Directed ratchet transport of cold atoms and fluxons driven by biharmonic fields: A unified view

PHYSICAL REVIEW E 104, 014120 (2021) Directed ratchet transport of cold atoms and fluxons driven by biharmonic fields: A unified view Ricardo Chacón 1and Pedro J. Martínez 2 1Departamento de Física Aplicada, E.I.I., Universidad de Extremadura, Apartado Postal 382, E-06006 Badajoz, Spain and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, E-06006 Badajoz, Spain 2Departamento de Física Aplicada, E.I.N.A., Universidad de Zaragoza, E-50018 Zaragoza, Spain and Instituto de Nanociencia y Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza, E-50009 Zaragoza, Spain (Received 16 March 2021; revised 20 May 2021; accepted 21 June 2021; published 16 July 2021) This paper discusses two retrodictions of the theory of ratchet universality which explain previous experimental results concerning directed ratchet transport of cold atoms in dissipative optical lattices in one case and of fluxons in uniform annular Josephson junctions in the other, both driven by biharmonic fields. It has to be emphasized that these retrodictions are in sharp contrast with the current standard explanation of such experimental results, and they offer optimal control of the ratchetlike motion of such entities. New experimental proposals with cold atoms and fluxons are discussed, providing additional tests for novel predictions from ratchet universality. DOI: 10.1103/PhysRevE.104.014120 I. INTRODUCTION Symmetry principles constrain the possible forms of the laws of nature by constituting a synthesis of those regularities that are independent of the specific dynamics. They usually play a deep and subtle role in the sense that many physical phenomena ultimately come to be explained in terms of mechanisms of symmetry breaking. A notable instance is the so-called ratchet effect [1–3], i.e., the possibility of generating directed transport from a fluctuating environment without any net external force. Indeed, it has been a fundamental research topic in diverse areas of science and technology since the end of the last century, partly because of its potential applications for manipulating such systems as coupled Josephson junctions [4] and molecular motors [5], as well as for designing microand nanodevices suitable for on-chip implementation. Directed ratchet transport (DRT) is now qualitatively understood to be a result of the interplay of nonlinearity, symmetry breaking [6], and nonequilibrium fluctuations including temporal noise [2], spatial disorder [7], and quenched temporal disorder [8]. The symmetry analysis alone, however, is insufficient to predict the direction and strength of the DRT. Recently, some of such fundamental aspects, including current reversals [9] and the quantitative dependence of DRT strength on the system’s parameters [10], have begun to be elucidated. In this regard, the theory of ratchet universality (RU) [11–13] predicts that there exists a universal force waveform that optimally enhances directed transport by symmetry breaking. The theory of RU refers to the criticality scenario that emerges when the generalized parity symmetry and the generalized time-reversal symmetry are broken, regardless of the nature of the dynamic equation in which the breaking of such symmetries results in DRT. For noiseless ratchets, the effectiveness of this theory has been demonstrated in diverse physical contexts in which the driving forces are chosen to be biharmonic, such as in the cases of topological solitons [8], Bose-Einstein condensates exposed to a sawtoothlike optical lattice potential [14], matter-wave solitons [10], one-dimensional granular chains [15], and Bose-Einstein condensates under an unbiased periodic driving potential [16]. Thus, the effectiveness of RU in quantum systems has been previously demonstrated, including the cases of directed transport of atoms in a Hamiltonian quantum ratchet (the values of the parameters used, which were chosen to maximize the directed transport, correspond to those of the universal biharmonic waveform; cf. Ref. [14]) and driven Bose-Einstein condensates (the authors show that the ratchet current is maximum for the values of the parameters that correspond to those of the ratchet potential associated with the universal biharmonic waveform; cf. Ref. [16]). There have also been quantitative explanations in coherence with the degree-of-symmetry-breaking mechanism, as predicted by the theory of RU [11,12], of the interplay between thermal noise and symmetry breaking in the DRT of a Brownian particle moving on a periodic substrate subjected to a homogeneous temporal biharmonic force [17–19], and of a driven Brownian particle subjected to a vibrating periodic potential [20]. Numerical analyses of a driven Brownian particle in the presence of non-Gaussian noise [21] and coupled Brownian motors with stochastic interactions in a crowded environment [22] have confirmed the RU predictions. Additionally, RU has recently been demonstrated in the bidirectional escape from a symmetric potential well [23]. This present paper discusses two retrodictions of the theory of RU which explain previous experimental results concerning DRT of cold atoms in dissipative optical lattices [24] in one case (Sec. II), and magnetic flux quanta (fluxons) in uniform annular Josephson junctions [25] in the other (Sec. III), both driven by biharmonic fields. It has to be emphasized that these retrodictions 2470-0045/2021/104(1)/014120(5) 014120-1 ©2021 American Physical Society CHACÓN AND MARTÍNEZ PHYSICAL REVIEW E 104, 014120 (2021) are in sharp contrast with the current standard explanation of such experimental results, and they suggest new aspects for experimental testing. Finally, we conclude with Sec. IV by summarizing our conclusions and discussing future work. II. COLD ATOMS Motivated by investigation into the mechanisms that yield directed diffusion in a symmetric periodic potential, Schiavoni et al. [24] carried out an experimental and numerical study of cold atoms in a one-dimensional dissipative optical lattice where directed motion appears as a result of the breaking of the system’s temporal symmetry after applying a biharmonic phase modulation to one of the lattice beams. In the accelerated reference frame, the atoms experience a stationary optical potential together with an inertial force F(t)=γ[Acos (ωt)+Bcos (2ωt−φ)],(1) where γis an amplitude factor, A=1−B, and the parameters B∈[0,1] and φ∈[0,2π] are the relative amplitude and initial phase difference of the two harmonics, respectively. Commenting on their experimental results, the authors claimed that, “By increasing Bfrom the zero value the atoms are set into directed motion, and a maximum for the c.m. velocity is reached for B≃0.5, i.e., for about equal amplitudes of the even and odd harmonics.” This statement has had the unfortunate consequence that most subsequent publications citing Ref. [24] have considered B=1/2 to be the condition that maximizes ratchet transport in systems subjected to a biharmonic temporal force. It will be shown below that the maximum center-of-mass (c.m.) velocity is reached for B=1/3, as predicted by the theory of RU [11–13]. Indeed, it has been demonstrated for temporal and spatial biharmonic forces that optimal enhancement of DRT is achieved when maximally effective (i.e., critical) symmetry breaking occurs, which implies the existence of a particular universal waveform [11–13]. Specifically, the optimal value of the relative amplitude Bcomes from the condition that the amplitude of the odd harmonic must be twice that of the even harmonic in Eq. (1), i.e., 1 −Bopt = 2Bopt ⇒ Bopt =1/3. Notice that this means that the contributions of the amplitudes of the two harmonics to the directed motion of the atoms are not independent, which in Ref. [24] is solely taken into account in the estimate of the optical pumping rate (escape rate) ∼sin2kz, with z∼A2Bbeing the displacement of the center of oscillation of the atoms in a potential well from the well center, after the substitution A=1−B. In such a case, one has that =(B) presents a single maximum at Bopt =1/3 for which the asymmetry between the escape rates toward the left and right wells is maximal, and hence one again expects a maximal nonzero current of atoms for B=1/3, as is indeed confirmed by the experimental results [24](seeFig.1). RU predicts that the strength of the nonzero mean current, v, has the functional dependence v∼S(B)p(φ)(2) [12], where S(B) accounts for the degree of breakage of the shift symmetry F(t+T/2) =−F(t), while the 2π-periodic function p(φ) accounts for the degree of breakage of the FIG. 1. Velocity of the center of mass of the atomic cloud for φ= π/2 (dots are the experimental data from Fig. 3 in Ref. [24]), and the curves Sφ=π/2=C1Fφ=π/2(B) (solid line), Sφ=0=C2Fφ=0(B) (dashed line), and C3(1 −B)2B(dotted line) as functions of the relative amplitude B[see the text; Eq. (1)]. Fixed parameters: C1= 6.6,C2=1.95,C3=6.9[26]. time-reversal symmetry F(−t)=F(t) and has two extrema at the optimal values φopt ={π/2,3π/2}. This dependence on φis indeed confirmed by the experimental results shown in Fig. 2of Ref. [24]. Also, S(B) presents features similar to those of the normalized version of the biharmonic force [Eq. (1)], F∗(t), for any value of φ(see Ref. [12] for additional details). Figure 1shows plots of the optical pumping rate (B) and the function S(B) for two limiting cases of the initial phase difference: one of the optimal values (φopt =π/2) and one of the least favorable values (φ=0) [26]. These curves fit the experimental data reasonably well, and they present a single maximum at B=1/3, as expected [11–12]. This retrodiction therefore indicates that the results of Ref. [24] provide FIG. 2. Maximum amplitude of the rectified dc voltage vs relative power dB ≡10 log10[P1(2)/P2(1)] of the two harmonics (experimental data from Fig. 4 in Ref. [25]; P1=const (squares), P2= const (circles). Vertical dotted lines indicate the values dB =±6.02 (see the text). 014120-2 DIRECTED RATCHET TRANSPORT OF COLD ATOMS AND … PHYSICAL REVIEW E 104, 014120 (2021) experimental proof of RU in the context of cold atoms in optical lattices. Further experiments with cold atoms could readily discriminate between the RU and the harmonic-mixing perturbation theory predictions by considering the inertial force γ[α(1 − B) cos(ωt)+Bcos(2ωt−φ)], with α>0, instead of that givenbyEq.(1). In such a case, one obtains that (B)∼ α2(1 −B)2Bpresents again a single maximum at Bopt =1/3, irrespective of the particular value of parameter α, while RU predicts an α-dependent maximal current of atoms for Bopt =α/(2+α).(3) III. MAGNETIC FLUX QUANTA Motivated by investigation into the mechanisms that yield DRT of solitons in long Josephson junctions, Ustinov et al. [25] studied experimentally the rectified dc voltage induced by the ratchetlike motion of a fluxon driven by a biharmonic microwave force of zero mean, applied to a spatially uniform long Josephson junction. The dynamics of the superconducting phase difference across the junction is described by the perturbed sine-Gordon equation ϕtt −ϕxx +sin ϕ=−αϕt+γ+γ(t), γ(t)≡γ1sin (t)+γ2sin (2t+θ),(4) with the boundary conditions ϕ(l)=ϕ(0) +2π,ϕx(l)= ϕx(0), where αis the dissipation constant due to quasiparticle tunneling current, γand γ1,γ2are the dc and ac normalized amplitudes (bias current densities), respectively, θis the initial phase difference of the two microwave harmonics, and the ac power levels P1and P2satisfy the scaling γi∼√Pi,i=1,2. The breakage of the shift symmetry of the ac field γ(t) leads to the ratchetlike motion of a fluxon, which is manifested in the nonzero rectified voltage across the junction at zero bias current. Commenting on their experimental and numerical results, the authors claimed that, “The results of the experimental measurements and numerical simulations [of Eq. (4)] are in good correspondence with the results of the first order (point-particle approximation) soliton perturbation theory.... Using this approach, the mean fluxon velocity (in the absence of dc bias) can be computed...as follows”: v∼γ2 1 γ2sin (θ+θ0)∼P1√P2sin (θ+θ0),(5) where θ0=arctan{2(/α)/[3 +(α/)2]}. The authors assumed a sufficiently high dissipation constant αsuch that θ0≈0, and they found that the rectified voltage presents a 2π-periodic dependence on θwith two extrema at the optimal values θopt ={π/2,3π/2}(cf. Fig. 3 in Ref. [25]). This dependence on θis indeed predicted from RU for the biharmonic field γ1sin(t)+γ2sin(2t+θ) when the dissipation phase θ0+π/2 reaches its highest value, i.e., when θ0→0[11]. Remarkably, their experimental results (see Fig. 2) indicate that the maximum amplitude of the rectified voltage, Vmax, presents, as a function of the relative power dB =dB1(2) ≡ 10 log10[P1(2)/P2(1)], a maximum at some optimum value of the power level P1(P2) while keeping the other power level P2(P1) constant. The following remarks would now seem to be in order. First, the theoretical prediction given by Eq. (5) indicates that Vmax should present a monotonous behavior as 0 3 6 9 12 -20 -15 -10 -5 0 5 10 15 20 mean flux velocity relative power (dB) FIG. 3. Mean fluxon velocity as a function of the relative power dB =±20 log10{η/[α(1 −η)]}[see the text; Eq. (7)] for θ=θopt ≡ π/2andα={1,2,3}[curves S(dB1)=C1L(dB1)andS(dB2)= C2L(dB2) with maxima at dBmax =−6.02 and 6.02, respectively]. Vertical dashed lines indicate the values dB =±6.02. Fixed parameters: C1=50,C2=60 [27]. a function of dB for both data series, thus failing to explain the experimental results shown in Fig. 2. Second, RU predicts again that optimal enhancement of DRT of fluxons requires that the amplitude of the odd harmonic must be twice that of the even harmonic in Eq. (4), i.e., γ1=2γ2⇒ P1∼4P2, and hence that Vmax should present an absolute maximum at dBmax ∼6(−6) when the power level P2(P1) is kept constant (see Fig. 2), thereby explaining the overall behavior of the experimental data. Notice that the values dB ∼±6 are independent of the particular values of the power levels which are kept constant (not stated in Ref. [25]) in each of the two data series. This retrodiction, therefore, indicates that the results of Ref. [25] provide experimental proof of RU in the context of fluxons in annular Josephson junctions driven by biharmonic fields. Finally, we propose additional experimental tests with fluxons by considering the ac field γ(t)≡γ[ηsin (t)+α(1−η)sin (2t+θ)](6) instead of that given by Eq. (4), where γis an amplitude factor, α>0, and η∈[0,1]. This means that P1∼γ2η2,P2∼ α2γ2(1 −η)2, and hence dB1≡20 log10{η/[α(1 −η)]}= −dB2. Now, the theoretical prediction given by Eq. (5) reads v∼αγ3η2(1 −η)sin(θ+θ0), which presents a single maximum at ηopt =2/3, irrespective of the particular value of parameter α, and hence the amplitude of the rectified voltage should present single maxima at dBmax = ±20 log10(2/α), thus indicating an explicit dependence on parameter α. In contrast, RU predicts once again that optimal enhancement of DRT of fluxons requires that the amplitude of the odd harmonic must be twice that of the even harmonic in Eq. (6), i.e., ηopt =2α/(1 +2α)⇒ P1∼4P2, and hence that the amplitude of the rectified voltage should present again absolute maxima at dBmax ∼±6, irrespective of the particular value of parameter α. Also, RU predicts that the mean fluxon 014120-3 CHACÓN AND MARTÍNEZ PHYSICAL REVIEW E 104, 014120 (2021) velocity has the functional dependence v∼S(dB)p(θ)(7) [12], where p(θ)isa2π-periodic function, while S(dB) presents features similar to those of the normalized version of the biharmonic field [Eq. (6)], γ∗(t), for any value of θ(see Ref. [12] for additional details). Figure 3shows plots of the functions S(dB1,2) for one of the optimal values of the initial phase difference (θopt =π/2) and three values of α[27]. One finds indeed that the respective curves are identical for any value of αand present maxima at dBmax ∼±6, as predicted by RU, while resembling the corresponding experimental data series shown in Fig. 2. IV. SUMMARY AND OUTLOOK In conclusion, we have discussed two retrodictions of the theory of RU which explain previous experimental results concerning DRT of cold atoms in dissipative optical lattices in one case, and fluxons in uniform annular Josephson junctions in the other, both driven by biharmonic fields (rocking ratchets). We expect that the theory of RU will explain other different implementations of the ratchet effect, such as gating ratchets [28], directed transport in coupled systems without external bias [29], and ratchets without any periodic substrate potential [30]. Our current work is aimed at exploring these cases [31]. ACKNOWLEDGMENTS The authors thank F. Renzoni for the help with the experimental data recovery and interchanges about this issue. R.C. acknowledges financial support from the Junta de Extremadura (JEx, Spain) through Project No. GR18081 cofinanced by FEDER funds. 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