thanks to rapid developments in the fabrication and self - assembly techniques , the electrical transport through nanoscale quantum systems such as a single quantum dot , multiple quantum dots , atoms or molecules coupled to metallic electrodes has been an interesting subject in recent years [ 1 - 6 ] . in the nanoscale quantum systems , the electrical transport is ballistic , while the phase coherence of the electrons is preserved . especially , the quantum interference effects in an ab ring including a quantum dot have been reported . the results showed that fano effect with asymmetric parameters was a good probe to quantum interference effects in the nanoscale systems . the transport properties of a quantum ring consisting of two parallel - coupled quantum dots sandwiched between two metallic electrodes have been also studied theoretically and experimentally in the last few decades [ 7 - 22 ] . for example , an ab interferometer including two coupled quantum dots with each quantum dot inserted in each arm was presented , and an oscillating electric current was detected experimentally . such a double - quantum - dot model consisting of the parallel - coupled double - dot system has been studied extensively in some previous theoretical works [ 13 - 18 ] . when the interdot coupling is considered , a bonding molecular state and an antibonding molecular state are developed . a swap effect can be found in the quantum system by tuning the magnetic flux threading the quantum ring , which may be used in the future quantum computations . recently , the transport properties of multi - parallel - coupled quantum dots have attracted considerable attention due to their potential applications and abounding physics [ 23 - 31 ] . zeng et al . studied the ab effects in a quantum ring consisting of four quantum dots sandwiched between two metallic electrodes , and a fano dip is developed when the energy levels of quantum dots in two arms are mismatched . offered a quantum model describing multi - parallel - coupled quantum - dot molecule , and fano effects in the quantum system were studied in detail . more recently , li et al . studied the electrical transport through a triple - arm ab interferometer consisting of three parallel quantum dots with electron - electron interactions under an applied electric field . in this work , we study the quantum interference effects in a double - aharonov - bohm interferometer consisting of five quantum dots sandwiched between two metallic electrodes as shown in fig . 1 . the left quantum ring consisting of the quantum dot 1 , the quantum dot 2 , quantum dot 3 and the left metallic electrode encloses the magnetic flux . the right ring consisting of the quantum dot 3 , the quantum dot 4 , the quantum dot 5 , and the right metallic electrode encloses the magnetic flux . for simplicity , we consider that only one energy level exists in each quantum dot , and the energy levels of all quantum dots can be tuned by the voltages applied on the quantum dots . in this work , we studied in detail the quantum interference effects in the double - quantum - ring structure in the case of symmetric dot - electrode couplings . the results show that the linear conductance peaks at zero temperature can be effectively tuned by using the intermediate quantum dot 3 . as the energy level of the quantum dot 3 changes , a three - peak structure may evolve into a two - peak structure in the linear conductance spectrum . when the energy levels of quantum dots in arms are not aligned with one another two fano peaks can be effectively modified by tuning the energy level of quantum dot 3 . our results show that the ab oscillating behavior depends strongly on the difference between the magnetic fluxes threading the left and right rings . in addition to the general ab oscillation with 2-period , an ab oscillation with -period can be developed when the difference between the two magnetic fluxes is ( 2n + 1)(n = 0 , 1 , 2 ,  ) . schematic plot of a double - aharonov - bohm interferometer consisting of five quantum dots sandwiched between two metallic electrodes . the left ring encloses a magnetic flux , and the right ring encloses a magnetic flux the parallel - coupled quantum - dot structure that we consider is illustrated in fig . the interdot and intradot coulomb interactions are neglected , and the total hamiltonian describing the quantum system is written as where the first term hleads in eq . ( 1 ) describes the left and right electrodes in the noninteracting electron approximation where   denotes the creation(annihilation ) operator for an electron with energy k and momentum k in electrode . the second term in eq . ( 1 ) describes the dynamics of the five parallel - coupled quantum dots , which can be modeled by using the following five - site hamiltonian where   creates ( annihilates ) an electron with the energy j in the jth quantum dot , and the ab phase   and the flux quantum . tij describes the interdot tunneling coupling between the dot i and dot j , for convenience , which is regarded as a real number . the third term in eq . ( 1 ) describing the tunneling coupling between the quantum dots and the electrodes is divided into two parts   describes the tunneling coupling between the isolated quantum dots and the electrode  a phase factor is attached to vj in the presence of the magnetic flux , and they can be written as , , , . in order to study the transport through the double - quantum ring , we need know the retarded ( advanced ) green s function . the retarded green s function is defined as applying the equation of motion method , we obtain a series of equations , four new green s functions arising from the tunneling couplings between the quantum dots and metallic electrodes appear in the above eqs . ( 816 ) , the retarded green s function   can be calculated by the following 5    5 matrix where   and ) ( j = 1 , 2 for  = l ; j = 4 , 5 for  = r ) . the linear conductance of the double quantum ring at zero temperature can be calculated by the landauer formula , the linewidth matrices   and   are given in the case of the symmetrical dot - electrode tunneling couplings , the linear conductance has the following form , where   and . from the above equation , we note that the linear conductance includes the contributions from four different electron tunneling paths ( 13 4 , 135 , 23 4;235 ) and interference terms among them . in this section , the dependence of the linear conductance  on system parameters is discussed numerically and analytically . the coupling strength between the quantum dots and the metallic electrodes   is taken as the energy unit . through this paper , all energy levels in quantum dots , tunneling couplings and fermi energy are measured by . we first study the linear transport properties of the double - ab interferometer consisting of five parallel - coupled quantum dots with the same energy levels ( j = 0 ,   j = 1 , 2 , 3 , 4 , 5 ) in the absence of the magnetic fluxes ( l = r = 0 ) . figure 2a shows the linear conductance  of the double - quantum - ring structure as a function of the fermi level ef in the case of symmetrical interdot tunneling couplings . the same interdot tunneling coupling strengths are chosen as t13 = t23 = t34 = t35 = t. there are total five molecular states in center five - dot quantum system , but three molecular states of which with zero energy are degenerate . other two molecular states are located at   2 t and 2 t , respectively . so we find that there are three resonance peaks with  = 2e / h appearing at 0 , 2 t and   2 t , respectively . with t increasing , the position of the center conductance peak has no changing , while the other two conductance peaks move in the opposite direction as shown in fig 2b and 2c , we display the linear conductance  as a function of the fermi energy ef in the case of asymmetrical interdot tunneling couplings . in fig . 2b , we assume the interdot tunneling coupling strengths as   and t34 = t35 . in this case , three conductance peaks are located at 0 , , and , respectively . as t34 decreases , the conductance peaks are suppressed and become narrower as shown in fig . 2c , we examine the transport properties of the double - quantum - ring structure by tuning t23 and t34 . here a fano dip ( zero conductance point ) appears at ef = 0 when t23 ( t34 ) is different from t13(t35 ) . the behind reason is the destructive interference effects between the electron waves directly transmitting through the three coupled quantum dots in series and side - coupled quantum dots ( see the inset in fig . 2c ) . as t23 and t34 decrease , a three - peak structure in the linear conductance spectrum evolves into the four - peak structure . when t23 = t34 = 0 , the quantum device becomes a three quantum - dot array sandwiched between two metallic electrodes with side - coupled quantum dots as shown in the inset of fig . 2c . the fano dip is located at the energy levels of the side - coupled quantum dots ( 2 = 4 = 0 ) . it is noted that two peaks become closer and narrower as t23 and t34 decrease . the energy levels of the quantum dots are chosen as 1 = 2 = 3 = 4 = 5 = 0 figure 3 displays the linear conductance  as a function of ef when the energy level of the third quantum dot 3 is not aligned with those of the other four quantum dots . five molecular states appear at , respectively . when 3 = 0 , three conductance peaks with  = 2e / h are centered at 0 ,   and , respectively . the linear conductance  as a function of ef under the different values of 3 is plotted in fig . the left two conductance peaks evolve into a single conductance peak , and the right conductance peak moves in the right direction in the case of 3 > 0 . we also find the height of the left conductance peak is suppressed . in the case of 3 the right two conductance peaks evolve into a single conductance peak , and the height of the conductance peak is suppressed . the left conductance peak moves in the left direction , and the height of the conductance peak has no changing . in order to explore the dependence of the linear conductance on the energy level of the intermediate quantum dot 3 , we plot the linear conductance as a function of 3 under several different interdot couplings in fig . the linear conductance first increases and reaches the maximum value at the certain value 3 m . once the interdot couplings are enough small , the resonant conductance peak is almost pined at ef . the transport properties of the double - quantum ring are similar with these of a single quantum dot coupled straightly to metallic electrodes . linear conductance  as a function of the fermi energy level ef in a the case of 3 > 0 ; b the case of 3 < 0 . 2 linear conductance  as a function of the energy level of the quantum dot 3 in the presence of different interdot tunneling coupling strengths . other system parameters are chosen as in fig . 2 in fig . 5 , we study the transport properties of the double - quantum ring when the same interdot coupling strengths ( t13 = t23 = t34 = t35 = t = ) are considered , and the energy levels of quantum dots parameters are chosen as ,   and , respectively . since the degenerate states are opened in this case , five conductance peaks appear in the linear conductance spectrum . the linear conductance  consists of three breit - wigner peaks and two sharp asymmetric fano peaks . the linear conductance in the absence of magnetic flux can be written as linear conductance  as a function of the fermi energy ef under several different values of the quantum dot 3 . other quantum - dot energy levels are chosen as , ,   and , respectively using eq . ( 17 ) , we arrive at from the above equation , we see clearly  = 0 for   or . the linear conductance spectrum has mirror symmetry around 3 in the case of 3 = 0 . with increasing 3 , the left fano peak is suppressed , while the right fano peak is firstly suppressed , then it is enhanced . the periodic oscillation in the linear conductance for the magnetic flux is key features when the phase coherence of the electrons is preserved . in this section , we start with the study of dependence of the linear conductance on the magnetic fluxes through the double - quantum ring . the dependence of the linear conductance on the fermi energy in the presence of the same magnetic flux through the left and right rings ( l = r =  ) is shown in fig . 6 . the same interdot tunneling couplings   and the same quantum - dot energy levels 1 = 2 = 3 = 4 = 5 = 0 = 0 are considered . when the magnetic flux  is presented , a fano dip is developed when the fermi energy is aligned with 0 . with  increasing , the three - peak structure disappears , while a four - peak structure in the linear conductance spectrum appears . when the magnetic fluxes are further increased , the four - peak structure evolves into a double - peak structure . after some algebra , we derive the linear conductance  in the presence of the magnetic flux as the following analytical form linear conductance  as a function of the fermi energy ef under several different magnetic fluxes threading the left and right rings l = r =  from the above equation , we see clearly that the linear conductance disappears when ef = 0 , and four molecular states are located around   and , respectively . with  increasing , the two middle conductance peaks are suppressed obviously , while the outside conductance peaks become shaper as shown in fig . when  approaches  , we can obtain an approximate formula for the linear conductance where 1 and 2 represent two small quantities . equation ( 26 ) shows two narrower conductance peaks are centered at   as shown in fig . when  = n(n is odd number ) , the linear conductance disappears everywhere for any value of ef . figure 7 shows the images of the linear conductance  as functions of l and the fermi energy ef . denotes the difference between magnetic fluxes through the left quantum ring and right quantum ring . when , a general 2-periodic ab oscillation for l is obtained as shown in fig . when , a -periodic ab oscillation appears in the quantum system as shown in fig . the above results can be explained as following expressions . when the magnetic fluxes through the double - quantum ring are considered , using eq . ( 24 ) , we arrive at when , the above equation can be simplified as images of the linear conductance  as functions of the fermi energy ef and l for a , b , and c , respectively . d linear conductance  as a function of magnetic flux l for   [ solid ( black ) line ] ,   [ dashed ( red ) line ] , and   [ dotted ( blue ) line ] , respectively . 6 so an ab oscillation with -period is developed as shown in fig . 7c and it is well known that an oscillating current in the ab interferometer has been detected experimentally . for a single quantum ring consisting of two parallel - coupled quantum dots sandwiched between two metallic electrodes , 2-periodic oscillation of the linear conductance  is reported in the pervious works . the linear conductance  as a function of l under the different energy levels in the quantum dot 3 or several different values of r is shown in fig . the fermi energy ef is fixed at , and other system parameters are chosen as in fig . the ab oscillation for l in the presence of different energy levels of the quantum dot 3 is potted in fig . a series of shaper resonant peaks appear at 2n (n = 0 , 1 , 2  ) , and the linear conductance reaches a minimum (  = 0 ) when l approaches ( 2n + 1 ) (n = 0 , 1 , 2  ) . when the energy level in quantum dot 3 is not aligned with other quantum - dot levels , the single conductance peak around 2n splits into the two conductance peaks . with 3 moving away from zero energy point , it is noted that the double - peak structure around 2n disappears slowly as the magnetic flux threading the right quantum ring increases . a ab oscillations for l in the presence of several different energy levels of the quantum dot 3 with r = 0 ; b ab oscillations for l in the presence of several different magnetic fluxes r with the fixed . other system parameters are chosen as 1 = 2 = 4 = 5 = 0 , , and we first study the linear transport properties of the double - ab interferometer consisting of five parallel - coupled quantum dots with the same energy levels ( j = 0 ,   j = 1 , 2 , 3 , 4 , 5 ) in the absence of the magnetic fluxes ( l = r = 0 ) . figure 2a shows the linear conductance  of the double - quantum - ring structure as a function of the fermi level ef in the case of symmetrical interdot tunneling couplings . the same interdot tunneling coupling strengths are chosen as t13 = t23 = t34 = t35 = t. there are total five molecular states in center five - dot quantum system , but three molecular states of which with zero energy are degenerate . other two molecular states are located at   2 t and 2 t , respectively . so we find that there are three resonance peaks with  = 2e / h appearing at 0 , 2 t and   2 t , respectively . with t increasing , the position of the center conductance peak has no changing , while the other two conductance peaks move in the opposite direction as shown in fig . , we display the linear conductance  as a function of the fermi energy ef in the case of asymmetrical interdot tunneling couplings . in fig . 2b , we assume the interdot tunneling coupling strengths as   and t34 = t35 . in this case as t34 decreases , the conductance peaks are suppressed and become narrower as shown in fig . 2c , we examine the transport properties of the double - quantum - ring structure by tuning t23 and t34 . here a fano dip ( zero conductance point ) appears at ef = 0 when t23 ( t34 ) is different from t13(t35 ) . the behind reason is the destructive interference effects between the electron waves directly transmitting through the three coupled quantum dots in series and side - coupled quantum dots ( see the inset in fig . 2c ) . as t23 and t34 decrease , a three - peak structure in the linear conductance spectrum evolves into the four - peak structure . when t23 = t34 = 0 , the quantum device becomes a three quantum - dot array sandwiched between two metallic electrodes with side - coupled quantum dots as shown in the inset of fig . 2c . the fano dip is located at the energy levels of the side - coupled quantum dots ( 2 = 4 = 0 ) . it is noted that two peaks become closer and narrower as t23 and t34 decrease . the energy levels of the quantum dots are chosen as 1 = 2 = 3 = 4 = 5 = 0 figure 3 displays the linear conductance  as a function of ef when the energy level of the third quantum dot 3 is not aligned with those of the other four quantum dots . five molecular states appear at , respectively . when 3 = 0 , three conductance peaks with  = 2e / h are centered at 0 ,   and , respectively . the linear conductance  as a function of ef under the different values of 3 is plotted in fig . 3a and 3b . the left two conductance peaks evolve into a single conductance peak , and the right conductance peak moves in the right direction in the case of 3 > 0 . we also find the height of the left conductance peak is suppressed . in the case of 3 the right two conductance peaks evolve into a single conductance peak , and the height of the conductance peak is suppressed . the left conductance peak moves in the left direction , and the height of the conductance peak has no changing . in order to explore the dependence of the linear conductance on the energy level of the intermediate quantum dot 3 , we plot the linear conductance as a function of 3 under several different interdot couplings in fig . the linear conductance first increases and reaches the maximum value at the certain value 3 m . once the interdot couplings are enough small , the resonant conductance peak is almost pined at ef . the transport properties of the double - quantum ring are similar with these of a single quantum dot coupled straightly to metallic electrodes . linear conductance  as a function of the fermi energy level ef in a the case of 3 > 0 ; b the case of 3 < 0 . 2 linear conductance  as a function of the energy level of the quantum dot 3 in the presence of different interdot tunneling coupling strengths . other system parameters are chosen as in fig . 2 in fig . 5 , we study the transport properties of the double - quantum ring when the same interdot coupling strengths ( t13 = t23 = t34 = t35 = t = ) are considered , and the energy levels of quantum dots parameters are chosen as ,   and , respectively . since the degenerate states are opened in this case , five conductance peaks appear in the linear conductance spectrum . the linear conductance  consists of three breit - wigner peaks and two sharp asymmetric fano peaks . the linear conductance in the absence of magnetic flux can be written as linear conductance  as a function of the fermi energy ef under several different values of the quantum dot 3 . other quantum - dot energy levels are chosen as , ,   and , respectively using eq . ( 17 ) , we arrive at from the above equation , we see clearly  = 0 for   or . the linear conductance spectrum has mirror symmetry around 3 in the case of 3 = 0 . with increasing 3 , the left fano peak is suppressed , while the right fano peak is firstly suppressed , then it is enhanced . the periodic oscillation in the linear conductance for the magnetic flux is key features when the phase coherence of the electrons is preserved . in this section , we start with the study of dependence of the linear conductance on the magnetic fluxes through the double - quantum ring . the dependence of the linear conductance on the fermi energy in the presence of the same magnetic flux through the left and right rings ( l = r =  ) is shown in fig . 6 . the same interdot tunneling couplings   and the same quantum - dot energy levels 1 = 2 = 3 = 4 = 5 = 0 = 0 are considered . when the magnetic flux  is presented , a fano dip is developed when the fermi energy is aligned with 0 . with  increasing , the three - peak structure disappears , while a four - peak structure in the linear conductance spectrum appears . when the magnetic fluxes are further increased , the four - peak structure evolves into a double - peak structure . after some algebra , we derive the linear conductance  in the presence of the magnetic flux as the following analytical form linear conductance  as a function of the fermi energy ef under several different magnetic fluxes threading the left and right rings l = r =  from the above equation , we see clearly that the linear conductance disappears when ef = 0 , and four molecular states are located around   and , respectively . with  increasing , the two middle conductance peaks are suppressed obviously , while the outside conductance peaks become shaper as shown in fig . when  approaches  , we can obtain an approximate formula for the linear conductance where 1 and 2 represent two small quantities . equation ( 26 ) shows two narrower conductance peaks are centered at   as shown in fig . figure 7 shows the images of the linear conductance  as functions of l and the fermi energy ef . denotes the difference between magnetic fluxes through the left quantum ring and right quantum ring . when , a general 2-periodic ab oscillation for l is obtained as shown in fig . when , a -periodic ab oscillation appears in the quantum system as shown in fig . the above results can be explained as following expressions . when the magnetic fluxes through the double - quantum ring are considered , using eq . ( 24 ) , we arrive at when , the above equation can be simplified as images of the linear conductance  as functions of the fermi energy ef and l for a , b , and c , respectively . d linear conductance  as a function of magnetic flux l for   [ solid ( black ) line ] ,   [ dashed ( red ) line ] , and   [ dotted ( blue ) line ] , respectively . 6 so an ab oscillation with -period is developed as shown in fig . 7c and 7d . it is well known that an oscillating current in the ab interferometer has been detected experimentally . for a single quantum ring consisting of two parallel - coupled quantum dots sandwiched between two metallic electrodes , 2-periodic oscillation of the linear conductance the linear conductance  as a function of l under the different energy levels in the quantum dot 3 or several different values of r is shown in fig . the fermi energy ef is fixed at , and other system parameters are chosen as in fig . the ab oscillation for l in the presence of different energy levels of the quantum dot 3 is potted in fig . appear at 2n (n = 0 , 1 , 2  ) , and the linear conductance reaches a minimum (  = 0 ) when l approaches ( 2n + 1 ) (n = 0 , 1 , 2  ) . when the energy level in quantum dot 3 is not aligned with other quantum - dot levels , the single conductance peak around 2n splits into the two conductance peaks . with 3 moving away from zero energy point , it is noted that the double - peak structure around 2n disappears slowly as the magnetic flux threading the right quantum ring increases . a ab oscillations for l in the presence of several different energy levels of the quantum dot 3 with r = 0 ; b ab oscillations for l in the presence of several different magnetic fluxes r with the fixed . other system parameters are chosen as 1 = 2 = 4 = 5 = 0 , , and in summary , the transport properties and quantum interference effects in a double - ab interferometer in series consisting of five quantum dots in the case of symmetric dot - electrode tunneling couplings are studied by using green s function equation of motion method . the energy levels of all quantum dots can be tuned by the voltages applied on the quantum dots in experiments . the linear conductance can be effectively modified by the intermediate quantum dot 3 . as the energy level in the quantum dot 3 changes , a three - peak structure in the linear conductance spectrum evolves into a two - peak structure . when the quantum - dot levels in arms are not aligned with one another the ab oscillation for the magnetic flux in the double - ab interferometer is also studied in this work . the results show that the ab oscillating behavior depends strongly on the difference between the magnetic fluxes threading the left and right quantum rings . an ab oscillation with -period for the magnetic flux threading the left quantum ring is developed when the difference between the two magnetic fluxes is ( 2n + 1)(n = 0 , 1 , 2 ,  ) . the authors thank the supports of the national natural science foundation of china ( nsfc ) under grant no . 10947130 and the science foundation of the education committee of jiangsu province under grant no . 09kjb140001 . this article is distributed under the terms of the creative commons attribution noncommercial license which permits any noncommercial use , distribution , and reproduction in any medium , provided the original author(s ) and source are credited . this article is distributed under the terms of the creative commons attribution noncommercial license which permits any noncommercial use , distribution , and reproduction in any medium , provided the original author(s ) and source are credited .
we investigate quantum interference effects in a double - aharonov - bohm ( ab ) interferometer consisting of five quantum dots sandwiched between two metallic electrodes in the case of symmetric dot - electrode couplings by the use of the green s function equation of motion method . 
 the analytical expression for the linear conductance at zero temperature is derived to interpret numerical results . 
 a three - peak structure in the linear conductance spectrum may evolve into a double - peak structure , and two fano dips ( zero conductance points ) may appear in the quantum system when the energy levels of quantum dots in arms are not aligned with one another . 
 the ab oscillation for the magnetic flux threading the double - ab interferometer is also investigated in this paper . 
 our results show the period of ab oscillation can be converted from 2 to  by controlling the difference of the magnetic fluxes threading the two quantum rings .