Two coupled Josephson junctions: dc voltage controlled by biharmonic current
We study transport properties of two Josephson junctions coupled by an external shunt resistance. One of the junction (say, the first) is driven by an unbiased ac current consisting of two harmonics. The device can rectify the ac current yielding a dc voltage across the first junction. For some values of coupling strength, controlled by an external shunt resistance, a dc voltage across the second junction can be generated. By variation of system parameters like the relative phase or frequency of two harmonics, one can conveniently manipulate both voltages with high efficiency, e.g., changing the dc voltages across the first and second junctions from positive to negative values and vice versa.
pacs:05.60.-k, 74.50.+r, 85.25.Cp 05.40.-a,
Transport phenomena in periodic structures by harvesting the unbiased external time–periodic stimuli and thermal equilibrium fluctuations have been one of the hottest topic in nowadays science. Examples range from biology and biophysics  to explain directed motion of biological motors  and particle transport in ion channels , or to design new separation techniques , to meso– and nano–physics covering newest and up-to-date experiments with optical lattices , persistent currents in quantum rings  and Josephson junctions , to mention only a few. The physics of the latter has been studied for almost five decades now . Devices with Josephson junctions constitute a paradigm of nonlinear systems exhibiting many interesting phenomena in classical and quantum regimes. Yet new and interesting phenomena arise as researchers are able to use an advantage of powerful computer simulations followed by real experiments on junctions proving theoretically predicted findings [9, 10, 11]. In this paper we study a system of two coupled Josephson junctions driven by an external ac current having two harmonics. Such driving has been considered as a source of energy pumped into the system and as an agent transferring the system to non-equilibrium states. Transport phenomena in non-equilibrium states are of prominent interest from both theoretical and experimental point of view. For example, transport processes driven by biharmonic driving have been intensively studied in variety of systems and in various regimes. We can quote: Hamiltonian systems , systems in the overdamped regime [12, 14] and for moderate damping [15, 16], dissipative quantum systems , electronic quantum pumps  and quantum control of exciton states , cold atoms in the optical lattices [5, 20, 21], soliton physics  and driven Josephson junctions . Here we intend to consider another aspect of transport in the system which consists of two coupled elements (subsystems). In particular, we take into consideration two coupled Josephson junctions. In a more general context it is a system which can be described by two degrees of freedom (it can consist of two interacting particles or one particle moving on a two-dimensional substrate) . The question is: What new properties of transport can be induced by interaction between two elements and under what conditions can transport be generated in the second element if the driving is applied only to the first element.
The layout of the paper is as follows: In section 2, we detail the system of two Josephson junction coupled by an external shunt resistance. Next, in section 3, we specify external driving applied to one of the junctions. In section 4, we analyze generic properties of dc voltages across the first and second junctions. In section 5, we elaborate on the voltage control. Section 6 provides summary and conclusions.
2 Model and its dynamics
Let us consider a system of two coupled resistively shunted Josephson junctions. The equivalent circuit representation of the system is illustrated in figure 2. The junctions are characterized by the critical currents , resistances and phases (more precisely, the phase differences of the Cooper pair wave functions across the junctions). The junctions are externally shunted by the resistance and driven by external currents and . We also include into our considerations Johnson thermal noise sources and associated with the corresponding resistances and .
From the Kirchhoff current and voltage laws, and two Josephson relations one can obtain the following equations
where the dot denotes a derivative of with respect to time and . We assume that all resistors are at the same temperature and the thermal equilibrium noise sources are represented by zero-mean Gaussian white noises which are delta-correlated, i.e. for . Note a symmetry property of equations (2) with respect to the change .
The limitations of the model (2) and its range of validity are discussed e.g. in Ref. . In particular, we work within the semi-classical regime of small junctions for which a spatial dependence of characteristics can be neglected, a displacement current accompanied with the junction capacitance is sufficiently small and can be ignored, and photon-assisted tunneling phenomena do not contribute.
The dimensionless variables and parameters can be introduced in various way. Here, we follow Ref.  and define the dimensionless time as:
are the the characteristic voltage and averaged critical current, respectively. The corresponding dimensionless form of equations (2) reads
where now the dot denotes a derivative with respect to dimensionless time and all dimensionless currents are in units of . E.g., . The parameter
is the coupling constant between two junctions and can be controlled by the external resistance . The parameter . The noises and are zero-mean –correlated Gaussian white noises, i.e. for and are linear combinations of noises . The dimensionless noise strength is , where is the Boltzmann constant.
3 System driven by biharmonic current
The dynamic behavior of two Josephson junctions described by equations (2) is analogous to the overdamped dynamics of two interacting Brownian particles moving in spatially symmetric periodic structures and driven by external time-dependent forces. However, it is not a potential system. If we think of and as position coordinates of two particles, then and correspond to the velocity of the first and second particles, respectively. In this mechanical analogue, junction voltage is represented by particle velocity and time-dependent current is represented by external force. We can use this analogue to visualize junction dynamics. The main transport characteristics of such mechanical systems are: the long-time averaged velocity of the first particle and the long-time averaged velocity of the second particle. The question is under what circumstances one can induce transport characterized by directed motion of both particles in the stationary regime, i.e. when and . In terms of the Josephson junction system (2) it corresponds to the dimensionless long-time averaged dc voltages and across the first and second junction, respectively.
Transport in symmetric periodic systems like (2) can be trivially induced by biased external drivings or/and . A non-trivial case is when zero-mean (non-biased) driving can induce transport. An example of zero-mean driving is the simplest harmonic signal . From the symmetry property of equations (2) it follows that such a current source is unable to induce non-zero averaged voltages, for details see [14, 16, 21, 27, 28, 29]. However, when the system is driven by external currents having two harmonics, i.e. when 
then transport can be induced within tailored parameter regimes.
In the paper, we will study a particular case when the biharmonic ac current (6) is applied only to one (say, the first) junction, and has the form
It is known that without coupling to the second junction (i.e. when ), the averaged dc voltage across
the second junction is zero while the dc voltage across the first junction is non-zero in
some regions of the parameters space. We can formulate several fundamental questions: (a) What
is the influence of coupling on the voltage ? (b) Can the voltage across
the second junction be induced by coupling (5) and how it depends on its strength?
(c) Can the voltages reversal be obtained by changing the control parameters? (d) Can the
voltages and have opposite signs?
Properties of the ac current determine whether
the voltages and display the above mentioned features.
One can distinguish two special cases of the two-harmonics ac current :
(i) The first case is when there is such that . It means that the driving is symmetric or invariant under the time-inversion transformation. It is the case when , see dashed and dotted-dashed lines in figure 2.
(ii) The second case is when there is such that . This is the case of the antisymmetric driving realized for the phases , see solid and dotted lines in figure 2.
The analysis of a similar system biased by a dc current and driven by an unbiased harmonic ac signal has been recently presented in Ref. . The main difference between these two set-ups is a constant dc current which in turn may lead to the phenomenon of the negative mobility (resistance) . Here, there is no such constant current applied to the system. However, by applying asymmetric ac signals we hope to be able to control the behavior of the first and second junctions. We would like to convince reader that transport across the junction can be qualitatively controlled just by adjusting the shape of the external signal applied to one junction only.
4 Voltage response to biharmonic current
To reduce a number of parameters of the model, we consider a special case of two identical junctions. In such a case and equations (2) take the form
where the driving current is given by the relation (7).
Now, we make four general conclusions about the dc voltages as a consequence of the symmetry properites of the ac driving current :
(A) – Let us consider the voltages and as functions of the amplitude of the second harmonic. If we make the transformation to equations (4), then it follows that
These relations yield and we conclude that
and when the second harmonic is zero, i.e. when .
(B) – The sign of voltages and can be controlled by the phase . Indeed, if one changes the phase then and from the relations (9) one gets
(C) – Because the driving is the same for and for , the symmetry relations
The above set of equations (4) cannot be handled by standard analytical methods used in ordinary differential equations. Therefore we have carried out comprehensive numerical simulations. We have employed Stochastic Runge–Kutta algorithm of the order with the time step of . We have chosen initial phases and equally distributed over one period . Averaging was performed over different realizations and over one period of the external driving . All numerical calculations have been performed using CUDA environment on desktop computing processor NVIDIA GeForce GTX 285. This gave us a possibility to speed the numerical calculations up to few hundreds times more than on typical modern CPUs .
We begin the analysis of transport properties of the system (4) by some general comments about its long-time behavior. In the long time limit, the averaged voltages can be presented in the form of a series of all possible harmonics, namely,
where is a dc (time-independent) component and are time-periodic functions of zero average over a basic period.
When both and are small, the dc components and of averaged voltages are zero in the long time limit. It can be inferred from the structure of equations (4): at least one of the amplitudes of the driving current should be greater than the amplitude of the Josephson suppercurrent. Also for high frequency , the averaged dc voltages are zero: very fast positive and negative changes of the driving current cannot induce the dc voltage and only multi-harmonic components of the voltages can survive. For smaller values of the frequency and higher values of amplitude, one can observe a stripe-like structure of non-zero values of both voltages and .
We can identify four remarkable and distinct parameter regimes where:
(I) and ,
(II) and ,
(III) and ,
(IV) and .
The regimes (I) and (II) dominate in the parameter space. If the regime (I) is detected then the regime (II) can be obtained from the relations (10) by changing the relative phase of the biharmonic current . Likewise, if the regime (III) is found then the regime (IV) can be determined from the relations (10).
4.1 Dominated regime: the same sign of dc voltages and
In figure 3 we present the regime (I) in the parameter plane which illustrate how voltages and depend on the amplitude and frequency of the biharmonic current when other parameters are fixed. In this regime, both voltages always take non-negative values.
The dependence of voltages and on the current amplitude is depicted in panels (a) and (b) of figure 4. We observe that for smaller amplitudes of external ac current the transport is activated only on the first junction while the voltage across the second junction is zero. We can identify non-zero response of the first junction at the amplitude around . The second junction awakes for larger values of amplitudes (the threshold is a little bit less than ). One can note non–monotonic behavior of all presented curves. After initial burst the amplitude of the voltage of the first junction decreases, revealing next enhancement together with the appearance of the non–zero voltage excited on the second junction (green dashed lines). For larger amplitude of the ac driving, voltages of both junctions decrease. In panels (a) and (b) of this figure, the phase of the second harmonic of the external signal is different: in panel (a) and in panel (b). The inspection of the results reveals that the change of the phase from to reduces the dc voltages more than twice.
In panel (c) of figure 4, the role of coupling is illustrated: for small values of the coupling constant the voltage is zero. There is a threshold value of coupling above which the voltage across the second junction can be generated. In the regime presented in figure 4, the threshold value . From equation (5) it follows that the external shunt resistance can be a good control parameter which decides whether takes non-zero or zero values. For fixed remaining system parameters, one can induce non-zero voltage across the second junction by decreasing the resistance to values . For the voltage diminishes.
In panel (d) of figure 4, the influence of the relative phase of two harmonic signals
on the dc voltages is depicted. It is a numerical evidence for our consideration on symmetry relations (9)-(12). We can deduce two conclusions:
(i) The maximal absolute values of voltages and are generated by the symmetric ac current, i.e. when or ;
(ii) Both dc voltages and are zero when the ac current is antisymmetric, i.e. when or .
From Refs [14, 28] it follows that for small amplitudes of both harmonics of the driving current (7) the dc voltages depend on the phase as . Such a behavior is universal and does not depend on details of systems or models. E.g. it has been observed in transport of flux quanta in superconducting films, see panel (b) in figure 1 of Ref. .
As follows from equations (10), the regime (II) can be obtained from the regime (I) by change of the phase . Then the corresponding figures can be obtained from figures 3 and 4 by the reflection with respect to the horizontal axis. Therefore we do not discuss this regime and we do not show corresponding figures.
4.2 Regime of the opposite sign of dc voltages and
Now, we address the question of whether we can identify parameter regimes (III) and (IV) where the dc voltages and take opposite sign. An illustrative example of such a regime is depicted in figure 5. We can identify a fine stripe-like structure of regions of non-zero dc voltages sparsely distributed in the parameter space. Outside the stripes, in large regions of parameter space, the voltages and are negligible small or zero.
The inspection of figures 5 suggests to perform a more accurate search for the horizontal section and the vertical section . In panels (a) and (b) of figure 6 we present results for the section . One can note the unique feature of the occurrence of windows of the frequency where the voltage takes the opposite sign to the voltage . The voltages as a function of display the non-monotonic dependence exhibiting maxima and minima. We can detect the interval where absolute values of voltages operate synchronously: simultaneous increase and decrease both of them. For higher temperature and within the presented range of the base frequency the voltages across both junctions alternate showing very interesting and rare curves of opposite voltage reversals . However, if we consider ten times lower temperature, i.e. , then only one window of the states working contrariwise survives. It means that other windows are created by thermal equilibrium fluctuations.
Two panels (c) and (d) in figure 6 illustrate the section and the section to show the influence of the amplitude of the ac driving. The voltages as a function of are non-monotonic and local maxima and minima can be detected.
The influence of the relative amplitude of the second harmonic is depicted in figure 7 for two values of the driving frequency (a) and (b). In the vicinity of we can notice previous findings where the voltages have the opposite sign for both junctions. However, if we increase then both voltages diminish and next take negative values following by positive signs for large , see panel (a) for details. For faster current the situation is opposite - for larger voltages become positive, next negative and end up with locked state for relatively large . It is worth to mention that relates to the situation where the second harmonic is zero resulting in a pure symmetric driving with no possible net transport. On the other hand, if then both harmonics of the signal (7) are equally strong and we are not able to say which part of it influences the dynamics of junctions more effectively.
In panel (c) we show the influence of interaction between two junctions on the voltage characteristics. The first observation is that there are optimal values of the coupling strength for which absolute values of voltages are locally maximal. The second observation is that variation of the coupling strength can lead to the phenomenon of the voltage reversal : the voltage changes its values from positive (negative) values to negative (positive) ones. The third observation is the same as presented in panel (c) of figure 4: there is a threshold value of coupling (or the resistance ) such that for (or for ) the voltage is reduced to zero.
In panel (d) we depict the influence of temperature. Most remarkably, transport is solely induced by thermal noise. Indeed, in the deterministic case (i.e. for ), when no thermal fluctuations act, the dc voltages vanish. With increasing temperature, the voltage starts to increase to positive-valued local maximum. Next, it decreases crossing zero and reaches a negative-valued local minimum. For further increase of temperature, the voltage tends to zero. In turn, the voltage takes negative values reaching minimal value as temperature start to increase. Next, it increases crossing zero and reaches local maximum. Finally, it monotonically decreases toward zero. We observe that optimal temperature occurs at which the voltage is maximal. There is also another optimal temperature at which in turn the absolute value of is maximal. Moreover, one can identify the voltage reversal phenomenon: the both voltages change their sign as temperature is varied.
From the presented analysis it follows that a biharmonic drive allows one to conveniently manipulate transport with high efficiency by changing the system parameters.
This paper presents the detailed study of the system of two coupled noisy Josephson junctions which undergoes the influence of the external ac biharmonic current applied to one of the junctions only. The dependence of voltages across the first and second junctions upon ac driving exhibits a rich diversity of transport characteristics. In particular, it is possible to control both junctions to operate simultaneously with positive or negative voltages of the same sign or, more interestingly, with the opposite sign. Moreover, the phenomenon of the voltage reversals is revealed: the voltages change the sign as one of the parameters is varied. Our findings can be experimentally verified in an accessible set-up of two Josephson junctions coupled by an external resistance. The work can open the perspective of a new type of electronic elements which is tunable between positive and negative dc voltages via external control parameters like the amplitudes and relative phases of an ac current. The straightforward extension of the ideas presented here would cover an increasing number of coupled junctions to three (or more) as well as study of other types of driving.
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