Entanglement via a three-well Bose-Hubbard system and via an optical beamsplitter
aa r X i v : . [ qu a n t - ph ] O c t Entanglement via a three-well Bose-Hubbardsystem and via an optical beamsplitter
M. K. Olsen
The University of Queensland, School of Mathematics and Physics, Brisbane,Queensland 4072, Australia
Abstract.
We compare and contrast the entangling properties of a three-well Bose-Hubbardmodel and an optical beamsplitter. The coupling between the different modes islinear in both cases, and we may identify two output modes. Obvious differences arethat our Bose-Hubbard model, with only the middle well initially occupied, does nothave a vacuum input port, there is no equivalent of a collisional, χ (3) nonlinearitywith the beamsplitter, and the results of the Bose-Hubbard model show a time-dependence. In the non-interacting case, we obtain analytic solutions and showthat, like a beamsplitter, the Bose-Hubbard system will not produce entanglementfor classical initial states. We also show that whether inseparability or entanglementare detected depends sensitively on the criteria measured, with different criteria givingcontradictory predictions.PACS numbers: 03.75.Gg,03.65.Ud,67.85.Hj Submitted to:
J. Phys. B: At. Mol. Phys.
E-mail: [email protected] ose-Hubbard Mode Splitter
1. Introduction
In this article we extend previous work which combined the two fields of quantuminformation and ultra-cold bosons to propose a method for the fabrication of spatiallyisolated entangled atomic populations [1]. We do this by comparing the performanceof a three-well Bose-Hubbard system [2, 3] to that of an optical beamsplitter for theproduction of spatially separated output modes, using well-known criteria [4, 5, 6, 7, 8].In particular, we consider the quadrature based criteria [6, 7, 8] which were notconsidered in the earlier paper.The field of ultra-cold bosons has seen much experimental and theoreticalinvestigation since the successful Bose condensation of bosonic atoms. For atomstrapped in an optical lattice, one investigative technique uses the Bose-Hubbard model.This model, from condensed matter physics, was originally shown by Jaksch et al [9] toprovide an accurate description of bosonic atoms trapped in a deep optical lattice. In thiswork we use a three well Bose-Hubbard model to propose and analyse the entanglingproperties of a quantum atom optical mode splitter and recombiner. We show thatthis can split an initial condensate in the central well into two separated entangledcondensates, with the detection of the entanglement being sensitive to both the initialquantum state of the condensate in the central well, and to the actual criteria used.We then examine and compare a quantum optical beamsplitter with one vacuum inputwith regard to the same correlations and input quantum states.The area of continuous-variable entanglement is very active [10, 11], with manycriteria having been developed to signify the presence of inseparability and entanglement,especially in bipartite systems. Many of these only apply fully to Gaussian systems andGaussian measurements. The most commonly used measurements are those developedby Duan et al [6] and Simon [7], using combinations of quadrature variances. Morerecently, Teh and Reid have shown the degree of violation of these inequalities thatis necessary to demonstrate not just inseparability, but genuine entanglement [12], asthese are only necessarily the same property for pure states. The criteria we use in thiswork fall into two categories. Those in the first category were developed by Hillery andZubairy [4] and expanded on by Cavalcanti et al [5] to cover multipartite entanglement,steering, and violations of Bell inequalities. As shown by He et al [13], the Hilleryand Zubairy criteria are well suited to number conserving processes such as those ofinterest here. The second category are quadrature based criteria, originally developedby Duan et al [6] and Simon [7] for inseparability and entanglement, and by Reid [8]for demonstrations of the Einstein-Podolsky-Rosen (EPR) paradox [14].Multi-mode entanglement in Bose-Einstein condensates (BEC) has been predictedand examined in the processes of molecular dissociation [15], four-wave mixing in anoptical lattice [16, 17, 18], and in the Bose-Hubbard model [19]. In the latter casethe separation of the modes is produced by the tunneling between wells, in both thecontinuous [20, 21, 13] and pulsed tunneling configurations [22, 23]. The quantumcorrelations necessary to detect entanglement can in principle be measured using the ose-Hubbard Mode Splitter et al [27].
2. Physical model, Hamiltonian and equations of motion
We will follow the approach taken by Milburn et al [28], generalising this to threewells [29, 30], and solving either the Heisenberg equations of motion or the fullyquantum positive-P phase space representation [31] equations, depending on whetherthere is a collisional interaction present or not. We consider these to be the mostsuitable approaches here because they are both exact, allow for an easy representationof mesoscopic numbers of atoms, can be used to calculate quantum correlations, andcan simulate different quantum initial states [32]. Just as importantly, both calculationsscale linearly with the number of sites and can in principle deal with any number ofatoms. One disadvantage of the positive-P representation is that the integration canshow a tendency to diverge at short times for high collisional nonlinearities [33]. Aslong as the procedures followed to derive the Fokker-Planck equation for the positive-Pfunction are valid [34], the stochastic solutions are guaranteed to be accurate whereverthe integration converges. With all the results shown here, the solutions were foundwithout any signs of divergences.The system is very simple, with three potential wells in a linear configuration. Eachof these can contain a single atomic mode, which we will treat as being in the lowestenergy level. Atoms in each of the wells can tunnel into the nearest neighbour potential,with tunneling between wells 1 and 2, and 2 and 3. With all the population initiallyin the middle well, the system acts as a time dependent mode splitter and recombiner.With the ˆ a j as bosonic annihilation operators for atoms in mode j , J representing thecoupling between the wells, and χ as the collisional nonlinearity, we may now write ourHamiltonian. Following the usual procedures [28], we find H = ~ X j =1 χ ˆ a † j ˆ a j + ~ J (cid:16) ˆ a † ˆ a + ˆ a † ˆ a + ˆ a † ˆ a + ˆ a † ˆ a (cid:17) . (1) For the case where the collisional interaction between the atoms is set to zero, wefind that an analytical solution of the Heisenberg equations of motion for the system ose-Hubbard Mode Splitter ddt ˆ a ˆ a † ˆ a ˆ a † ˆ a ˆ a † = − iJ iJ − iJ − iJ iJ iJ − iJ iJ × ˆ a (0)ˆ a † (0)ˆ a (0)ˆ a † (0)ˆ a (0)ˆ a † (0) . (2)This set of linear operator equations is readily solved, having the solutionsˆ a ( t ) = 12 (cos Ω t + 1) ˆ a (0) − i √ t ˆ a (0) + 12 (cos Ω t −
1) ˆ a (0) , ˆ a † ( t ) = 12 (cos Ω t + 1) ˆ a † (0) + i √ t ˆ a † (0) + 12 (cos Ω t −
1) ˆ a † (0) , ˆ a ( t ) = − i √ t ˆ a (0) + cos Ω t ˆ a (0) − i √ t ˆ a (0) , ˆ a † ( t ) = i √ t ˆ a † (0) + cos Ω t ˆ a † (0) + i √ t ˆ a † (0) , ˆ a ( t ) = 12 (cos Ω t −
1) ˆ a (0) − i √ t ˆ a (0) + 12 (cos Ω t + 1) ˆ a (0) , ˆ a † ( t ) = 12 (cos Ω t −
1) ˆ a † (0) + i √ t ˆ a † (0) + 12 (cos Ω t + 1) ˆ a † (0) , (3)where we have made the substitution Ω = √ J for reasons of notational elegance. Theseequations allow us to find analytical expressions for all the correlations of interest, aswe shall do further on in the article.We can also solve the Heisenberg equations in terms of ˆ X i and ˆ Y i , the quadratureoperators. Settingˆ X i = ˆ a i + ˆ a † i and ˆ Y i = − i (cid:16) ˆ a i − ˆ a † i (cid:17) , (4)we find ˆ X ( t ) = 12 (cos Ω t + 1) ˆ X (0) + 1 √ t ˆ Y (0) + 12 (cos Ω t −
1) ˆ X (0) , ˆ Y ( t ) = 12 (cos Ω t + 1) ˆ Y (0) − √ t ˆ X (0) + 12 (cos Ω t −
1) ˆ Y (0) , ˆ X ( t ) = 1 √ t ˆ Y (0) + cos Ω t ˆ X (0) + 1 √ t ˆ Y (0) , ˆ Y ( t ) = − √ t ˆ X (0) + cos Ω t ˆ Y (0) − √ t ˆ X (0) , ˆ X ( t ) = 12 (cos Ω t −
1) ˆ X (0) + 1 √ t ˆ Y (0) + 12 (cos Ω t + 1) ˆ X (0) , ˆ Y ( t ) = 12 (cos Ω t −
1) ˆ Y (0) − √ t ˆ X (0) + 12 (cos Ω t + 1) ˆ Y (0) , (5)which then allow us to find solutions for any correlations written in terms of thesequadratures. Examples of these are the quadrature squeezing [35] and Reid EPRcorrelations [8]. ose-Hubbard Mode Splitter In this case ( χ = 0), it is not obvious how to solve the equations of motion analytically.We will therefore use the positive-P representation [31], which allows for exact solutionsof the dynamics arising from the Hamiltonian of Eq. 1, in the limit of the average of aninfinite number of trajectories of the stochastic differential equations in a doubled phase-space. In practice we obviously cannot integrate an infinite number of trajectories, buthave used numbers large enough that the sampling error is within the line thicknessesof our plotted results. Following the standard methods [35], the set of Itˆo stochasticdifferential equations [34] are found as dα dt = − iχα +1 α − iJ α + q − iχα η ,dα +1 dt = 2 iχα + 21 α + iJ α +2 + q iχα + 21 η ,dα dt = − iχα +2 α − iJ ( α + α ) + q − iχα η ,dα +2 dt = 2 iχα + 22 α + iJ (cid:0) α +1 + α +3 (cid:1) + q iχα + 22 η ,dα dt = − iχα +3 α − iJ α + q − iχα η ,dα +3 dt = 2 iχα + 23 α + iJ α +2 + q iχα + 23 η , (6)where the η j are standard Gaussian noises with η j = 0 and η j ( t ) η k ( t ′ ) = δ jk δ ( t − t ′ ).As always, averages of the positive-P variables represent normally ordered operatormoments, such that, for example, α mj α + nk → h ˆ a † n ˆ a m i . We also note that α j = ( α + j ) ∗ onlyafter taking averages, and it is this freedom that allows classical variables to representquantum operators.
3. Quantum correlations
As well as the populations in each well, we can also calculate any type of operatorproducts that we desire, analytically in the case without interactions. Beginning withonly the middle well occupied, we find the analytic non-interacting solutions for thenumbers in each well, h ˆ a † ( t )ˆ a ( t ) i = h ˆ a † ( t )ˆ a ( t ) i = 12 sin Ω t h ˆ a † (0)ˆ a (0) i , h ˆ a † ( t )ˆ a ( t ) i = cos Ω t h ˆ a † (0)ˆ a (0) i . (7)On the scale of Fig. 1, which shows numerical solutions, the above are indistinguishablefrom the stochastic solutions for χ = 0.The next class of correlations we calculate are the number variances, including thenumber difference between the populations of wells 1 and 3. In terms of the operators, ose-Hubbard Mode Splitter N j N N Figure 1. (Colour online) The populations in each well as a function of time, for J = 1, χ = 10 − , and N (0) = 200, with N (0) = N (0) = 0. The atoms in the centrewell begin in a Fock state, although an initial coherent state leads to indistinguishableresults. The results shown are the average of 1 . × stochastic trajectories. Thenon-interacting analytical results are indistinguishable on this scale. The quantitiesplotted in this and subsequent plots are dimensionless. these are V ( ˆ N j ) = h ˆ a † j ˆ a j ˆ a † j ˆ a j i − h ˆ a † j ˆ a j i ,V ( ˆ N − ˆ N ) = h (cid:16) ˆ a † ˆ a − ˆ a † ˆ a (cid:17) i − h ˆ a † ˆ a − ˆ a † ˆ a i . (8)In the non-interacting case and with only the middle well initially occupied, we find V ( ˆ N ) = V ( ˆ N ) = 14 n sin (Ω t ) V ( ˆ N (0)) + (1 − cos (Ω t )) h ˆ N (0) i o ,V ( ˆ N ) = cos(Ω t ) V ( ˆ N (0)) + 14 sin t h ˆ N (0) i ,V ( ˆ N − ˆ N ) = 12 sin Ω t (cid:0) Ω t (cid:1) h ˆ N (0) i . (9)These results, for initial Fock and coherent states in the middle well, are shown in Fig. 2and Fig. 3.The second correlation is an entanglement measure adapted from an inequalitydeveloped by Hillery and Zubairy, who showed that, considering two separable modesdenoted by i and j [4], |h ˆ a † i ˆ a j i| ≤ h ˆ a † i ˆ a i ˆ a † j ˆ a j i , (10) ose-Hubbard Mode Splitter Jt V a r i an c e s V(N )V(N ) V(N -N ) Figure 2. (Colour online) The number variances of Eq. 9, for an initial Fock stateof 200 atoms in the middle well. We see that all variances are periodic in the non-interacting case. Jt V a r i an c e s V(N -N )V(N ) V(N ) Figure 3. (Colour online) The number variances of Eq. 9, for an initial coherent stateof 200 atoms in the middle well. We see that the maximum variances are much largerthan in Fig. 2, and that all variances are periodic in the non-interacting case. ose-Hubbard Mode Splitter et al [5] have extended this inequality to provide indicators of EPR steering [14, 36, 37]and Bell violations [38]. We now define the correlation function ξ = h ˆ a † ˆ a ih ˆ a ˆ a † i − h ˆ a † ˆ a ˆ a † ˆ a i , (11)for which a positive value reveals entanglement between modes 1 and 3. We easily seethat ξ gives a value of zero for two independent coherent states and a negative resultfor two independent Fock states. This inequality, and the EPR-steering development ofit, have been shown to detect both inseparability and asymmetric steering in a three-well Bose-Hubbard model under the process of coherent transfer of atomic population(CTAP) [22, 23]. In our non-interacting case, with all population initially in well 2, wefind the analytic result ξ = 14 sin Ω t h h ˆ N (0) i − V ( ˆ N (0)) i , (12)so that this measure detects entanglement whenever the initial population in the middlewell is in a sub-Poissonian state, with the measure being maximised for a number state.The signature of entanglement identically vanishes for an initial coherent state, whichis to be expected since our system is somewhat analogous to a beamsplitter, with linearcouplings between the modes [27].Cavalcanti et al [5] further developed the work of Hillery and Zubairy to findinequalities for which the violation denotes the possibility of EPR-steering and Bellstates. The EPR-steering inequality for two modes is written as |h ˆ a i ˆ a † j i| ≤ h ˆ a † i ˆ a i (ˆ a † j ˆ aj + 12 ) i , (13)while the Bell state inequality is written as |h ˆ a i ˆ a † j i| ≤ h (ˆ a † i ˆ a i + 12 )(ˆ a † j ˆ aj + 12 ) i , (14)Calling on the overworked Alice and Bob, if Alice measures mode i and Bob measuresmode j a violation of the inequality (13) signifies that Bob would be able to steer Alice,and vice versa for a swapping of the modes. These inequalities allow us to define acorrelation function which signifies the presence of EPR-steering when it has a value ofgreater than zero,Σ ij = h ˆ a i ˆ a † j ih ˆ a † i ˆ a j i − h ˆ a † i ˆ a i (ˆ a † j ˆ aj + 12 ) i , (15)and another for which a positive value signifies the presence of Bell correlations, ζ ij = h ˆ a i ˆ a † j ih ˆ a † i ˆ a j i − h (ˆ a † i ˆ a i + 12 )(ˆ a † j ˆ aj + 12 ) i . (16)For the EPR-steering correlation, we can solve the Heisenberg equations to findΣ = Σ = 14 sin Ω t (cid:0) sin Ω t − (cid:1) h ˆ N (0) i − V ( ˆ N (0)) sin Ω t, (17) ose-Hubbard Mode Splitter V ( ˆ X ( t )) = 14 (cos Ω t + 1) V ( ˆ X (0)) + 12 sin Ω t V ( ˆ Y (0)) + 14 (cos Ω t − V ( ˆ X (0)) ,V ( ˆ Y ( t )) = 14 (cos Ω t + 1) V ( ˆ Y (0)) + 12 sin Ω t V ( ˆ X (0)) + 14 (cos Ω t − V ( ˆ Y (0)) ,V ( ˆ X ( t )) = 12 sin Ω t V ( ˆ Y (0)) + cos Ω t V ( ˆ X (0)) + 12 sin Ω t V ( ˆ Y (0)) ,V ( ˆ Y ( t )) = 12 sin Ω t V ( ˆ X (0)) + cos Ω t V ( ˆ Y (0)) + 12 sin Ω t V ( ˆ X (0)) ,V ( ˆ X ( t )) = 14 (cos Ω t − V ( ˆ X (0) + 12 sin Ω t V ( ˆ Y (0)) + 14 (cos Ω t + 1) V ( ˆ X (0)) ,V ( ˆ Y ( t )) = 14 (cos Ω t − V ( ˆ Y (0)) + 12 sin Ω t V ( ˆ X (0)) + 14 (cos Ω t + 1) V ( ˆ Y (0)) . In experimental quantum optics, these quadrature variances are measured via homodynedetection, which is not as simple for massive particles, although at least two methodshave been proposed [24, 25].We now investigate the Duan-Simon correlations between wells 1 and 3, withinseparability being detected when V ( ˆ X ± ˆ X ) + V ( ˆ Y ∓ ˆ Y ) < , (18)where for simplicity of expression we have dropped the time variable. To express theseparticular correlations, we also need the quadrature covariances, which are found as V ( ˆ X , ˆ X ) = 14 (cid:0) cos Ω t − (cid:1) h V ( ˆ X (0)) + V ( ˆ X (0)) i + 12 sin Ω t V ( ˆ Y (0)) ,V ( ˆ Y , ˆ Y ) = 14 (cid:0) cos Ω t − (cid:1) h V ( ˆ Y (0)) + V ( ˆ Y (0)) i + 12 sin Ω t V ( ˆ X (0)) . (19)Setting DS ± = V ( ˆ X ± ˆ X ) + V ( ˆ Y ∓ ˆ Y ), we find DS + = V ( ˆ Y (0)) + V ( ˆ Y (0)) + cos Ω t h V ( ˆ X (0)) + V ( ˆ X (0)) i + 2 sin Ω t V ( ˆ Y (0)) ,DS − = V ( ˆ X (0)) + V ( ˆ X (0)) + cos Ω t h V ( ˆ Y (0)) + V ( ˆ Y (0)) i + 2 sin Ω t V ( ˆ X (0)) , (20)which can then be minimised with respect to time. We find that the minimum foreither correlation is at 4, so that inseparability for our initial conditions is not foundby this measure, irrespective of the initial quantum state of the atoms in well 2. Wealso found that the Reid EPR inequalities [8] showed no evidence of the EPR paradox.This is despite the fact that entanglement is present according to the Hillery-Zubairycriteria [4], and emphasises the importance of using the correct correlations to detectcontinuous-variable entanglement in a given system. ose-Hubbard Mode Splitter Jt V a r i an c e s -N )V(N ) V(N ) Figure 4. (Colour online) The number variances as a function of time, for J = 1, χ = 10 − , and N (0) = 200 in a Fock state, with N (0) = N (0) = 0. The resultsshown are the average of 1 . × stochastic trajectories. The non-interactinganalytical results are only the same at short times and we see that the amplitudesof the oscillations grow with time, despite the fact that a χ (3) nonlinearity preservesthe number statistics in an isolated well. In the interacting case, our method of choice is to use numerical stochastic integrationto find solutions of the full positive-P representation equations [1]. This allows us tocalculate the expectation values of any operator moments that can be written in normalorder. Taking into account the normal ordering, the number variances are written as V ( ˆ N j ) = α + 2 j α j + α + j α j − α + j α j ,V ( ˆ N − ˆ N ) = V ( ˆ N ) + V ( ˆ N ) − V ( ˆ N , ˆ N ) , = V ( ˆ N ) + V ( ˆ N ) − (cid:16) α † α α † α − α +1 α × α +3 α (cid:17) . (21)All of these give values of zero for uncorrelated Fock states or vacuum. Wheneverone of the variances is less than the mean population of the corresponding mode, wehave suppression of number fluctuations below the Poissonian coherent state level. Theindividual quadrature variances are found as V ( ˆ X i ) = 1 + 2 α + i + α i + α + 2 i − α i + α + i ,V ( ˆ Y i ) = 1 + 2 α + i − α i − α + 2 i − − i ( α i − α + i ) , (22)with the combined quadrature variances and covariances needed for the Duan-Simonand Reid correlations being the obvious extensions of these. ose-Hubbard Mode Splitter Jt ξ -50-40-30-20-1001020304050 Fock coherent Figure 5. (Colour online) The Hillery-Zubairy criteria as a function of time, for J = 1, χ = 10 − , and N (0) = 200 in initial Fock state and coherent states, with N (0) = N (0) = 0. The results shown for the initial Fock state are the average of1 . × stochastic trajectories and those for the initial coherent state are averagedover 1 . × trajectories. The non-interacting analytical results are only the sameat short times and we see that the correlations degrade with time, with the coherentstate correlation showing no entanglement at any time. For our results in the interacting case, we have chosen a nonlinearity of χ = 10 − ,again with either a Fock or coherent state with an average of 200 atoms in themiddle well. These different quantum states are simulated using the methods foundin Olsen and Bradley [32]. We have simulated results for the numbers in each well(Fig. 1), the number variances (Fig. 4), the Hillery-Zubairy criteria (Fig. 5) and someof the various quadrature variance correlations canonically used to detect continuous-variable entanglement and EPR-steering. Those we present here are the Duan-Simoncriteria [6, 7] of Eq. 18 and the Reid EPR inequalities [8].The number variances for an initial Fock state are shown in Fig. 4, from whichwe can see that they take the same periodic form as in the non-interacting case ofFig. 2, but the amplitude of the oscillations grows in time. The results for an initialcoherent state follow the same pattern as in Fig. 3, but again with the maxima of theoscillations increasing with time. The increase in these variances is purely a result of thelinear coupling between the wells, since the collisional nonlinearity in an isolated modepreserves the number statistics. Although of the same strengths, the coupling betweenthe wells is independent, so that we see the statistics of ˆ N − ˆ N are initially Poissonian.The interaction of the collisional nonlinearity and the couplings causes the statistics to ose-Hubbard Mode Splitter Jt D uan - S i m on Figure 6. (Colour online) The Duan-Simon inseparability criteria as a functionof time, for J = 1, χ = 10 − , and N (0) = 200 in a coherent states, with N (0) = N (0) = 0. The solid line is V ( ˆ X − ˆ X )+ V ( ˆ Y + ˆ Y ) and the dash-dotted lineis V ( ˆ X − ˆ X ) + V ( ˆ Y + ˆ Y ) We see that inseparability is only indicated for short timesand that the violation of the inequality is not large. These results are the average of9 . × trajectories. become super-Poissonian with increasing interaction time.In Fig. 5 we show the Hillery-Zubairy criterion ξ for the detection of entanglementbetween wells 1 and 3. We see that an initial Fock state in the centre well means that thiscorrelation becomes periodically positive at early times, but that the entanglement signalis degraded over time. For an initial coherent state, this measure gives no indicationof entanglement. This is in contradiction with the results of Fig. 6 where consider aninitial coherent state and find a violation of Duan-Simon inequalities at short times.We note that using time dependent quadrature angles can maximise the violations,as shown previously for Kerr-squeezed optical states mixed on a beamsplitter [39, 40],but we have not considered this here since the inseparability signal for the canonicalquadratures is so weak. In any case, even an optimisation of the quadrature angles stillfinds no violation of the inequalities after a short time. We also calculated the ReidEPR criteria between wells 1 and 2 and 1 and 3, and found no evidence that EPR-steering is present in this system. In this case, the quadrature measures agree with thephase-independent measures of Eq. 13. ose-Hubbard Mode Splitter
4. The beamsplitter
Since our three well system, with one input mode and two output modes with linearcouplings, can loosely be compared to a beamsplitter with one non-zero input, itis informative to compare the performance of a standard optical beamsplitter usingthe same correlations. The equations relating the inputs and outputs of a losslessbeamsplitter can be written asˆ a out = √ η ˆ a in + p − η ˆ b in , ˆ b out = − p − η ˆ a in + √ η ˆ b in , (23)where √ η is the amplitude reflectivity. For reasons of simplicity, we will treat only abalanced beamsplitter, with η = 1 / h ˆ b † in ˆ b in i = 0. It is then trivial to see thattotal number is conserved, as it must be, and as happens for our three well system.One obvious difference is that the beamsplitter has two inputs and two outputs and thetransmission is not time dependent, but our main interest here is in the linear coupling.This is present in both systems.Examining firstly the correlation of Eq.11, we find that the left hand side is h ˆ a † out ˆ b out ih ˆ b † out ˆ a out i = 14 h ˆ a † in ˆ a in i , (24)and the right hand side of the expression is h ˆ a † out ˆ a out ˆ b † out ˆ b out i = 14 h h (ˆ a † in ˆ a in ) i − h ˆ a † in ˆ a in i i . (25)Combining these, we find ξ ab = 14 h h ˆ a † in ˆ a in i − V ( ˆ Na in ) i , (26)which, apart from the time dependence, is the same as the result of Eq. 12. This againshows that this measure will detect entanglement for any input state a for which thenumber fluctuations are less than Poissonian. Two coherent states will give a value ofzero, and therefore will not lead to entangled outputs. The two possible equivalents ofEq. 13 for the beamsplitter giveΣ ab = Σ ba = − V ( ˆ Na in ) , (27)which can obviously never be positive, so that these measures do not detect EPR-steering.We can also consider the Duan-Simon quadrature correlations [6, 7] using the sameapproach. We write the output quadratures in terms of the inputs asˆ X outa = √ η ˆ X ina + p − η ˆ X inb , ˆ Y outa = √ η ˆ Y ina + p − η ˆ Y inb , ˆ X outb = √ η ˆ X inb − p − η ˆ X ina , ˆ Y outb = √ η ˆ Y inb − p − η ˆ Y ina , (28)which alllows us to calculate the necessary quadrature moments analytically. ose-Hubbard Mode Splitter η = 1 /
2, and find V ( ˆ X outa ± ˆ X outb ) = V ( ˆ X ina ) + V ( ˆ X inb ) ± h V ( ˆ X inb ) − V ( ˆ X ina ) i ,V ( ˆ Y outa ∓ ˆ Y outb ) = V ( ˆ Y ina ) + V ( ˆ Y inb ) ∓ h V ( ˆ Y inb ) − V ( ˆ Y ina ) i , (29)so that the Duan-Simon correlations are V ( ˆ X outa + ˆ X outb ) + V ( ˆ Y outa − ˆ Y outb ) = 2 h V ( ˆ X inb ) + V ( ˆ Y ina ) i ,V ( ˆ X outa − ˆ X outb ) + V ( ˆ Y outa + ˆ Y outb ) = 2 h V ( ˆ X ina ) + V ( ˆ Y inb ) i . (30)For a squeezed amplitude input in mode a with variance V ( ˆ X ina ) = e − r and vacuum in b , the second of these gives a value of 2(1 + e − r ), therefore demonstrating inseparability.However, for an input Fock state in mode a and vacuum in b , these correlations predicta value of 4 Na in + 4, immediately contradicting the prediction of Eq. 26, showing onceagain the importance of using the correct inequalities for a given system.Using these analytic results, we can also find values for the Reid EPRcorrelations [8], for which V inf ( ˆ X outj ) V inf ( ˆ Y outj ) < j , we find for inputs of a squeezed state and vacuum,Γ a = Γ b = 21 + cosh r , (32)showing that the paradox is demonstrated and steering is possible as soon as we have asqueezed input. On the other hand, for inputs of a Fock state | N i and vacuum, we findΓ a = Γ b = ( N + 4 N + 2) N + 4 , (33)which has a minimum value of 1. We therefore see that the Reid measure does notsignify the presence of the EPR paradox in this case, in agreement with Eq. 27.
5. Conclusions
We have shown that our three-well Bose Hubbard system produces entanglementbetween the atoms in two non-adjacent wells, but not at a sufficient level to demonstratethe EPR paradox via the measures we have investigated here. As we have demonstrated,different measures can lead to different indications as to whether inseparability andentanglement are present. Those based on the Hillery-Zubairy results perform betterfor an initial Fock state than for an initial coherent state in the middle well, for which thequadrature based correlations have a superior performance. This is in agreement withthe claim that the Hillery-Zubairy measures are superior for processes which conservenumber. Due to the sufficient but not necessary nature of the inequalities used, wecannot say that a demonstration of EPR-steering is impossible with this system, only ose-Hubbard Mode Splitter
Acknowledgments
This research was supported by the Australian Research Council under the FutureFellowships Program (Grant ID: FT100100515).
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