# Semiclassical theory of a quantum pump

###### Abstract

In a quantum charge pump, the periodic variation of two parameters that affect the phase of the electronic wavefunction causes the flow of a direct current. The operating mechanism of a quantum pump is based on quantum interference, the phases of interfering amplitudes being modulated by the external parameters. In a ballistic quantum dot, there is a minimum time before which quantum interference can not occur: the Ehrenfest time . Here we calculate the current pumped through a ballistic quantum dot if is comparable to the mean dwell time . Remarkably, we find that the pumped current has a component that is not suppressed if .

###### pacs:

73.23.-b,05.45.Mt,73.50.Pz## I Introduction

A suitably chosen periodic perturbation of an electronic device may result in a direct current, even when the system is not biased. In the adiabatic limit, when the applied perturbation is slow in comparison to the escape rate to external contacts, the electronic state is the same after each period while a finite charge is transferred through the device during each cycle. Such devices are called charge pumps. Charge pumps have been proposed as current sources and minimal-noise current standards,Pothier et al. (1992); Martinis et al. (1994); Avron et al. (2001) as well as diagnostic tools to monitor how mesoscopic devices respond to changing external parameters.Thouless (1983); Fal’ko and Khmelnitskii (1989); Bykov et al. (1989); Niu (1990); Spivak et al. (1995); Switkes et al. (1999); Ebbecke et al. (2004)

In this article, we consider a so-called ‘quantum charge pump’. In a quantum pump, the external perturbation affects the phases of the electron wavefunctions only, not the classical dynamics of the electrons.Switkes et al. (1999); Thouless (1983); Fal’ko and Khmelnitskii (1989); Spivak et al. (1995); Brouwer (1998); Altshuler and Glazman (1999); Zhou et al. (1999) Such a scenario can be realized in a ballistic quantum dot, where small variations of shape-defining gate voltages or of an applied magnetic field are known to significantly affect the electronic phase while leaving classical trajectories unaffected. Such a device, with two shape-distorting gate voltages to drive the current, was built by Switkes et al.Switkes et al. (1999) Although the pumped current in the original experiment of Ref. Switkes et al., 1999 was obscured by rectification effects,Brouwer (2001) ‘quantum pumping’ was demonstrated in a later experiment, be it not in the adiabatic regime.DiCarlo et al. (2003)

Microscopically, the current in a quantum pump is the result of quantum interference: two classical trajectories initially a microscopic distance apart (a Fermi wavelength) split and rejoin, with a phase difference that depends on the applied perturbation. Indeed, if the mean dwell time in the device exceeds the dephasing time , the pumped current is suppressed.Cremers and Brouwer (2002); Moskalets and Bütiker (2001) Aleiner and Larkin have pointed out that, in a ballistic quantum dot (as in any quantum system with a well-defined classical counterpart for which the dynamics is chaotic), there is another time that governs quantum interference:Aleiner and Larkin (1996, 1997) the Ehrenfest time . The Ehrenfest time is the time during which two classical trajectories initially a Fermi wavelength apart are separated to a macroscopic distance (the width of the contacts or the size of the quantum dot) by the chaotic classical dynamics.Zaslavsky (1981); Larkin and Ovchinnikov (1968) The Ehrenfest time is given by

(1) |

where is the Lyapunov exponent of the classical dynamics in the quantum dot and denotes the total number of channels in the contacts. The constant in Eq. (1) is independent of . While the dephasing time governs the suppression of interference for long dwell times, the Ehrenfest time is the minimum time needed for the appearance of interference effects.

Being an interference effect, the current pumped through a ballistic chaotic quantum dot is a random function of any parameter that affects quantum phases, such as the shape of the dot or the applied magnetic field (taken at a reference point during the cycle). Therefore, one typically considers averages and fluctuations of the current, taken with respect to an ensemble of dots with slightly varying shape or with respect to a range of magnetic field strengths. Since the ensemble-averaged current , the magnitude of the pumped current is measured through the second moment . With a few exceptions,Zhou et al. (1999); Martinez-Mares et al. (2004) random matrix theory has been the preferred framework for a statistical theory of quantum-dot based charge pumps.Brouwer (1998); Shutenko et al. (2000); Vavilov et al. (2001); Polianski et al. (2002); Polianski and Brouwer (2003); Vavilov et al. (2005) Random matrix theory describes the regime , in which quantum effects occur on a time scale much shorter than the mean dwell time in the quantum dot. In this article, we are interested in a quantum pump in the regime in which and are comparable. Since, typically, , this requires that we consider the classical limit .

A remarkably wide range of -dependences has been reported in the literature for various quantum interference effects in ballistic quantum dots. Weak localization, the quantum interference correction to the ensemble-averaged conductance in the presence of time-reversal symmetry, is suppressed if is large.Aleiner and Larkin (1996); Adagideli (2003); Rahav and Brouwer (2005); Jacquod and Whitney (2006) On the other hand, conductance fluctuations have no dependence,Tworzydlo et al. (2004a); Jacquod and Sukhorukov (2004); Brouwer and Rahav (2005) whereas the quantum correction to the spectral form factor of a closed quantum dot with broken time-reversal symmetry exists for finite only,Tian and Larkin (2004); Brouwer et al. (2006) being absent in the random matrix limit .Berry (1985) The main result of the present article is that current pumped in a quantum pump has yet another -dependence,

(2) |

where is the prediction of random matrix theory.

We calculate the Ehrenfest-time dependence of the pumped current using a semiclassical theory in which the pumped current is expressed as a sum over classical trajectories in the quantum dot. Our theory goes beyond a previous semiclassical theory of quantum pumps by Martinez-Mares et al.,Martinez-Mares et al. (2004) who did not consider the role of the Ehrenfest time. Whereas Ref. Martinez-Mares et al., 2004 calculated the pumped current in the ‘diagonal approximation’, in which only pairs of identical classical trajectories are considered, we also include the leading off-diagonal terms in the sum over classical trajectories. In this respect, our theory builds on previous work by Richter and SieberRichter and Sieber (2002) and Heusler et al.,Heusler et al. (2006) who developed a systematic way to include off-diagonal terms in trajectory sums of the weak localization correction to the dot’s conductance and other quantum interference corrections to transport. Unlike the diagonal approximation, which is known to violate current conservation, the present version of the semiclassical theory is fully current conserving.

In Sec. II we express the pumped current in terms of a sum over classical trajectories. This section closely follows Ref. Martinez-Mares et al., 2004. The sum over classical trajectories, which gives the final result (2), is then performed in Sec. III. We compare the theoretical predictions to numerical simulations in Sec. IV and conclude in Sec. V.

## Ii Semiclassical theory of a quantum pump

The geometry under consideration is shown in Fig. 1. For definiteness, we consider a ballistic chaotic quantum dot coupled to two electron reservoirs via ballistic point contacts. The quantum dot has two holes. Slow periodic variations of the magnetic flux through each of the holes change the phases of wavefunctions, not the classical trajectories. Both holes are ‘macroscopic’: their size is large in comparison to the Fermi wavelength and their boundary is smooth.

We assume that the variation of the fluxes is adiabatic, with period much larger than either the dwell time or the Ehrenfest time. In the adiabatic regime, the time-averaged current through the left contact can be expressed in terms of the scattering matrix of the quantum dot,Brouwer (1998)

(3) |

Here is the flux through hole , , measured in units of the flux quantum , and the integral in Eq. (3) is taken over the area enclosed in -space in one pumping cycle, see Fig. 1. The number of channels in the left and right contacts are and , respectively, and . In Eq. (3) we assumed zero temperature. The current through the right-contact reads

(4) |

Current conservation implies

(5) |

so that

(6) |

The key point of the semiclassical approach is to express the scattering matrix as a sum over classical trajectories ,Jalabert et al. (1990); Baranger et al. (1993a, b)

(7) |

where starts at a contact with transverse momentum compatible with lead mode and ends with transverse momentum compatible with mode . (The modes are in the left lead and the modes are in the right lead.) Further is the classical action (which includes Maslov phases) and is the stability amplitude. The classical action is modified by the presence of the fluxes and as

(8) |

where is the winding number of the trajectory around the hole , . We’ll be interested in the regime , and , for which the discreteness of the winding numbers does not play a role. Substituting Eqs. (7) and (8) into Eqs. (3) and (4), we arrive atMartinez-Mares et al. (2004)

(9) | |||||

where both and are compatible with modes and upon entrance and exit, respectively.

In the remainder of this article we limit ourselves to bilinear response, . For bilinear response, the fluctuations of the kernels and can be neglected when performing the integral over the enclosed area in -space, so that it is sufficient to calculate the average and variance of the kernels and in order to find the average and variance of the pumped current ,

(10) | |||||

(11) |

In the semiclassical framework, this means that we can neglect the dependence of the classical actions on the variations of the parameters and in the sines in Eq. (9).

Before we can perform the summation over classical trajectories necessary to calculate we must specify the statistics of the winding numbers and . For a quantum dot with chaotic classical dynamics, one may assume that the two winding numbers and are statistically uncorrelated, and that the average winding number is zero,

(12) |

For the variance of the winding number we takeBerry and Robnik (1986)

(13) |

where the coefficients depend on the size and shape of the quantum dot and on the location of the hole , , but not on the Ehrenfest time and is the duration of the trajectory . For a chaotic quantum dot, one may also assume that the winding numbers and of different trajectories and are uncorrelated, , except if there exist strong classical correlations between the two trajectories. Strong classical correlations exist if the two trajectories are within a phase space distance , where is a cut-off below which the chaotic classical dynamics in the quantum dot can be linearized. If two trajectories are correlated for only a part of their duration, as shown schematically in Fig. 2, only the duration of the correlated segment contributes to the average of the product of the winding numbers,

(14) |

Although our theory is formulated for a quantum pump where the current is driven by time-dependent magnetic fluxes, the mathematical framework to describe other geometries, e.g. a pump in which current is driven by shape changes, is identical.Martinez-Mares et al. (2004) For generic perturbations of a chaotic quantum dot the parameters still enter the classical action as a Gaussian random process, and the actions of different trajectories will have statistically uncorrelated dependences on the parameters. In that sense, our theory applies to arbitrary perturbations, not only time-dependent magnetic fluxes. (The only exception is a pump in which a gate voltage is changed uniformly in the quantum dot. A similar separation between ‘generic’ perturbations and a uniform gate voltage shift exists in the random matrix description of quantum pumps.Brouwer (1998); Vavilov et al. (2001); Polianski and Brouwer (2003))

## Iii Summation over classical trajectories

The semiclassical framework and the statistics of the winding numbers outlined in the previous section is the same as that used in a previous semiclassical theory of quantum pumps by Martinez-Mares, Lewenkopf, and Mucciolo.Martinez-Mares et al. (2004) In Ref. Martinez-Mares et al., 2004, the fourfold sum over classical trajectories required to calculate the current variance is evaluated in the ‘diagonal approximation’, in which the four trajectories contributing are taken pairwise equal. Although Ref. Martinez-Mares et al., 2004 reports agreement between semiclassics and random matrix theory, the use of the diagonal approximation to calculate the current variance is problematic because it violates the unitarity relation (6): In the diagonal approximation, one has , while . The diagonal approximation also fails to account for the fact that the variance of the pumped current does not depend on the presence or absence of time-reversal symmetry.Brouwer (1998); Shutenko et al. (2000)

Problems with the diagonal approximation are not limited to the semiclassical theory of a quantum pump. In fact, the diagonal approximation is well known to fail at providing a current-conserving description of other quantum interference effects, such as the weak localization correction to the conductance and universal conductance fluctuations.Stone (1995) For weak localization, this problem was solved by Richter and Sieber, who were able to include the relevant off-diagonal configurations of classical trajectories into the trajectory sum.Richter and Sieber (2002) Below, we use a generalization of Richter and Sieber’s technical innovationHeusler et al. (2006); Braun et al. (2005); Brouwer and Rahav (2005, 2006) to perform the trajectory sums required for a current-conserving semiclassical theory of a quantum pump.

Since the two winding numbers and are statistically uncorrelated and have zero average, the trajectory sum in Eq. (9) immediately gives , and hence

(15) |

In order to calculate the average of the squared current, it is technically most convenient to use the second equality in Eq. (11). Using the semiclassical expression (9), one has

(16) | |||||

The non-vanishing contributions to the current variance are from quadruples of trajectories for which the two action differences in the sine functions are almost identical, resulting in a non-oscillatory contribution to the trajectory sum. However, this alone does not ensure a finite contribution to : In addition, the ensemble averages of the winding numbers should not vanish. This means that the trajectory pairs and , and and should be correlated for (at least) part of their duration. However, since these trajectories exit the dot in different contacts, they can not be identical.

The configurations of classical trajectories that meet these requirements are the same as the trajectories that contribute to conductance fluctuations.Brouwer and Rahav (2005) They are shown in Fig. 3. The first type (labeled “A”) consists of two pairs of trajectories that undergo two separate small-angle encounters. Between the encounters and the contacts and , and and are pairwise equal (up to quantum uncertainties). Between the two encounters, is paired with or , while is paired with the remaining trajectory, or . The second type of trajectories (labeled “B”) contains a closed loop, which is traversed one (extra) time by two of the four trajectories. Below we discuss both contributions separately. Since the calculations closely follow the calculations of the conductance fluctuations, we refer to Ref. Brouwer and Rahav, 2005 for details of the formalism, and here restrict ourselves to those parts of the calculation where we differ from Ref. Brouwer and Rahav, 2005.

### iii.1 contributions of type A

Within the configurations of type A, one distinguishes trajectories for which the first small-angle encounter fully resides inside the quantum dot, such that the entrance of the pairs trajectories and , and and is uncorrelated, and a configuration of trajectories in which the first small-angle encounter touches the entrance contact, so that all four trajectories enter the dot together. The two situations are shown in Fig. 4. Since and exit through the left contact, whereas and exit through the right contact, the second encounter must fully reside inside the quantum dot in all cases. We refer to the two cases shown in Fig. 4 as “A1” and “A2”.

The durations of the encounters, defined as the time during which the phase space distance between the trajectories is less than a classical cut-off scale , are denoted and , as depicted in Fig. 4. The durations of the uncorrelated stretches between the encounters are denoted by and . We then expand

(17) |

and find quadruples of trajectories , , , and for which the exponents in Eq. (17) are small. Between the encounter regions and the contacts, and , and and are paired, resulting in only a small contribution to the action differences in Eq. (17). In order to ensure that the contribution from the stretch between the encounters is small as well, one pairs with and with between the encounters for the first term in Eq. (17), and with and with for the second term. In the former case and . In the latter case, one has and . At each of the encounters we take a Poincaré surface of section. The action differences or that appear in the exponents in Eq. (17) are then expressed in terms of the phase space distances and between the two solid trajectories in Fig. 3 along the stable and unstable directions in phase space at each of the encounters, .Spehner (2003); Turek and Richter (2003) The encounter durations and are expressed in terms of these coordinates as

(18) |

Calculating the trajectory sum as in Refs. Heusler et al., 2006; Braun et al., 2005; Brouwer and Rahav, 2005, 2006, we then find

(19) | |||||

The factors and in the denominator in Eq. (19) cancel a spurious contribution from the freedom to take the Poincaré surface of section at an arbitrary point in the encounter region.

We then make the variable change , . In terms of the new variables, one has . Performing the integrals over , , , and , one then finds

(20) |

where we abbreviated .

We now turn to compute contributions of type A2. Because the first encounter touches the leads, the encounter duration is no longer determined by the phase space coordinates and , but becomes an integration variable itself. The integration range for is ,Brouwer and Rahav (2006) so that

(21) | |||||

The second encounter time is still given by Eq. (18) above. As before, we perform the variable change , , . Upon integration over , , , , and we then find

(22) | |||||

Here we reverted to the notation in order to make contact with the result for obtained previously. The second term between the square brackets in the second line of Eq. (22) leads to a rapidly oscillating function of and will be neglected. Combining the remaining terms with Eq. (20) and taking the limit () while keeping the ratio fixed, we find

(23) | |||||

where the Ehrenfest time is defined as

(24) |

Note that the limit at fixed , taken above, can be realized by narrowing the lead openings such that the dwell time increases. While this procedure may change the trajectories contributing to pumping, their statistical properties should be (almost) unaffected. In particular, the Lyapunov exponent will converge to its value in the closed cavity.

### iii.2 contributions of type B

The trajectory configurations of type B provide the dominant contributions to the pumped current at large Ehrenfest times. Each pair and consists of a ‘short’ trajectory and a ‘long’ trajectory, where the long trajectory differs from the short trajectory by winding once around a periodic orbit, see Fig. 3. (Strictly speaking, one should say that the long trajectory winds one extra time around the periodic orbit, because all trajectories involved will wind multiple times around the periodic orbit if is much longer than the period of the periodic orbit.) In the trajectory sum (9), each trajectory can be the short one. Following Ref. Brouwer and Rahav, 2005, we parametrize these configurations by Poincaré surfaces of sections which measure the phase space coordinates and ) of two short trajectories with respect to the stable and unstable manifolds of the periodic orbit.

It will be useful to define several relevant times. The two short trajectories in the configuration are depicted in Fig. 5. The time for which the short trajectories are correlated with the periodic trajectory (phase space distances smaller than the cutoff ) is denoted [for the short trajectory of the pair ] and [for the short trajectory of the pair ]. By our construction, the longer trajectory in the pair (not drawn in Fig. 5) will be correlated with the periodic trajectory for a time , , where is the period of the periodic orbit. Since the trajectories may leave the neighborhood of the periodic trajectory together, the correlations may extend away from the periodic trajectory. If applicable, we denote the time for which the trajectories are correlated with each other before they arrive to the periodic orbit by . Similarly, denotes the eventual correlation time after the trajectories leave the periodic orbit. If , the encounter region extends away from the periodic orbit and may touch the entrance contact. Hence, we distinguish contributions of type B1 and type B2, where type B1 refers to those trajectories for which the encounters fully reside in the quantum dot, and type B2 refers to configurations of trajectories with correlated entry (encounter that touches the entrance lead opening). Since the trajectory pairs and exit through different contacts, one does not have to consider the possibility that the encounter region touches the exit lead opening. The Poincaré surfaces of section are taken at points where the short trajectories are correlated with the periodic orbit. The distance between the two Poincaré surfaces of section is measured by the travel time along the periodic orbit, where we require .

Even the shortest trajectory in a pair can wind around the periodic trajectory several times. Thus, it is possible that . In order to separate out full revolutions, we write

(25) |

where is a non-negative integer and . The use of complicates some of the calculations since this time is not a continuous function of the surface of section coordinates. On the other hand, this time appears naturally due to the random Gaussian character of the winding numbers .

Without loss of generality (but with inclusion of a combinatorial factor two), we take to be one of the short trajectories. Expanding the product of sines as in Eq. (17), we then see that smallness of the the action difference requires to be the other short trajectory for the first term in Eq. (17), whereas is the other short trajectory in the second term in Eq. (17). In the former case, one has

The time is the overlap of the ‘remainders’ and , see Fig. 6. In the latter case, one has

The resulting contribution of type B1 then reads

(26) | |||||

The contributions of type B2 are calculated in a way very similar to the one used for type A2,

(27) | |||||

Adding both contributions, we then obtain

(28) | |||||

In App. A it is shown that the contribution from the combination between the curly brackets is of order , which can be neglected with respect to . The contribution from the term is a rapidly oscillating function of and is omitted. The only nonzero contribution arises from the term , which is the only term that is multiplied by both and . This factor can be taken out of the integral over the phase space variables. The resulting integral was computed in Ref. Brouwer and Rahav, 2005, with the result

(29) | |||||

In Eq. (29) we omitted terms proportional to . Typical periodic trajectories contributing to Eq. (29) have duration . In the limiting procedure discussed below Eq. (24) these contributions scale as , justifying this approximation. The integral over the period of the periodic trajectory is easily calculated, resulting in

(30) |

Combining both contributions together, we find that the total variance of the pumped current is

(31) |

Equation (31) agrees with random matrix theory in the limit .Brouwer (1998); Shutenko et al. (2000); Vavilov et al. (2001) At finite , the variance of the pumped current is reduced below the random matrix value, but the reduction is not more than a factor two.

When the pumped current is dominated by trajectories of type A. For the pumped current is carried by trajectories of type B. Since such trajectory configurations involve a closed loop they are associated with fluctuations of the density of states. Although this scenario is very similar to that of the Ehrenfest-time dependence of the conductance fluctuations, the pumped current has a -dependent part, whereas the conductance fluctuations in a chaotic quantum dot are fully independent. The difference occurs, because the mean total duration of the trajectories involved in the internal loop for trajectories of type A is twice the mean duration of the internal loop for trajectories of type B. Conductance fluctuations are insensitive to the loop duration (as long as the trajectories stay inside the quantum dot), but the pumped current is not.

Equation (31) is valid for systems with and without time reversal symmetry. This is well known in terms of random matrix theory.Brouwer (1998); Shutenko et al. (2000) We have verified this explicitly in the semiclassical approach, see App. B. We also verified that calculation of the current variance using the correlator , while technically more involved, gives the same result for the pumped current, as required by unitarity.

## Iv Numerical simulation

Despite the remarkable similarity of the semiclassical calculation of the pumped current and the semiclassical calculation of the conductance fluctuations, the former shows a dependence on the Ehrenfest time, whereas the latter does not. In this section we report numerical simulations of the pumped current and compare these with our theoretical predictions.

Because the computational cost of numerical simulation of two-dimensional quantum dots with large Ehrenfest times is prohibitive, Jacqoud, Schomerus, and Beenakker suggested to simulate one dimensional chaotic maps instead.Jacquod et al. (2003) The phenomenology of chaotic maps is identical to that of chaotic cavities,Fishman et al. (1982); Izrailev (1990) but the computational cost of simulating a map is significantly lower than that of simulating a cavity. Chaotic maps have been successfully used to study the Ehrenfest time dependence of a range of properties of chaotic quantum dots.Jacquod et al. (2003); Tworzydlo et al. (2003); Jacquod and Sukhorukov (2004); Tworzydlo et al. (2004b, c); Rahav and Brouwer (2005, 2006); Schomerus and Jacquod (2005)

The quantum map propagates a finite state vector of size in time,

(32) |

where is the Floquet operator of the map. Assigning two consecutive sets of and elements of the vector to contacts, the map can be used to construct a -dimensional scattering matrix according to the ruleFyodorov and Sommers (2000); Jacquod et al. (2003)

(33) |

where is a matrix projecting on the lead sites, , and is the quasi energy.

The map used in our simulations is the ’three-kick quantum rotator’, Tworzydlo et al. (2004b) which is specified by the Floquet operator,

(34) |

where

(35) |

In this model is even, but not a multiple of . The kick potential is given by

(36) |

where breaks the parity symmetry of the model. Blümel and Smilansky (1992) The parameter plays the role of a magnetic field resulting in the breaking of time reversal symmetry. This model was used in Refs. Tworzydlo et al., 2004b and Rahav and Brouwer, 2006 to compute the weak localization correction of the conductance.

There are two parameters in this model: The chaoticity parameter and the ’magnetic field’ . We will use and as the time-dependent parameters that drive the quantum pump. Although both and appear in the classical map and thus affect classical trajectories, the variations used in our simulations are sufficiently small that the changes of the classical dynamics can be neglected within the mean dwell time .

In the simulations we took equal channel numbers in both contacts, . The semiclassical limit is taken by increasing the dimension of the system, while keeping the dwell time fixed. This means that is scaled proportional to . This way we are certain that there are no variations in the classical dynamics while decreasing . With these definitions the Ehrenfest time is estimated to be

(37) |

plus an -independent constant. The Lyapunov exponent is calculated independently from a simulation of the classical counterpart of the map.

The pumping strength depends on the two parameters and that describe the rate at which variations of and affect transport. (In the semiclassical theory, these parameters were called and .) In order to calculate and , we first performed separate numerical simulations of the quantities

(38) |

and compared these to the result of a semiclassical calculation,

(39) |

We then calculated the kernels and numerically, and considered the ratio

(40) |

The prediction of the semiclassical theory is

(41) |