃ C Q appears to be larger than that guaranteed bythe rigorous result of ref. [8] – analogous conclusions havebeen made on the basis of explicit solutions of other mod-els [8].Now we turn to a more general case when the force F N can vanish with the system size, but not faster than 1 /N .In this case one needs to consider separately two rangesof Q :log F Q = − ξ (cid:18) Qk F (cid:19) × log N, Q ≪ F N k F v s , log (cid:16) N F N k v s (cid:17) , F N k F v s ≪ Q ≪ k F , (30)where ξ is given by eq. (27).One can see from the first line of eq. (30) that forsmall momenta the result coincides with those of eq. (26)and, in fact, again can be obtained by the method of ref.[8] without considering dynamics, see Appendix C. Solv-ing the time-dependent Schr¨odinger equation explicitlyis mandatory for obtaining the expression for larger mo-menta (the second line of eq. (30)). Observe that thelatter expression manifestly depends on the force.One can easily find the adiabatic mean free path, Q ∗ , from eq. (30): Q ∗ k F = 1 ξ (log N ) − , F N k v s ≫ √ log N , (cid:16) log N F N k v s (cid:17) − , N ≪ F N k v s ≪ √ log N . (31)Again, solving the time-dependent Schr¨odinger equationis essential for obtaining the second line of this expres-sion, while the first line is obtained in ref. [1] withoutconsidering dynamics.Eq. (31) implies that F N = O (1 /N ) (32)is a necessary asymptotic condition for the many-bodyadiabaticity. While the scaling F N = O (1 /N ) is beyondthe range of validity of eq. (31), the second line of thisequation indicates that the condition (32) can be alsosufficient for adiabaticity. This is indeed the case, as isproven in Appendix D. Thus, we establish eq. (32) asa necessary and sufficient condition for adiabaticity inthe driven impurity-fluid system in the large system sizelimit.At this point it is worth to comment on the results ofref. [1] where we have studied adiabaticity breakdownin the integrable impurity-fluid system by the methodof ref. [8]. Integrability allowed us to calculate C Q ex-plicitly and establish a necessary adiabatic condition F N ≤ O (1 / log N ). Obviously, the result (32) of thepresent paper is much stronger. However, the analysisof a microscopic integrable model retains its value sinceit underpins the effective model (8) and illustrates howthe phenomenological parameters of the effective modelare related to microscopic parameters. We present thisanalysis in Appendix E. A particularly interesting con-clusion from this analysis is that the result of calculationof C Q within the integrable model and within the effectivemodel (8) coincide for all Q , not just within the conser-vative validity range (12). This indicates that the actualvalidity range of the effective treatment can be larger. B. Experimental implication
Damped oscillations of an impurity particle driventhrough a 1D quantum fluid were observed in a recent ex-periment [12], where a quantum fluid consisted of N ≃ / k F throughout the whole range of experi-mental conditions.One may wonder whether it is possible to maintainthe many-body adiabaticity for at least a few cycles ofoscillations by e.g. applying a smaller force. Our analysisindicates that this can be challenging. The reason is thatin practice the spatial amplitude of oscillations is limitedby the size of the quasi-1D optical cigar-shaped trap. Itis easy to see that this amplitude is inversely propor-tional to the force. This constrains the force to satisfy F N & k / (2 πm ∗ N ) , where m ∗ is the effective mass ofthe impurity in the fluid. This can be only marginallyconsistent with the adiabatic condition (32). Thereforemaintaining adiabaticity for several periods of oscilla-tions would require an extremely careful choice of thedriving force – not too high to sustain adiabaticity butnot too low to keep the spatial amplitude of oscillationswithin the trap size. VII. SUMMARY AND CONCLUDINGREMARKS
To summarize, we have analyzed the dynamics of themany-body adiabatic fidelity, F Q , in a one-dimensionalimpurity-fluid system where a force applied to the im-purity pulls the latter through the fluid. We have em-ployed an effective low-energy theory which enabled usto find explicit expressions for F Q and the adiabaticmean free path, Q ∗ , in terms of the size of the sys-tem and the effective parameters of the fluid and theimpurity-fluid coupling. Our results imply that the statevector of the impurity-fluid system completely departsfrom the instantaneous ground state already for acquiredmomenta which are vanishingly small in thermodynamiclimit, unless the force scales down with the system sizeas 1 /N or faster. This dramatically improves the nec-essary adiabatic condition F N ≤ O (1 / log N ) obtainedpreviously [1].It is remarkable that quantum adiabaticity breaksdown already at small acquired momenta. When the ac-quired momentum reaches the vicinity of πρ , our quan-titative results may be inapplicable since the vicinity of Q = πρ can be beyond the range of validity of the effec-tive model employed. However, the adiabaticity is any-way gone at this point (unless F N ≤ O (1 /N )). It is worthnoting that crossing the Q = πρ point is potentially themost dangerous part of the oscillation cycle with respectto preserving adiabaticity [16, 17]. This means that theadiabatic condition can become only more stringent be-yond the range of validity of our approach.It should be emphasized that while the many-body adi-abaticity would be sufficient to ensure quasi-Bloch os-cillations in an arbitrary one-dimensional impurity-fluidsystem, adiabaticity breakdown is not necessarily fatalfor oscillations. This has been confirmed in the experi-ment of ref. [12]. On the theoretical side, there is a con-sensus that quasi-Bloch oscillation of a sufficiently heavyimpurity can occur in the thermodynamic limit for a forcewhich is finite in the thermodynamic limit but sufficiently small as compared to other intensive (i.e. independent onthe system size) quantities of the system [9–11, 13, 15].The latter condition defines a notion of a thermodynamic adiabaticity, as contrasted to the genuine quantum many-body adiabaticity studied in the present paper. It is amatter of a debate whether thermodynamic adiabaticityis sufficient for quasi-Bloch oscillations of a light impu-rity [9–11, 13–18]. The present paper does not contributeto this debate.
ACKNOWLEDGMENTS
We thank M. Zvonarev for useful discussions. Thework was supported by the Russian Science Foundationunder the grant N o Appendix A: Diagonalization for a fixed Q Here we present details on diagonalization of theHamiltonian H ( Q ) at a given Q .
1. Unitary transformation
Unitary transformation generated by W defined by eq.(15) with arbitrary coefficients α qQ yields e ˆ W Q ˆ a q e − ˆ W Q = ˆ a q − α qQ e − iqX (A1)and e ˆ W Q ˆ P e − ˆ W Q = ˆ P + X q q ( α qQ ˆ a † q e − iqX + h.c. ) − ∆ P, (A2)where the constant ∆ P reads∆ P = X q q | α qQ | . (A3)As a consequence, e ˆ W Q ˆ P tot e − ˆ W Q = ˆ P tot (A4)and e ˆ W Q ˆ He − ˆ W Q = ˆ H d + C − (A5) − X q | q | ( α qQ ∆ v q + δ qQ p πL | q | ∆ v qQ ) ˆ a † q e − iqX + h.c. ! An instructive demonstration of the subtlety of interplay betweenthe two concepts was recently presented in ref. [28], where animpurity-fluid system with a time-dependent coupling constant(but in the absence of a force) was considered. It appeared thatthe outcomes of the thermodynamically adiabatic and the gen-uinely quantum-adiabatic evolutions could be identical or dra-matically different in the same system, depending on the choiceof the initial state. with ˆ H d given by eq. (14) and the constant C given by C = v s X q | q || α qQ | + 1 √ πL X q p | q | δ qQ ∆ v qQ ( α qQ + α qQ ) . (A6)The last term in eq. (A5) vanishes when one choosescoefficients α qQ according to (16). In this case∆ P = k F (2 π ) (cid:16) ( δ + Q ) − ( δ − Q ) (cid:17) . (A7)
2. Ground state
Eigenstates of the diagonalized Hamiltonian (14) donot depend on Q and read |{ n q } , K i ≡ |{ n q }i ⊗ | K i , (A8)where |{ n q }i is an eigenstate of the oscillator part of(14) with n q bosons for each q , while | K i is the state ofthe impurity with momentum K . The eigenvalues of thehamiltonian and the total momentum read, respectively E { n q } ,K = v Q K + v s X q | q | n q , + C and (A9) P tot { n q } ,K = K + X q q n q . (A10)We wish to find the minimal eigenenergy E { n q } ,K fora given total momentum P tot { n q } ,K . To this end we in-troduce P ± ≡ X ± q> | q | n q . (A11)and rewrite eqs. (A9) and (A10) as, respectively, E { n q } ,P tot = v Q K + v s ( P + + P − ) + C and (A12) P tot { n q } ,K = K + P + − P − . (A13)This leads to E { n q } ,K = vP tot + C + ( v s − v ) P + + ( v s + v ) P − . (A14)Since two last terms in this equation are nonnegativewhile C is the same for all { n q } and K , the r.h.s. of eq.(A14) has minimum at P + = P − = 0. Hence the groundstate of ˆ H d for a given value of the total momentum reads | vac , P tot i , where | vac i is |{ n q }i with all n q = 0, and theground state Φ Q of ˆ H Q is given by eq. (17).We note that h Φ Q | ˆ P | Φ Q i = P tot − ∆ P. (A15) Appendix B: Asymptotic analysis
From eqs. (25) and (23) one getslog F ( Q ) = − Z Q Z Q dQ ′ dQ ′′ ∂ Q ′ δ + Q ′ π ∂ Q ′′ δ + Q ′′ π Σ + ( Q ′′ − Q ′ ) − n δ + Q → δ − Q , Σ + → Σ − o . (B1)withΣ ± ≡ M − X n =1 n e − inφ ± = −
12 log(2 − φ ± ) + Ci( M φ ± )(B2)and φ ± ≡ δkF N Z Q ′′ Q ′ d e Q ( v s ∓ v Q ) . (B3)Here Ci stands for the cosine integral function and weuse an arbitrary momentum cutoff u ≡ ( M − δk ∼ k F . (B4)If one takes u = k F , as we do in the rest of the paper,then M = ( N + 1) /
2. Note that φ ± ≪ N F N ≫ M φ ± ≪ , (B6)which is equivalent to Q ≪ F N / ( v s k F ). In this case onegetsΣ ± = log M + γ E + O (cid:0) ( φ ± M ) (cid:1) + O (cid:18) N (cid:19) = log N + γ E + log MN + O (cid:18) Qv s k F F N (cid:19) ! (B7)Observe that only the leading term is cutoff-independent.The second limit is M φ ± ≫ , (B8)which is equivalent to Q ≫ F N / ( v s k F ). In this case thecutoff-dependent part of the r.h.s. of eq. (B2) is vanish-ingly small, Ci( M φ ± ) = O (cid:18) M φ ± (cid:19) . (B9)We find it convenient to further expand the cutoff-independent part of eq. (B2) in φ ± to obtainΣ ± = − log φ ± + O ( φ ± ) + O (cid:18) N φ ± (cid:19) . (B10)Now we can substitute the asymptotic expansions (B7)and (B10) to eq. (B1). Since the adiabaticity breaksdown already for Q/k F = o (1), it is reasonable to furtherexpand all functions in the integrand in eq. (B1) over Q ′ and Q ′′ . This finally leads to eq. (30). Appendix C: Orthogonality catastrophe andadiabaticity1. Orthogonality catastrophe
From eqs. (17) and (15) one can can calculate theorthogonality overlap C Q ≡ |h Φ | Φ Q i| = exp − X q | α qQ − α q | ! . (C1)This is done with the help of the relation e α ˆ a † − α ′ ˆ a e − α ′ ˆ a † + α ′ ˆ a = e ( α − α ′ ) ˆ a † e − ( α ∗ − α ′ ) ˆ a e −| α − α ′ | / − i Im αα ′ (C2)valid for arbitrary α and α ′ .We calculate log C Q explicitly by substituting the ex-pression (16) for α qQ in eq. (C1), expanding the latter in Q and performing the sum over q . This way we obtainin the leading order the r.h.s. of eq. (26).
2. Relation between the adiabaticity and theorthogonality catastrophe
We wish to establish that C Q ≃ F Q in the limit of large N . The rigorous form of this relation was proven in ref.[8]. In the considered case it reads |C Q − F Q | ≤ F N Q Z dQ ′ q h Ψ | ˆ H Q ′ | Ψ i − h Ψ | ˆ H Q ′ | Ψ i (C3)The integral in this equation is cutoff-dependent but doesnot diverge with N . Therefore the r.h.s of the inequal-ity can be made vanishingly small, i.e. o (1) in the limit N → ∞ , whenever Q = o ( F N ). This way we reproduceeq. (26) and the first line of eq. (30) without solvingthe Schr¨odinger equation, with the use of a shortcut in-troduced in ref. [8]. The price is a superficially reducedrange of validity of the results, Q = o ( F N ) instead of Q = O ( F N ) obtained by solving the dynamical problem. Appendix D: Proof that F N = O ( N ) is sufficient foradiabaticity For the sake of such a proof we take the integral ineq. (23) by parts. Importantly, this produces the factor F N /q which eventually does the job. This way for q > | β qQ − α qQ | ≤ F N πq L (cid:16) A ± Q (cid:17) , (D1) where A ± Q k F v s ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ Q δ ± Q v s − v Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ Q δ ± Q (cid:12)(cid:12)(cid:12) Q =0 v s − v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + Z Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ Q ′ δ ± Q ′ v s − v Q !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dQ ′ (D2)does not depend on q and is finite in the thermodynamiclimit. Here the sign in A ± Q and δ ± Q is the same as the signof q . The sum over positive q in eq. (25) is bounded fromabove according to X q> | β qQ − α qQ | ≤ ζ (3) A + Q F N N π k v s ! , (D3)and the sum over negative q is constrained analogously.Here ζ (3) is the Riemann zeta function. This proves thatwhenever F N ≤ ǫN π k v s s ζ (3) (cid:18)(cid:16) A + Q (cid:17) + (cid:16) A − Q (cid:17) (cid:19) (D4)the adiabatic fidelity is bounded from below, F Q ≥ e − ǫ . (D5) Appendix E: Integrable model: consistency check
Here we consider a microscopic integrable impurity-fluid model. The model consists of N fermions and asingle impurity particle with a mass m equal to the massof a fermion. Fermions do not interact with each otherbut couple to the impurity via the repulsive contact po-tential. The Hamiltonian readsˆ H Q = ( − i ∂ X + Q ) m − N X j =1 m ∂ x j + g N X j =1 δ ( x j − X ) , (E1)where Q = F t is the impulse of the force, X and x j are the coordinates of the impurity and the j ’th fermionrespectively and g > Q the model (E1) is integrable as shown byMcGuire [27]. In fact, this model is one of the simplestmodels solvable via the Bethe ansatz: Its eigenfunctioncan be expressed through ( N + 1) × ( N + 1) Slatter-like determinants [29, 30]. For this reason it has beenpossible to obtain a wealth of analytical results and togain a number of deep insights into the physics of themodel [28–34]. Although this model is a special caseof the Yang-Gaudin model [35, 36], it might deserve aseparate name – McGuire model – due to its conceptualimportance.The integrability of the model enabled us to apply thetechnique of ref. [8] and relate the adiabaticity break-down to the orthogonality catastrophe in ref. [1]. Herewe focus on the relation between the microscopic model(E1) to the effective model (8). Our aim is to underpinthe effective model (8) and better understand its valid-ity range. In what follows the notations and conventionsfollow refs. [28, 34].First, we would like to relate the effective scatteringphases in eq. (8) to the microscopic scattering phases ofthe Bethe ansatz [28], δ ± BA Q = π − arctan (cid:18) Λ k F ∓ v s g (cid:19) , (E2)where v s is the Fermi velocity in the present context andthe parameter Λ can be found from the equation [28] Qk F = g πv s (cid:20) (cid:18) Λ + 2 v s g (cid:19) arctan (cid:18) Λ + 2 v s g (cid:19) − (cid:18) Λ − v s g (cid:19) arctan (cid:18) Λ − v s g (cid:19) + 12 ln 1 + (2 v s /g − Λ) v s /g + Λ) (cid:21) . (E3)To do this we consider the overlap e C Q ≡ (cid:12)(cid:12)(cid:12) h e Φ Q | Φ Q i (cid:12)(cid:12)(cid:12) (E4)between the ground state Φ Q and the noninteracting ground state e Φ Q of the impurity-fluid system. e C Q can becalculated both in the microscopic model (E1) [31] and the effective model (8), with the leading order resultslog e C Q = − δ +BA Q π ! + δ − BA Q π ! log N (E5)and log e C Q = − δ + Q π ! + δ − Q π ! log N, (E6)respectfully. These two equations are compatible when δ ± Q = 2 δ ± BA Q . (E7)Now we turn to the orthogonality overlap of interest, C Q . We have calculated it within the McGuire model ina similar manner as in ref. [31], with the resultlog F Q = − δ +BA Q − δ +BA 0 π ! + δ − BA Q − δ − BA 0 π ! × (cid:0) log N + O (1) (cid:1) , (E8)where O (1) refers to the limit of N → ∞ .After accounting for eq. (E7) and expanding in small Q this result agrees with eq. (26). It should be empha-sised that this agreement is not limited to small Q buttakes place in the whole Brillouin zone, − k F < Q < k F .This indicates that the actual range of validity of the ef-fective model can span well beyond the conservative con-dition (12). This agreement also suggests that the bosonoperators a q , a † q in fact do not depend on Q ( cf. footnote3), at least in the impurity-fluid system described by themicroscopic Hamiltonian (E1). [1] Oleg Lychkovskiy, Oleksandr Gamayun, andVadim Cheianov, “Quantum many-body adia-baticity, topological thouless pump and drivenimpurity in a one-dimensional quantum fluid,”AIP Conf. Proc. , 020024 (2018).[2] Max Born, “Das adiabatenprinzip in der quantenmee-hanik,” Zeitschrift f¨ur Physik , 167 (1926).[3] Max Born and Vladimir Fock, “Beweis des adiabaten-satzes,” Zeitschrift f¨ur Physik , 165–180 (1928).[4] Lev D Landau, “Zur theorie der energieubertragung. ii,”Physics of the Soviet Union , 28 (1932).[5] Clarence Zener, “Non-adiabatic crossing of energy lev-els,” in Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences , Vol.137 (The Royal Society, 1932) pp. 696–702.[6] Tosio Kato, “On the adiabatictheorem of quantum mechanics,”Journal of the Physical Society of Japan , 435–439 (1950).[7] Tameem Albash and Daniel A. Lidar, “Adiabatic quan-tum computation,” Rev. Mod. Phys. , 015002 (2018).[8] Oleg Lychkovskiy, Oleksandr Gamayun, and Vadim Cheianov, “Time scale for adiabaticity breakdown indriven many-body systems and orthogonality catastro-phe,” Phys. Rev. Lett. , 200401 (2017).[9] D. M. Gangardt and A. Kamenev, “Bloch os-cillations in a one-dimensional spinor gas,”Phys. Rev. Lett. , 070402 (2009).[10] M. Schecter, D. M. Gangardt, and A. Kamenev, “Dy-namics and Bloch oscillations of mobile impurities in one-dimensional quantum liquids,” Annals of Physics ,639–670 (2012).[11] M. Schecter, A. Kamenev, D. M. Gangardt,and A. Lamacraft, “Critical velocity of a mo-bile impurity in one-dimensional quantum liquids,”Phys. Rev. Lett. , 207001 (2012).[12] Florian Meinert, Michael Knap, Emil Kirilov, KatharinaJag-Lauber, Mikhail B Zvonarev, Eugene Demler, andHanns-Christoph N¨agerl, “Bloch oscillations in the ab-sence of a lattice,” Science , 945–948 (2017).[13] O. Gamayun, O. Lychkovskiy, and V. Cheianov, “Ki-netic theory for a mobile impurity in a degenerate Tonks-Girardeau gas,” Phys. Rev. E , 032132 (2014). [14] O. Gamayun, “Quantum Boltzmann equation for a mo-bile impurity in a degenerate Tonks-Girardeau gas,”Phys. Rev. A , 063627 (2014).[15] O. Lychkovskiy, “Perpetual motion and driven dy-namics of a mobile impurity in a quantum fluid,”Phys. Rev. A , 040101 (2015).[16] Michael Schecter, Dimitri M. Gangardt, and AlexKamenev, “Comment on “Kinetic theory for a mo-bile impurity in a degenerate tonks-girardeau gas”,”Phys. Rev. E , 016101 (2015).[17] O. Gamayun, O. Lychkovskiy, and V. Cheianov,“Reply to “Comment on ‘Kinetic theory for a mo-bile impurity in a degenerate Tonks-Girardeau gas’ ”,”Phys. Rev. E , 016102 (2015).[18] Michael Schecter, Dimitri M Gangardt, and AlexKamenev, “Quantum impurities: from mobile joseph-son junctions to depletons,” New Journal of Physics ,65002–65019 (2016).[19] R. Balian, J. F. Gregg, and D. ter Haar, From micro-physics to macrophysics (Springer, 2007).[20] Yasumasa Tsukamoto, Tatsuya Fujii, and NorioKawakami, “Critical behavior of tomonaga-luttinger liq-uids with a mobile impurity,” Phys. Rev. B , 3633(1998).[21] A Kamenev and LI Glazman, “Dynamics of a one-dimensional spinor bose liquid: A phenomenological ap-proach,” Phys. Rev. A , 011603 (2009).[22] MB Zvonarev, VV Cheianov, and Thierry Giamarchi,“Edge exponent in the dynamic spin structure factor ofthe yang-gaudin model,” Phys. Rev. B , 201102 (2009).[23] T. Giamarchi, Quantum Physics in One Dimension (Ox-ford University Press, 2003).[24] FDM Haldane, “’luttinger liquid theory’of one-dimensional quantum fluids. i. properties of theluttinger model and their extension to the general 1dinteracting spinless fermi gas,” Journal of Physics C:Solid State Physics , 2585 (1981).[25] Adilet Imambekov, Thomas L. Schmidt, andLeonid I. Glazman, “One-dimensional quantumliquids: Beyond the luttinger liquid paradigm,” Rev. Mod. Phys. , 1253–1306 (2012).[26] Gregory W Moore, “A comment on Berry connections,”arXiv preprint arXiv:1706.01149 (2017).[27] J. B. McGuire, “Interacting fermions in one dimension.I. Repulsive potential,” J. Math. Phys. , 432 (1965).[28] Oleksandr Gamayun, Oleg Lychkovskiy, EvgeniBurovski, Matthew Malcomson, Vadim V. Cheianov,and Mikhail B. Zvonarev, “Impact of the injec-tion protocol on an impurity’s stationary state,”Phys. Rev. Lett. , 220605 (2018).[29] Christian Recher and Heiner Kohler, “From hardcorebosons to free fermions with painlev´e v,” Journal of Sta-tistical Physics , 542–564 (2012).[30] Charles JM Mathy, Mikhail B Zvonarev, and EugeneDemler, “Quantum flutter of supersonic particles in one-dimensional quantum liquids,” Nature Physics , 881–886 (2012).[31] H. Castella and X. Zotos, “Exact calculationof spectral properties of a particle interact-ing with a one-dimensional fermionic system,”Phys. Rev. B , 16186–16193 (1993).[32] E. Burovski, V. Cheianov, O. Gamayun, andO. Lychkovskiy, “Momentum relaxation of a mo-bile impurity in a one-dimensional quantum gas,”Phys. Rev. A , 041601(R) (2014).[33] Oleksandr Gamayun, Andrei G Pronko, and Mikhail BZvonarev, “Impurity Green’s function of a one-dimensional fermi gas,” Nuclear Physics B , 83(2015).[34] Oleksandr Gamayun, Andrei G Pronko, and Mikhail BZvonarev, “Time and temperature-dependent correlationfunction of an impurity in one-dimensional fermi andtonks–girardeau gases as a fredholm determinant,” NewJournal of Physics , 045005 (2016).[35] CN Yang, “Some exact results for the many-body prob-lem in one dimension with repulsive delta-function inter-action,” Phys. Rev. Lett. , 1312 (1967).[36] M Gaudin, “Un systeme a une dimension de fermions eninteraction,” Physics Letters A24