Duality relation for quantum ratchets
J. Peguiron1,2and M. Grifoni2 1
Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 2
Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
共Received 9 July 2004; revised manuscript received 28 September 2004; published 19 January 2005兲
A duality relation between the long-time dynamics of a quantum Brownian particle in a tilted ratchet potential and a driven dissipative tight-binding model is reported. It relates a situation of weak dissipation in one model to strong dissipation in the other one, and vice versa. We apply this duality relation to investigate transport and rectification in ratchet potentials: From the linear mobility we infer ground-state delocalization for weak dissipation. We report reversals induced by adiabatic driving and temperature in the ratchet current and its dependence on the potential shape.
DOI: 10.1103/PhysRevE.71.010101 PACS number共s兲: 05.40.⫺a, 05.30.⫺d, 05.60.Gg, 73.23.⫺b
Periodic structures with broken spatial symmetry, known as ratchet systems关1兴, present the attractive property of al-lowing transport under the influence of unbiased forces. The interplay of dissipative tunneling 关2兴 with unbiased driving enriches the quantum ratchet effect with features absent in its classical counterpart like, e.g., current reversals as a function of temperature关3,4兴. Quantum ratchet systems have only re-cently been experimentally realized in semiconductor关4兴 and superconductor关5兴 devices. Also from the theory side there are still few works 关3,6–10兴 which, with the exception of
关7,8兴, are restricted to the regime of moderate-to-strong
damping. After the semiclassical work 关3兴, further progress towards a quantum description was made in关9兴, where the role of the band structure in ratchet potentials sustaining few bands below the barrier was investigated. Recently, a quan-tum Smoluchowski treatment 关10兴 added to the available methods. In this paper, we generalize to an arbitrary ratchet potential a duality relation put forward in关11兴 for a cosine potential. It provides a tight-binding description of quantum Brownian motion in a ratchet potential, and leads to an ex-pression for the ratchet current valid in a wide parameter range including weak dissipation and nonlinear adiabatic driving. We apply this method to discuss rectification and ground-state delocalization occurring for weak dissipation in ratchet potentials. Our results encompass correctly the clas-sical limit.
We consider the Hamiltonian HˆRof a quantum particle of mass M in a one-dimensional periodic potential V共q+L兲 = V共q兲 tilted by a time-dependent force F共t兲,
HˆR共t兲 = pˆ2
2M+ V共qˆ兲 − F共t兲qˆ. 共1兲 The potential assumes in Fourier expansion the form
V共qˆ兲 =
兺
l=1
⬁
Vlcos共2lqˆ/L −l兲, 共2兲
and can take any shape. Apart from special configurations
兵Vlsin共l− l1兲=0∀l其 of the amplitudes Vl and phases l,
this potential is spatially asymmetric and describes a ratchet system. The interaction of the system with a dissipative
ther-mal environment is modeled by the standard Hamiltonian HˆB of a bath of harmonic oscillators whose coordinates are bi-linearly coupled to the system coordinate qˆ关2兴. The bath is fully characterized by its spectral density J共兲. We consider strict Ohmic damping J共兲=, which reduces to instanta-neous viscous damping共viscosity兲 in the classical limit. In such a system, the ratchet effect is characterized by a nonva-nishing average stationary particle current vR⬁ = limt→⬁t−1兰0
tdt
⬘
v共t
⬘
兲 in the presence of unbiased driving, characterized by limt→⬁t−1兰0tdt
⬘
F共t⬘
兲=0, switched on at timet
⬘
= 0. In this paper, we shall consider the particular case of unbiased bistable driving switching adiabatically between the values ±F. We report a method to evaluate the stationary velocity vDC⬁ 共F兲 in the biased situation of time-independent driving F, which is also of experimental interest 关4,5兴. The ratchet current in the presence of adiabatic bistable driving can be expressed asvR⬁=vDC⬁ 共F兲+vDC⬁ 共−F兲.The whole information on the system dynamics is con-tained in the reduced density matrixˆ共t兲=TrBWˆ 共t兲, obtained from the density matrix Wˆ 共t兲 of the system-plus-bath Hˆ =HˆR+ HˆB, with time-independent driving F, by performing the trace over the bath degrees of freedom. To evaluate the evolution of the average position具q共t兲典=TrR兵qˆˆ共t兲其, the di-agonal elements P共q,t兲=具q兩ˆ共t兲兩q典 of the reduced density matrix are needed, and can be obtained by real-time path integrals techniques关2兴. The velocity follows by time differ-entiation. At initial time t
⬘
= 0, we assume a preparation in a product form where the bath is in thermal equilibrium with the ratchet system Wˆ 共0兲=ˆ共0兲e−HˆB关TrBe−HˆB兴−1. The bath temperature is fixed by T = 1 /kB. This leads to a double path integral P共qf,t兲 =
冕
dqi冕
dqi⬘具qi兩ˆ共0兲兩qi⬘典冕
qi qf Dq ⫻冕
qi⬘ qf D*q⬘
A关q兴A*关q⬘
兴F关q,q⬘
兴 共3兲 on the continuous coordinates q and q⬘
. Here A关q兴 = exp兵−共it/ប兲HˆR其 is the propagator of the ratchet system for PHYSICAL REVIEW E 71, 010101共R兲 共2005兲RAPID COMMUNICATIONS
a path q共t
⬘
兲, and F关q,q⬘
兴 the Feynman-Vernon influence functional of the bath inducing nonlocal-in-time Gaussian correlations between the paths q共t⬘
兲 and q⬘
共t⬘
兲 关2兴. Due to the nonlinearity of the potential V共q兲, these path integrals cannot be performed explicitly. For a cosine potential, Fisher and Zwerger关11兴 introduced an exact expansion in the propaga-tor A关q兴 which transforms the path integrals into Gaussian ones that can be performed. Generalizing this idea for the arbitrary periodic potential共2兲, we find the expansionexp
再
− i ប冕
0 t dt⬘
V共q共t⬘
兲兲冎
=兺
m=0 ⬁兺
兵lj其兿
j=1 m冉
− i⌬lj ប冊
冕
0 t dtm冕
0 tm dtm−1. . . ⫻冕
0 t2 dt1exp再
− i ប冕
0 t dt⬘
共t⬘
兲q共t⬘
兲冎
, 共4兲 where共t⬘
兲=共2ប/L兲兺mj=1lj␦共t⬘
− tj兲, and ⌬l= Vl 2e il for l⬎ 0, ⌬ −l=⌬l * . 共5兲The physical meaning of these new quantities will be dis-cussed later. For each term of the sum on m in共4兲 we have introduced m intermediate times tj, and corresponding
indi-ces lj taking any value among 兵±1, ±2, ...其. The sum 兺兵lj其
runs on all configurations of these indices. A similar expan-sion is performed for the propagator A*关q
⬘
兴, involving a new set of m⬘
times t⬘
j and indices l⬘
j being used to define⬘
共t⬘
兲 similarly to共t⬘
兲. This enables us to rewrite the average po-sition 具q共t兲典=兰dqqP共q,t兲 in terms of a series in the ampli-tudes Vlof the potential.Though still intricate, the resulting expression becomes easier to treat in the long-time limit we are interested in. Quantitatively, the measurement time t should be very long on the time scale␥−1=共/ M兲−1set by dissipation. A second approximation is necessary to proceed: we neglect terms e−␥tj, e−␥t⬘j, e−␥共t−tj兲, e−␥共t−tj⬘兲, e−Btj, e−Btj⬘, e−B共t−tj兲, and
e−B共t−t⬘j兲, whereB= 2kBT /ប, in the integrands involved in
the series expression for具q共t兲典. We shall refer to this assump-tion as the rare transiassump-tions共RT兲 limit and discuss its validity later. Generalizing关11兴, we consider a Gaussian wave packet centered at position q0= TrR关qˆˆ共0兲兴 and momentum p0 = TrR关pˆˆ共0兲兴 as initial preparation for the ratchet system. We obtain the important result
具q共t兲典 ⬃ t→⬁ RT q0+ p0 + Ft −具qTB共t兲典TB. 共6兲 Parts of the series expression for 具q共t兲典 has been summed, yielding the first three terms. The rest can be identified with the series expression for the expectation value of the position operator qˆTB= L˜ 兺n=−⬁ ⬁n兩n典具n兩 of a driven tight-binding 共TB兲
system, described by the Hamiltonian
HˆTB=
兺
n,l=−⬁ ⬁ 共⌬l兩n + l典具n兩 + ⌬l *兩n典具n + l兩兲 − Fqˆ TB, 共7兲 and bilinearly coupled to a different bath of harmonic oscil-lators. The spectral density of this bath JTB共兲=J共兲/关1 +共/␥兲2兴 is still Ohmic but presents a cutoff at the frequency␥ set by dissipation. At initial time t
⬘
= 0 the TB system is prepared in the state兩n=0典. The calculation shows that the ⌬lintroduced in共5兲 are identified with the couplings of the TB system 共7兲. We stress that the lth harmonic of the original potential results in a coupling to the lth neighbors in the dual TB system as sketched in Fig. 1. One can easily show that the spatial symmetry condition on the phaseslis the same
in both systems. The first three terms on the right-hand side of共6兲 reproduce exactly the classical solution for the average position 具q共t兲典 of a free system, V共q兲⬅0, at long times. In this linear case, the quantum and classical solutions should be identical, due to Ehrenfest theorem, and they are, because the TB average 具qTB共t兲典TB vanishes in the absence of the potential V共q兲. We expect the same result when the potential is present but unimportant, e.g., for large driving F and/or high temperatures T.
The series expression for the diagonal elements of the reduced density matrixˆTBof the TB system, which leads to the series expression for具qTB共t兲典TB, can be written in terms of pairs of TB trajectories qTB共t
⬘
兲=−1兰0
t⬘dt
⬙
共t⬙
兲 关with共t⬘
兲introduced above Eq.共5兲兴, and qTB
⬘
共t⬘
兲 defined similarly in terms of⬘
共t⬘
兲. From that one extracts the spatial periodicity L˜ of the TB system, yielding L˜=L/␣, where␣=L2/ 2ប is the dimensionless dissipation parameter of the original sys-tem. These pairs of trajectories combine in discrete paths in the q − q
⬘
plane parametrized by pairs of integers 共n,n⬘
兲. Each path starting in the diagonal element共0,0兲 and ending at time t in共m,m兲 contributes to 具m兩ˆTB共t兲兩m典. Each transi-tion in the path brings a corresponding factor⌬land all pathsinvolve at least two transitions 共cf. Fig. 2兲. Written in this form, the diagonal elements of the reduced density matrix are a solution of a generalized master equation关12兴 in terms of transition rates⌫m from the TB site共n,n兲 to the site 共n + m , n + m兲. Consequently, these rates are expressed in power series of all the couplings⌬l, starting from second order. As
the times tj, t
⬘
j introduced in共4兲 are identified with thetran-sition times in the TB representation, the rates⌫mgive also a FIG. 1. Dual relation between a dissipative ratchet system and a tight-binding共TB兲 model sketched for a two-harmonics ratchet po-tential共thick curve兲. Each harmonic 共thin curves兲 generates cou-plings to different neighbors in the TB system, according to Eqs.共5兲 and 共7兲. The periodicity L˜ of the TB model is determined by the viscosityin the original model.
J. PEGUIRON AND M. GRIFONI PHYSICAL REVIEW E 71, 010101共R兲 共2005兲
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way to control our assumption of rare transitions. It corre-sponds to neglect those paths which involve transitions on a time scale max共␥−1,B−1兲 after the initial time t
⬘
= 0 or before the final time t⬘
= t. As transitions in the TB model happen on a time scale⌫m−1, the duality relation will be valid when the transitions are rare on the time scale max共␥−1,B −1兲, i.e., when all rates satisfy⌫mⰆmin共␥,B兲. This condition is con-trolled by the dissipation through␥=/ M and the tempera-ture throughB= 2kBT /ប.
Due to the change of periodicity length between the two systems, the dissipation parameter␣and the energy drop per unit cell⑀= FL become␣˜ = 1 /␣and˜ =⑀ ⑀/␣in the TB system. Thus, weak dissipation in one system maps to strong dissi-pation in the other one although the viscosity in the spec-tral density does not change. The asymptotic dynamics is usually described by the nonlinear mobility = limt→⬁v共t兲/F. With these notations, the duality relation 共6兲
can be rewritten in the form
共␣,⑀兲→ RT0
−TB共1/␣,⑀/␣兲, 共8兲 where0= 1 / is the mobility of the free system, V共q兲⬅0. In the special case of a cosine potential, this relation was already obtained in关11兴 for the dc mobility. It it also inter-esting to notice that it was also derived in关13兴 for the linear ac mobility in a cosine potential. However, we did not com-pletely succeed in generalizing Eq. 共8兲 in the presence of time-dependent driving.
We shall now focus on the evaluation of the stationary velocityvDC⬁ 共F兲. By solving the generalized master equation mentioned above, one finds the stationary velocity vTB⬁ = L˜ 兺mm⌫min the dissipative TB system. The duality relation 共6兲 can then be used to obtain
vDC⬁ 共F兲 = F/−共L/␣兲
兺
m=1⬁
m共⌫m−⌫−m兲. 共9兲 As discussed above, the rates ⌫m are power series in the couplings⌬l starting from second order. For a given m⫽0,
there are only two possible second-order contributions to⌫m,
which, after use of Eq.共5兲, sum up to 关2兴
⌫m共2兲=
Vm2 4ប2␥
冕
−⬁⬁
de−m2␣˜Q共兲+im共FL˜/ប␥兲. 共10兲 The influence of the dissipative environment enters through the dimensionless bath correlation function Q共兲
= 2兰0⬁d关coth共/ 2兲共1−cos兲+i sin兴/共1+2兲 with = kBT /ប␥. At zero bias F = 0 and in the scaling limit ប␥
ⰇkBT, the rates show a power-law dependence on tempera-ture⌫m共2兲⬀T2m2␣˜−1. The linear mobility
TBis thus dominated by the rate⌫1共2兲at low temperatures, and vanishes at T = 0 for
␣⬍1, which corresponds to free dynamics=0in the dual weak-binding system关14兴. This suggests that the occurrence of a delocalization to localization transition at␣= 1 for the ground state of a cosine potential关11,15兴 would not be af-fected in more general potentials共see also Fig. 3兲.
In the remainder of the paper, we focus on the ratchet current induced by adiabatic bistable driving vR⬁=vDC⬁ 共F兲 +vDC⬁ 共−F兲. The second-order rates obey ⌫m
共2兲共−F兲=⌫
−m
共2兲共F兲 and therefore cancel out in the expression for the ratchet current. Hence, we have to focus on contributions of at least third order to the rates ⌫m. Here we neglect higher orders.
This is known to provide a good approximation in TB sys-tems with large dissipation parameter␣˜ and/or high tempera-ture关2兴. For simplicity we also consider a potential consist-ing of only two harmonics. There is no problem of principle to include more harmonics关16兴. We find, with m= ±1, ±2,
⌫m共3兲= V12V2 4ប3␥2Im
冋
冕
−⬁ ⬁ dG兩m兩共3兲共兲eim共FL˜/ប␥兲−i sgn共m兲册
, 共11兲where we have introduced=2− 21, and G1共3兲共兲 = −
冕
0 ⬁ de−2␣˜Q共兲关e−2␣˜Q共+兲+␣˜Q共+2兲 + e−2␣˜Q共−兲+␣˜Q共−2兲兴, G2共3兲共兲 =冕
0 ⬁ de␣˜Q共兲−2␣˜Q共+/2兲−2␣˜Q共−/2兲. 共12兲 At third order the rates obey⌫m共3兲共F,兲=⌫−m共3兲共−F,−兲, which is a consequence of parity. The dependence of the ratchet current on the potential parameters is then up to third order in the potential amplitudevR⬁⬀ V1 2V
2sin共2− 21兲. 共13兲 The ratchet current vanishes for a symmetric potential sin共2− 21兲=0 as it should.
FIG. 2. Representation of some of the second-order共a,b兲 and third-order 共c兲 paths contributing to the diagonal elements of the reduced density matrix of the tight-binding model, and the
corre-sponding dependence on the couplings⌬l. FIG. 3. Ratchet current and stationary velocity共inset兲 as func-tion of temperature for the potential of amplitude⌬V depicted in Fig. 1. Weak dissipation is chosen with ␣=0.2 and ប␥=0.76⌬V. Driving is set to FL = 0.57⌬V.
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The behavior of the particle and ratchet currents as func-tion of temperature and driving is shown in Figs. 3 and 4 for a two-harmonics potential. In Fig. 3, the driving is set to FL = 0.57⌬V, whereas in Fig. 4, the temperature is fixed to kBT = 0.23⌬V. With V1= 4V2, the untilted potential, depicted in Fig. 1, has a barrier height ⌬V=2.2V1. We choose ␣ = 0.2 and ប␥= 0.76⌬V. It means that the typical action is
冑
2M⌬VL2⬇2ប, and the dissipation rate ␥=/ M is about one-fourth of the classical oscillation frequency ⍀0= 2
冑
V1/ ML2in the untilted potential共weak dissipation兲. In this numerical application, none of the rates exceeds 0.05␥ and 0.08B, which means that the duality relation is valid for this system. Moreover, the third-order rates stay at least one order of magnitude below the second-order ones. The ratchet current presents several reversals as a function both of the driving and the temperature. As expected for the small values of driving and dissipation used in Fig. 3, the stationary ve-locity is very close to the value of a free systemv0= F /at T = 0, which corresponds to localization vTB⬁ ⬇0 in the TB system关17兴. Accordingly, vR⬁⬇0 in this regime. The station-ary velocity also tends tov0共dashed line in Fig. 4兲 for driv-ing or temperatures much higher than the potential barrier, and the ratchet current vanishes correspondingly. If observed in experiments, this linear behavior would provide a direct estimation of dissipation.In conclusion, we obtained a duality relation yielding a tight-binding description of Brownian motion in a tilted ratchet potential. We demonstrated its application to investi-gate rectification of adiabatic driving and ground-state delo-calization for weak dissipation.
We thank U. Weiss for seminal discussions. This work was supported by the Dutch Foundation FOM.
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关16兴 A potential with few harmonics also comes naturally in
experi-ments with arrays of Josephson junctions关5兴.
关17兴 We obtain the opposite behavior vTB→⬁ at low temperatures for␣=1.26.
FIG. 4. Stationary velocity and ratchet current共inset兲 as a func-tion of driving for the potential of amplitude⌬V depicted in Fig. 1. The dashed line is the classical solution in the absence of potential. Weak dissipation is chosen with␣=0.2 and ប␥=0.76⌬V. Tempera-ture is set to kBT = 0.076⌬V.
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