Entanglement Concentration for two atomic ensembles using an effective atom-light beamsplitter
aa r X i v : . [ qu a n t - ph ] A ug Entanglement concentration for two atomicensembles using an effective atom-light beamsplitter
R Tatham and N Korolkova
School of Physics and Astronomy, University of St Andrews, Fife, ScotlandE-mail: [email protected], [email protected]
Abstract.
We present a protocol for increasing the entanglement between twoentangled atomic ensembles based on applying an approximate atom-light beamsplittertransformation to both ensembles. The effective asymmetric atom-light beamsplitter iscreated via a double-pass quantum non-demolition interaction between polarized lightand a spin polarized atomic ensemble, derived from the linearised dipole interaction.The entanglement concentration protocol itself uses the procrustean method, similar tothat first devised for light by Browne et al [2003
Phys. Rev. A Submitted to:
J. Phys. B: At. Mol. Opt. Phys. ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter
1. Introduction
Quantum communication schemes, from quantum key distribution to teleportation anddense coding, offer better ways to exchange information in a network. Experimentsinto these areas are abundant and have yielded lots of successes over small distances.However, over larger distances, optical losses, phase diffusion, and mixing with thermalstates cause the signals to decohere over some finite transmission length. In fact, theerror probability scales exponentially with the length of the channel.The most obvious way to overcome this would be to create a series of signalamplifiers to be spaced out between the source and destination where the signal couldbe stored onto a quantum memory device and read out again. Such quantum repeaterprotocols have been devised [1] with slight variances between schemes, but are in generalreliant on entanglement distillation procedures and quantum memory devices that canbe used to store a signal for a brief period of time.Due to the long lifetimes of atomic states, the most promising protocol so far usesthe coherent spin states of cesium atoms to serve as a memory device. The quantumsignal to be transmitted is contained in the polarization state of the incoming light modeand this information is written onto the macroscopic coherent spin state of the atomicensemble. Julsgaard et al [2] successfully showed that the coherent spin states of twoatomic ensembles could become entangled in such a way as to be analogous to the twomode squeezed state for light, written in the number basis | TMSS i = √ − λ ∞ X n =0 λ n | n i | n i (1)where n is the photon number and λ quantifies the reduction (squeezing) of the quantumuncertainty of the global state. We will discuss the particular meaning of these quantitiesin the case of atomic ensembles in Section 3. The better entangled the ensembles, thebetter the outcome of the quantum repeater protocol. However the entanglement of twoatomic ensembles is at present limited and yields teleportation fidelities of < . ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter purification . That is, they begin with amixed entangled state containing some non-Gaussian noise and aim at reducing thatnoise thus increasing the degree of entanglement. In contrast, the original procrusteanscheme [8, 11, 12], the schemes based on the Kerr nonlinearity [5, 13] and the atomicscheme presented here deal with entanglement concentration . That is, the aim is theincrease of entanglement content in an initially pure Gaussian partially entangled state.In Section 2 the quadrature system and interaction used in the scheme are described.In Section 3 the entanglement concentration is shown with losses taken into account inSection 4. In Section 5 we conclude with discussion of the performance of the scheme.
2. The system and interactions
The standard QND Hamiltonian couples one of the two effective quadrature fieldoperators of the strongly polarized light mode (re-scaled polarization variables) withone of the effective quadratures of the spin-polarized atomic ensemble, that is with acertain component of the re-scaled collective spin operator (for review see [14]). Thepolarization variables are defined as follows. We assume the light beam to be stronglypolarized in the x -direction so that the actual polarization variable, the Stokes operatorˆ S x , can be replaced by its expectation value h S x i . The Stokes operators can then bedefined, e.g. for a pulse travelling in the z-direction, byˆ S x = c Z T (cid:0) ˆ a † x ˆ a x − ˆ a † y ˆ a y (cid:1) dτ ≈ h S x i = A x , (2)ˆ S y = c Z T (cid:0) ˆ a † x ˆ a y + ˆ a † y ˆ a x (cid:1) dτ, (3)ˆ S z = c i Z T (cid:0) ˆ a † x ˆ a y − ˆ a † y ˆ a x (cid:1) dτ (4)where ˆ a x,y ≡ ˆ a x,y ( z, t ) are the annihilation operators for photons linearly polarized inthe x - and y - directions respectively with h ˆ a i ( z, t ) , ˆ a † j ( z ′ , t ) i = δ ij δ ( z − z ′ ) δ ( t − t ′ ) /c and A x is the real expectation value of ˆ a x and ˆ a † x when highly polarized. T is theinteraction time and τ = t − z/c . With this in mind we can define continuous variablequadratures for the light state ˆ X L and ˆ P L asˆ X L = ˆ S y p h S x i , ˆ P L = ˆ S z p h S x i . (5)Here h ˆ X L , ˆ P L i = i as with conjugate position and momentum and we have set ~ = 1.Note that these quadratures describe the quantum polarization of light and are not theusual amplitude and phase quadratures.A similar description for atoms is possible, also using appropriate, re-scaled“quadratures”. Let us first define the relevant variables. The collective angularmomentum of the atomic ensembles ˆ J , which has components ˆ J j , j = x, y, z obey the ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter N a cesium atoms at room temperature with a ground state degeneracy asused in Julsgaard et al [2]. There, in a homogenous magnetic field the cesium atomswere pumped into the | F = 4 , m F = 4 i state in the first cell and | F = 4 , m F = − i in the second cell to form coherent spin states oriented in the + x and − x directionsrespectively. In this way, the collective angular momentum in the x -direction can also bereplaced by the expectation value and D ˆ J x E = − D ˆ J x E with subscripts 1 , σ µ,ν = | µ i h ν | the operators canbe written as ˆ J x = N a X m F m F ˆ σ m F ,m F , (6)ˆ J y = N a X m F C ( F, m F ) (ˆ σ m F +1 ,m F + ˆ σ m F ,m F +1 ) , (7)ˆ J z = N a i X m F C ( F, m F ) (ˆ σ m F +1 ,m F − ˆ σ m F ,m F +1 ) (8)where C ( F, m F ) = p F ( F + 1) − m F ( m F + 1) . (9)As the atomic ensemble is spin-polarized, conjugate position and momentumquadratures for atoms (subscript A ) can be defined as (e.g.[14],[15]):ˆ X A = ˆ J y p h J x i , ˆ P A = ˆ J z p h J x i . (10)The interaction between light and atoms is represented by the linearised dipoleinteraction with far-off detuning (off-resonant interaction):ˆ H = X j − d j · E ( R j ) (11)where d j = − er j is the dipole operator for the j th atom and R j is the location of the j th atom. If, for example, the polarized light propagates in the z -direction through anatomic ensemble, the linearised interaction Hamiltonian can be written asˆ H = a Z T ˆ S z ( t ) ˆ J z ( t ) dt ≈ κ ˆ P L ˆ P A (12)where a is a coupling constant and κ = a p h S x i T h J x i . Any higher order coupling termsare negligible if the laser beam is far detuned from the transition frequencies.Equation (12) is an example of a Quantum Non-Demolition (QND) Hamiltonianwhich would result in a phase shift in the ˆ X L quadratures of the light by + κ ˆ P A as itpasses through the atoms. There is a corresponding back action on the distribution of ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter κ ˆ P L in the ˆ X A quadratureof the atomic ensemble. Physically, the interaction causes the polarization of the lightto rotate about the axis of propagation, dependant on the quadrature distribution ofthe atoms. The back action effect on the atoms is to rotate the macroscopic spin statearound the axis of propagation.Whereas a single pass of a light pulse through an atomic medium corresponds tothe simple QND interaction described above, multiple passes open the possibility for alarger design freedom for the effective Hamiltonian, as for each pass a particular form ofthe underlying QND interaction can be adjusted (see e.g. [16]). A double pass scheme[17] can be used for the generation of polarization squeezed light by optical Faradayrotation. Another double pass scheme was suggested and thoroughly studied in thecontext of quantum memory for light modes based on macroscopic atomic ensembles atroom temperature, as well as for the generation of entanglement between light and atoms[18]. There, the interaction Hamiltonian has been shown to include two main parts,one equivalent to the beamsplitter interaction, and the other to two-mode squeezing.Depending on the geometry of the setup, either of the two underlying dynamics canbe selected. In this paper we exploit the fact that the double-pass Hamiltonian canapproximate, with high fidelity, an actual beamspitter transformation, the quality ofthe approximation being dependent not only on the interaction strength, but also onthe particular quantum states of the interacting modes [19]. Tuning the interactionparameter, a highly asymmetric atom-light beamsplitter can be realized. In whatfollows, we describe the entanglement concentration scheme based on such an atom-light beamsplitter and analyse the performance of the scheme.
3. Entanglement Concentration
In the continuous variable regime, to increase the entanglement between two quantumobjects with Gaussian quadrature distributions a non-Gaussian element is required.The measurement process offers this opportunity. Analogous to the photon subtractionschemes examined in [20] and [8] for increasing the entanglement in two mode squeezedstates of light, a photon count heralds an increase in entanglement between twomacroscopic atomic ensembles.In the case of the two mode squeezed state for light, the scheme is fairly simple, thetwo correlated beams are sent to separate highly transmissive beamsplitters, upon whichthey combine with another mode (usually the vacuum). There is a small probabilitythat a photon will be subtracted from the main beam and proceed to be detectedby a photon counter. If the counters at both beamsplitters register the presence ofphotons, then the entanglement of the two mode squeezed state increases. We wish fora similar “photon subtraction” scheme for our light and atoms in order to increase theentanglement between the two atomic ensembles as represented in Figure 1.The two atomic ensembles are prepared in an entangled two mode squeezed stateas described in [2] . Previous works [2, 15, 21] have detailed the methods for entangling ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter Figure 1.
Procrustean entanglement concentration for atomic ensembles: theprinciple. The entanglement concentration protocol works similar to the light scheme[11, 12]. The two ensembles are entangled like a TMSS. Then x -polarized light interactswith the atoms (number basis | i ) and non-Gaussian measurements are done to seewhether the polarization of the photons has altered. A positive response from bothdetectors heralds an increase in entanglement between the atomic ensembles. The twosquare boxes symbolize the effective atom-light beamsplitter based on the double QNDinteraction. Figure 2.
Procrustean entanglement concentration for atomic ensembles: details ofatom-light interactions. Light is strongly polarized in the x -direction. It interactswith the spin-polarized atomic ensemble in a double-pass interaction equivalent tothe effective atom-light beamsplitter which probabilistically accomplishes photonsubtraction in a light beam. A y -polarized photon detection at the output impliesthat the atomic entanglement has increased. the collective atomic spins of the atomic ensembles using equations of the form (12). Bysending a polarized light mode through the atomic ensembles and taking a homodynemeasurement, it is possible to collapse the atomic states into an entangled two modesqueezed state with variance ∆ (cid:16) ˆ X A − ˆ X A (cid:17) = ∆ (cid:16) ˆ P A + ˆ P A (cid:17) = e − r where r isdependent on κ and given by r = 12 ln (cid:0) κ (cid:1) . (13)Further, it is assumed that any displacement of the joint atomic quadrature distributions ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter λ = tanh( r ) is the squeezing parameter, dependent on the interaction strength κ between a light mode and atomic ensemble during the initial entanglement processvia the relation (13). The number basis is considered to be the basis generated by theannihilation operator ˆ a A = ˆ X A + i ˆ P A up to normalisation. On consideration the number n represents the number of atoms in the upper excited spin state in the basis of the ˆ J x operator. That is, n = 1 corresponds to the superposition of all possible combinationsof atomic spins of the atoms in the ensemble for which a single atom is excited.If the light is initially prepared in a symmetric Gaussian state centred at X L = 0, P L = 0 then the light can be considered to be in a vacuum state. That is, there are nophotons polarized in the ˆ S y or ˆ S z directions despite a steady base stream of linearly x -polarized photons so that h S y i = h S z i = 0. In the number basis generated by operatorsˆ S + = ˆ S y + i ˆ S z = p h S x i ( ˆ X L + i ˆ P L ) and ˆ S − = ˆ S † + it is clear that a Fock state is simplythe number of photons polarized linearly in the y -direction.In the number basis, the functionality of the photon subtraction scheme can bereadily explained. If, after the beamsplitter-like interaction, one or more photons aredetected behind the polarized filter, then the interaction has rotated the polarization ofsome photons in the light mode. The corresponding back action on the atomic ensemblesis to flip the spin of one or more of the atoms in the system (see Fig. 2).The initial state of the light and atoms is given by | Ψ I i LA = √ − λ ∞ X n =0 λ n | n i | n i | i | i (14)where the first and second atomic modes are subscripted 1 and 2 respectively andthe light vacuum modes are subscripted 3 and 4. The light vacuum modes 3 and 4(i.e. modes that are only polarized in the x -direction), are sent through the atomicensembles 1 and 2 respectively and made to interact twice with the ensembles via thebeamsplitter interaction described in [19]. That is, in one atom-light “beamsplitter”two QND interactions are performed. The first is of the form ˆ H = φ ˆ P L ˆ X A and theoutgoing light modes are redirected back into the atomic ensembles to interact via asecond interaction, ˆ H = − φ ˆ X L ˆ P A . We use the reasonable approximations on thequadratures that x l ≈ p − φ x l and (1 − φ ) x a ≈ p − φ x a for small φ with similarconditions on p l and p a [19]. This double-pass scheme then performs the role of abeamsplitter transformation and is treated accordingly in what follows. The state afterthe beamsplitter interaction can be described by | Ψ F i LA = ∞ X n =0 √ − λ λ n × n X k ,k =0 s(cid:18) nk (cid:19)(cid:18) nk (cid:19) × φ k + k (cid:0) − φ (cid:1) n − k − k × | n − k i | n − k i | k i | k i (15)where φ is the strength of the interaction. ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter x -polarized photons and allowing only photons whose polarizationshave been rotated by the interaction until y -polarized to proceed (Fig. 2). A detectorthen registers whether y -polarized photons are present. As detectors are still relativelyinefficient at counting photons, it is instead assumed that the detectors used can, toa high degree of efficiency, detect simply the presence of one or more photons (“on-offdetector”). The state thus becomes: | Ψ A i out = r − λ S ∞ X n =0 ∞ X u,v =1 λ n s(cid:18) nu (cid:19)(cid:18) nv (cid:19) × φ u + v (cid:0) − φ (cid:1) n − u − v | n − u i | n − v i (16)(17)where S is the probability of getting an affirmative measurement at both detectors: S = 1 − λ − λ (cid:16) φ + (1 − φ ) (cid:17) − (cid:0) − λ (cid:1) − λ (1 − φ ) (cid:16) φ + (1 − φ ) (cid:17) + 1 − λ − λ (1 − φ ) . (18)(19) The amount of entanglement in the two ensembles is quantified by the negativity andlogarithmic negativity [22] of the state defined by N ( ˆ ρ ) = 12 Tr (cid:16)p ( ˆ ρ P T ) − ˆ ρ P T (cid:17) = (cid:13)(cid:13) ˆ ρ P T (cid:13)(cid:13) − , (20) E N ( ˆ ρ ) = ln(1 + 2 N ( ˆ ρ )) = ln( (cid:13)(cid:13) ˆ ρ P T (cid:13)(cid:13) ) (21)where k·k denotes the trace-norm, i.e. p ˆ ρ ˆ ρ † , and ˆ ρ P T is the partial transpose of thedensity matrix ˆ ρ .It is simple to show that for a two mode squeezed state, the negativity andlogarithmic negativity are given as N (TMSS) = λ − λ , (22) E N (TMSS) = ln(1 + λ ) − ln(1 − λ ) . (23)For projections onto an exact photon state (e.g., for a single photon detection), thenegativity and logarithmic negativity can be calculated analytically. However, for on-off type measurements the entanglement measures must be calculated numerically dueto the infinite sums over u and v . Fortunately, these sums appear to converge quicklyand so we can set a reliable truncation point. The calculation is done in a similar way ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter et al [20]. Firstly, the density matrix of the state isexpanded as | Ψ A i out h Ψ A | = ∞ X a,b,c,d ρ a,b,c,d | a i h c | ⊗ | b i h d | (24)where ρ a,b,c,d = ( h a | h b | ) | Ψ A i out h Ψ A | ( | c i | d i )= (1 − λ ) S ∞ X u,v =1 λ a + u λ c + u × s(cid:18) a + uu (cid:19)(cid:18) a + uv (cid:19)(cid:18) c + uu (cid:19)(cid:18) c + uv (cid:19) × ( φ ) u + v ) (cid:0) − φ (cid:1) a + c + u − v ) δ a − b,v − u δ c − d,v − u . (25)(26)The partial transpose of this state is given by( | Ψ A i out h Ψ A | ) P T = ∞ X a,b,c,d ρ a,d,c,b | a i h c | ⊗ | b i h d | (27)and the elements are zero unless the total Fock number of the entangled state, N = a + b = c + d , is non-zero. This follows from the delta functions in ρ a,d,c,b .Operator (27) is block diagonal and we can write it as a direct sum of each N -dependentsubmatrix: ( | Ψ A i out h Ψ A | ) P T = ⊕ ∞ N =0 ( | Ψ A i out h Ψ A | ) P T ( N ) (28)where ( | Ψ A i out h Ψ A | ) P T ( N ) is the ( N + 1) × ( N + 1) N th submatrix.The negativity of the partially transposed state is then computed by numericallydiagonalising each block individually to obtain the eigenvalues of each submatrix andadding up the absolute value of all negative eigenvalues. A cut-off, N max , must beintroduced that is large enough compared to the mean number of excited spins. Thereis of course a trade-off between N max and the length of time needed to perform thecalculation. For the purposes of this calculation, a value of N max = 100 gave veryprecise results. That is, at N max = 100 the numerical values of logarithmic negativityconverge to at least 7sf and to increase N max beyond this is not beneficial. The resultsare shown in Figure 3.The entanglement between the two atomic ensembles is increased for all valuesof initial squeezing except for very high λ . This increase in E N is not very large butcomparable to when light modes are used in place of atomic ensembles [20], as arethe probabilities shown in Figure 4. As can be seen, a trade-off is required betweeninteraction strength and probability of success. As interaction strength increases, sodoes the probability of success, but the validity of the beamsplitter approximationdecreases. Note, however, that the beamsplitter approximation has a very high fidelityof approximately 0.99 even for interaction strengths as high as φ ≈ .
35. For moderate ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter l L og a r it h m i c N e g a ti v it y Figure 3.
A plot of the logarithmic negativity against λ for (i) standard twomode squeezed state (solid) and for the output of the photon subtracted schemefor the beamsplitter like interaction between light and atomic ensemble when(ii) φ = 0 . φ = 0 .
01 (dotted). See text for discussion. initial squeezing, E N is largely unaffected by the interaction strength. As λ approaches ≈ .
95, the concentration procedure ceases to bring further benefit. However, this onlyoccurs at exceptionally high squeezing of the atoms, which is not experimentally viable.
4. Modelling Detector Inefficiencies
The largest contribution to loss in the photon subtraction scheme for light modes comesfrom detector inefficiency. For the atomic ensemble scheme the efficiency of the detectorswill also play a crucial role. The detectors here have a reduced number of photonsto detect due to the polarization filter used to stop the base stream of x -polarizedphotons. The inefficiency of the detector can be modelled as an ideal detector behinda beamsplitter of transmittivity η = ν . The light mode is combined with a vacuumon a beamsplitter and the vacuum is traced out before the projection measurementis performed. For a Fock state | k i combining with a vacuum, this amounts to thetransformation | k, i → k X s =0 s(cid:18) ks (cid:19) ν s (cid:16) √ − ν (cid:17) k − s | s, k − s i . (29) ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter . . . . . . . . l P r ob a b ilit y Figure 4.
Probability of success against λ for (i) interaction strength φ = 0 . φ = 0 .
05 (dashed) and (iii) φ = 0 .
01 (dotted). The probabilities ofsuccess are small but comparable with the light scheme.
Directly before the detection is performed, the state of the density matrix is given by ρ = (cid:0) − λ (cid:1) ∞ X m,n =0 λ m + n n X k ,k =0 m X j ,j =0 s(cid:18) nk (cid:19)(cid:18) mj (cid:19)(cid:18) nk (cid:19)(cid:18) mj (cid:19) × φ j + k + j + k (cid:0) − φ (cid:1) n +2 m − j − k − j − k k X s =0 j X t =0 k X y =0 j X z =0 N k ,k ,j ,j s,y,t,z | n − k i h m − j | ⊗ | n − k i h m − j | ⊗ | s i h t | ⊗ | y i h z | (30)where N k ,k ,j ,j s,y,t,z = s(cid:18) k s (cid:19)(cid:18) j t (cid:19)(cid:18) k y (cid:19)(cid:18) j z (cid:19) (cid:16) √ − ν (cid:17) j + k + j + k − s − t − y − z × ν s + t + y + z δ k − s,j − t δ k − y,j − z . (31)Light modes 3 and 4 are subsequently measured for the presence or absence of photonsusing the operator ( − | ih | ) and traced out. The density matrix of the two remainingatomic modes can then be described by ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter ρ out ,η = ∞ X m,n =0 (cid:0) − λ (cid:1) λ m + n min( m,n ) X k ,k =0 s(cid:18) nk (cid:19)(cid:18) mk (cid:19)(cid:18) nk (cid:19)(cid:18) mk (cid:19) φ k + k ) × (cid:0) − φ (cid:1) n +2 m − k − j h − (1 − η ) k − (1 − η ) k + (1 − η ) k + k i | n − k i h m − k | ⊗ | n − k i h m − k | (32)and the probability of success, taking into account detector losses, is given by S η = 1 − λ − λ (cid:2) φ + (1 − φ ) (cid:3) − − λ )1 − λ (cid:2) φ (1 − η ) + (1 − φ ) (cid:3) (cid:2) φ + (1 − φ ) (cid:3) + 1 − λ − λ (cid:2) φ (1 − η ) + (1 − φ ) (cid:3) . (33)(34)The effect that detector inefficiency has on the logarithmic negativity of the atomicensembles can be shown in Figure 5.As expected, the entanglement concentration becomes less pronounced for lowdetector efficiency. The positive message is that even for efficiencies as low as η = 0 . λ values for which entanglement is increased (although theprobability of success is quite low in this case). This range is experimentally accessibleand so it is good news that the entanglement concentration protocol is more robustagainst imperfections.
5. Conclusions
In this paper, we have employed a beamsplitter-like QND interaction between light andatomic ensembles based on the linearised dipole interaction between strongly polarizedlight and atomic levels for increasing the entanglement between two atomic ensembles.The entangled atomic ensembles in the initial two mode squeezed state interact witha highly polarized light mode in a quantum vacuum state via effective atom-lightbeamsplitter. The output light modes are subsequently detected using on-off detectors.This is analogous to the procrustean entanglement concentration scheme [11, 12] basedon photon subtraction for distilling entanglement in light modes. Similar to their scheme,in our protocol detector clicks at both light outputs herald the successful “photonsubtraction” (spin-flip in the atomic ensembles) and thus successful entanglementdistillation. To assess the performance of the atomic entanglement distillation scheme,we have calculated the logarithmic negativity for the output quantum state of thetwo atomic ensembles and shown that it can increase indicating that the entanglementconcentration procedure has been successful. The probability of success is very smallbut comparable to the probabilities for the corresponding light schemes that have been ntanglement concentration for two atomic ensembles using an effective atom-light beamsplitter l L og a r it h m i c N e g a ti v it y Figure 5.
Dependence of the logarithmic negativity on the efficiency of the detectors.The solid line depicts the initial two mode squeezed state of the entangled atomicensembles. Then the logarithmic negativity is shown for η = 1 (dashed line), η = 0 . η = 0 . η = 0 . φ = 0 . demonstrated experimentally [8]. The resulting atomic states are non-Gaussian and itremains to be seen whether they can be used for any teleportation procedures withoutdevising a way to re-Gaussify the system. Use of the beamsplitter approximation allowsus to closely mimic the entanglement concentration scheme that exists for light and toget an idea of how the light-atom protocol performs in comparison. The next step wouldbe to remove the approximation and to assess the scheme for stronger interactions buta different approach has to be applied in this case. Acknowledgments
The research has been supported by the EU STREP project COMPAS FP7-ICT-2007-C-212008 under the FET-Open Programme, by the Scottish Universities Physics Alliance(SUPA) and by the Engineering and Physical Sciences Research Council (EPSRC).
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