Asymptotic theory of quasiperiodically driven quantum systems

PHYSICAL REVIEW E 97, 062139 (2018) Asymptotic theory of quasiperiodically driven quantum systems David Cubero1,*and Ferruccio Renzoni2,† 1Departamento de Física Aplicada I, EUP, Universidad de Sevilla, Calle Virgen de África 7, 41011 Sevilla, Spain 2Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom (Received 18 February 2018; published 21 June 2018) The theoretical treatment of quasiperiodically driven quantum systems is complicated by the inapplicability of the Floquet theorem, which requires strict periodicity. In this work we consider a quantum system driven by a biharmonic driving and examine its asymptotic long-time limit, the limit in which features distinguishing systems with periodic and quasiperiodic driving occur. Also, in the classical case this limit is known to exhibit universal scaling, independent of the system details, with the system’s reponse under quasiperiodic driving being described in terms of nearby periodically driven system results. We introduce a theoretical framework appropriate for the treatment of the quasiperiodically driven quantum system in the long-time limit and derive an expression, based on Floquet states for a periodically driven system approximating the different steps of the time evolution, for the asymptotic scaling of relevant quantities for the system at hand. These expressions are tested numerically, finding excellent agreement for the finite-time average velocity in a prototypical quantum ratchet consisting of a space-symmetric potential and a time-asymmetric oscillating force. DOI: 10.1103/PhysRevE.97.062139 I. INTRODUCTION Quantum systems driven by a periodic perturbation are ubiquitous in physics [1–3]. Their theoretical treatment is greatly simplified by the Floquet theorem, which reduces the time-dependent problem to an essentially time-independent one. Quasiperiodically driven systems are also of great interest for the unique features they exhibit with respect to their periodic counterpart. In this case the theoretical treatment is much more cumbersome, due to the fact that the use of Floquet theorem is strictly limited to periodic systems, and cannot be directly applied to a quasiperiodically driven system. In this work we consider quantum systems driven by a biharmonic quasiperiodic drive, though the results can be easily generalized to include drives with more harmonics. We specifically examine the long-time response of the system to the driving, as it is in this limit that a quasiperiodic system exhibits features distinct from its periodic counterpart. We derive an expression for the finite-time response of the system, valid in the long-time limit and based on periodically driven states, which precisely captures all essential features of the response of the system. The theoretical framework introduced here has a broad validity and is of direct relevance to a system consisting of a Bose-Einstein condensate driven by a biharmonic force, a system which has been extensively explored theoretically [4,5] and experimentally [6–8]inthe case of a periodic forcing. This work is organized as follows. In Sec. II we introduce the model and definitions. In Sec. III we revisit the asymptotic theory for classical systems [9,10] which will be useful as an introduction to the quantum case, fully discussed in Sec. IV. *[email protected] †[email protected] II. THE MODEL We consider a particle of mass m, subject to a spatially periodic potential V(x) with period L:V(x+L)=V(x)for all x. The particle is driven by a force depending on two frequencies, ω1and ω2, F(t)=f(ω1t;ω2t+θ),(1) where f(x1,x2)=f(x1+2π,x2)=f(x1,x2+2π) (it is periodic in both arguments) and θis the driving phase, a parameter that can be tuned to control the system’s response. We are interested in the time average of some quantity describing the particle’s state, namely, the particle’s average velocity vTs vTs=1 Tst0+Ts t0 dt v(t),(2) where v(t) is the instantaneous velocity at time t,Tsis the observation time, and t0is the initial time. Our focus is on the asymptotic limit, in which universal scaling is expected to occur; thus the particle’s velocity is computed in the infinite time limit Ts→∞. III. ASYMPTOTIC THEORY FOR CLASSICAL DRIVEN SYSTEM In this section, we show how for the classical case, in the asymptotic limit, the finite-time average velocity of the particle under a driving with two generic—not necessarily commensurate—frequencies ω1and ω2can be expressed in terms of the particle’s infinite-time velocity under a periodic driving. In the following, we denote as vpq the particle’s infinite-time velocity—Eq. (2) with Ts→∞—when the driving is periodic, i.e., ω2=ω1p/q, with pand qinteger coprimes. The asymptotic theory, first derived in Ref. [9], 2470-0045/2018/97(6)/062139(7) 062139-1 ©2018 American Physical Society DAVID CUBERO AND FERRUCCIO RENZONI PHYSICAL REVIEW E 97, 062139 (2018) then will be extended to the quantum regime in the following section. Specifically, we show below that in the asymptotic limit given by Ts→∞ (ω2−ω1p/q)Ts=const,(3) i.e., ω2→ω1p/q, the finite time current vTscan be approximated by vTs∼1 δω2Tsθ0+δω2Ts θ0 d θv pq( θ),(4) where δω2=ω2−ω1p/q (5) and θ0=θ+δω2t0.(6) Equation (4) provides the dependence of the finite-time current on the second (or first) driving frequency, in terms of the infinite-time velocity under periodic driving. To derive (4), we divide the interval between t0and t0+ Tsinto large subintervals of length  T, each containing many driving cycles, and thus we can express the finite-time average velocity as vTs=1 Tst0+Ts t0 dt v(t)∼ T Ts j 1  Tt0+(j+1) T t0+j T dt v(t).(7) Here  Tis chosen as  T=NT, with Na large number, and T=2πq/ω1, the driving period when ω2=ω1p/q. We can take Nas an integer, since in that case the deviation of vTsand vN Ttends to zero for large Ts. Also note that in the limit (3)of interest here, we have δω2Ts=const, and thus δω2→0 and δω2T→0. In this limit we will take N→∞, albeit with the condition δω2 T=δω2NT →0, which implies NT Ts→0.(8) This could be satisfied with, e.g., N∼(Ts/T)1/2,orN∼ log(Ts/T). By writing t=t0+j T+ t, with 0 ⩽ t⩽ T,in each integral appearing in the right side of (7), the external action driving the instantaneous velocity v(t) can be written as F(t)=fω1t;p qω1t+δω2t+θ =fω1t;p qω1t+ θj+δω2 t,(9) where  θj≡θ0+(δω2 T)j. Since |δω2| t<|δω2| T→0, for small enough |δω2|, the driving (9) in the time interval tj⩽ t⩽tj+ T, with tj=t0+j T, can be well approximated by fω1t;p qω1t+ θj,(10) i.e., the driving associated to the periodic case vpq( θj). In the case the velocity does not depend on the initial condition, then vpq(θ) is well defined, and for large enough Ts(i.e., large  T), we can approximate the finite-time integral with its value for periodic driving, 1  Ttj+ T tj dt v(t)∼vpq( θj).(11) This expression is then exact in the limit considered here, Ts→ ∞,δω2Ts=const, leading to vTs∼1 δω2Ts j  θv pq( θj)∼1 δω2Tsθ0+δω2Ts θ0 d θv pq( θ), (12) where  θ= θj− θj−1=δω2 T. This completes the proof for the case in which the particle’s velocity is independent of the initial conditions. Equation (4) thus holds in dissipative or noisy systems where the infinite-time current vpq is well defined, that is, independent of the initial conditions. For a Hamiltonian classical system, additional averaging over initial conditions would be required [2]. IV. QUANTUM SYSTEMS AND FLOQUET THEOREM In nondissipative quantum systems, there is usually a strong dependence on the initial conditions. Thus, the procedure used in the classical case does not in general apply. In particular, the approximation (11), which requires the infinite-time current under periodic driving to be independent of the initial conditions, does not necessarily hold—and in consequence neither does the expression (4) for the finite-time current under generic driving. In this section we show how the results for the classical case can be extended to the quantum regime. This will require taking the initial conditions into account. A. Bloch states in driven systems In a spatially periodic system, Bloch states form a basis of the Hilbert space. Thus we can restrict our analysis to initial conditions consisting of a single Bloch state. The quantum dynamics for the general case can then be determined by invoking the superposition principle. The system of interest here is the extension to the quantum case of the classical system considered previously. The quantum dynamics is described by the Hamiltonian H=p2 2m+V(x)−F(t)x, (13) where pis the particle’s momentum operator, V(x)isthe space-periodic potential, with period L, and F(t) is an arbitrary time-dependent drive, not necessarily periodic. The Hamiltonian (13) does not commute with the translation operator TL, TLψ(x,t)=ψ(x+L,t),for all ψ(x,t),(14) because of the presence of the time-dependent drive F(t). Thus, Bloch states are not eigenfunctions of the Hamiltonian. Following a standard approach [5], we perform the gauge transformation  ψ(x,t)=e−iA(t)x/¯hψ(x,t) (15) 062139-2 ASYMPTOTIC THEORY OF QUASIPERIODICALLY DRIVEN … PHYSICAL REVIEW E 97, 062139 (2018) with A(t)=t t0 dtF(t),(16) which yields a transformed Hamiltonian  H,  H=[p+A(t)]2 2m+V(x),(17) which commutes with TL. Therefore, the system wave functions  ψ(x,t) can be written as superpositions of Bloch states  ψk(x,t) with quasimomentum ¯hk, that is, of states with the form  ψk(x,t)=eikxuk(x,t),where uk(x+L,t)=uk(x,t) for all x. (18) The wave number kcan always be taken within the first Brillouin zone: −π/L ⩽k⩽π/L. Finally, we recall the important result that if the initial condition ψ(x,t0) is a Bloch state with wave number k0, then at any time the system state ψ(x,t) is also a Bloch state (i.e., an eigenstate of TL), but with a modified wave number given by k(t)=k0+1 ¯ht t0 dtF(t).(19) This result will be used for the calculation of the finite-time current in a quantum system. It can be readily proven by using the fact that the gauge-transformed Hamiltonian  Hcommutes with TL. B. Floquet-Bloch states For a time-periodic driving, indicated here by Fpq, with period T,Fpq(t+T)=Fpq(t) for all t, the Floquet theorem allows one to simplify the theoretical treatment. While this does not directly apply to the case of quasiperiodic driving of interest here, Floquet states will come into play in the derivation of the velocity under quasiperiodic driving. We thus recall the main concepts which will be used in the following. The Floquet theorem states that a periodically driven quantum system has a complete set of solutions of the form ψ(x,t)=exp(−it/¯h)ϕ(x,t), with ϕ(x,t +T)=ϕ(x,t) for all t, where are the quasi-energies, which can be taken within the First Brillouin zone, −¯hω/2<<¯hω/2, with ω=2π/T . Let us assume that the periodic driving force is unbiased, t0+T t0 dt Fpq(t)=0.(20) Then the gauge transformation (15) yields a time-periodic function A(t), and thus the transformed Hamiltonian  His also a time-periodic, in addition to being space-periodic. We are then entitled to write a complete set of solutions, Bloch-Floquet states, of the form  ψk,n(x,t)=e−in(k,θ)t/¯hϕk,n(θ,x,t), ϕk,n(θ,x,t)=eikxφk,n(θ,x,t),(21) where φk,n(θ,x +L,t)=φk,n(θ,x,t) =φk,n(θ,x,t +T),for all xand t, (22) and nis an additional quantum number needed to label the states. Note that the initial condition is the same for the gauge-transformed problem and the original one,  ψ(x,t0)= ψ(x,t0). The functions φk,n(x,t) are space-periodic, so we will assume them to be normalized within a single period, L 0dx|φk.n(x,t)|2=1. C. Asymptotic finite-time velocity For a periodic driving, the particle velocity can be written in terms of Floquet-Bloch states, with the average current vn(k,θ) in the (n,k) Floquet-Bloch state written as vn(k,θ)=1 Tt0+T t0 dt L 0 dx ψk,n(x,t)∗ ×−¯h im ∂ ∂xψk,n(x,t),(23) where we highlighted the dependence on the driving phase θ. We now consider a generic biharmonic driving, not necessarily periodic. We fix the frequency ω1and consider a frequency ω2in the neighborhood of ω1p/q. We thus restrict the analysis to a small driving frequency deviation δω2= ω2−ω1p/q, that is, within the asymptotic limit defined by Eq. (3). The derivation for quasiperiodically driven quantum systems follows the same strategy as for the classical case: the interval Tsis divided into many large subintervals of length  T, with the driving approximated by a periodic one within each subinterval. For the sake of simplicity, we are going to assume in the following that the initial condition ψ(x,t0) is a Bloch state with wave number k0, the superposition principle allowing for generalization. At a later time tj=t0+ Tj, where  Tis a large multiple of T(though much smaller than Ts), we know from (19) that ψ(x,tj) is a Bloch state with wave number kj=k0+tj t0 dtF(t)/¯h. (24) During the time interval  Tafter tj, the driving force F(t) can be well approximated by its periodic counterpart, Fpq(t)=fω1t;p qω1t+ θj,(25) corresponding to the driving phase  θj=θ+δω2tj.(26) Therefore, by using the gauge transformation Aj(t)=t tj dtFpq (t),(27) the state within the interval  Tcan be described as a superposition of Floquet-Bloch states of the form (21) with k=kj and θ= θj. Note also that since the periodic driving force is unbiased, Aj(t)=t t0dtFpq (t)=A(t). If  Tis large enough (see Ref. [5] and Appendix A), the average current within that time interval can be approximated 062139-3 DAVID CUBERO AND FERRUCCIO RENZONI PHYSICAL REVIEW E 97, 062139 (2018) by their Floquet values, without interference between FloquetBloch states, 1  Ttj+ T tj dt v(t)∼ n|Cn|2vnk0+tj t0 dtF(t)/¯h, θj, (28) where Cn( θj)=ϕkj,n(tj)|ψ(tj)is the projection of the system wave function on the (n,kj) Floquet-Bloch state of the periodic case with driving phase  θj. In Appendix Awe show that interference effects between Floquet-Bloch states in time-periodic systems decay as the inverse of the observation time. Furthermore, we prove in Appendix Bthat the absolute square of the coefficients Cnremains invariant with time in the asymptotic limit considered here: |Cn( θj)|2∼|Cn(θ0)|2for all j(δω2→0).(29) In other words, in this limit, the probability to be in a certain Floquet state is not affected by the probability of being in other states, remaining constant in time. The expressions (28)–(29) are the quantum case equivalent of the approximation (11) derived in the classical case. The contribution from the different subintervals can then be summed up, in complete analogy with the classical case, to give vTs∼1 δω2Tsθ0+δω2Ts θ0 d˜ θ n|Cn|2 ×vnk0+t0+T˜ θ−θ δω2T t0 dtF(t)/¯h, ˜ θ,(30) where T=2πq/ω1and |Cn|2=|ϕk0,n(t0)|ψ(t0)|2. Therefore, in the quantum driven system, the finite-time current can be described by the current of Floquet states, with a population |Cn|2that depends only on the distribution of Floquet states at initial time. Equation (30) is for a system that starts with ψ(x,t0) being a Bloch state with wave number k0. For an arbitrary initial condition we need only to replace Cnby Ck0,n and add a sum over all initial wave numbers k0, vTs∼1 δω2Tsθ0+δω2Ts θ0 d˜ θ k0,n |Ck0,n|2 ×vnk0+t0+T˜ θ−θ δω2T t0 dtF(t)/¯h, ˜ θ.(31) V. NUMERICAL VALIDATION The validity, in the asymptotic limit, of the expressions (30) and (31) was confirmed by comparison with the numerical solution of the Schrödinger equation. In the numerical calculations, we have considered the periodic potential V(x)=V0cos(2πx/L) (32) and the biharmonic driving force F(t)=F1cos(ω1t)+F2cos(ω2t+θ).(33) -2 -1.5 -1 -0.5 00.5 11.5 2 (ω2−2ω1)Ts 0.06 0.08 0.1 0.12 0.14 0.16 〈v〉 Ts / T = 103 Ts / T = 104 Ts / T = 105 FIG. 1. Finite-time average velocity as a function of the second driving frequency ω2for a spatially symmetric (32) quantum ratchet subject to the time-asymmetric biharmonic force (33), with V0=1, F1=F2=ω1=2, and θ=−π/2, starting from ψ(x,0) =const, i.e., a momentum eigenstate with wave number k0=0. The solid line is the prediction of the asymptotic expression (30) based on Floquet states for the periodic case ω2=2ω1, and the points correspond to different observation times. Reduced units are defined such that m=2π/L =5¯h=1, and initial time fixed to t0=0. The time-dependent Schrödinger equation is solved numerically, for each initial wave number k0, using a standard spectral algorithm [11], with a spatial mesh of 128 grid points. In all cases, the asymptotic prediction (30)or(31)is computed numerically by calculating the Floquet states at each  θj, with a driving phase step  θj=0.01. The Floquet states are determined by finding the eigenstates of the evolution operator. Figure 1shows an example for a quantum ratchet starting from a uniform wave function, i.e. a momentum eigenstate with wave number k0=0. In Fig. 2we show the results for a quantum particle that starts from a Gaussian wave packet with a momentum distribution about ¯h¯ k0, with ¯ k0=−0.3, and a momentum dispersion of σp=¯hσk, with σk=0.01. More specifically, the initial condition is a superposition of N0=10 momentum eigenstates between ka=−0.35 and kb=−0.26 (for reference, the minimum of the Brillouin zone is kmin =−π/L = −0.5), ψ(x,0) =A0 N0  j=1 eikjx √2πexp −(kj−¯ k0)2/4σ2 k,(34) where kj=ka+(j−1)k,k =(kb−ka)/(N0−1), and A0is a normalization constant, given by A0=k/(2πσ2 k)1/4. The top panel of Fig. 2shows the spatial distribution. Despite using only 10 plane waves, the deviation from a Gaussian |ψ(x,0)|2=exp[−x2/2σ2 x]/2πσ2 x, with σx=1/(2σk), is not appreciable in the plot. For both cases of an initial single Bloch state, and of a superposition state, the numerical results are in excellent agreement with the theoretical prediction, with the agreement improving for increasing interaction time, as expected. 062139-4 ASYMPTOTIC THEORY OF QUASIPERIODICALLY DRIVEN … PHYSICAL REVIEW E 97, 062139 (2018) -2 -1 0 1 2 (ω2−2ω1)Ts 0.001 0.0012 0.0014 0.0016 〈v〉 Ts / T = 104 Ts / T = 105 -300 -200 -100 0 100 200 300 position x 0 0.002 0.004 0.006 0.008 |ψ(x,0)|2 FIG. 2. Finite-time average velocity as a function of the second driving frequency ω2for a system with the same parameters as in Fig. 1, starting from a Gaussian wave function (34) (the top panel shows the initial probability density in space) with average initial momentum ¯hk0=−0.06. The solid line is the prediction of the asymptotic expression (31), and the points correspond to different observation times. The results of Figs. 1and 2prove the validity of the asymptotic expressions (30) and (31), thus validating our approach for the theoretical treatment of quasiperiodically driven quantum systems. These results also extend to the quantum regime the universal asymptotic scaling already identified in the classical case. VI. CONCLUSIONS In this work we considered a quasiperiodically driven quantum system, with the driving consisting of two harmonics with incommensurate frequencies. Due to the nonperiodic nature of the driving, the standard theoretical framework of Floquet states does not directly apply. We consider the asymptotic long-time limit of the system’s dynamics, as it is in this limit in which a quasiperiodic system exhibits features distinct from its periodic counterpart. Also, it is known from the classical case that in such a limit we can expect universal scaling, independent of the system specific details. A theoretical framework, based on Floquet states of nearby periodic situations, which applies to quasiperiodically driven quantum systems in such a limit is developed, and an expression for the asymptotic scaling of relevant quantities for the system at hand is derived. These expressions were tested numerically, finding excellent agreement for the finite-time average velocity in a prototypical quantum ratchet consisting of a space-symmetric potential and a time-asymmetric oscillating force. ACKNOWLEDGMENTS We acknowledge financial support from the Ministerio de Economía y Competitividad of Spain, Grant No. FIS201680244-P. APPENDIX A: FLOQUET STATES EXPRESSION OF THE FINITE-TIME ASYMPTOTIC VELOCITY UNDER PERIODIC DRIVING In this Appendix we show that for a periodically driven quantum system the asymptotic velocity can be expressed as the weighted sum of the average velocity associated with the different Floquet states, each weighted with the relative population, and without interference between them. This result wasfirstdiscussedinRef.[5]. In addition, we show that the interference terms decay as the inverse of the observation time, in agreement with the observation of no interference in the infinite-time limit. We restrict ourselves here to the periodic case, F(t+T)= F(t), T=2πq/ω1, and consider the finite-time current vTs=1 Tst0+Ts t0 dt p(t) m,(A1) where p(t)=ψ(t)|ˆ p|ψ(t),ˆ p=−¯hi∂/∂x.Thesolution |ψ(t)can be expanded in Floquet states |ψα(t)= exp(−iαt/¯h)|ϕα(t), where αis a set of quantum indexes, e.g., α={k,n}, and |ϕα(t+T)=|ϕα(t)(see Sec. IV B): |ψ(t)= α Cα|ψα(t).(A2) Thus vTs=1 Ts α α C∗ αCαt0+Ts t0 dt ×exp −it(α−α) ¯hpαα(t) m,(A3) where pαα(t)=ϕα(t)|ˆ p|ϕα(t)=pαα(t+T). In the limit Ts→∞, the integral in (A3) becomes the Fourier transform of a periodic function, which is different from zero only at the frequencies (α−α)/¯hthat are an integer multiple of ω=2π/T . Since the quasienergies are restricted to the first Brillouin zone, |α−α|<¯hω, the integral in (A3) is nonzero only when α=α. If we assume that there is no degeneracy in the distribution of Floquet states, we find vTs∼ αC2 αvα(Ts→∞),(A4) where vαis the average current associated with the Floquet state α. This is the result used in the main body of the present work. Note that there is usually no degeneracy for a fixed quasimomentum ¯hk. Degeneracy due to states with different k does not produce interference, because states with different quasimomentum are perpendicular (they are eigenstates with different eigenvalues of a Hermitian operator, a translation operator), and the action of the momentum operator produces another Bloch state with the same quasimomentum, ˆ pψ ,nk(x,t)=ˆ pekxiuk,n(x,t) =¯h iekxikiuk,n(x,t)+∂ ∂xuk,n(x,t),(A5) 062139-5 DAVID CUBERO AND FERRUCCIO RENZONI PHYSICAL REVIEW E 97, 062139 (2018) because uk,n is space periodic, i.e., uk,n(x+L,t)=uk,n(x,t), and thus so is its spatial derivative. Therefore, pαα(t)=0for αand αdenoting Bloch states with different quasimomentum. Moreover, explicit analytical expressions can be easily found for the interference terms in the periodic finite-time current (A3). For two states αand αwith different quasienergy, taking advantage of the periodicity of the functions pαα(t), and using Ts=NT, we find 1 Tst0+Ts t0 dt exp −it(α−α) ¯hpαα(t) m =1 Ts N−1  j=0t0+T(j+1) t0+Tj dt exp −it(α−α) ¯hpαα(t) m =1 Tt0+T t0 dt exp −it(α−α) ¯hpαα(t) mDα,α,(A6) where Dα,α=1 N N−1  j=0 exp −iT (α−α) ¯hj (A7) is just a geometric series, and thus easy to compute, yielding Dα,α=1 N 1−exp [−iT N(α−α)/¯h] 1−exp [−iT (α−α)/¯h].(A8) Therefore, in the limit N→∞, we obtain, from (A8), |Dαα|∼1/N, and thus the integral (A6) decays as 1/Ts as Ts→∞.Evenif(α−α)T/¯his very small, by expanding the exponential in the denominator, we arrive at |Dαα|2(N(α−α)T/¯h)−1, and thus the condition (α− α)Ts/¯h1 guarantees the absence of interference between those two Floquet states. APPENDIX B: ASYMPTOTIC TIME INDEPENDENCE OF THE FLOQUET STATE DISTRIBUTION UNDER QUASIPERIODIC DRIVING We show here that the projection of the wave function ψ(x,tj) on the Floquet states ϕkj,n(x,tj), Cn( θj)= ϕkj,n(tj)|ψ(tj), remains constant in time in absolute value, provided we are within the stated asymptotic limit, δω2 T→0. As we show below, this feature can be traced back to a highly oscillating behavior of Cnas a function of  θj. First, we write the evolution over a time step using the evolution operator Uover the interval tj<t<t j+1, |ψ(tj+1)=U(tj+1,tj)|ψ(tj).(B1) Since in this interval the driving force F(t) can be well approximated by its periodic counterpart Fpq(t), Eq. (25), we can write the evolution operator in the lowest orders in δω2 T= θas U(tj+1,tj)=(1 + θW j)Upq(tj+1,tj)+O( θ2),(B2) where Upq is the evolution operator associated with the periodic driving Fpq(t), and Wjan operator that depends on  θj. It is easy to show that the fact that Uis an unitary operator, U†U=U†U=1, implies that Wjis anti-adjoint: W† j=−Wj.(B3) Note that the Floquet theorem implies |ϕk,n(θ,tj)= |ϕk,n(θ,t0). We recall that |ψ(tj)is a Bloch state with a wave vector kjthat depends on time, as per Eq. (19)or(24), and thus kjcan be expressed as a function of the phase θjvia (26). After the appropriate gauge transformation Aj(t), as per (27), we deal with Floquet-Bloch states with θ= θj. We aim to express the Floquet-Bloch states |ϕkj+1,n( θj+1,t0)at tj+1in terms of the states |ϕkj,n( θj,t0)at time tjvia a Taylor expansion in the first order in  θ. Since the states ϕk(θ),n(θ,x,t0) are expected to vary smoothly with θ, we can Taylor expand the Floquet-Bloch states at tj+1, |ϕkj+1,n( θj+1,t0) =|ϕkj,n( θj,t0)+ θ|ϕ kj,n( θj,t0)+O( θ2),(B4) where the prime denotes the derivative with respect to θ, |ϕ k,n(θ,t0)= ∂ ∂θ |ϕk(θ),n(θ,t0),(B5) being in (B4) calculated at θ= θj. By differentiating the orthonormalization condition ϕk(θ),n(θ,t0)|ϕk(θ),n(θ,t0)=δn,n,(B6) we find ϕ k,n(θ,t0)|ϕk,n(θ,t0)=−ϕ k,n(θ,t0)|ϕk,n(θ,t0)∗.(B7) Combining (B2) and (B4)intoCn( θj+1) yields Cn( θj+1)=Cn( θj)e−n( θj) Ti ¯h+ θ n an,n( θj) ×Cn( θj)e−n( θj) Ti ¯h+O( θ2),(B8) where an,n( θj)=ϕkj,n( θj)Wjϕkj,n( θj)+ϕ k,n(θ,t0)|ϕk,n(θ,t0). (B9) Equations (B3) and (B7)imply an,n( θj)=−an,n( θj)∗.(B10) The quantities n( θj) T/¯happearing in the exponentials of Eq. (B8) are not small, indeed, they are responsible for large phase oscillations in the coefficients Cnat every step. In order to obtain a smoother equation for small  θ, we need first to transform to the following set of coefficients  Cn, which differ from Cnon a phase:  Cn( θ)=Cn( θ)expi θ θ0 dθn(θ) ¯hδω2,(B11) yielding  Cn( θj+1)= Cn( θj)+ θ n an,n( θj) Cn( θj) ×e− θj+ θ θ0dθin(θ)−n(θ) ¯hδω2+O( θ2).(B12) Equation (B12) shows that  Cn( θj+1)→ Cn( θj) in the limit  θ→0. Therefore, unlike Cn, the coefficients  Cnvary 062139-6 ASYMPTOTIC THEORY OF QUASIPERIODICALLY DRIVEN … PHYSICAL REVIEW E 97, 062139 (2018) smoothly at each step. After taking the limit  θ→0in(B12), we arrive at ∂ ∂θ  Cn( θ)= n an,n( θ)e−i θ θ0dθn(θ)−n(θ) ¯hδω2 Cn( θ).(B13) It can be readily checked that the solution of (B13)isgivenby  Cn( θ)= n (eS( θ))n,nCn(θ0),(B14) where S( θ)n,n= θ θ0 dθan,n(θ)e−iθ θ0dθ n(θ)−n(θ) ¯hδω2,(B15) and we have used the fact  Cn(θ0)=Cn(θ0). From (B10), it is clear that Sis anti-adjoint, Sn,n( θ)=−Sn,n( θ)∗.(B16) Following the standard procedure used for Hermitian matrices, it is then easy to show that all eigenvalues of Sare imaginary, and its eigenvectors are, or can be chosen as, orthonormal. The expressions (B14)–(B16) are the central results in this Appendix. In the limit δω2→0, the exponential in (B15) is a highly oscillatory function that makes off-diagonal terms of Snegligible against the diagonal terms, where the exponential is absent. Since the diagonal terms are purely imaginary [Eq. (B16)], we arrive at the simple prediction |Cn( θ)|2∼|Cn(θ0)|2for all  θ(δω2→0).(B17) This means that the distribution of Floquet-Bloch states, i.e., the populations |Cn( θ)|2, remain constant throughout the process, being unaffected by the population of different Floquet-Bloch states, though present as a phase factor in Cn( θ). This is due to the highly oscillatory phases in the off-diagonal terms of Sin the limit δω2→0. [1] D. Cubero and F. Renzoni, Brownian Ratchets: From Statistical Physics to Bio and Nano-motors (Cambridge University Press, Cambridge, 2016). [2] P. Reimann, Brownian motors: Noisy transport far from equilibrium, Phys. Rep. 361,57 (2002). [3] P. Hänggi and F. Marchesoni, Artificial Brownian motors: Controlling transport on the nanoscale, Rev. Mod. Phys. 81,387 (2009). [4] S. Denisov, L. Morales-Molina, and S. Flach, Quantum resonances and rectification in ac-driven ratchets, Europhys. Lett. 79,10007 (2007). [5] S. Denisov, L. Morales-Molina, S. Flach, and P. Hänggi, Periodically driven quantum ratchets: Symmetries and resonances, Phys.Rev.A75,063424 (2007). [6] T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Directed transport of atoms in a Hamiltonian quantum ratchet, Science 326,1241 (2009). [7] T. Salger, S. Kling, S. Denisov, A. V. Ponomarev, P. Hänggi, and M. Weitz, Tuning the Mobility of a Driven Bose-Einstein Condensate via Diabatic Floquet Bands, Phys.Rev.Lett.110, 135302 (2013). [8] C. Grossert, M. Leder, S. Denisov, P. Hänggi, and M. Weitz, Experimental control of transport resonances in a coherent quantum rocking ratchet, Nat. Commun. 7,10440 (2016). [9] J. Casado-Pascual, D. Cubero, and F. Renzoni, Universal asymptotic behavior in nonlinear systems driven by a two-frequency forcing, Phys.Rev.E88,062919 (2013). [10] D. Cubero, J. Casado-Pascual, and F. Renzoni, Irrationality and Quasiperiodicity in Driven Nonlinear Systems, Phys. Rev. Lett. 112,174102 (2014). [11] P. Bader and S. Blanes, Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schrödinger equations, Phys. Rev. E 83,046711 (2011). 062139-7