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mesoscopic circuits with multiterminal electron wave guides have interesting and useful physical properties . figure [ fig : setup ] shows an example of a circuit where a y - branch wave guide acts as a coherent beam splitter for electrons injected from a normal or superconducting reservoir . such a circuit with normal electron reservoirs ( nyn ) has been first suggested to test statistical properties of the fermi electrons by measuring current - current correlations in the arms of the splitter.@xcite this correlation is negative for the case of fermi statistics , which was indeed observed in the experiment.@xcite however , the correlations may change the sign and become positive ( similar to the case of bose particles ) when the electrons are injected from a superconducting reservoir ( nys).@xcite this effect is related to the cooper pairing in the superconductor . another kind of interference effect , namely a phase dependent conductance , has been predicted@xcite for a circuit geometry ( syn ) where the arms of the electron beam splitter are connected to a superconductor . in this case , the electrons undergo andreev reflection from the ns interfaces@xcite , picking up the superconducting phase at the connection points . the phase difference at the connection points is created by a supercurrent flowing along the surface of superconducting electrode , e.g. due to a presence of a magnetic field ( see fig . [ fig : setup ] ) . the phase dependence of the conductance of such ns interferometers has been observed in a large number of experiments.@xcite    in this paper , we investigate the properties of current - voltage characteristics of sns interferometers , i.e. sys - circuits where the arms of the y - branch splitter and the injection lead are connected to superconducting electrodes , as shown in fig . [ fig : setup ] . as is well known , the current between two superconductors at applied voltage @xmath2 ( @xmath3 is the superconducting gap ) is governed by the mechanism of multiple andreev reflections ( mar).@xcite the fingerprint of this transport mechanism is the subharmonic gap structure ( sgs ) at @xmath4.@xcite in sns interferometers one should expect the sgs to be significantly modified and become sensitive to the superconducting phase difference ( see fig . [ fig : setup ] ) . this conclusion is supported by the following qualitative argument . one may consider the interferometer as an sns junction ( @xmath5 ) which is coupled to the injection electrode @xmath6 via the y - branch wave guide . the sns junction contains bound andreev levels which are broadened due to the open connection to the injection lead . the andreev levels , the energy of which depends on the superconducting phase difference , produce resonances in the current transport between the superconducting reservoirs . these resonances will significantly modify the sgs , similar to the effect of superconducting bound states in long sns junctions.@xcite however , in the sns interferometers the resonance structures will be phase dependent.@xcite    the problem has become particularly interesting due to recent experiments on a three - terminal sns interferometer by kutchinsky et al . , @xcite in which phase dependence of the conductance structures has been observed . in this experiment , the normal region of the interferometer was fabricated with diffusive two - dimensional metallic film . the effect will be more pronounced with a ballistic normal region , especially in the quantum transport regime , when the y - wave guide supports a small amount of conducting modes . in practice , such devices could be fabricated with etched or gated ballistic 2d electron gas in a multilayered semiconducting structure.@xcite quantum transport in ns interferometers has been investigated in refs . , and the resonance enhancement of the two - particle andreev current has been discussed . in sns interferometers the situation is more complex , since the interplay between the andreev bound states and mar will lead to resonances appearing in all multiparticle currents . the purpose of our study is to develop mar theory for sns interferometers , i.e. , to include the interference effect and the effect of andreev resonances in the mar calculation scheme@xcite and to analyze the phase dependent current - voltage characteristics ( cvc ) . the paper is structured as follows . in section ii we present the model of the interferometer and derive equations for the mar scattering amplitudes . in section iii , we present the results of numerical calculation of the cvc of the interferometer , and in section iv , we present analytical perturbative analysis of the resonant current structures in the cvc . we consider the sns interferometer shown in fig . [ fig : setup ] , which consists of a normal - electron y - branch quantum wire , fabricated e.g. with etched or gated 2d electron gas , which is connected to two bulk superconducting electrodes . the superconducting electrode connected to the two arms of the splitter is ring shaped , so that magnetic flux can be sent through the ring to induce a superconducting phase difference @xmath7 between the connection points @xmath8 and @xmath9 . the current is sent through the interferometer by applying voltage @xmath10 between the electrode @xmath6 and the ring . our aim is to calculate the injected current as a function of the applied voltage and the phase difference , @xmath0 . to this end we shall apply the theory of coherent multiple andreev reflections ( mar ) , which is based on the calculation of scattering states created by incoming quasiparticles from the three superconducting terminals.@xcite for the sake of simplicity , we adopt a single mode description of the wires and model the splitter by a 3@xmath113 scattering matrix @xmath12 ( see eq . ( [ s0 ] ) below ) .    solving a coherent mar problem includes the following steps : first the recurrence for the scattering amplitudes is formulated by employing the boundary conditions at the scatterer and at the ns interfaces . then a solution for the recurrence can be constructed , from which the current is calculated . in the present case of a three - terminal junction standard mar technique can not be used directly . to solve this problem we will describe the andreev reflections from the two interfaces of the interferometer , seen by the injector @xmath6 , as a complex reflection from a single effective interface ; by doing this we reduce the 3-terminal problem to an equivalent 2-terminal problem , on which we can apply standard mar technique for a 2-terminal junctions developed in ref .. reflection from the interferometer ( left interface ) is then represented by a non - trivial transfer matrix that contains information about andreev states in the interferometer . to more clearly understand the effect of the phase difference on the injection current , we will neglect the length of the normal wires and assume all ns interfaces to be completely transparent . this will remove unnecessary complications due to normal - electron resonances @xcite and superconducting resonances @xcite associated with the saint - james - de gennes bound states.@xcite at the same time , the resonant property of the junction as well as the phase sensitivity of the current will persist due to the presence of andreev states in the interferometer with energy @xmath13 , @xmath14    in the mar regime , the time - dependent scattering states consist of a superposition of scattered waves with energies @xmath15 ( @xmath16 is an integer ) shifted with respect to the energy of the incoming wave @xmath17 . in each branch of the wire the wave functions of the sideband @xmath16 consists of a superposition of electron and hole plane waves moving in both directions , the corresponding amplitudes being labeled with @xmath18 according to fig . [ fig : setup2 ] . in the normal wires , the scattering state wave function has the form , @xmath19 where @xmath20 is the wave function in the @xmath21-th wire at the left side ( @xmath22 ) , @xmath23 is a coordinate along the wire , @xmath24 is the wave function at the right side and @xmath25 is the normal wave vector ( @xmath26 ) . in the superconducting electrodes , the corresponding wave functions , including six possible source terms , are : @xmath27 \nonumber \\ \psi_{l(2)}^s & = & e^{-i\sigma_z(\frac{ev}2t+\frac{\phi}4)}\bigl[(\delta_{3j } \psi_0^+ e^{i \tilde{k}^+_0 { x}}+ \delta_{4j } \psi_0 ^ - e^{-i \tilde{k}^-_0 { x}})e^{-iet } \nonumber \\ & + & \sum_{-\infty}^{\infty}(c_n \psi_n^+ e^{-i \tilde{k}^+_n { x } } + f_n \psi_n^- e^{i \tilde{k}^-_n { x}})e^{-ie_nt}\bigr ] \nonumber \\ \psi_{r}^s & = & e^{i\sigma_z\frac{ev}2t}\bigl[(\delta_{5j } \psi_0^+ e^{-i \tilde{k}^+_0 x}+ \delta_{6j } \psi_0 ^ - e^{i \tilde{k}^-_0 x})e^{-iet } \nonumber \\ & + & \sum_{-\infty}^{\infty}(k_n \psi_n^+ e^{i \tilde{k}^+_n x } + l_n \psi_n^- e^{-i \tilde{k}^-_n x})e^{-ie_nt}\bigr ] \label{ansatzs},\end{aligned}\ ] ] where @xmath28 are constant coefficients and @xmath29 are elementary solutions to the bogoliubov - de gennes equation @xcite , @xmath30 where @xmath31 and @xmath32 is the pauli matrix . the different source terms are distinguished by the index @xmath33 corresponding to injection of an electron / hole from each of the electrodes . injections from the left and from the right generate essentially different scattering states and currents , and to keep track of the side of injection we will introduce an additional side index @xmath34 corresponding to injection from the left electrodes and right electrode , respectively , @xmath35    the scattering in the y - branch splitter is described by a unitary scattering matrix : @xmath36 where the scattering matrix , @xmath37 is chosen to be symmetric , @xmath38 , with a real transmission amplitude from right to left , @xmath39 , and also to be independent of energy and thus the same for electrons and holes , @xmath40 the latter approximation is reasonable since the scale of the energy dispersion of the scattering matrix is given by the fermi energy , while we are interested in a much smaller energy interval comparable to the superconducting gap , @xmath41 . the unitary conditions for @xmath12 are expressed through the equations , @xmath42 , @xmath43 , where @xmath44 , @xmath45 and @xmath46 , and @xmath47 the transmission coefficient @xmath48 concerns the transparency of the sns junction of the interferometer and controls the position of the andreev levels , eq . ( [ al ] ) , while the transmission coefficient @xmath49 describes coupling of the sns junction to the injection lead .    by means of a canonical transformation the scattering between the injection electrode and the interferometer can be separated from the scattering between the arms of the interferometer . to this end we introduce new scattering amplitudes , @xmath50 , and similarly for @xmath51 , and @xmath52 , and rewrite eq . ( [ scatt ] ) on the form , @xmath53 @xmath54 the scattering equations for holes have a similarly form . it is clear from eq . ( [ scatt2 ] ) that the transport through the interferometer , from the right electrode to the left electrodes , is independent of the ( -)-coefficients . thus , we can treat the junction as an effective two - terminal junction with transparency @xmath55 . it is convenient for the further calculations to introduce vector notations , @xmath56 and similar for the other scattering amplitudes , and to rewrite eq . ( [ scatt2 ] ) through a transfer matrix , @xmath57 where @xmath58 . the same equation also holds for the hole coefficients . it is convenient to gauge out the reflection phase @xmath59 from the transfer matrix , which can be done by a transformation of the scattering amplitudes , @xmath60 , whereby @xmath61 we now proceed with a derivation of the recurrence relation and first consider the right ns - interface , where we match the wave functions in eq . ( [ ansatzn ] ) and ( [ ansatzs ] ) , @xmath62 this is the recurrence relation for an ideal sn - interface where the andreev reflection amplitude is given by @xmath63 and the source term , @xmath64 corresponds to the injection from the right electrode , @xmath65 . similar relations hold for each of the two sn - interfaces at the left side . in terms of the vectors @xmath66 and @xmath67 introduced in eq . ( [ trans ] ) , they have the form , @xmath68 for @xmath69 . for @xmath70 there is also a source term describing injection from the left electrodes ( @xmath71 ) . now the vectors @xmath72 and @xmath73 can be eliminated by means of eq . ( [ x1 ] ) for electrons and for holes , which yields @xmath74 where @xmath75 @xmath76 . the source term , @xmath77 , for @xmath78 has the form @xmath79,\;\ ; \nu=-1 , \label{pnl}\end{aligned}\ ] ] where @xmath80 . the source term for the injection cases @xmath81 has form similar to eq . ( [ pnl ] ) with @xmath82 . the matrix @xmath83 describes complete reflection from the interferometer for energies inside the energy gap , @xmath84 , where it obeys a standard transfer matrix equation , @xmath85 . for @xmath86 this is purely andreev reflection , identical to the one described by eq . ( [ rec1 ] ) , which implies that in this case the interferometer works as a single ideal sn interface . in the general case , @xmath87 , the reflection consists of both andreev and normal reflections , the probability of andreev reflection being given by the matrix element @xmath88 . thus , at @xmath1 the probability turns to zero , and therefore the andreev transport through the interferometer is blocked . furthermore , the transfer matrix @xmath83 contains information about andreev bound states in the interferometer . assuming for a moment the sns junction disconnected from the injection lead , @xmath89 and considering a stationary version of eq . ( [ rec2 ] ) , @xmath90 , in combination with eq . ( [ t0 ] ) we get the solvability condition for these equations on the form @xmath91=0 $ ] , which gives the andreev level in eq . ( [ al ] ) .    in principle the equations ( [ rec1 ] ) , ( [ rec2 ] ) and ( [ rec3 ] ) provide a complete set of recurrences for the mar amplitudes . however , following ref . , it is convenient to introduce new amplitudes , @xmath92 , which allow us to get rid of redundant mar amplitudes and to unify notation for different injection cases , @xmath93 then the recurrence relations eq .  ( [ rec1 ] ) , ( [ rec2 ] ) and ( [ rec3 ] ) can be written on a compact form , @xmath94 where @xmath95 and the index @xmath96 specifies the particular form of the transfer matrices @xmath97 and @xmath98 for different side bands and for different injection directions . equations ( [ c])-([ut ] ) realize a mapping of the mar problem on the problem of wave propagation along the energy axis . the wave amplitudes for the propagation in the upward and downward directions are given by the upper , @xmath99 , and lower , @xmath100 , components of the vector @xmath101 , @xmath102 the probability current flowing along the energy axis is defined in the usual way as @xmath103 this probability current is conserved within the superconducting gap , @xmath104const for @xmath84 , due to the unitarity properties of the matrices @xmath105 and @xmath106 . violation of the unitarity for @xmath107 outside the gap indicates leakage of the probability current into the reservoirs . we notice that in real space , the probability current is carried alternatively by electrons and holes . the charge current through the interferometer is determined by the probability currents of all the side bands , @xmath108 in this equation , the spectral current @xmath109 consists of the sum over the scattering states and all the sidebands at the injection side of the junction , @xmath110 ; the factor @xmath111 is the superconducting density of states , and @xmath112 is the fermi distribution function within the superconducting reservoirs . for @xmath113 , the free solution of eqs . ( [ rec ] ) and ( [ recx ] ) can be written on the form , @xmath114 while the solution for negative @xmath16 , @xmath115 , reads @xmath116 at the injection point @xmath70 , the mar coefficients are found from eq . ( [ rec ] ) , @xmath117 this equation imposes two relations between the four unknown coefficients , @xmath118 ; another two relations are imposed by the boundary conditions at infinity in energy space @xmath119 which fix the ratios @xmath120 . to implement the boundary conditions , we introduce effective reflection coefficients , @xmath121 , which characterize the intensity of the back scattering at specific points of the mar ladder . the definition reads , @xmath122 where @xmath123 with help of the vectors @xmath124 the solution in eq . ( [ mnm ] ) can be presented in a compact form ( using the equality @xmath125 ) , @xmath126 where the brackets denote a scalar product of two - component vectors , and @xmath127 the corresponding expression for negative @xmath16 is @xmath128 the form of solution in eqs . ( [ c>0])- ( [ c<0 ] ) is particularly useful for the calculation of the sgs : the essential processes which contribute to the sgs involve transmission through the energy gap region . this transmission contains resonances which are included in the transfer matrix @xmath129 . the latter is straightforward to calculate because it does not require excursions to infinity . on the other hand , the calculation of the quantities @xmath121 includes such excursions which , however , go outside the energy gap , and the corresponding recurrences rapidly converge .    to calculate the sgs , it is convenient to separate out the current associated with the scattering across the gap , which is most important for the sgs , from the current of thermal excitations which involves transitions between states below or above the gap . to this end , we first rewrite the spectral current in eq . ( [ jdef ] ) through leakage currents into the electrodes defined as , @xmath130 . by using the equality @xmath131 following from eq . ( [ recx ] ) , we get @xmath132 ( here we explicitly write the scattering state index @xmath133 , introduced in eq . ( [ ansatzs ] ) ) . we note that the conservation of the probability current inside the energy gap implies that the leakage current is zero inside the gap , @xmath134 . having made this observation , we write the total current in eq . ( [ jdef ] ) on the form , @xmath135 @xmath136 \tanh{e\over 2k_bt } . \label{isep}\ ] ] the first two terms in this equation correspond to the current of thermal excitations : they cancel each other when the temperature approaches zero due to detailed balance,@xcite @xmath137 for the partial currents defined as @xmath138 keeping the last term in eq . ( [ isep ] ) , we get for the current at low temperature , @xmath139 , the following equation , @xmath140 the current @xmath141 in eq . ( [ jnu1 ] ) can be presented after some algebra on the form which is useful for analytical study , @xmath142 where @xmath143\hat r_{0-}),\\ i_{n,\nu}(e)&= & ( \hat r_{n+},[\hat{u}_n^{\dagger-1}\sigma_z\hat{u}_n^{-1}-\sigma_z]r_{n+ } ) \label{in}\nonumber.\end{aligned}\ ] ] in this section , we present the results of numerical calculation of the current given by eq . ( [ itot ] ) , where the spectral current is given by eq . ( [ jx ] ) . in order to numerically evaluate the effective reflection amplitudes @xmath121 we use the equations , the definitions eq . ( [ rdef ] ) , using eq . ( [ mnm ] ) and eq . ( [ lim ] ) , @xmath144 which follow from the definitions in eq . ( [ rdef ] ) , and also from eq . ( [ mnm ] ) and eq . ( [ lim ] ) . for zero phase difference , the current - voltage characteristic ( cvc ) in fig . [ fig : iv0 ] shows the well known subharmonic gap structure for short junctions with current structures at @xmath4 , corresponding to thresholds for the @xmath16-particle currents@xcite . these current structures are associated with the usual gap edge singularities,@xcite and they are most pronounced if the transparency @xmath55 of the junction is small , @xmath145 . the cvcs of the interferometer for different values of the phase difference at zero temperature are shown in fig . [ fig : iv1 ] . increasing the phase difference , the structure changes and the positions of the current structures can no longer be associated with the subharmonics of the energy gap . instead , we observe enhancement of the subgap current and the appearance of large current peaks which move downwards in voltage with increasing phase difference . if the transparency is small , as in fig . [ fig : iv1 ] , the peak positions reflect the andreev spectrum of eq . ( [ al ] ) : the peaks appear at @xmath146 and @xmath147 , as shown in fig .  [ fig : closeup ] . with further increase of the phase difference , the current structures decrease and completely disappear at @xmath1 . at the same time , the excess current at large voltage becomes large and negative , turning into a deficit current . the cvc is @xmath148-periodic and symmetric around @xmath149 . it is illuminating to plot the individual n - particle currents , given by eq . ( [ itotn ] ) . in fig . [ fig : iv2 ] , the first four n - particle currents are shown for @xmath150 and for the same interferometer parameters as in fig . [ fig : iv1 ] . higher order currents are too small to be visible in the same picture , and their contribution to the current may be neglected . for comparison , the inset shows the corresponding cvc when @xmath86 . it follows from the plots for @xmath87 , that the peak at low voltage originates exclusively from the four - particle current . it exists within the voltage interval @xmath151 and within a certain window of phase difference . the next peak , at larger voltage , results from a combination of two structures : the peak of the three - particle current , and the overshoot of the onset of the pair current . the onset of a single - particle current at @xmath152 is reduced in comparison with the @xmath153 case , which leads to significant reduction of the current at large voltage . the reason is spectral intensity transfer from the continuum to bound andreev states . the large deficit current at high voltage is apparently the result of this reduction as well as of the above mentioned reduction of the multiparticle currents related to the suppression of andreev reflections by the interferometer . the further analysis shows that the single- and three - particle currents ( and in general odd @xmath16-particle currents ) injected from the electrode @xmath6 and from the electrodes @xmath154 are identical . in contrast , the two- and four - particle currents possess large structures only for injection from the electrode @xmath6 while currents injected from the interferometer electrodes @xmath154 are negligible small . in the next section , we will proceed with the interpretation of these features . in the case of weak coupling of the @xmath5 junction in the interferometer to the injection lead @xmath6 , @xmath145 , it is possible to perform a perturbative analysis of the n - particle currents in order to explain the origin of the phase dependent current structures in the cvc . in this limit , the current is dominated by the contribution of the andreev resonances , which we will consider in the lowest order with respect to the small parameter @xmath145 . we will proceed with a detailed analysis of the four lowest - order n - particle currents in order to explain the major structures in the ivc . let us start the discussion with the approximation for the effective reflection amplitudes at specific levels in the mar ladder , @xmath124 , in eq . ( [ rm ] ) . in the lowest order approximation with respect to @xmath49 , these reflection amplitudes are contributed by single back scattering , @xmath155 since the denominator @xmath156 in expression eq . ( [ jx ] ) for the spectral current contains a factor @xmath157 which comes from the product of transfer matrices @xmath105 in eq . ( [ mnm ] ) , we can neglect all terms except the zero order one in eq . ( [ in ] ) , whereby @xmath158 the denominator @xmath156 in eq . ( [ jx ] ) has to be considered separately for different currents , @xmath16 , and injection cases , @xmath110 . the single - particle current involves only a single crossing of the tunnel barrier , schematically shown in fig . [ fig : mar]a . thus , it can not be resonant , and the denominator @xmath159 can be considered in the lowest - order approximation . the expression for the single particle current , derived from eqs . ( [ itotn ] ) , ( [ jx ] ) , and ( [ iappn ] ) then reads , @xmath160 where @xmath161 the form of eq . ( [ i1appr ] ) is a standard one for a tunnel current given by the tunnel hamiltonian model were @xmath162 is the density of states ( dos ) in the injection lead , while @xmath163 has the meaning of the effective phase dependent dos in the interferometer . an important property of the dos of the interferometer is the disappearance of the singularities at the gap edges for nonzero phase difference . the reason is that the resonances ( andreev states ) move down into the gap , according to eq . ( [ al ] ) ( see fig .  [ fig : dos ] ) . consequently , the onset of the single - particle current at @xmath164 , which is of the order @xmath49 when @xmath86 , is reduced at finite phase difference ( see fig . [ fig : iv2 ] ) .    in order to calculate the pair current we will first inspect the corresponding transfer matrix , @xmath165 . for @xmath65 ( injection from the right ) this transfer matrix has zeros in the limit @xmath89 at the energies of the andreev bound states in the interferometer , @xmath166 . the expansion of the matrix in powers of @xmath49 has the form @xmath167 + o(t^2 ) , \label{m20}\end{aligned}\ ] ] where we have used the identity , @xmath168 , derived from the unitarity condition eq . ( [ unit ] ) and @xmath76 , in the limit @xmath89 . equation ( [ m20 ] ) clearly demonstrates the resonant behavior . the first term is real and proportional to the deviation of the energy @xmath169 for the andreev reflection from the energy of the andreev state . this term determines the position of the resonance . the second , imaginary , term is proportional to @xmath49 and determines the width of the resonance . both the position and width of the resonance depend on the phase difference . at @xmath145 the spectral current is dominated by the andreev resonance , and neglecting the non - resonant current we arrive at the expression @xmath170 where @xmath171 @xmath172 . a factor of 2 appears in the current expression , in addition to the two - particle factor @xmath173 in eq . ( [ jx ] ) , since there are two andreev resonances , at @xmath174 and @xmath175 . the integrand in eq . ( [ i2appr ] ) has the form of the transmission coefficient for an asymmetric quantum mechanical double barrier structure . the difference in the tunnel rates @xmath176 is due to the different dos for the injection and exit energies , but also due to the different effective transparencies of the y - branch splitter , @xmath177 , for electrons and holes . the resonant transparency of the junction for the pair current is proportional , according to eq . ( [ i2appr ] ) , to the product @xmath178 which turns to zero at @xmath1 . thus the resonant pair current is blocked at @xmath1 , in agreement with our earlier general conclusion about the suppression of andreev reflection at the interferometer in this case . evaluation of the integral over energy in eq . ( [ i2appr ] ) under the condition , @xmath179 , yields the pair current @xmath180 the magnitude of this current is generally of order @xmath49 ; however , in the vicinity of the resonant onset , @xmath181 for deviations @xmath182 , the rate @xmath183 increases due to the enhancement of the dos in the right superconductor , @xmath184 , which yields the overshoot seen in fig . [ fig : iv2 ] . at lower voltage , @xmath185 , the pair current is non - resonant and proportional to @xmath186 , which is also the case for the pair current @xmath187 injected from the interferometer . the resonant three - particle current can be analyzed in a similar manner . we construct the corresponding transfer matrix using the pair current matrix , eq .  ( [ m20 ] ) , and for @xmath65 we get @xmath188 . this equation has resonances at the same energies as the pair current , @xmath166 , but nevertheless the resonant current is small , @xmath189 , since the mar path , fig . [ fig : mar]c , includes an additional crossing of the tunnel barrier . the magnitude of the current is considerably enhanced in the vicinity of voltage @xmath190 when the second andreev reflection occurs at the gap edge of the right electrode , @xmath191 ( see fig . [ fig : mar]c ) . the singular dos at the gap edge in this electrode appears in the mar calculation as a quasibound state situated at the gap edge . thus , this mar process can be interpreted as an overlap of two resonances , one at the andreev bound state at energy @xmath175 , and another at the quasibound state at the gap edge . a similar analysis applies to the current injected from the interferometer , @xmath71 : in this case , the double resonance is formed by the andreev state at @xmath174 and the quasibound state at lower gap edge . the height of the double resonance peak is @xmath192 . a double resonance involving two andreev bound states occurs in the four - particle current injected from the right as shown in fig . [ fig : mar]d . the corresponding mar path is selected by the conditions @xmath193 and @xmath194 . the resulting peak is situated at @xmath195 and has the hight @xmath196 . in the vicinity of the peak , @xmath197 , the expression for the resonant four - particle current becomes , @xmath198 ^ 2+e_2 ^ 2{\tilde \gamma}_-^2 } , \label{i4}\ ] ] where @xmath199 the current peak under consideration is a full - scale current structure in a region of the ivc which in point contacts is dominated by non - resonant four - particle current , which is of the order @xmath200 . as any andreev resonance , the four - particle current peak strongly depends on the phase difference ; it disappears at @xmath1 . the peak exists within the voltage interval @xmath151 and within the phase difference window @xmath201 . we note that the positions of all discussed current structures are proportional to the order parameter @xmath3 and therefore temperature dependent and scale with @xmath3 . we end this section with the discussion of cvc at large voltage , @xmath202 . the current at large voltage results exclusively from the single particle and pair currents . it is straightforward to calculate the single particle current at large voltage from eq . ( [ i1appr ] ) , @xmath203 the first term in this equation is the current through the normal junction , while the second term is the single - particle contribution to the excess current , @xmath204 this excess current is large , @xmath205 , and negative , which reflects the suppression of the current onset at @xmath164 . the pair current contribution to the excess current is dominated by the resonance and is given by eq . ( [ i2a ] ) . the total excess current to leading order in @xmath49 has the form @xmath206 the excess current in the interferometer is large and negative , in sharp contrast to quantum point contacts , where the excess current is small , @xmath207 , and positive.@xcite however , it turns to zero at @xmath86 , in agreement with our observation that in this case the interferometer behaves as a quantum point contact . the excess current achieves its minimum value at @xmath1 because of blockade of the pair current . we have investigated the phase dependence of the dc current - voltage characteristics ( cvc ) in sns interferometers consisting of two superconducting reservoirs connected by a y - shaped normally conducting quantum wire . we have developed a generalization of the existing coherent mar theory , which fully incorporates the effect of interference , as well as resonance effects due to andreev bound states in the interferometer . the phase dependent current - voltage characteristic @xmath0 has been numerically investigated for single - mode wires with length smaller than the superconducting coherence length . the analytical study of the current was performed using a resonance approximation in the limit of weak coupling of the interferometer to the injection lead . we found significant enhancement of the subgap current at finite phase difference compared to the case of zero phase difference . the enhancement is accompanied by a shift of the onset of the current at @xmath208 to smaller voltages , @xmath209 , and appearance of current peaks at @xmath174 and @xmath210 . these features are produced by resonant andreev reflection of the injected particles by the s@xmath211ns@xmath212 junction of the interferometer : they are most pronounced for weak coupling of the s@xmath211ns@xmath212 junction to the injection lead . the positions of all current structures are temperature dependent and scale with @xmath3 . we further found strong suppression of the current when the phase difference approaches @xmath213 : in this region all subgap current structures disappear and , moreover , the excess current at large voltage becomes negative ( deficiency current ) . the magnitude of the deficiency current is considerably large at @xmath1 . this effect results from the suppression of the andreev reflections from the s@xmath211n and s@xmath212n interfaces due to the interference in the arms of the beam splitter ( cf . a similar effect in ns interferometers@xcite ) . we conclude our discussion with some brief comments on the connection of the studied model to more realistic experimental devices based on 2d electron gas structures . in such devices , the normal region has a typical size which exceeds the superconducting coherence length and it contains several conducting electronic modes . furthermore , the ns interfaces are not perfectly transparent . however , this effect is less important since the interface reflection coefficient in practice can be rather small ( @xmath214 ) . increasing the length of the arms of the interferometer will result in the appearance of new phase dependent current peaks due to increasing number of andreev states in the sns junction of the interferometer ( cf . ref .  ) . similarly , increasing the length of the injection lead will result in the appearance of current resonances which do not depend on the phase difference , due to @xmath7-independent superconducting bound states in this lead.@xcite in interferometers with multimode wires , the resonant current peaks will be smeared ; 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a theory of coherent multiple andreev reflections ( mar ) is developed for superconductor - normal metal - superconductor ( sns ) interferometers . 
 we consider a y - shape normal electron beam splitter connecting two superconducting reservoirs , where the two connection points to the same superconductor can have different phase . 
 the current is calculated in the quantum transport regime as a function of applied voltage and phase difference , @xmath0 . 
 mar in interferometers incorporates two features : interference in the arms of the splitter , and interplay with andreev resonances . 
 the latter feature yields enhancement of the subgap current and current peaks with phase - dependent positions and magnitudes . 
 the interference effect leads to suppression of the subgap current and complete disappearance of the current peaks at @xmath1 . 
 the excess current at large voltage decreases and changes sign with increasing phase difference .