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atomtronics is an emerging area of investigation in which analogues of electronic circuits and devices are constructed using ultra - cold bosonic atoms rather than electrons as in conventional electronics  @xcite . the conventional way to construct an atomtronic device is to use cold atoms trapped in an optical lattice , which has a description in terms of either the mott - hubbard model transferred from condensed matter physics  @xcite for fermionic atoms , and the bose - hubbard model for bosonic atoms  @xcite . these models can represent either insulating behaviour , in the mott insulator regime , or conducting behaviour , in the superfluid regime . in this work we consider only bosonic atoms . shortly after the realisation of trapped bose - einstein condensates ( bec ) , jaksch _ et al . _ @xcite showed that the bose - hubbard model can provide an accurate description of bosonic atoms trapped in a deep optical lattice . the basics of this model have been used to investigate a wide variety of atomtronic devices  @xcite , including one with diode - like behaviour  @xcite , a single - atom switching transistor  @xcite , and one which uses a triple well configuration to mimic a field effect transistor  @xcite . more recent proposals include circuits of diodes and transistors  @xcite , and an atomtronic battery  @xcite . proposed atomtronic transistors with the idea of combining them into gates for quantum computers  @xcite . we have no doubt that there will be more proposals in the future , and what we will show here is that a full quantum description of both the dynamics and the initial state needs to be taken into account to guarantee accurate descriptions of the dynamics of such devices . various theoretical methods have been used to analyse these devices up until the present . for small numbers of atoms and sites , direct quantum calculations using a master equation are often possible . et al . _ developed a quantum master equation to treat systems which interact with sources and sinks , based largely on methods which have been extremely successful in quantum optics . with this method , they have analysed electronic diodes , field - effect transistors , bipolar junction transistors , and an analogy to a logic gate  @xcite . the master equation was then solved numerically , giving fully quantum solutions . this method allows for the investigation of systems with relatively small numbers of atoms and lattice sites , with the authors typically treating @xmath0 or @xmath1 sites , each with @xmath2 or @xmath3 atoms . used coupled nonlinear schrdinger equations in one dimension to calculate the phase evolution of eigenstates , with two distinguishable atoms in a triple well  @xcite . for the small systems considered by these authors , their methods are perfectly adequate . in the mesoscopic regime , when the numbers of atoms can make density matrix calculations complicated , the one - body schrdinger equation can be used for non - interacting atoms , or the mean - field gross - pitaevskii equations ( gpe ) for interacting atoms . both of these have been used to model coherent atomic transport in a three - well potential  @xcite and each has disadvantages . the one - body schrdinger equation , being linear , can not include the interactions required for the mott insulator regime . the gpe approach can not describe any quantum statistical features , such as the actual quantum states or any entanglement properties . the three - well coherent transport model has previously been analysed using the fully quantum positive - p representation  @xcite , which was used to note the differences caused by different initial quantum states on the dynamics , the entanglement properties  @xcite and the quantum steering properties  @xcite .    in what follows we base ourselves on the approach taken by milburn _ et al . _ @xcite , generalisng this to four wells  @xcite , and using the fully quantum positive - p phase space representation . we consider this to be the most suitable approach here because it is exact , allows for an easy representation of mesoscopic numbers of atoms , and can simulate different quantum states  @xcite . just as importantly , the positive - p calculations scale linearly with the number of sites and can in principle deal with any number of atoms . these are powerful advantages when we wish to consider mesoscopic numbers of atoms in arbitrary numbers of potential wells . one disadvantage of the positive - p representation is that the integration has a tendency to diverge for high collisional nonlinearities  @xcite , although it often converges for long enough to show marked differences from mean - field predictions  @xcite and also allows for the calculation of quantum correlations  @xcite . the four - well system is as shown in the schematic of fig . [ fig : setup ] , where the circles represent the wells or lattice sites , each of which contains a single atomic mode . the @xmath4 are bosonic annihilation operators for atoms in mode @xmath5 , the @xmath6 are the ground state single - atom energies of the wells , and @xmath7 represents the coupling between the wells . we assume that there is no coupling between well @xmath1 and wells @xmath2 and @xmath0 . we also assume that any atoms at each site are initially in their ground state . the basic idea is that atoms from the outside wells will tunnel into the centre well and from there to well @xmath1 . because of the wave nature of the condensed atoms , we expect that the initial phase difference , @xmath8 between the atomic modes at sites @xmath2 and @xmath0 will affect the rate of tunnelling into the middle and hence into well @xmath1 . we would hence have a type of phase sensitive gate , where the phase of one mode mode can be used to control the occupation of another .     and @xmath0 have nearest neighbour couplings , and well @xmath1 is coupled with well @xmath3 . the @xmath4 are bosonic annihilation operators for atoms in mode @xmath5 , the @xmath6 are the ground state single - atom energies of the wells , and @xmath7 represents the coupling between the wells . @xmath8 represents a phase - shift which is applied to mode @xmath0 at the beginning of the evolution , when the populations of @xmath2 and @xmath0 will be equal , with @xmath3 and @xmath1 initially being empty . ] following the usual procedures  @xcite , with @xmath9 as the s - wave collisional term , we write the hamiltonian as @xmath10 where @xmath5 runs from @xmath2 to @xmath1 . starting from this hamiltonian , our first step is to find the semi - classical mean - field equations in the gross - pitaevskii approach . we will use the solutions of these for comparison purposes , since it is well known that they are not always accurate for the bose - hubbard model , even for calculation of the mean fields  @xcite . using the variables @xmath11 to represent the mean - fields , we find @xmath12     for the numbers in each well as a function of dimensionless time , @xmath13 . the initial phase difference , @xmath8 , between wells @xmath2 and @xmath0 is zero , and the numbers in these two wells are equal . wells @xmath3 and @xmath1 are initially empty . we see that atoms pass through the middle well and populate the fourth . the parameters used in this and all subsequent plots are @xmath14 , @xmath15 , @xmath16 , @xmath17 , and @xmath18 . ]    to solve the full quantum equations , we use the positive - p representation  @xcite , which allows for exact solutions of the dynamics arising from the hamiltonian of eq .  [ eq : ham ] in terms of normally - ordered operator averages , in the limit of the average of an infinite number of trajectories of stochastic differential equations in a doubled phase - space . this method also allows for the representation of different quantum states in the initial conditions  @xcite . the positive - p representation does have the disadvantage that it can suffer from divergence problems in cases with a largish colliisional nonlinearity , often limiting its utility to short time dynamics  @xcite . in this letter , it is exactly this regime we are analysing , so that the positive - p representation is perfectly adequate . following the standard methods  @xcite , the set of it stochastic differential equations  @xcite are found as @xmath19 where the @xmath20 are standard gaussian noises with @xmath21 and @xmath22 . as always , averages of the positive - p variables represent normally ordered operator moments , such that , for example , @xmath23 . in order to analyse and compare the effects of different initial states , we begin with a total of @xmath24 atoms , evenly distributed between wells @xmath2 and @xmath0 and analyse the classical evolution given by eq .  [ eq : gpe ] . the main quantity of interest is @xmath25 , the number of atoms in well @xmath1 as a function of time . with the initial phases equal , we see a @xmath26 transfer of atoms into the fourth well at certain times , as shown in fig .  [ fig : classzero ] . when we set @xmath8 to @xmath27 , this fourth well remains unpopulated , showing that the device can act semi - classically as a phase - dependent gate . with @xmath28 , as seen in fig . [ fig : classpihalf ] , we see that a little less than half the atoms appear in the fourth well . having thus benchmarked the device , we will now proceed to see how it performs with initial coherent and fock states in wells @xmath2 and @xmath0 .     for the numbers in each well for the same parameters and initial conditions as fig . [ fig : classzero ] except that @xmath28 . we see that the maximum population of well four is approximately half that seen in fig . [ fig : classzero ] . ]    the initial quantum states are modelled as in ref . @xcite , with the positive - p representation equations being numerically integrated over a large number of stochastic trajectories . the immediate difference is that , because we can calculate close approximations to normally - orderd operator expectation values , we are also able to calculate the standard deviations about the average solutions for the atom numbers . we begin with coherent states with the same average occupation numbers as used classically . the important differences are that the initial physical number distribution is now poissonian ( close to gaussian as the number of quanta increases ) and quantum noise is included in the calculations . we immediately find that the average solutions for the intensities are the same as the semi - classical mean - field solutions , for both @xmath29 and @xmath30 . the only difference is that there is a distribution about these solutions , as shown by the lines plotted at plus and minus one standard deviation . thus the semi - classical treatment gives the most likely solutions , although we must remember that most individual experiments will give something different . how important this is in the case of initial coherent states will depend on how precisely we wish a particular atomtronic device to function .     and the solutions are the average of @xmath31 stochastic trajectories of the positive - p equations . ]     and @xmath32 are equal . the initial conditions are the same as for fig . [ fig : czero ] , but with @xmath30 , and the solutions are the average of @xmath33 stochastic trajectories of the positive - p equations . ]    when we begin the simulations with fock states of definite atom number in wells @xmath2 and @xmath0 , we find that the device loses its phase dependence completely . the average results are almost identical to those found semi - classically for @xmath28 , and do not change irrespective of the initial phase . this is not unexpected since fock states can be roughly thought of as having indeterminate phase . as can be seen in fig . [ fig : fzero ] , however , the standard deviation around the mean solution is significant . in practice this means that the device would be totally inappropriate as a phase sensitive gate if the initially occupied wells are in number states . as this is usually thought of as the natural state for mott insulators , this would be problematic .     and the solutions are the average of @xmath34 stochastic trajectories of the positive - p equations . the solutions for initial fock states are identical for all values of @xmath8 . ] although we have considered the two extreme cases of coherent and fock states here , it is worth noting that there are other possible initial states . as an example , the collisional nonlinearity of condensed atoms has been shown to give a  crescent " state , where the wigner function is stretched in the phase space  @xcite . this initial state has been previously shown to affect the dynamics of molecular association  @xcite and would be expected to have an effect with our device as well . the advantage of the positive - p representation is that it allows us to consider arbitrary initial quantum states , as well as the calculation of any correlations that can be expressed as normally - ordered operator expectation values . in conclusion , we have shown that the positive - p representation is a useful tool for analysing the performance of atomtronic devices where mesoscopic numbers of sites and atoms are involved . it is easily extended to much greater numbers of both than we have considered here . we have also shown that mean - field claculations can be very misleading for such systems and it is therefore important to undertake full quantum calculations . we have considered two markedly different initial states , showing that the coherent states , as expected , give results close to the semi - classical predictions . on the other hand , if the atoms are initially in fock states , the semi - classical predictions are almost useless . as shown previously , correlations between the atomic modes at each site can also be calculated with this representation , which will open the door for fully quantum atomtronics in which use can be made of properties such as entanglement and epr steering . this will definitely open up regimes that are not readily accessible with standard electronics . this research was supported by the australian research council under the future fellowships program ( grant i d : ft100100515 ) , the new zealand marsden fund and a rutherford discovery fellowship from the royal society of new zealand .
it is known that a semi - classical analysis is not always adequate for atomtronics devices , but that a fully quantum analysis is often necessary to make reliable predictions . while small numbers of atoms at a small number of sites are tractable using the density matrix , a fully quantum analysis is often not straightforward as the system becomes larger . 
 we show that the fully quantum positive - p representation is then a viable calculational tool . 
 we postulate an atomtronic phase - gate consisting of four wells in a bose - hubbard configuration , for which the semi - classical dynamics are controllable using the phase of the atomic mode in one of the wells . 
 we show that the quantum predictions of the positive - p representation for the performance of this device have little relation to those found semi - classically , and that the performance depends markedly on the actual quantum states of the initially occupied modes . 
 we find that initial coherent states lead to closest to classical dynamics , but that initial fock states give results that are quite different . 
 a fully quantum analysis also opens the door for deeply quantum atomtronics , in which properties such as entanglement and epr ( einstein - podolsky - rosen ) steering become valuable technical properties of a device .