aa r X i v : . [ qu a n t - ph ] J a n Modular quantum computing andquantum-like devices
R.Vilela Mendes ∗ CMAFCIO, Faculdade de Ciˆencias, Universidade de Lisboa
Abstract
The two essential ideas in this paper are, on the one hand, thata considerable amount of the power of quantum computation may beobtained by adding to a classical computer a few specialized quan-tum modules and, on the other hand, that such modules may beconstructed out of classical systems obeying quantum-like equationswhere a space coordinate is the evolution parameter (thus playing therole of time in the quantum algorithms).
Keywords: Quantum computation, Quantum Fourier transform, Oracles,Fiber and wave-guide optics
Classical, probabilistic and quantum computing are three computing modal-ities which, adopting a Turing Machine-like scheme [1] [2], may be brieflydescribed in the following way:Let M be a states machine with one working tape with alphabet Γ andan input tape with alphabet Σ. At each time the machine configuration c isthe content of the working tape, the position of two pointers (in the inputand working tapes) and the current state. Let C ( x ), of cardinality N , be theset of all possible configurations when the input is x . ∗ [email protected]; [email protected]; http://label2.ist.utl.pt/vilela/
1t each time step the machine, in a state q ∈ Q , reads a symbol σ ∈ Σ inthe input tape and the current symbol γ ∈ Γ in the working tape, changes toa state q ′ ∈ Q , prints a symbol γ ′ ∈ Γ in the working tape and the pointersmove right ( R ) or left ( L ) in the respective tapes. The probability of theseoperations is controlled by a mapping T from C ( x ) into a space ST : Q × Σ × Γ × Q × Γ × { L, R } → S This mapping is called the transition function from which the transition prob-ability between c i and the next c i +1 configuration p ( c i , c i +1 ) = F ( T ( c i , c i +1 ))may be obtained. In all cases it is assumed that the internal state of the ma-chine is not observed except at the final time of the calculation. The mapping T defines a matrix in the space of configurations C . T (cid:16) q j , γ j , p (1) j p (2) j | q i , γ i , p (1) i p (2) i (cid:17) ⊜ T ( c j , c i )The three computation models correspond to different choices of T andof p ( c i , c i +1 ) = F ( T ( c i , c i +1 )) Classic deterministic computation: S = { s : s = 0 , } p ( c j , c i ) = T ( c j , c i ) = s (1)Only one element in each line of the transition matrix T is different fromzero. Classical probabilistic computation: S = { s : s ∈ [0 , } p ( c j , c i ) = T ( c j , c i ) = s (2)with the condition X j T ( c j , c i ) = 1 (3) T is a stochastic matrix preserving the L norm in the space of configurations. Quantum computation: S = (cid:8) s ∈ C : | s | = 1 (cid:9) p ( c j , c i ) = | T ( c j , c i ) | = | s | (4)2ith the condition X j | T ( c j , c i ) | = 1 (5)that is, T is a unitary matrix preserving the L norm in the space of config-urations.In all cases the transition probabilities between initial and final states arepositive and normalized. The difference between the three computationalmodels is the method used to find the transition probabilities.Physical implementations of the computational models require physicalelements for coding , interaction between the elements to perform the writ-ing and change of states and finally an evolution process to represent thetransition function. Coding, interaction and evolution . And, in each case,the evolution should be such as to satisfy the constraints (1) or (3) or (5).Some quantum systems, when sufficiently isolated from the environment,because their coherent time-evolution is unitary, provide physical modelsof quantum computation. However, quantum computation is not quantummechanics. Any other system, that provides coding, interaction and a changeof states compatible with (4) (5), may also provide a model of quantumcomputation. In particular the state evolution of these systems should beunitary. Such systems have been called quantum-like.In Ref.[3] it has been proposed that classical paraxial light propagation,being ruled by a Schr¨odinger-like equation may also provide a model of quan-tum computation. There is, of course, no contradiction with the physicalrules of quantum mechanics because in the classical paraxial system thepropagation is along a space coordinate which plays the same role as timein the quantum mechanical Schr¨odinger equation. As a consequence thetransfer function may be implemented by the unitary propagation of infor-mation along a space coordinate. Considering the coding and interactionrequirements, a good candidate for this implementation seems to be fiber orwave-guide optics.The idea of using quantum-like systems for quantum computation andsimulation of quantum effects has been later explored (see for example [4] -[12]) by several authors.
Although it has not yet been rigorously proven that
BP P ( BQP , thatis, that quantum circuits cannot be efficiently simulated in a bounded-error3robabilistic machine, the quantum oracle algorithms, that have been de-veloped, provide circumstantial evidence that quantum computing is indeedmore efficient than classical computing.The power of quantum computing hinges both on the capacity to dealwith superpositions of many different states (quantum parallelism) and onthe enhancement of particular computational paths (quantum interference).The following three resources are responsible for the efficiency of the knownquantum algorithms:(i) Preparation of a linear superposition of all possible basis states P x | x i ;(ii) Call to a reversible oracle operation X x | x i | ψ i → X x | x i | f ( x ) ⊕ ψ i = X x | x i U f ( x ) | ψ i the target qubit(s) | ψ i being usually chosen to be eigenstates of the controlledunitary operations U f ( x ) with eigenvalues e iα ( x ) ;(iii) Use of the e iα ( x ) phases (kicked back to | x i ) to enhance, by interfer-ence, particular computational paths.The oracle is the quantum subroutine that contains the information spe-cific to each particular problem. The way the oracle is chosen to act (in par-ticular the choice of the target qubit as an eigenstate of U f ( x ) ) implies thatthe natural interference device is the quantum Fourier transform (QFT). Onthe other hand, the QFT, operating on the state | · · · i , also generates asuperposition of all the basis states. This suggests that most of the powerof quantum computing may be obtained by adding to a classical computer afew basic modules, namely:(i) A quantum Fourier transform module(ii) Programmable oracle modules.In theoretical discussions the oracle is considered to be a subroutine call,invocation of which only costs unit time. However, one should not forgetthat it is an operation acting in all basis states and therefore, to benefit fromquantum parallelism the practical requirements for its implementation arenot very different from those of the quantum Fourier transform.Quantum computing requires the coding, manipulation and detection ofentangled qubits. Nuclear spins, atom states, flux units, Cooper pairs orsingle photon polarizations have been proposed and used to encode qubitsand exhibit quantum logic operations. Qubits encoded in such fundamentalmatter units might indeed be the ultimate building blocks of future quantum4omputers. For practical computing applications, a scalable tensor productstructure is required to avoid an exponential demand for physical resources.However, this seems difficult to achieve with the prototype quantum gatesthat have been developed. Therefore a search for alternative implementationsseems appropriate.Section 2, improving and extending a previous proposal [3], discussesan implementation of quantum computing operations in classical systemsthat propagate according to a Schr¨odinger equation with a space coordinateplaying the role of time. Here one tries to make a concrete proposal for theimplementation of the theory using fiber or wave-guide optics, the qubitsbeing robustly coded in particular modes or on their polarizations, with theresult of the (unitary) operations being read off at particular locations of theoptical systems. Fiber or planar wave-guide optics implementations benefitfrom the large amount of technological sophistication already developed forcommunications. Therefore, the emphasis is on the construction of quantumgates using devices and techniques currently available in this field. As thesophisticated optical elements developed so far have been done mostly fortelecommunication purposes one also clarifies the implementation progressneeded to make them appropriate for the quantum computation purposes. Aswell as the issues of coding and gate implementation, also polarization effects,signal coupling and the notions of mixing, entanglement and coherence arediscussed in this setting.Finally, Section 3 discusses how these quantum-like elements might scale-up to construct a quantum Fourier transform module as well as programmableoracles. In optical fibers or planar wave-guides, mode propagation may be well ap-proximated by a Schr¨odinger equation with the longitudinal z -coordinateplaying the role of time. Ref.[3] follows a reasoning similar to the Leontovich-Fock [13] description of paraxial beams in the parabolic approximation. Hereone generalizes the derivation in [3] by explicitly including polarization ef-fects. 5rom the Maxwell equations, with ρ = J = M = 0, one obtains theHelmholz equation for a space-varying dielectric constant ▽ (cid:18) ε E · ▽ ε (cid:19) + △ E = εµ ∂ E∂t (6)Consider now a fixed frequency transversal mode E ( x, y, z, t ) = E ( x, y, z ) exp ( iωt )and an index of refraction profile ε ( x, y, z ) = n ( x, y, z ) = n ( z ) − V ( x, y ) (7)where n ( z ) is the index of refraction at the fiber axis and V ( x, y ) << n ( z ).With this last condition and neglecting terms in ( ▽ V ) and V ▽ V , oneobtains, for a transversal electric mode (cid:18) ∂ ∂z + △ (cid:19) E + n k E − n ▽ ( E · ▽ V ) ≃ k = ωc and λ = πk is the wavelength in vacuum.Introduce the slowly varying (in z ) complex vectorial function ψ ( x, y, z ) E ( x, y, z ) = ψ ( x, y, z ) exp (cid:18) ik Z z n ( ζ ) dζ (cid:19) (9)For slow variation of the index of refraction along the fiber axis over distancesof the order of one wavelength λ n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) dn ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) << ψ along z and derivatives of n ( z ) and end up with iλ ∂∂z (cid:18) ψ x ψ y (cid:19) = λ n ( z ) (cid:18) −△ −△ (cid:19) + πn ( z ) (cid:18) V ( x, y ) 00 V ( x, y ) (cid:19) + λ πn ( z ) (cid:18) ∂ x V + ∂ x V ∂ x ∂ xy V + ∂ y V ∂ x ∂ xy V + ∂ x V ∂ y ∂ y V + ∂ y V ∂ y (cid:19) (cid:18) ψ x ψ y (cid:19) (10)which is a quantumlike version of the Schr¨odinger-Pauli equation. The roleof time in this equation is played by the spatial (longitudinal) coordinate ofthe light beam, the role of Planck’s constant is played by the light wavelength6nd the role of potential energy by the index of refraction of the medium.Thus, a beam of light, a purely classical object, obeys equations formallyidentical to those of quantum mechanics.The unitary z -evolution of the electromagnetic complex amplitude is de-scribed by the evolution operator ˆ U ( z )ˆ U ( z, z ) ψ ( x, y, z ) = ψ ( x, y, z ) , (11)associated to the Hamiltonianˆ H ( z ) = (cid:18) ˆ p x p y (cid:19) n ( z ) + Γ( x, y, z ) . (12)with ˆ p x = − iλ ∂∂x , ˆ p y = − iλ ∂∂y and a potential function Γ( x, y, z ) which,for general V ( x, y ), has local and nonlocal terms mixing the polarizations asfollows from Eq.(10). Manipulation of the polarization will play an importantrole in this quantumlike computation approach. As seen from the last termin Eq.(10) it is obtained by engineering the index of refraction profile.Other quantumlike systems are reviewed in [14] [15] [16]. They includesound-wave propagation in acoustic waveguides, charged-particle beams andlight beams inside diode lasers. Full implementation of quantum algorithmsmight also be obtained in these systems. For each unitary operation a steadystate is to be established and the result of the computation is read at theappropriate space location. The notion of preservation of time coherence needed to define the reliability and maximum number of operations in quan-tum computation is here replaced by space coherence of the steady state thatis established in the device.Before discussing practical implementations of the quantumlike represen-tation, I add two speculative remarks:(i) Abrams and Lloyd [17] have shown that were quantum mechanicsnonlinear, more computational power could still be achieved. There is noevidence indicating that actual quantum mechanics is nonlinear. However, inthe quantumlike scheme it is quite simple to implement a nonlinear Schr¨odingerequation evolving in the z − coordinate. Therefore, quantumlike nonlinearcircuits might provide an adequate framework to test Abrams and Lloyd’sideas.(ii) Brun [18] has pointed out that hard problems could in principle besolved, even by a classical computer, if it had access to a closed timelike curve.Except maybe in extreme cosmological conditions, closed timelike curves are7ot readily available. However, if in a computational scheme (both classicaland quantum) time is replaced by space, simulation of closed timelike curvesis not unthinkable.Before proceeding it should be pointed out that other implementation ofsome features of quantum algorithms by linear optical methods have beenproposed by several authors (see for example [19] [20] and references therein).To obtain the entanglement needed for universal quantum computation, theproposed optics implementations use either:(i) Kerr nonlinearities, which are hard to achieve at the single-photonlevel or(ii) a probabilistic scheme based on the nonlinearity implicit in the selec-tion by single-photon detectors.What is proposed here and in Ref.[3] is a more radical proposal in thesense that, instead of setting up a time sequence of optical events as theimplementation of the quantum algorithm, one uses the fact that, in opticalfibers, mode propagation is well approximated by a Schr¨odinger equationwith the z -coordinate along the fiber playing the role of time. In Ref.[3], several ways to code qubits on a fiber, using either discrete orcontinuous variables, were already discussed. Here simpler implementationsare proposed which might be robustly obtained with the materials availablefor optical communication applications.Consider three types of qubit codings in two types of fibers:(a) In single-mode (double-polarization) fibers a qubit would correspondto the two polarizations directions of the LP mode [21] (Fig.1).(b) In single-mode fibers a qubit might also be associated to the am-plitudes of a particular polarization in two distinct fibers, one of the fibersassociated to | i and the other to | i .(c) In fibers with normalized frequency allowing for LP and LP modes,a qubit may be associated to the two distinct LP modes, without distin-guishing polarization states (Fig.1). Counting the polarizations one has fourdegrees of freedom associated to the LP mode, which allows for the codingof two qubits and the implementation of a two-qubit gate in a single fiber(see below).These codings are the simplest ones for optical fiber implementations.Notice however that the reliability of fiber optics techniques allows for reliable8igure 1: Two distinct qubit coding choicesmanipulation and separation of many other modes. For example for a fiberwith normalized frequency 5 . < πr co λ p n co − n cl < . LP may be excited, allowing for 40 different quantumlike degrees of freedomin a single fiber. ( r co , n co , n cl denote the core radius and the index of refractionof core and cladding) Universal quantum computation requires one-qubit gates performing arbi-trary unitary transformation and, at least, a two-qubit gate performing aunitary transformation in the four-dimensional tensor space which, togetherwith the one-qubit transformations, generates the unitary group in four di-mensions. For the one-qubit gates, two schemes seem appropriate: (1) Polarization coding
Isotropic single-mode fibers support two degenerate polarization modeswhich propagate with the same constants β i = k n i . However it is rela-tively easy to make the fibers to behave as linearly birefringent or circularlybirefringent media [22] [23]. The birefringence of the fiber is conventionallycharacterized by a Jones matrix J [24] [25] [26] which defines the amountof transformation of the phase induced by the difference ∆ β L = β x − β y (or∆ β C = β R − β L for circular polarization). Because a global phase associatedto k R z n ( ζ ) dζ is already taken into account, the Jones matrix is what de-fines the phase rotation of ψ ( x, y, z ) in Eq.(9). For linear birefringence theJones matrix relating the output and input phase of a fiber of length L , is J L (∆ β L ) = (cid:18) e iL ∆ β L / e − iL ∆ β L / (cid:19) (13)9n this expression it is assumed that the fast axis, that is, the one with thelargest β , is in the x − direction. If the fast axis is at an angle θ relative tothe x − direction the Jones matrix would be J L (∆ β L , θ ) = (cid:18) cos L ∆ β L + i cos 2 θ sin L ∆ β L i sin 2 θ sin L ∆ β L i sin 2 θ sin L ∆ β L cos L ∆ β L − i cos 2 θ sin L ∆ β L (cid:19) (14)Any U (2) matrix may be decomposed into U ( α, θ, β ) = (cid:18) e iα/ e − iα/ (cid:19) (cid:18) cos θ i sin θ i sin θ cos θ (cid:19) (cid:18) e iβ/ e − iβ/ (cid:19) (15)hence it follows from (13) and (14) that any U (2) transformation may beobtained on linearly birefringent fibers.Linear birefringence is easily obtained by elliptical cores, lateral stress,bending or application of an electrical field. For fixed one qubit gates themost robust method is probably the use of cooling induced stress [27]. Cir-cular birefringence is obtained by geometrical twisting (spun fibers) or axialmagnetic fields (Faraday rotation). Adjustment of the intensity of theseproperties by the variation of applied electromagnetic fields is a potentiallyuseful feature for the construction of programmable modules.Engineering the birefringency properties is a very flexible way to obtainone qubit gates using single mode double-polarization fibers. For example,in the cases above one has assumed that the fast axis is fixed along the fibersegment. If instead one has a continuously rotating fast axis, a more complexJones matrix is obtained J L (∆ β L , ξ ) = (cid:18) cos δ + i L ∆ β L δ sin δ Lξδ sin δ − Lξδ sin δ cos δ − i L ∆ β L δ sin δ (cid:19) with δ = q ( L ∆ β L ) + 4 ( Lξ ) and ξ = dθdz the constant rate of rotation ofthe fast axis along the z − coordinate.Also, for a simple circularly birefringent fiber the Jones matrix is J L (∆ β C ) = (cid:18) cos L ∆ β C sin L ∆ β C sin L ∆ β C cos L ∆ β C (cid:19) and for a fiber that is both linearly and circularly birefringent (for examplea linearly birefringent spun fiber or a linearly birefringent one with an axial10agnetic field) the Jones matrix is J (∆ β C , α ) = cos L ∆ β C − i − α α sin L ∆ β C α α sin L ∆ β C − α α sin L ∆ β C cos L ∆ β C + i − α α sin L ∆ β C ! with α = γn x − n y + q ( n x − n y ) γ and γ being the nondiagonal term in the relativedielectric constant tensor n x iγ − iγ n y
00 0 n z .Linear and circular birefringence allow for the implementation of any U (2) transformation in the polarization-encoded qubits. Preparation andmeasurement of the polarization-encoded qubits is obtained by polarizingfibers and polarizing beam-splitters. (2) LP coding For a fiber with a parabolic index profile, the LP modes may be ap-proximated by the first harmonic excitations along the x and y directions.Denoting by a † and b † the corresponding creation operators, one has thefollowing correspondence | i ↔ LP ; (cid:0) a † | i , b † | i (cid:1) ↔ LP (16)The SU (2) group, operating irreducibly in the 2-dimensional space (cid:0) a † | i , b † | i (cid:1) ,is the following subgroup of the Weyl-symplectic group in 2-dimensions J + = a † bJ − = b † aJ = (cid:0) a † a − b † b (cid:1) (17)As explained in Ref.[3] and as follows from Eq.(10) in Sect. 2.1, changing theindex profile along x and y as well as the coefficient of the Laplacian one hasaccess to all generators of the two-dimensional Weyl-symplectic group andin particular to those of the SU (2) subgroup. Therefore, by engineering theindex profile, all unitary rotations may be implemented on the LP − encodedqubits.Requiring a precise adjustment of the index profile, an unitary manipu-lation of the LP − encoded qubits is more complex than the correspondingoperation on polarization-encoded qubits. Therefore this encoding might beonly recommended for control qubits.11n the quantumlike scheme one deals not with single photon events, butwith steady-state beams. Therefore conversion between the two encodings isrelatively easy using standard optical techniques. To obtain universal computation, in addition to one-qubit gates performingarbitrary unitary transformations, one needs at least one entangling gate.This is a gate that, together with one-qubit gates, generates all U (4) trans-formations. The CNOT, CS (controlled sign) or CP (controlled phase) gatesare such gates, but there are many others (Appendix A). coding Here one shows how to obtain a controlled (entangling) gate using the twoqubit codings discussed before. On a fiber carrying LP modes, the LP mode has four degrees of freedom, two of them associated to the two possibleorientations of the mode (see Fig.1) and the other two to the polarization.Let the two orientations of the LP mode code the control qubit and thepolarization code the target qubit. For later convenience the codes for the | i and | i qubits will be V, H (vertical, horizontal) for the polarizations(target) and a, b for the positions (control) of the LP modes. If the fiber isconstructed in such a way that the | i sectors in the LP mode are linearlybirefringent and the | i sectors are isotropic (see Fig.2), a phase gate isobtained corresponding to the matrix M = e iθ (18)in the basis ( | i , | i , | i , | i ), the first entry being the control qubit andthe second the target qubit. θ is the additional phase that the | i t target qubitobtains in the | i c sector of the control qubit. In all cases there is a globalphase that should be taken into account arising from the z − propagationin the gate. With different choices of the birefringence distribution otherentangling U (4) matrices may be obtained.Suppose that at the input of the gate the beam is a superposition of the LP modes polarized on the x, y plane ( α | i t + α | i t ) and that it is the12igure 2: Coding of a controlled gate, using LP modes for the control qubitand polarization for the target qubitposition (control) mode a that is active. Then in the sector a of the fiber theoutput is | i c ⊗ (cid:0) α | i t + α e iθ | i t (cid:1) = α | i + α e iθ | i , (19)whereas in the b sector the target qubit is unchanged. That is, the degreesof freedom of the beam are entangled.The nature of this entanglement is what has been called local entangle-ment in the sense that it refers to the degrees of freedom carried by the samephysical entity. For a more general control qubit ( β | i c + β | i c ) one has β | i c ⊗ ( α | i t + α | i t ) + β | i c ⊗ (cid:0) α | i t + α e iθ | i t (cid:1) (20)which would be faithfully implemented in the LP gate. The target qubitchanges but only in the sector a of the gate.The usual statement that entangling two-qubit gates requires a nonlineareffect, actually refers to the tensor product in (20), which here is obtained by Some authors have claimed that the notion of entanglement should include otherfeatures in addition to non-separability. Here entanglement is simply used in the sense ofnon-separability. nonlocal entanglement in quantum mechanics. In quantummechanics two photons may become entangled if they have interacted in thepast, in general because they were produced by a common source . Theythen share a common wavefunction and, in this sense they are also partsof the same physical system. They only become independent entities if thewavefunction decoheres, and then entanglement is gone. So local and nonlo-cal entanglement are not so very different as it might seem. On this opticalentanglement of the beam degrees of freedom there is another parallel withquantum mechanics. In quantum mechanics the more noteworthy featureof entanglement is the fact that correlation between the photons remains ifat a later time they are well separated in space. Here the role of time isplayed by the longitudinal z − coordinate of the fiber and the entanglementthat occurs in the gate may be observed at a later z . This, of course, if noiseor the fiber imperfections do not destroy space coherence. Like in quantummechanics. In short, entanglement requires interaction and remembrance ofthe interaction effects along the propagation path.In some quantum computing applications, for example in quantum Fouriertransform (QFT) as will be seen later, the full entangled output of the phasegates is not used. Instead, in each line of the output of the QFT one wouldwant to find β ( α | i t + α | i t ) + β (cid:0) α | i t + α e iθ | i t (cid:1) , that is, a partial trace over the control qubit is effectively done.If instead of linearly birefringency the | •i region is circularly birefringent,also entangling gates may be constructed. Here the two-bit gate is based onthe four degrees of freedom of the LP modes of a circular fiber. A similarconstruction might done using the T E, T M −
12 modes of a rectangular fiber.Modern fiber optics technology is also able to handle multimode fibers whichwould provide entangling gates for many more qubits.Instead of a single fiber carrying LP modes, one may use two fibers (orlight wave guides on a chip) one for the control position code a and the otherfor the code b . Each one of the light guides might carry the full polarizationinformation or the a − fiber might only contain the vertical ( V ) component They may also be entangled by entanglement swapping which involves measurement,a nonlinear operation. b − fiber the horizontal ( H ) component. For future reference all theseequivalent possibilities will be denoted as a G − gate.Depending on its position on the quantum circuits, qubits may play therole of target or control qubits. Therefore to each qubit one associates twosynchronous wave guides, to carry both position and polarization informa-tion. While one of the lines carries optically the full polarization, the othermight well be electrical, with the interaction of polarization ( V, H ) and posi-tion ( a, b ) modes carried out by optical or electro-optical means. Notice alsothat conversion of polarization to position and vice versa is easily obtainedby polarizing beam splitters and polarization preserving fibers. The mainchallenge in this dual coding scheme is to preserve linearity in the gate. In acontrolled phase gate only the b − line needs to enter the gate, the polariza-tion coming from the target line being established in this line which is thenpassed through the appropriate retarder. A different alternative for the construction of two-qubit gates would be touse only one type of coding, for example polarization coding. In this casethe tensor product of control and target qubits is not achieved by the cou-pling position-polarization, but it requires an interaction between the twopolarized beams, which only occurs through interaction with an optical ac-tive medium. Fig.3 sketches the required mechanism. After being split bya polarizing beam splitter (PBS) the V component of the target beam isfurther split by another unit (controlled beam splitter, CBS) that is con-trolled by the V component of the control beam. One of the branches isthen passed through a phase retarder ( θ ) to implement the controlled phase2 − qubit gate. This implements the operation in Eq.(21). The essential el-ement is the controlled beam splitter (CBS) which can be achieved by adynamical holography mechanism. A grating, dynamically created on a ma-terial by interaction of the control and a reference beam, splits the targetbeam. Optically and electro-optically controlled beam splitters have beendiscussed and constructed before (see for example [28] - [32] and referencesin [33],[34]). However they operate mostly in an ON-OFF regime and here,as seen in Eq.21, one needs linear operation. In Appendix B, the basic the-ory of one such device is discussed as well as the requirements and challengesfaced to obtain linear operation. 15igure 3: Optical two qubit phase gate with polarization coding. PBS =polarizing beam splitter; CBS = controlled beam splitter; R = referencebeam. θ ′ = θ + π α | H i t + α | V i t β | H i c + β | V i c (cid:27) → β α | H c H t i ⊕ β α | H c V t i ⊕ β α | V c H t i ⊕ β α e iθ | V c V t i (21) As a general remark on the optical implementation of the operations ofquantum-like computing, it should be pointed out that one is in a morefavorable position than in the usual one-photon quantum computing imple-mentation. Here one deals with light beams and therefore nonlinear effectsare much easier to obtain. Furthermore one deals not with a transient tem-poral phenomenon, but with the establishment, in a optical network, of asteady state phenomenon. The initial state at the input of the quantum-likecircuit must be established by a coherent source which also acts as a refer-ence beam at other points of the circuit. The role of time being played by aparticular space coordinate, all the interference and gate operations are per-formed until a steady state configuration is established in the network, thefinal result of the calculation being read-off at some well defined coordinate.This also means that, as long as all superposition and interference phe-nomena are implemented by optical waves, some intermediate gate operationsmight be performed by electro-optical means. For example in a controlledphase gate the amplitude and phase of the vertical polarizations of control16nd target beams may be measured by heterodyning with the reference beamand then, with the result of the gate operation computed by electronic means,the same reference beam might by the appropriate retarders generate the op-tical output beams. Also at intermediate points of the network the signalsmay even be split, examined or amplified as long as the phase is preserved orthe phase change is duly taken into account. Of course all-optical operationof the gates and of the whole circuit is desirable and a goal to be achieved.There is, in these intermediate measurements, no conflict with the nocloning theorem of quantum information. In the usual proof of the no cloningtheorem, one assumes that an unitary operator U exists such that U | ψ > = | ψψ > for all ψ and then, by applying U to γ = αφ + βφ obtain U | γ > = α | φ φ > + β | φ φ > = | γγ > , a contradiction. No cloning means that, givenan unknown quantum state, no measurement can find out what was exactlyits wave function before the measurement. By contrast given a beam of lightone can split it in a polarization basis by a polarizing beam splitter and thenby heterodyning it with a coherent reference beam find the amplitude andphase of each one of the components. Given that knowledge, and because thephase is defined module 2 π , the beam may then be synchronously reproduced.In conclusion: the possibility to measure and then reproduce the quantum-like signal, means that it will not be appropriate for cryptography purposes.However, because it may have interference, parallelism, (local) entanglementand unitary propagation along a (computing) coordinate, it may be used forcomputation purposes. In the previous subsections the emphasis has been on linear gates, becausethey are the ones most useful for computation purposes. However quan-tum technology is not only quantum computing and nonlinear quantum (orquantum-like) effects are also of interest. The electric field associated to a sin-gle photon is very weak. This poses a major problem for all-optical quantumoperations using single photons, because significant, medium-mediated, non-linear interactions would be required between two photons. A very strongcooperative effect of atoms would be required to perform interaction of single-photon signals. The Kerr effect at the one photon level might be enhancedby choosing frequencies near resonances of the material, but then appreciableloss effects would be expected.In the optical quantum-like approach the signals, being coded not with17ingle photons but with light beams, nonlinear effects are much easier toobtain. In particular, a great development has already been achieved withnonlinear effects for switching purposes in classical all-optical networks. Di-rectional couplers are used as optical switches, as power dividers or com-biners, multiplexers, demultiplexers and intensity modulators. On-off logicgates based on the Kerr effect have also been proposed by several authors.First studied by Jensen [35] the nonlinear directional coupler is a robustdevice exploring the Kerr effect. In spite of its nonlinear nature, by explor-ing the role of constants of motion, an analytic solution may be obtainedfor the input-output transfer function of the device [36]. Therefore a pre-cise quantitative control of the transfer function is obtained. For the readerconvenience, the main equations and parameters of the coupler are summa-rized in the Appendix C. Denoting by −→ E (1) (0) , −→ E (2) (0) , −→ E (1) ( L ) , −→ E (2) ( L )the transversal electric fields at the input and output of the two ports of acoupler (1 and 2) of length L , one has a transfer function E (1) j ( L ) = (cid:18) e iL − β (+) M (+) + e iL − β ( − ) M ( − ) (cid:19) jk E (1) k (0)+ (cid:18) e iL − β (+) M (+) − e iL − β ( − ) M ( − ) (cid:19) jk E (2) k (0) E (2) j ( L ) = (cid:18) e iL − β (+) M (+) − e iz − Lβ ( − ) M ( − ) (cid:19) jk E (1) k (0)+ (cid:18) e iL − β (+) M (+) + e iL − β ( − ) M ( − ) (cid:19) jk E (2) k (0) (22)where the matrices M (+) , M ( − ) and the propagation factors β (+) , β ( − ) asso-ciated to the symmetric and asymmetric modes are completely specified bythe material parameters of the coupler (Eqs. 57, 58). Through the constantsof motion they have a nonlinear dependence on the coupler medium and onthe intensity of the beams. Of course in the linear case M (+) and M ( − ) areunit matrices.For practical purposes one should notice that propagating through thecoupler each beam suffers changes of phase and polarization rotations dueboth to itself and to the signal in the other beam, this latter action being theone that is more relevant for the computational effect of the device. Manydifferent nonlinear operations may be obtained by the appropriate choice ofthe parameters. 18 Quantum modules
A very important element in the quantum algorithms is the quantum Fouriertransform (QFT). For n qubits and N = 2 n the QFT is y k = 1 √ N N − X l =0 x l e i πlk/N , (23)the N number sets { y } and { x } being coded by the n qubits as follows x = ( j , j , · · · , j n ) = j n − + j n − + · · · + j n (24)The QFT may be looked at as an unitary transformation in the computa-tional basis of n qubits, implementing the transformation [37] ⌊ j j · · · j n i→ n/ (cid:26)(cid:16) ⌊ i + e i π jn ⌊ i (cid:17) (cid:18) ⌊ i + e i π (cid:16) jn − + jn (cid:17) ⌊ i (cid:19) · · · (cid:16) ⌊ i + e i π ( j + j + ··· + jn n ) ⌊ i (cid:17)(cid:27) (25)This decomposition of the QFT leads directly to the quantum circuit (for 4qubits) in Fig.4 where H and R k are the Hadamard and the controlled phasegates H = (cid:18) − (cid:19) ; R k = (cid:18) e i π/ k (cid:19) (26)This circuit has n ( n + 1) / O (2 n ) steps.There are however more efficient wirings [38] - [40].Griffiths and Niu [41] have proposed a semiclassical approach to the quan-tum Fourier transform. It is semiclassical in the sense that it requires a mea-surements of the output qubits to obtain a signal to control the gates. In thetime evolution approach to quantum computing this scheme would only beapplicable when the QFT is the final step in the quantum circuit. Howeverin the quantum-like approach because, as discussed before, measured beamsmay be fully restored, the Griffiths and Niu configuration may be used atany point in the circuit. 19igure 4: A O (2 n ) quantum Fourier transform circuit for 4 qubitsWhen using a single coding in the optical gates, for example polarizationcoding, the QFT circuits for quantum-like computation would be identicalto the classical ones. However, when the LP coding scheme (with one ortwo wave guides) is used, the configuration might be slightly different. Fig.5displays one such implementation. In each input, except the first, the inputqubits are duplicated, assigned both to the polarization modes of single mode( LP ) fiber and to position LP modes. The H − modules are Hadamardgates implemented by one-qubit gates with LP modes polarization. Boththe polarization ( V, H ) and the position ( a, b ) information are fed to the gate.There the a, b information and the polarization (
V, H ) are used to generatea polarized LP signal which is fed to a partially birefringent fiber, whichimplements a two-qubit phase gate, as described in Section 2.4.1. Notice thatwhereas the polarization information is naturally carried in a LP mode (thefine lines in Fig.5) the position information (the thick lines) for the LP modemay be carried to the gate electronically or by a LP fiber, whatever is moreconvenient. At the end of the polarized LP fiber in the gate, the outputpolarization is obtained by merging the a and b modes into a LP polarizedmode. All gates are identical, differing only on the length of the birefringent LP fiber section. The form of the quantum Fourier transform (Eq.23) is formally identical tothe classical discrete Fourier transform (DFT). In this sense, what the QFTdoes is a DFT on the amplitudes of the quantum state. On the other hand20igure 5: Quantum-like Fourier transform using LP codingit is known that the Fourier transform may be obtained from a light front,representing the function, by observation of the far field (or the focused farfield) at several angles. This led to several optical proposals for the DFTby, for example, passing a coherent light through a zero or π phase maskand observing the far field in the focus plane of a lens. These purely opticalapproaches that have also been proposed [42] - [45] for the QFT with singlephotons may, even more easily, be adapted to the light beam quantum-likeapproach. Oracles [46] [47] are functions f : (0 , m → (0 , n which, typically, are needed both for the preparation of the input signal tothe quantum circuit and for queries about the final state. In terms of a po-larization coding of beams in the quantum-like approach, these are functions f : ( H, V ) m → ( H, V ) n Such functions may be implemented by linear couplers, beam splitters, in-terferometers, phase rotaters and the two-qubit gates discussed before. It isdesirable to use electro-optical control in these units to have programmableflexibility of the oracles. 21
Conclusions
1) In this paper (and in [3]) by identifying a Schr¨odinger-like evolution alonga space coordinate of a classical system, we have concluded that quantumcomputation might be carried out both by quantum systems evolving in timeand by a classical wave system evolving along a space coordinate. This stealsthe primacy of quantum systems to execute quantum computing operations.Even more, one might say that quantum computing is more general thanquantum mechanics or simply that in quantum mechanics Nature is doingquantum computing along the time direction.2) There is, of course, a difference in these two modalities of quantumcomputing due to the particular nature of our observer status in the universefor which, to look at a timeline (at a particular space) has properties distinctfrom looking at a spaceline (at a particular time). When looking at a timeline,after the operation the same time is no longer there, in contrast with thetimely permanence of a spaceline. As a result if a measurement is made witha projection filter in the space evolution, the same results are obtained asin quantum mechanics, but on the other hand there are alternative ways toobserve which give complete access to the value of the wave function.3) The optical implementations of the one and two qubit gates in thispaper have been kept are simple as possible, using only LP and LP modes.However with the growing sophistication on handling multimode fibers it isconceivable that, using this optical quantum-like approach, it will be possibleto obtain high degrees of circuit compactness and parallelism. Of particularinterest for the development of interesting quantum-like devices are the recenttechnological advances in space light modulators (SLM) [48] [49].4) The current and potential applications of quantum technology are notrestricted to quantum computing, other promising uses are in fields of controland communications. Whereas it seems that in quantum computing thelinear gates are the most useful, nonlinear gates are expected to be potentiallyuseful in other applications. This was the main motivation to discuss atsome length in section 2 and in the appendix C the analytical aspects of thenonlinear circuits.5) As stated before, there are, in addition to light waves, other systemswhich display quantum-like behavior when its evolution along a space coor-dinate is observed. Not all of them will be as appropriate as light to performcomputations, in particular because of the need to maintain coherence in theevolution. Nevertheless a case that might deserve some attention is the case22f spin waves [50] [51]. It is known [52] that arbitrary one-qubit gates together with a two-qubitCNOT are capable of universal quantum computation. It the follows that,more generally, any two-qubit gate capable of generating, together with theone-qubit gates, the full U (4) group would also be universal. Such two-qubitgates have been called entangling (or imprimitive) gates, because they mapdecomposable states into indecomposable ones. A gate that is not entanglingis called primitive [53].Let e ij be a 4 × e ij ) mn = δ im δ jn (27)Then, the 16 Lie algebra generators of U (4) are I ij = i ( e ij − e ji ) J ij = e ij + e ji i = je ii (28)They are related to the Lie algebra generators of U (2) ⊗ U (2) by σ µ ⊗ σ ν = P i e ii J + J − I − I e − e + e − e J + J J + J − I + I J − J − I − I − I − I J − J − I + I e + e − e − e J − J − I + I e − e − e + e (29)where σ µ = { σ ≡ , σ , σ , σ } are the identity 2 × ⊗ σ ν and σ µ ⊗ , are the algebraic elements associated to one-qubit operations. The remaining 9 elements in (29) are of the form σ i ⊗ σ j ( i, j = 1 , , ⊗ σ i , σ a ⊗ σ b ] = σ a ⊗ [ σ i , σ b ] (30)[ σ i ⊗ ,σ a ⊗ σ b ] = [ σ i , σ a ] ⊗ σ b it follows that, given any one of the 9 elements σ i ⊗ σ j it is possible to generatethe full U (4) algebra by commutation with the (one-qubit) generators ⊗ σ ν σ µ ⊗ . These 9 elements are therefore a basis for the imprimitive (entan-gling) elements of the algebra. Linear combinations of these elements as wellas linear combinations with one-qubit transformations are also entangling. Many controllable beam splitters have been proposed in the past. They useeither mechanical displacement of metasurfaces [29], electro-optical modula-tors and a Mach-Zehnder interferometer [30], optical bistability by surfaceplasmons [31], etc.Optically controlled beam splitters have been discussed. For example[28] uses a grating made of polymer slices alternated with layers of alignednematic liquid crystal. When the liquid crystal is aligned the input light beamis split into a transmitted and a refracted component, however when anotherpump beam is turned on, the liquid crystal suffers a nematic to isotropicphase transition, the refractive index contrast vanishes and the structurebecomes transparent to the incoming light. Because fine-tuning of the indexcontrast seems difficult, this interesting device is mostly suited for an ON-OFF operation mode. The same applies to electro-optic operated liquidcrystal devices [32].The ON-OFF behavior of the controlled beam splitters is appropriate fordigital communication purposes, but for analog or quantum-like computingapplications a smoother, linear or quasi-linear, dependence on the controlsignal is desirable. Quantum-like applications are even more demanding be-cause information on the phase of the control signal should be taken intoaccount.The propagation of a transversal electric field in a nonlinear media isdescribed by the equation △ E − µ ε ∂ E ∂t = µ ∂ P L ∂t + µ ∂ P NL ∂t (31)For the nonlinear contribution to the refraction index one considers either aKerr or a photorefractive medium. Let E = −→ E ( x, z ) e iωt (32)24 x, z ) being the coordinates of the propagation plane of field. Consider a fixedthick sinusoidal grating along the x coordinate (Fig.6), and the propagationof a light wave on this grating, that is △−→ E ( x, z )+ µ ε ω −→ E ( x, z ) = − µ ε χ (1) ω −→ E ( x, z ) − µ ε χ ( NL ) cos ( Qx ) ω −→ E ( x, z )(33)Figure 6: A thick gratingPassing to the Fourier transform on the x coordinate and writing −→ E ( q, z ) = −→ ψ ( q, z ) e ikz (34)with −→ ψ ( q, z ) having a slow variation on z , one obtains (cid:0) − k − q (cid:1) −→ ψ ( q, z )+2 ik∂ z −→ ψ ( q, z )+ α −→ ψ ( q, z )+ β (cid:16) −→ ψ ( Q + q, z ) + −→ ψ ( Q − q, z ) (cid:17) ≃ α = µ ε χ (1) ω β = µ ε χ ( NL ) ω (36)being the linear and nonlinear refractive coefficients. In Eq.(35) one hasneglected the term ∂ z −→ ψ .With q and Q >
0, constructive interference of the diffractive componentsin the thick grated slab requires k + q = αk + ( Q − q ) = α (37)25hen only the 0 th and the 1 st diffraction orders are non-evanescent, Q =2 q and one has the following equations for the transmitted and diffractedcomponents 2 ik ∂ −→ ψ ( q, z ) ∂z = − β −→ ψ ( Q − q, z )2 ik ∂ −→ ψ ( Q − q, z ) ∂z = − β −→ ψ ( q, z ) (38)with solution −→ ψ ( q, z ) = −→ ψ ( q,
0) cos (cid:18) β k z (cid:19) −→ ψ ( Q − q, z ) = i −→ ψ ( q,
0) sin (cid:18) β k z (cid:19) (39)In conclusion: the amount of splitting of the beam by the thick grating iscontrolled by the nonlinear contribution to the refractive index.Now, to have the splitting of the beam controlled by another light beam,the grating should not be fixed but created by the intensity of the otherbeam. Because one wants to have the tuning to be also a function of thephase, consider 3 light beams (target, control and reference, t, c, R ) −→ E t = (cid:12)(cid:12)(cid:12) −→ E t (cid:12)(cid:12)(cid:12) −→ ǫ t e i ( ωt −−→ k t ·−→ x + θ t ) −→ E c = (cid:12)(cid:12)(cid:12) −→ E c (cid:12)(cid:12)(cid:12) −→ ǫ c e i ( ωt −−→ k c ·−→ x + θ c ) −→ E R = (cid:12)(cid:12)(cid:12) −→ E R (cid:12)(cid:12)(cid:12) −→ ǫ R e i ( ωt −−→ k R ·−→ x + θ R ) (40)The intensity of the sum of the three signals is (cid:12)(cid:12)(cid:12) −→ E t −→ + E c + −→ E R (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) −→ E t (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) −→ E c (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) −→ E R (cid:12)(cid:12)(cid:12) + −→ ǫ t · −→ ǫ c (cid:12)(cid:12)(cid:12) −→ E t (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) −→ E c (cid:12)(cid:12)(cid:12) (cid:16) e i ( θ c − θ t + ( −→ k t −−→ k c ) ·−→ x ) + c.c. (cid:17) + −→ ǫ t · −→ ǫ R (cid:12)(cid:12)(cid:12) −→ E t (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) −→ E R (cid:12)(cid:12)(cid:12) (cid:16) e i ( θ t − θ R + ( −→ k R −−→ k t ) ·−→ x ) + c.c. (cid:17) + −→ ǫ c · −→ ǫ R (cid:12)(cid:12)(cid:12) −→ E R (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) −→ E c (cid:12)(cid:12)(cid:12) (cid:16) e i ( θ c − θ R + ( −→ k R −−→ k c ) ·−→ x ) + c.c. (cid:17) (41)26ith linearly polarized signals it is always possible to have −→ ǫ t · −→ ǫ c = −→ ǫ t · −→ ǫ R = 0 −→ ǫ c · −→ ǫ R = 0 (42)Then, only the last of the mixed terms is nonvanishing and, (cid:12)(cid:12)(cid:12) −→ E t −→ + E c + −→ E R (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) −→ E t (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) −→ E c (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) −→ E R (cid:12)(cid:12)(cid:12) + −→ ǫ c ·−→ ǫ R (cid:12)(cid:12)(cid:12) −→ E R (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) −→ E c (cid:12)(cid:12)(cid:12) (cid:16) e i ( θ c − θ R ) e i ( −→ k R −−→ k c ) ·−→ x + c.c. (cid:17) (43)By the Kerr effect or on a photorefractive material, one may use this intensityto create a grating along the −→ k R − −→ k c direction. This holographic-like patterncarries the information on the intensity and phase of the target signal, whichwith the choice (42) is not contaminated by the interaction with the targetsignal nor by the interaction of the reference beam with the target. Theintensity of the reference beam, in general larger than the one of the othersignals defines the amplitude of the grating effect.For Kerr materials P NL = χ (3) (cid:12)(cid:12)(cid:12) −→ E t −→ + E c + −→ E R (cid:12)(cid:12)(cid:12) (cid:16) −→ E t −→ + E c + −→ E R (cid:17) (44)therefore the β factor in Eq.(39) is proportional to −→ ǫ c · −→ ǫ R (cid:12)(cid:12)(cid:12) −→ E c (cid:12)(cid:12)(cid:12) e iθ c , that is,the splitting of the target beam would be directly controlled by the controlbeam. For small β this action is approximately linear on the amplitude of thecontrol, however deviations from linearity occur for large β . The situationmight be improved by manipulation of the −→ ǫ c · −→ ǫ R term, that is, makingthe control beam pass through a medium that rotates the polarization as afunction of the intensity. Alternatively one might act on the intensity of thecontrol beam by electro-optical means to obtain β = sin − (cid:16) α (cid:12)(cid:12)(cid:12) −→ E c (cid:12)(cid:12)(cid:12)(cid:17) . For photorefractive materials the change of the refractive index is propor-tional to the space derivative of the intensity (see for example [33] ch. 21.4)and the mechanism is quite similar.
Directional couplers are useful devices currently used in fiber optics commu-nications. Because of the interaction between the two input fibers, power fed27nto one fiber is transferred to the other. The amount of power transfer iscontrolled by the coupling constant, the interaction length or the phase mis-match between the inputs. If, in addition the material in the coupler regionhas strong nonlinearity properties, the power transfer will also depend on theintensities of the signals [35] [54]. A large number of interesting effects takeplace in nonlinear directional couplers [55] [56] [57] [58] with, in particular,the possibility of performing all classical logic operations by purely opticalmeans [59].Here one summarizes how, by exploring the constants of motion of thecoupler equation, explicit analytical solutions are obtained for both the linearand nonlinear couplers, as used in Sect.2 for two-qubit gates. Further detailsmay be found in Ref. [36].Consider two linear optical fibers coming together into a coupler of non-linear material. The equation for the electric field is △ E − µ ε ∂ E∂t = µ ∂ P L ∂t + µ ∂ P NL ∂t , (45) P L ( r, t ) = ε χ (1) E ( r, t ) being the linear polarization of the medium, P NL ( r, t ) = ε χ (3) | E ( r, t ) | E ( r, t ) the nonlinear polarization in the instantaneous non-linear response approximation and transversal dependence of χ (1) and χ (3) have been considered negligible.Separating fast and slow (time) variations E ( r, t ) = {E ( r, t ) e − iω t + c.c. } P NL ( r, t ) = {P NL ( r, t ) e − iω t + c.c. } (46)one obtains for the e − iω t part of a transversal mode P NL , ( r, t ) = 3 ε χ (3) (cid:26) e − iω t (cid:20)(cid:18) |E , | + 23 |E , | (cid:19) E , + 13 E , E , E ∗ , (cid:21) + c.c. (cid:27) (47)the labels 1 and 2 denoting two orthogonal polarizations.The dependence on transversal coordinates ( x, y ) is separated by consid-ering E k ( r, t ) = g Ψ ( i ) k ( x, y, z ) e iβ i z e − iω t (48)with Ψ ( i ) k ( x, y, z ) being an eigenmode of the coupler with slow variation along z ∆ Ψ ( i ) k + (cid:18) ω c (cid:0) χ (1) (cid:1) − β ( i )2 (cid:19) Ψ ( i ) k = 0 (49)28 i ) denotes the mode index, k the polarization and ∆ = (cid:16) ∂ ∂x + ∂ ∂y (cid:17) .Neglecting ∂ Ψ ( i ) ∂z one obtains2 iβ ( i ) ∂ Ψ ( i )1 , ∂z = − ω c χ (3) (cid:26)(cid:18)(cid:12)(cid:12)(cid:12) Ψ ( i )1 , (cid:12)(cid:12)(cid:12) + 23 (cid:12)(cid:12)(cid:12) Ψ ( i )2 , (cid:12)(cid:12)(cid:12) (cid:19) Ψ ( i )1 , + 13 Ψ ( i )2 , Ψ ( i )2 , Ψ ( i ) ∗ , (cid:27) (50)In the directional couplers the propagating beams are made to overlap alongone of the transversal coordinates ( x ). Typically, in the overlap region of thedirectional coupler, the eigenmodes are symmetric (+) and antisymmetric( − ) functions on x , the amplitudes in each fiber at the input and output ofthe coupler being recovered byΨ (1) k = (cid:16) Ψ (+) k + Ψ ( − ) k (cid:17) Ψ (2) k = (cid:16) Ψ (+) k − Ψ ( − ) k (cid:17) (51)An explicit analytic solution, also for the nonlinear coupler equation (50),may be obtained by noticing that it has two constants of motion ∂∂z (cid:26)(cid:12)(cid:12)(cid:12) Ψ ( i )1 (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Ψ ( i )2 (cid:12)(cid:12)(cid:12) (cid:27) = 0 ∂∂z n Ψ ( i ) ∗ Ψ ( i )2 − Ψ ( i )1 Ψ ( i ) ∗ o = 0 (52)Therefore, defining (cid:12)(cid:12)(cid:12) Ψ ( i )1 (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Ψ ( i )2 (cid:12)(cid:12)(cid:12) = α ( i ) Ψ ( i ) ∗ Ψ ( i )2 − Ψ ( i )1 Ψ ( i ) ∗ = iγ ( i ) (53)one obtains for the electrical field of the eigenmodes i ∂E ( i )1 ∂z = − − β ( i ) E ( i )1 − i − k ( i ) E ( i )2 i ∂E ( i )2 ∂z = − − β ( i ) E ( i )2 + i − k ( i ) E ( i )1 (54)with − β ( i ) = β ( i ) + ω c χ (3) β ( i ) α ( i ) − k ( i ) = ω c χ (3) β ( i ) γ ( i ) (55)29otice that, through α ( i ) and γ ( i ) , − β ( i ) and − k ( i ) depend on the material prop-erties, on the geometry of the mode and also on its intensity. One maynow obtain, for each eigenmode, the input-output relation of the nonlinearcoupler E ( i )1 ( z ) = e i − β ( i ) z ( E ( i )1 (0) cos − k ( i ) z ! − E ( i )2 (0) sin − k ( i ) z !) E ( i )2 ( z ) = e i − β ( i ) z ( E ( i )1 (0) sin − k ( i ) z ! + E ( i )2 (0) cos − k ( i ) z !) (56)the nonlinearity being embedded into − β ( i ) and − k ( i ) − β ( i ) = β ( i ) + ω c χ (3) β ( i ) (cid:18)(cid:12)(cid:12)(cid:12) E ( i )1 (0) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E ( i )2 (0) (cid:12)(cid:12)(cid:12) (cid:19) − k ( i ) = ω c χ (3) β ( i ) Im (cid:16) E ( i ) ∗ (0) E ( i )2 (0) (cid:17) (57)To obtain the corresponding input-output relations in the two fibers onedefines a matrix M ( ± ) ( z ) = cos − k ( ± ) z ! − sin − k ( ± ) z ! sin − k ( ± ) z ! cos − k ( ± ) z ! (58)Eq.(56) is rewritten E ( ± ) ( z ) = e i − zβ ( ± ) M ( ± ) ( z ) E ( ± ) (0) (59) z being the interaction length of the directional coupler. Using (51) the fields30t the output of the coupler are related to the input fields by E (1) j ( z ) = (cid:18) e iz − β (+) M (+) + e iz − β ( − ) M ( − ) (cid:19) jk E (1) k (0)+ (cid:18) e iz − β (+) M (+) − e iz − β ( − ) M ( − ) (cid:19) jk E (2) k (0) E (2) j ( z ) = (cid:18) e iz − β (+) M (+) − e iz − β ( − ) M ( − ) (cid:19) jk E (1) k (0)+ (cid:18) e iz − β (+) M (+) + e iz − β ( − ) M ( − ) (cid:19) jk E (2) k (0) (60)For the linear coupler case the M ( ± ) ( z ) matrices are the unit matrices andthe coupling arises only from the difference in the propagation constants − β (+) , − β ( − ) of symmetric and antisymmetric modes. However in both cases,linear and nonlinear, explicit analytical expressions are obtained for the cou-pling as a function of the input intensities and the material properties. In − β ( i ) the nonlinear effect is a function of the energy of the incoming signalsand − k ( i ) has a geometrical interpretation as − k ( i ) = ω c χ (3) β ( i ) (cid:12)(cid:12) Ψ ( i ) ∗ × Ψ ( i ) (cid:12)(cid:12) Here it was assumed that the frequency of the two incoming signals tothe coupler is the same. If they have different frequencies ω and ω thecorresponding constants of motion, as a function of the associated fieldsΨ , Ψ , would be (cid:12)(cid:12) Ψ (cid:12)(cid:12) ; (cid:12)(cid:12) Ψ (cid:12)(cid:12) ; β ω Ψ ∗ × Ψ + β ω Ψ ∗ × Ψ However, in this case, these constants of motion do not seem to be sufficientto obtain an explicit analytical solution.
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