Introduction

The QPC of conduction electrons has been an active research field for decades in the mesoscopic transport systems with the measuring of magnetic-flux-dependent current through an Aharonov-Bohm (AB) interferometer1. An AB interference experiment in the Coulomb blockade regime has been reported utilizing a bare AB ring with a quantum dot (QD) embedded within one arm of the ring2. The coherent properties provide new insight into the state of current transport in the dot. The two-particle interference has been investigated as a direct result of quantum exchange statistics in the absence of two-particle interaction3,4,5. The experimental realization of two-electron interference5 reproduces the original Hanbury Brown and Twiss experiments6,7 and has become a central study of multiple-particle wave functions. The QPC also has a great influence on the shot-noise properties8,9,10 from the viewpoint of transport electron-correlation. In fact, the quantum coherence of conduction-electron through different channels is caused physically by the phase accumulation of spatial motion from the electrode to QD.

The QPC with two-body Coulomb interaction (TBCI), however, has not yet been explored in a many-electron system. We in this report consider a double QD system weakly coupled to two metallic electrodes with two magnetic flux-lines of opposite directions as shown in Fig. 1. In the absence of the TBCI and inter-dot hop**, the single-particle as well as the two-particle AB phase interference3,4,5 does not exist at all since the total magnetic flux embraced by the electrode-dots-electrode loop is zero. We predict in this report a novel phenomenon of the QPC induced by the TBCI, which generates the wave-function overlap between two flux-lines and thus leads to the phase interference.

Figure 1
figure 1

Schematic diagram of a double quantum-dot device with two antiparallel fluxes and tunable inter-dot Coulomb interaction.

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The new observation is demonstrated by the full counting statistics (FCS) for particle transport in mesoscopic systems. The FCS has attracted great attentions in recent years8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31 since the intrinsic properties of the mesoscopic system can be identified by the high-order cumulants of electron correlation besides the average transport-current itself32. It is believed that the high-order cumulants are useful to detect the zeros of generating function33 and the intrinsic multi-stability34. Moreover the fractional charge in a chiral Luttinger liquid35 is able to be extracted from the high-order cumulants and the Majorana bound-states in a nanowire as well. It is also well known that the high-order cumulants, e.g., the shot noise, the skewness, are more sensitive to the QPC effect than the average current in various types of QD systems17,31,36,37,38. Especially the fifteen-order21,22 and finite-frequency cumulants25 can be extracted from the real-time measurements in the electron transport through a semiconductor QD.

The main goal of this paper is to study the effect of TBCI on the QPC, which can be probed by the high-order cumulants of electron transport. A new type of QPC, which does not exist in the single- or two-electron transport in the absence of TBCI, is observed and well explained by the TBCI. We analyze the TBCI-induced QPC in the FCS based on the particle-number-resolved quantum master equation (QME), which provides a formulation to include the TBCI. We demonstrate how the QPC characterized by the flux-dependent oscillation of cumulants can be used to probe the TBCI.

We consider a double-QD weakly coupled to two metallic electrodes with two antiparallel magnetic flux-lines of Φ1 = ζΦ0 and Φ2 = −Φ1, which are assumed to be perpendicular to the plane of QD-electrode seen from Fig. 1. Here, Φ0 = ch/e is the quantum unit of magnetic flux and thus ζ is a dimensionless flux-number, which is a positive value when the magnetic field is pointed into the QD-electrode plan as that of the left-hand flux in Fig. 1. The reason for the arrangement of antiparallel magnetic fluxes is that the wave-function interference of single-particle or non-interacting two-particle can be avoided since the total flux embraced by the two-dot loop is zero. The TBCI induced QPC has not yet been reported to our knowledge. The Hamiltonian of the transport system is given by H = Hdot + Hleads + HT. Here the first term

represents the QD Hamiltonian, where (di) is the creation (annihilation) operator of an electron in the QD-i (i = 1, 2) and Œis the bare energy level of electron in the QD. U is the inter-dot Coulomb repulsion-strength between two electrons occupied respectively on two dots. In addition we assume the high intra-dot Coulomb interaction, so that the double-electron occupation in the same QD is prohibited. The electron transition between two dots, i.e. the inter-dot hop**, is neglected to avoid the AB interference around a single flux-line. In the considered case there exist total four occupation states of electrons on the two dots denoted respectively by with corresponding energy-eigenvalues . The term stands for the Hamiltonian of the two electrodes, with the energy dispersion εαk and (aαk) being the creation (annihilation) operators of wave vector k in the αth electrode. The last term

describes the tunneling coupling between the QD-i and the electrode-α. pαi characterizes the flux-induced phases for electron hop** from the QD-i to the electrode-α, which are seen to be , , , in the considered case, so that the total phase is zero from the electrode-L to R along one arm, for example, from the electrode-L through dot-1 to the electrode-R.

Results

We now explore the QPC in the viewpoint of the FCS of electronic transport with a particular focus on the TBCI induced effect. The symmetrical bias-voltage is assumed at the QD-electrode tunnel junctions with the energy unit meV in the typical experiment of transport39. In the numerical evaluations, the parameters are chosen as . Based on the QME the electron number distribution n(t) can be evaluated in principle. We then calculate the cumulants of n(t), which, we will see, are more sensitive to the transport mechanism. The first-order cumulant defined as the average of electron number distribution is related to average current obviously such that . The second-order cumulants is the mean square deviation and is related to the zero-frequency shot noise defined by . The third cumulant characterizes the skewness of the distribution. The over bar denotes the statistic average as usual that . In general the normalized second-order cumulant , which is called the Fano factor and the third cumulant are used to describe the shot noise and the skewness respectively. The electron-number resolved statistics is called the super-Possonian for and sub-Possonian otherwise. We are particularly interested in the QPC, which depends on the magnetic flux number ζ. To this end we in Fig. 2 display the flux-number dependence of the average current (upper panel), Fano factor Fa (middle panel) and skewness Sk (lower panel) at temperatures T = 0 K (a) and T = 8 K (β = 1.5) (b) with the symmetric dot-electrode coupling ΓL = ΓR. In the absence of TBCI (black solid line) i.e. U = 0, the current curve does not vary with ζ as it should be, since the total flux embraced by the electron path-loop is zero no matter whether the dots are single- or double-occupied. On the other hand, for ultrastrong inter-dot Coulomb interaction such that U = ∞ (blue dotted line), which blocks the two-electron occupation states in the finite bias voltage, the average current does not depend on the flux again as the U = 0 case (black solid line). The flux-independence of the cumulants is explicitly verified from the analytic solution in the method for both U = 0 and U = ∞. When the Coulomb interaction is finite with the interaction strength U = 10 in Fig. 2 (red dashed line), the current oscillates with the flux number ζ. This phase interference is a result of the TBCI, which as a matter of fact provides the overlap of two-particle wave function between two flux lines. Most importantly, we observe a characteristic oscillation period 1/2 for the TBCI induced QPC, which is also robust against the thermal fluctuation at low temperature seen from Fig. 2(a,b). The ζ-dependence is derived rigorously in terms of the QME (see the detailed analysis in the section of method). Here we may present a plausible interpretation for the strict period from the two-electron scattering picture. For the two-particle wave functions transported from the electrode L to the two dots the Coulomb interaction in the semiclassical limit becomes , where denotes the wave function of ith electron (i = 1, 2) transported from upper (u) and lower (l) paths to the dot-1, 2 respectively with x1,2 being the coordinates of dots. Notice that each dot can be occupied no more than one electron the Coulomb interaction term in the mean field approximation reduces to

Figure 2
figure 2

(upper panel), Fa (middle panel) and Sk (lower panel) versus flux-number ζ in the absence of inter-dot hop** with the symmetric dot-electrode coupling and , at temperatures T = 0 K (a) and T = 8 K (b).

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which varies with respect to ζ periodically and the period is precisely 1/2. For the electrons we may need the antisymmetric two-particle wave function, which does not affect the characteristic period-1/2. With the antisymmetric two-particle wave function

the two-electron Coulomb interaction becomes

which leads again to the characteristic period-1/2 for the ζ–dependence. The two-electron transport probability depends also on the two-electron Fermi function , which decreases with U. Thus the amplitude of current oscillation decreases with the increase of U and tends to zero when U → ∞, so that the ζ–dependence of current becomes a constant again (blue dotted line). The oscillations of Fano factor Fa and the skewness Sk are of the same period 1/2 and maximum values of Fa correspond to the minima of the current. This super-Possonian statistics of maximum values of Fa at zero temperature (a) is generated by the destructive interference of the TBCI induced QPC, which may be called the QPC blockade, namely, the effective bunching of the tunneling events. With the increase of temperature the Fano factor Fa and the skewness Sk decrease slightly seen from Fig. 2(b). The dot energy level, which is in Fig. 2, does not affect characteristic oscillation period of the ζ–dependence. The same plots, however, for are displayed in Fig. 3, in which it is seen that the dot levels can greatly change the pattern shape but not the period.

Figure 3
figure 3

(upper panel), Fa (middle panel) and Sk (lower panel) versus flux-number ζ in the absence of inter-dot hop** with the symmetric dot-electrode coupling at zero temperatures when , respectively for , (a), , (b).

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The characteristic oscillation period being 1/2 for finite Coulomb-interaction is also robust against the dynamic fluctuation, which may causes the asymmetry between electrode and dot couplings that . Figure 4 shows the ζ-dependence of (upper panel), Fa (middle) and Sk (lower) at finite temperature T = 8 K for asymmetric dot-electrode coupling (a), (b) and (c), respectively. Obviously, in these cases with the zero Coulomb-interaction (U = 0, black solid lines) and the ultrastrong interaction limit (U → ∞, blue dotted lines), , Fa and Sk are independent of the magnetic flux no matter whether the dot-electrode coupling is symmetric or not. However, in the presence of finite Coulomb interaction for U = 10 in Fig. 4, the evident oscillations are observed (red dashed lines). The oscillation period is 1/2 for the fixed bias voltage.

Figure 4
figure 4

(upper panel), Fa (middle panel) and Sk (lower panel) versus flux-number ζ in the absence of inter-dot hop** at and , for asymmetric dot-electrode coupling (a), (b), (c).

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We have demonstrated that in the absence of inter-dot hop** the QPC of the ζ-dependent oscillation is induced by the TBCI. To compare with the usual AB effect we now introduce the inter-dot hop** term . In this case (with the inter-dot hop**) the system thus becomes a AB interferometer of two series loops with each loop embracing a single flux-ζ. The energy eigenvalues and eigenstates of the coupled double dots are found as , , , , , , . Figure 5 displays the (upper panel), Fa (middle) and Sk (lower) as functions of flux number ζ with symmetric dot-electrode coupling at finite temperature T = 8 K for the inter-dot Coulomb interactions U = 0 (a), U = 10 (b) and U = ∞ (c). The oscillation of current curve with respect to the flux-number ζ is the typical AB phase interference with period 1, which is the characteristic quantity of AB effect. The amplitudes of oscillation increase with the inter-dot coupling (J = 2 black solid line; J = 1 red dashed line; J = 0.5 blue dotted line). The average current decreases with the increase of inter-dot Coulomb interaction seen from Fig. 5(a–c) for respectively due to the suppression of two-electron occupation probability. The Fano factor Fa is less than 1 for U = 0 (a) indicating sub-Possonian statistics and increases with U resulted from the Coulomb-interaction blockade (b) (c). For the finite interaction that U = 10 (b) the Coulomb interaction slightly changes the shape of interference pattern. The Fano factor Fa and skewness Sk are more sensitive to the phase interference. The Fa curves possess sharp peaks at half-integer values of ζ because of the QPC blockade while the Sk plots have corresponding sharp dips. The Coulomb interaction basically does not affect the oscillation period of average current with respect to ζ. However, additional peaks (dips) of Fa (Sk) arise at the integer number of flux for the case of U = ∞ (c). This can be interpreted by the Coulomb blockade at the high current value (upper panel). The Coulomb-blockade induced peak-height of Fa increases with U and reaches the highest value at the limit case U = ∞. It should be noticed also an interesting phenomenon that the number statistics of transport electrons is super-Possonian at the peak position while sub-Possonian at dips for nonvanishing Coulomb interaction (b) (c). It may be worthwhile to remark that the ζ-dependence of transport current with the inter-dot hop** for U = 0 [Fig. 4(a)] is, as a matter of fact, the same as that in the non-interacting two-particle interference resulted from the quantum exchange statistics3,4,5 and becomes the usual single-particle AB interference at U = ∞.

Figure 5
figure 5

(upper panel), Fa (middle panel) and Sk (lower panel) versus flux-number ζ in the presence of inter-dot hop** for the Coulomb interaction strength U = 0 (a), U = 10 (b) and U = ∞ (c) with , and , .

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Conclusion and Discussion

In conclusion, the TBCI induced QPC has been explored for the first time according to the FCS of transport electrons. The observed flux-dependence of cumulants is different from the single- or non-interacting (i.e. U = 0) two-particle AB interference, since the total flux embraced by the two arms of electrode-dot-electrode paths is zero. Both the TBCI strength U and two-electron occupation probability are the necessary conditions to generate the magnetic-flux dependent QPC of the current , Fano factor Fa and skewness Sk. In the strong interaction limit U → ∞ the phase interference does not appear as in the case of U = 0, since the two-electron occupation probability tends to zero reducing to the single-particle case. For the finite U, the period-1/2 oscillations are found in the absence of inter-dot hop** and can be interpreted by the TBCI, which actually provides the overlap probability of two-particle wave functions between two flux-lines leading to the nonvanishing phase coherence. This QPC is robust against dynamic asymmetry of coupling and also thermal fluctuation at low temperature. When the inter-dot hop** is introduced as a comparison the system becomes a AB interferometer with two series loops, for which the period of current oscillation is the typical value 1 of the AB interference and does not vary with the strength of Coulomb interaction U. However the higher order cumulant Fa (Sk) displays additional peaks (dips) at the integer number of flux due to the Coulomb blockade and can serve as a probe to detect the TBCI.

Methods

Cumulant generating function matrix

The QD-electrode coupling is assumed to be sufficiently weak, such that the sequential tunneling is dominant. Therefore, the quantum transport can be well described by a QME of the reduced density matrix in the basis of eigenstates of the QDs. Regarding the Hamiltonian HT as a perturbation, the second-order cumulant expansion leads to (under the Born and Markov approximations) the following particle-number-resolved QME for the reduced density matrix19,40,41

with

and

where denotes coupling strength relating the electrode-α and dots-i, j. , (fα is the Fermi function of the electrode α) and gα is the density of states in the electrode-α. The Liouvillian superoperator is defined as . describes the reduced density matrix of the double-QD for n electrons tunneling through the junction up to time t. Throughout this paper, we set ħ = e = 1. In order to facilitate the calculation of FCS, we introduce the following cumulant generating function (CGF)42

where χ is the counting field. The CGF F(χ) connects the particle-number-resolved density matrix by , where the trace is over the eigenstates of double-QD and the column matrix is defined by

Evidently, one can obtain e−F(χ) in terms of the column matrix . Since the particle-number-resolved density matrix Eq. (2) has the following form

thus, satisfies the equation

where A, C and D are three square matrices. We can obtain the explicit form of the matrix operator Lχ by processing a discrete Fourier transformation for the matrix element of Eq. (2).

The tunneling time through the QD system is much shorter than The counting time (i.e. the time of measurement) under the low frequency limit, thus F(χ) can be written as17,18,19,42,43,44,45

with λ(χ) being the eigenvalue of Lχ and tending to zero when χ → 0. Using the Eq. (7) one can obtain

Plugging Eq. (8) into the Secular equation

and extending the determinant in a series of (iχ)k, we can obtain the kth-order cumulant Ck by setting the coefficient of (iχ)k being zero. The first three cumulants C1, C2 and C3 are obtained directly. We have seen that the CGF matrix Lχ is the key quantity to obtain the cumulant. The particle-number-resolved QME for the reduced density matrix (2) can be evaluated in the state basis . The result is explicitly written as

In this paper, we mainly discuss the quantum transport properties at extremely low temperature. In order to obtain an analytic formula we focus on the limiting case of zero temperature.

For the case of U = 0, the corresponding Fermi functions are seen to be in the bias voltage region , , , Thus, we obtain the CGF matrix Lχ as

where

Performing the Taylor expansion of eiχ the secular equation reduces in the symmetric dot-electrode coupling case to a power series equation of m

here , , . From Eq. (11) we can find the cumulants C1, C2, C3, which are independent of the magnetic flux number ζ. The ζ-independence is easy to understand since the net phase embraced by the wave functions is zero for both one and two particles transported from upper and lower arms of the loop in the absence of TBCI.

On the other hand when U = ∞ the corresponding Fermi functions become , in the bias voltage region and thus the secular equation is seen to be

where , , . At zero temperature the the secular equation reduces to a simple form

with . We again find that the cumulants C1, C2, C3 are ζ-independent. The reason for the independence is also obvious since the double occupation of QDs is zero at finite bias voltage for U = ∞ and the system reduces to a single-particle interferometer with zero total flux.

However for the finite TBCI with U = 10 the corresponding Fermi functions become , in the bias region . The matrix operator is found as

with . The secular equation does not have a simple form of power series and the cumulants C1, C2, C3 are ζ-dependent resulted from the dot-electrode couplings , which are ζ-dependent including the phases . The TBCI provides the overlap of two-particle wave functions between two fluxes leading to the ζ-dependent phase coherence.

Additional Information

How to cite this article: Wang, Z. et al. Quantum Phase Coherence in Mesoscopic Transport Devices with Two-Particle Interaction. Sci. Rep. 5, 12873; doi: 10.1038/srep12873 (2015).