OPERATORS IN QUANTUM AND CLASSICAL OPTICS

Transcription

1 DRSTP/LOTN School, Driebergen, April 010 OPERATORS IN QUANTUM AND CLASSICAL OPTICS Gerard Nienhuis Huygens Laboratorium, Universiteit Leiden, Postbus 9504, 300 RA Leiden I. MAXWELL S EQUATIONS FOR RADIATION AND COULOMB FIELD A. Radiation and Coulomb modes of electromagnetic field We consider a closed system of charged particles and the electromagnetic field. We wish to express the equations of motion in a form that makes its quantization trivial. Maxwell s equations can be represented in two homogeneous equations and two inhomogeneous ones. The homogeneous equations do not contain sources, and they take the form B = 0 ; The inhomogeneous equations in vacuum and in the presence of point charges are B = 1 c ( t E + j ɛ 0 ) Sources of the fields are the charge density ρ and the current density j, defined by E + t B = 0. (1.1) ; E = ρ ɛ 0. (1.) ρ( r) = α q α δ( r r α ), j( r) = α q α rα δ( r r α ). (1.3) The homogeneous equations are automatically respected when one introduces a vector potential A and a scalar potential Φ, so that B = A, E = t A Φ, (1.4) The definition of mode functions is induced by the second inhomogeneous equation (1.) in terms of the potentials ( A) = 1 ( c t A Φ t + 1 ) j ɛ 0 Mode functions M ν are introduced by the eigenvector relation. (1.5) ( M ν ) = ω ν c M ν. (1.6) We define the inner product of two vector functions F and G as the integral d r F G. With this definition, the left-hand side of (1.6) can be viewed as the square of a hermitian operator..., which proves that its eigenvalues are non-negative. This definition (1.6) of modes remains valid in the presence of reflecting boundaries, as in a cavity. Then the mode functions M ν must obey appropriate boundary conditions. In free space, the modes form a continuum. For simplicity, we denote the modes as a discrete set. This can be enforced by the standard procedure of selecting a large rectangular quantization volume V, and impose periodic boundary conditions. Alternatively, discrete summations can be read as integration over a continuum of mode numbers. Electronic address: nienhuis@molphys.leidenuniv.nl

2 We separate the space of mode functions in the subspace with eigenvalue zero, and the subspace with positive eigenvalues. The subspace with eigenvalues zero spans the Coulomb space, with basis functions C µ. The subspace with non-zero eigenvalues is called the radiation space, with basis functions R λ. These two subspaces are mutually orthogonal, since they correspond to non-overlapping sets of eigenvalues of a Hermitian operator. We express the eigenvalue relations as ( C µ ) = 0, ( Rλ ) = ω λ c R λ (ω λ > 0). (1.7) When we take the divergence... of the last equality in (1.7), we find that the radiation modes R λ have vanishing divergence, whereas the first equation (1.7) shows that the Coulomb modes C µ have vanishing curl. Hence C µ = 0, Rλ = 0. (1.8) Since the combination of Coulomb modes C µ and radiation modes R λ span the whole of function space, any vector field F is expanded in a unique way in the mode functions. Its projection on Coulomb space is called F C, and its projection on radiation space is F R, with F C = µ C µ F µ, FR = λ R λ F λ with F µ = d r C µ F, F λ = d r R λ F. This separation of function space corresponds to a separation of the vector field F = F C + F R into a longitudinal (curl-free) part F C and a transverse (divergence-free) part F R. The present formulation proves that this separation is unique and complete. A similar separation is valid in the presence of macroscopic non-dissipative media. We use the freedom of gauge to make the vector potential divergence free, so that A = 0, and A = A R. The equations (1.4) can now be reexpressed as B = B R = A, ER = t A, EC = Φ. (1.9) This gives separate equations for the radiation and Coulomb parts of the electric field E, whereas the magnetic field B has no Coulomb part. B. Coulomb field The inhomogeneous Maxwell equations (1.) gives for the Coulomb part of E t E C = 1 ɛ 0 j C, EC = ρ ɛ 0. This shows directly that the charge and current density ρ and j C are related by the continuity equation ( t ρ = j C = ) j. Moreover, one notices that E C is just the Coulomb field corresponding to the instantaneous locations of the charged particles. Since the Coulomb field is curl-free, it lies indeed in Coulomb space. We conclude that the Coulomb contribution to E is fixed by the instantaneous positions of the charges, and it moves along with them. This demonstrates that the Coulomb field E C is not an independent degree of freedom. It is simply determined by the charges. This instantaneous relation between the charge positions and the Coulomb field indicates that the separation in radiation and Coulomb field is not relativistically invariant. C. Radiation field and equations of motion The radiation part of eq. (1.5) gives A = 1 c ( t A 1 ɛ 0 j R ), (1.10)

3 where we used that ( A) = A for the divergence-free vector potential. Apparently, the radiation part of the current j R is the source of A, which in turn determines the radiation field E R and B. The motion of the particles is determined by the Coulomb force and the Lorentz force ( m α rα = q E( rα α ) + r α B( r ) α ). (1.11) The independent degrees of freedom of the closed system of particles and fields are the particle positions { r α } and the radiation part of the vector potential A R ( r), specified by {A λ }. The state of the system is fully described by { r α }, { r α }, {A λ } and { A λ }. When the state is known at any instant of time, the equations of motion (1.10) and (1.11) completely determine the state in the future as well as in the past. 3 II. HAMILTONIAN DESCRIPTION AND QUANTIZATION A. Radiation Hamiltonian and normal variables As found above, the state of the radiation field at any instant of time is fully specified by the two variables A λ and A λ. Alternatively, these can be represented as the single complex normal variable ɛ0 a λ = (ω λ A λ + ia ω λ ). λ Conversely, the set {a λ } determines both A and E R by A = (a λrλ + a λr ɛ 0 ω λ), ER = ωλ A = i(a λrλ a λr λ ɛ λ). (.1) 0 λ λ Note that the proof of (.1) is not completely trivial. One may have to use that the vector potential A is real, and that the eigenfunctions R λ can be chosen real. The total energy of the system of particles and fields separates in two contributions: one from the radiation field and one from the particles, including the Coulomb energy. After use of eq. (1.9) for B and E R, the radiation-field contribution is written as H R = 1 d r(ɛ 0E R + 1 B ) = 1 ω λ (a µ 0 λa λ + a λ a λ). (.) The equations of motion for a mode of the free field are identical to those of a harmonic oscillator. The variables A λ and ɛ 0 A λ serve as generalized canonical coordinate and momentum for the mode. The remaining energy is the kinetic energy of the particles (which is the sum of m α r α/) and the Coulomb field energy (arising from E C ). With some straightforward algebra this can be expressed as H p = α λ 1 ( p α q αa( rα )) + VC. (.3) m α Here V C is the Coulomb interaction energy of the particles, which arises from the Coulomb field energy ɛ 0 d r E C /. The quantity p α = m α rα + q α A( rα ) serves as the canonical momentum of particle α in the radiation field. It is rewarding (and not trivial) to verify that the classical Hamilton equations ẋ β = H/ p β, ṗ β = H/ x β for the generalized coordinates and momenta of particles and fields with the total Hamiltonian H = H R +H p indeed reproduce the equations of motion (1.10) and (1.11). This shows that the total energy actually serves as the Hamitonian in terms of the generalized coordinates and momenta. Conservation of energy is then automatic. B. Quantization Now that we have reexpressed the equations of motion (Maxwell s equations for the fields, and Newton s law for the particles) in a canonical Hamiltonian form, quantization has become trivial: just treat the generalized canonical

4 coordinates and momenta as operators, with the commutation rules [ˆp β, ˆx β ] = δ ββ. For the particles this implies that ˆ p α = ( /i)( / r α ). The normal field variables turn into field operators, for which the canonical commutation rules produce the well-known rules [â λ, â λ ] = i δ λλ. Equations (.1), (.) and (.3) remain valid with the replacement a λ â λ. Specifically, the quantum operators for the vector potential becomes ˆ A = (â λrλ + â λr ɛ 0 ω λ), (.4) λ λ 4 while the electric and the magnetic field take the form ˆ E R = ωλ i(â λrλ â λr ɛ λ), 0 λ ˆ B = λ (â λ Rλ + â λ ɛ 0 ω R λ). (.5) λ The Hamiltonian of the radiation field is Ĥ R == 1 ω λ (â λâλ + â λ â λ ). (.6) λ In the Schrödinger picture, the evolution of the quantum system is governed by the Schrödinger equation ( /i)(d/dt) Ψ = Ĥ Ψ for the state vector Ψ. In the Heisenberg picture, any physical quantity G obeys the equation of motion (d/dt)ĝ = (i/ )[Ĥ, Ĝ]. The Heisenberg equations of motion for the operators ˆ p α, ˆ r α and â λ closely resemble the classical equations of motion for the corresponding classical variables, corresponding to Maxwell s equations for the fields, and Newton s law with the Coulomb-Lorentz force for the particles. For instance, the evolution of the field operators is found as d iω λ â λ + i q αˆ rα dtâλ ɛ 0 ω R λ(ˆ r α ), (.7) λ ) with ˆ rα = (ˆ pα q α ˆ A(ˆ rα ) /m α. This confirms that only the radiative part of the current is a source of the field. In the absence of sources, the Heisenberg equation (.7) is simply dâ λ /dt = iω λ â λ. It is remarkable that the coupling between the fields and the particles in the Hamiltonian arises only in the kineticenergy terms in (.3), which contain the products q α ˆ A ˆ pα /m α. Note that the argument ˆ r α of ˆ A is also a quantum operator. In free space, it is customary to choose the modes of the radiation field as plane-wave modes. Then the index λ defines a mode of the radiation field, which takes the form of a normalized vector function α R λ ( r) = 1 V e λ e i k λ r. (.8) The mode is a plane wave, with wave vector k, and a normalized polarization vector e λ that is normal to k λ. For each wave vector, there are two independent polarization vectors. This reflects the transverse nature of the radiation field. The wave vectors are discrete, and for a cubic quantization volume V = L 3 with side L they take the values kλ = π(n x, n y, n z )/L, with integer n x, n y and n z. The mode functions form a complete set of orthonormal transverse vector functions on the volume V. With this expression for the modes, the field operators (.4) and (.5) attain their standard form. III. SEPARATION OF ANGULAR MOMENTUM OF RADIATION FIELD A. Classical description From Maxwell s theory it is well-known that the electromagnetic field has a density of momentum ɛ 0 E B. The integrated contribution from the Coulomb field E C combined with the kinetic momentum contributes to the total canonical momentum α p α of the charged particles. The momentum density of the radiation field is ɛ 0 ER B.

5 From now on, we only consider the radiation field, and we shall suppress the index R on the electric field E and the angular momenta. The angular momentum of the radiation field, which is J = ɛ 0 d r r ( E B) = d r j. (3.1) This expression can be separated after expressing the magnetic field in the vector potential and applying partial integration [1]. This leads to the result J = L + S, (3.) 5 with L = ɛ 0 i d r E i ( r )A i, S = ɛ 0 d r E A. (3.3) Since A is the transverse vector potential, these quantities are independent of gauge. The contribution L varies with the choice of the origin, just as an orbital angular momentum, so that it has an extrinsic nature. Moreover, it is determined by the phase gradient of the field. On the other hand, the contribution S does not change for a different choice of the origin, and it is determined by the polarization of the field. This gives it the flavor of a spin angular momentum. B. Quantum operators The expressions (3.3) for the contributions L and S to the angular momentum of the radiation field are quite suggestive for their interpretation as orbital and spin parts. However, this interpretation is problematic. This is clear when we consider the quantized version of the system. It is convenient to choose circular polarization vectors e ± ( k) in the plane normal to the wave vector k. The helicity of the vector e + is parallel to k, whereas e has opposite helicity. The quantum operator for the quantity S is obtained by substituting the quantum operators ˆ A and ˆ E in the expression (3.3). The result can be put in the intuitively attractive form [, 3] ˆ S = k k ( â k +( k)â + ( k) â ( k)â ( ) k). (3.4) This simply illustrates that each photon with wave vector k and polarization vector e + ( k) contributes to S a unit, in the direction of k. A photon with the opposite circular polarization e gives the opposite contribution. An obvious property of the quantum operator S is that its three components Ŝx, Ŝ y and Ŝz commute, simply because the creation and annihilation operators for different modes commute. In fact, number states of all modes with circular polarization are common eigenstates of all three components. This implies that ˆ S cannot be viewed as a proper angular momentum operator, which generates rotations. On the other hand, the operator ˆ J is an angular momentum momentum operator, as is exemplified by the commutation rule [Ĵx, Ĵy] = i Ĵz, etc. As a result, the commutation rules for the components of the quantum operator ˆ L take the form [, 4] [ˆL x, ˆL y ] = i (ˆL z Ŝz), (3.5) etc. These remarkable commutation properties can be traced back to the fact that a rotation of the polarization of a radiation field without rotating the field pattern itself would violate the transversality of the field. The vector operator ˆ S is a proper quantum operator within the space of physical states. Its transformation properties resemble a rotation of the polarization pattern only insofar as it is allowed within the constraint of transversality [4]. IV. STATES OF FREE FIELD MODE A. Number states The states of a single mode of the radiation field are mathematically equivalent to the states of a free harmonic oscillator. A single mode is described by the Hamiltonian Ĥ = ω(â â + 1 ), with the commutation rule [â, â ] = 1.

6 (The mode index λ is suppressed.) From the commutation rules it follows that the energy eigenstates are the number states n, with n = 0, 1,..., so that â n = n n 1, â n = n + 1 n + 1, â â n = n n. These states are stationary, and therefore highly non-classical: they have a well-determined field amplitude, and a fully undetermined phase. The expectation values of the fields ˆ E, ˆ A and ˆ B are zero. The ground state 0 is also called the vacuum state. But the fluctuations E, A and B are non-zero, even in the vacuum state. The number n indicates the number of elementary excitations (photons) of the mode. Each photon represents an energy ω. Photons as elementary excitations of a single mode are just as delocalized as the mode function R. Localized single-photon states can be formed as superpositions of single-photon states in different modes, such as c λ λ, with λ the state n with n = 1 in the mode λ. 6 B. Coherent states The states z that correspond most closely to classical states have (average) field values as given in (.1) with â replaced by the complex number z. They are defined by the requirement z â z = z, z â â z = z. This implies the eigenvalue relation â z z z = 0, with the solution z = exp( 1 z ) n z n n! n. (4.1) According to Eq. (4.1), in a coherent state the probability distribution P n over the number states is z n z P n = e. (4.) n! This is a Poissonian distribution, with average value n = z. The variance of a Poissonian distribution is equal to its average, so that n n n = n = z. (4.3) The coherent states are normalized by definition. However, they are not orthogonal. Their overlap can be directly evaluated from the expansion (4.1), with the result z z = exp ( 1 z 1 ) z + z z, (4.4) so that the strength of the overlap has a simple Gaussian shape z z = exp ( z z ). Moreover, the coherent states are overcomplete: each state of the mode can be expanded in coherent states, but this expansion is not unique. One expansion can be found by applying the closure relation Î = 1 d z z z, (4.5) π where the integration extends over the complex plane. The operator Î is the unit operator for the mode. The uncertainty in a coherent state is best specified by the introducing quadrature operators ˆX = 1 (â + â ), Ŷ = 1 i (â â ), which obey the commutation relation [ ˆX, Ŷ ] = i, and therefore the uncertainty relation X Y 1. In a coherent state, X = Y = 1/, for all values of z. Hence the uncertainty is equally divided over the two quadratures, and X and Y have the same value as in the vacuum state. Coherent states can alternatively be described as a displaced vacuum state with z = ˆD(z) 0, (4.6) ˆD(z) = exp(zâ z â). (4.7) The displacement properties of ˆD(z) follow from the identity ˆD (z)â ˆD(z) = â + z.

7 7 U Y D X D Y X U amplitude squeezing phase squeezing FIG. 1: Illustration in the XY -plane of the shape of amplitude and phase squeezed states. These can be created by applying a displacement to a squeezed vacuum state. C. Squeezed states We introduce the unitary squeeze operator Ŝ(ξ) = exp( 1 ξ â 1 ξâ ), for arbitrary complex number ξ = ρ exp(iθ). It transforms the field operator â as The rotated quadrature operators Ŝ (ξ)âŝ(ξ) = â cosh ρ â exp(iθ) sinh ρ. ˆ X = ˆX cos θ + Ŷ sin θ, ˆỸ = ˆX sin θ + Ŷ cos θ then transform according to Ŝ (ξ) ˆ XŜ(ξ) = ˆ Xe ρ, Ŝ (ξ) ˆỸ Ŝ(ξ) = ˆỸ e ρ. (4.8) Hence Ŝ effectively multiplies the quadrature components by scalar factors. The squeezed vacuum state Ŝ(ρ) 0 for ξ = ρ real has a reduced uncertainty X, and an enhanced uncertainty Y, with X Y = 1 unmodified. Squeezed coherent states arise when a displacement operator is applied to the squeezed vacuum state, and we consider the free evolution of the initial state ψ(0) = ˆD(z)Ŝ(ξ) 0. For positive ξ, the long axis of the ellipse is in the Y -direction, so that initially the ellipse is vertically oriented. When z is taken real, the center of the ellipse lies on the X-axis at time zero. During free evolution, this ellipse rotates in the clockwise direction, at the angular velocity ω. This is illustrated in Figure 1 on the left. In this case, the fluctuations in ˆX are reduced at the times that the expectation value ˆX is maximal. This means that the amplitude fluctuations are reduced compared with the vacuum fluctuations, which are the same as in a coherent state. This is called amplitude squeezing. The reduction of the amplitude fluctuations is compensated by an enhancement of phase fluctuations. Conversely, when z is taken imaginary, the center of the ellipse lies initially on the Y -axis, and the uncertainty in a quadrature is maximal when its expectation value passes a maximum. This is the case of enhanced amplitude fluctuations, and phase squeezing. This situation is pictured in the Figure 1 on the right. These results show how a combination of squeezing Ŝ(ξ) (pumping that is quadratic in the ladder operators), displacement ˆD(z) (pumping that is linear in the ladder operators), and free evolution Û(t) acting on a vacuum state creates minimum-uncertainty states. The uncertainty ellipses can have arbitrary ellipticity (determined by ξ), arbitrary locations in phase space (determined by z), and arbitrary orientation (determined by the angle ωt). It is important to realize that the squeezed vacuum state is not a vacuum state, since it has a non-vanishing expectation value of the photon number. By using (4.8) one finds that 0 Ŝ (ρ) ˆNŜ(ρ) 0 = 1 0 ˆX e ρ + Ŷ e ρ 1 0 = 1 (cosh(ρ) 1) = sinh ρ. (4.9) Reduction in the fluctuations in a quadrature below the vacuum fluctuations is possible, but not without the creation of photons in a special way.

8 8 D. Phase and number operators For a mode of the radiation field, the number of photons is described by the number operator ˆN = â â. The exponential operator exp[ i ˆNφ 0 ] adds an amount φ 0 to the phase of the field, since for coherent states exp( i ˆNφ 0 ) z = ze iφ 0. Since the number operator generates phase shifts (justs as the momentum operator generates position shifts) this suggests that the phase is canonically conjugate to the photon number, and that its phase representation the number operator would take the form N = i / φ. This also suggests the existence of a phase operator ˆΦ, that should obey the commutation rule [ ˆN, ˆΦ] = i (as originally suggested by Dirac [5]). However, if we take matrix elements of this commutation relation between number states, we obtain (n n ) n ˆΦ n = iδ nn. This is an obvious contradiction: the l.h.s. disappears for n = n, whereas the r.h.s. vanishes only for n n. This problem is partly related to the fact that the phase is a periodic variable: it is defined only modulo π. Hence, it is more natural to consider the exponential operator Ê = exp( iˆφ), since the value of exp( iφ) on the unit circle defines the value of φ apart from additive factors π. From the expected commutator [ ˆN, ˆΦ] it follows that [ ˆN, Ê] = Ê, which gives the matrix elements (n n ) n Ê n = n Ê n. Hence n Ê n can only be non-zero for n n = 1. This is in line with the polar decomposition of the annihilation operator. Since Ê is expected to be unitary, we would expect the factorization iˆφ a = e ˆN = Ê ˆN. From the known properties â n = n n 1 we find then Ê n = n 1, which is equivalent to the expression Ê = n n 1 n. The operator Ê has indeed the expected commutator [ ˆN, Ê] = Ê with ˆN, and it shifts the photon number, in agreement with the expectation that the phase operator generates shifts in the photon number. The eigenstate of Ê with eigenvalues exp( iφ) is φ = n n e inφ. This state is not normalizable, just as eigenstates of position and momentum of a particle. exp( i ˆNφ 0 ) shifts the phase states according to Then the operator exp( i ˆNφ 0 ) φ = φ + φ 0, as expected. So everythings seems perfectly in line with what one would expect. However, Ê is not unitary: even though ÊÊ = 1, one finds Ê Ê = This deviation from unitarity arises from the fact that Ê 0 = 0, which does not conserve the norm of a state vector. Hence no Hermitian operator ˆΦ exists so that Ê = exp( iˆφ). On the other hand, quantummechanical observables are required to be Hermitian, to make sure that the eigenvalues are real, and the eigenstates are orthogonal: a measurement must have a real outcome, and the states corresponding to different outcomes must be distinct. This can be remedied formally by truncating the space of number states, so that n max is the highest value. If we then define Ê = exp( iˆφ) by Ê n = n 1 for 1 n n max, and Ê 0 = n max, the unitarity is restored. This method is discussed by Barnett and Pegg [6]. For n max larger than all relevant photon numbers in a particular problem, the specific value becomes immaterial. For arbitrarily large (but finite) n max, the number of eigenstates of the modified operator E, and of the correponding Hermitian operator ˆΦ, is n max + 1, and the set of n max + 1 eigenvalues φ are evenly distributed over the unit circle. Another possibility is to consider a specific measurement technique of the phase, and analyze the precise quantities measured. Often it is found that an observed quantity is of type  Â, with  = â + ˆbe iφ, â and ˆb annihilation operators of different modes. An example is the observation of an interference pattern, as function of a (classical) phase variable φ. E. Relative phase and number difference operators Just as the phase of a mode is (more or less) canonically conjugate to the photon number, the relative phase between two modes λ and λ is canonically conjugate to the difference in photon number. A natural candidate for

9 9 n λ 0 0 n λ FIG. : Sketch of the 9 different number states of the two modes with N = 8 photons. the exponential operator ˆF = exp( iˆφ λλ ) for two modes is defined by ˆF n λ, n λ = n λ 1, n λ + 1. Then ˆF leaves the total photon number unchanged, while changing the number difference by. The operator ˆF can be separately defined for each subspace corresponding to a given value of N = n λ + n λ. This subspace is spanned by the n + 1 states n, 0, n 1, 1,..., 0, n. An example of such a substate is indicated in Figure. The unitarity of the operator ˆF can be secured by defining its action on the minimal difference state as ˆF 0, n = n, 0. In the n + 1 dimensional space of states with n photons, the operator ˆF (and hence the phase-difference operator ˆΦ λλ ) has the n + 1 eigenstates 1 N N, φ k = N n, n e inφ k, N + 1 n=0 with eigenvalues exp( iφ k ) specified by φ k = πk/(n + 1), k = 0, 1,..., N. These states are also eigenstates of the total number operator â λ λa 1 + â λ â λ with eigenvalue N. The relative phase operator ˆΦ λλ is naturally defined as having these states as eigenstates, with the eigenvalues φ k. Only for large values of the total photon number N do the relative-phase eigenvalues have a dense spectrum of eigenvalues within the interval [0, π]. V. DENSITY MATRIX AND PHASE SPACE DISTRIBUTIONS FOR SINGLE SYSTEM A. Density matrix and quantum measurement In a classical picture of a measurement, a physical system is brought into contact with a measurement device (the meter). By the interaction the state of the meter is changed so that it reflects the value of an observable of the system. Ideally, the state of the system is not affected by the interaction. Therefore a repeated measurement on the same system can be used to enhance the measurement precision. The state of a classical system is specified by the values of the observables. In elementary quantum mechanics the state of a system is specified by a normalized vector ψ in a Hilbert space H, so that ψ ψ = 1. Observables are represented by Hermitian linear operators ˆQ acting on the Hilbert space of state vectors. Such an operator can be represented by its eigenvectors φ i and the corresponding eigenvalues q i, so that ˆQ φ i = q i φ i. When the observable Q is measured on the system in the state ψ, the outcome is any one of the eigenvalues q i of the corresponding operator ˆQ. The probability for the outcome q i is the overlap p i = φ i ψ. The expectation value of the measurement outcome is ˆQ = i p i q i = ψ ˆQ ψ. (5.1)

10 Immediately following the measurement with this outcome, the state of the system is the eigenstate φ i, so that when the measurement of ˆQ is repeated immediately, it returns the same eigenvalue q i with certainty. This standard picture of an instantaneous change of the state as a result of a measurement is known as the projection postulate. It implies that it is impossible to determine the state vector ψ of a single system, even by repeated measurements. The determination of a state vector is only possible when we have at our disposal an ensemble of identical systems in identical states. For later use it is convenient to slightly generalize the notation of the measurement process as described here. We introduce the projection operators on the eigenstates φ i as ˆP i = φ i φ i. (5.) Then the probability p i that the system is detected to be in the eigenstate φ i can be expressed as p i = ψ ˆP i ψ. (5.3) The normalized state of the system directly after the measurement can also be expressed in terms of the projection operator, as ψ after = ˆP i ψ / p i. (5.4) Next, we allow the system to be in a mixed state, where a density matrix is needed. When the system is not ideally prepared, a classical uncertainty exists as to the precise state vector. Let us assume that there is a probability r 1 that the (normalized) state vector is ψ 1, a probability r that the (normalized) state vector is ψ, etc. Then the density matrix takes the form 10 ˆρ = n r n ψ n ψ n. (5.5) The (real and non-negative) probabilities r n add up to 1, so that the density matrix is normalized in the sense that Trˆρ = 1. When the observable ˆQ is measured, the expectation value of the outcome is the average of the expectation values ψ n ˆQ ψ n, with the probabilities r n as weighting factors. This implies that ˆQ = Trˆρ ˆQ. (5.6) The probability for the measurement outcome q i, which is the same as the probability that the system is detected in the state φ i, is p i = Trˆρ ˆP i. (5.7) This is the average over the probabilities φ i ψ n for this measurement outcome for the system in the state ψ n. In the same spirit, the state of the system immediately after the measurement can be denoted as ˆρ after = ˆP i ˆρ ˆP i /p i. (5.8) It is important to notice that these results (5.5)-(5.8) are valid both for a pure state and for mixtures. In the special case of a pure state vector ψ, the density matrix is the simple projection operator ˆρ = ψ ψ. When the density matrix represents a pure state, it obeys the identities ˆρ = ˆρ, and Trˆρ = 1. For a mixed state, it obeys the inequality Trˆρ < 1. Whether the density matrix ˆρ represents a pure state or a mixture, the density matrix (5.8) after the measurement coincides with the projection operator on the state (5.4), so that it always corresponds to a pure state. One should notice that the pure states ψ n that compose the density matrix (5.5) are assumed to be normalized, but not necessarily orthogonal. When these states ψ n are not orthogonal, they are not eigenvectors, and the probabilities r n are not eigenvalues of ˆρ. In fact, in this common formulation of the measurement process in quantum mechanics it has been tacitly assumed that the system has a single degree of freedom, such as a single spin, or a single particle. In these notes we discuss the description of quantum measurements in the more general case of composite systems, which contain different degrees of freedom. These can refer to different properties (such as spin and translational state) of a single particle, or to different subsystems that may be spatially separated. Then a measurement on one subsystem does not specify the state completely. On the other hand, as a result of the measurement on one subsystem, the state of another subsystem can be modified.

11 11 B. Characteristic functions of density matrix The state of a single radiation mode is specified by the normalized density matrix ˆρ. characteristic functions of the complex variable λ We introduce the three χ N (λ) = e λ âe λâ = Trˆρ e λ âe λâ, (5.9) χ A (λ) = e λâ e λ â = Trρ e λâ e λ â, (5.10) χ S (λ) = e λ â+λâ = Trˆρ e λ â+λâ. (5.11) It does not matter whether ρ is a pure or a mixed state. The index N stands for normal ordering, the index A for antinormal ordering, and the index S for symmetric ordering. We use the operator identities eâ+ ˆB = e ˆBeÂe [Â, ˆB]/ = eâe ˆBe [Â, ˆB]/, (5.1) which hold when the commutator [Â, ˆB] is a scalar. These identities can be proven by differentiating the operator exp( ξ ˆB) exp[ξ(â + ˆB)] exp( ξâ) with respect to ξ. From eq. (5.1) it follows that e λ âe λâ = e λâ e λ âe λ = e λ â+λâ e λ /, (5.13) so that the three characteristic functions (5.9)-(5.11) are related by χ N (λ) = χ A (λ)e λ = χ S (λ)e λ /. (5.14) Hence, knowledge of any one of the three characteristic functions is sufficient to determine the other ones. Conversely, either one of the three characteristic functions (5.9)-(5.11) determines the density matrix, according to the identities ˆρ = 1 d λ χ N (λ)e λâ e λ â = 1 d λ χ A (λ)e λ âe λâ = 1 d λ χ S (λ)e λ â λâ, (5.15) π π π with the integrations over the complex λ plane. Indeed, these expressions for ˆρ lead to the correct expressions (5.9)-(5.11) for the characteristic functions, as can be shown with the identities Tr e λâ e λ âe µ âe µâ = Tr e λ âe λâ e µâ e µ â = Tr e λ â λâ e µ â+µâ = πδ (λ µ). (5.16) The validity of (5.16) can be proven directly by inserting the closure relation (4.5) in the first expression. Equations (5.15) expand the density matrix either in terms of normally (N), antinormally (A) or symmetrically (S) ordered products of annihilation and creation operators. Normal ordering means that annihilation operators are placed on the right side of the creation operators, and antinormal ordering implies the reverse order. In symmetrically ordered products of powers of â and â, terms as â n (â ) m occur only in symmetric combinations of all orderings, such as they arise when one evaluates the product (â + â ) n+m. In the same way one can prove that any operator ˆF can be represented in normal, antinormal, or symmetric form with ˆF = 1 π d λ φ N (λ) e λâ e λ â = 1 π d λ φ A (λ) e λ âe λâ = 1 π φ N (λ) = e λ âe λâ = Tr ˆF e λ âe λâ, φ A (λ) = e λâ e λ â = Tr ˆF e λâ e λ â, d λ φ S (λ) e λ â λâ, (5.17) φ S (λ) = e λ â+λâ = Tr ˆF e λ â+λâ. (5.18) Furthermore, we introduce the three functions of the complex variable z, which follow by substituting a by z, and a by z in the expressions (5.17) for ˆF, so that f N (z) = 1 d λ φ N (λ) e λz +λ z, π f A (z) = 1 d λ φ A (λ) e λz +λ z, π f S (z) = 1 d λ φ S (λ) e λz +λ z. (5.19) π

12 These functions have the same functional form (of z and z ) as the operator ˆF (as function of â and â ), in the properly ordered form. On the other hand, the relations (5.19) have the nature of two-dimensional Fourier transforms, which may be inverted to give φ N (λ) = 1 d z f N (z) e λz λ z, π φ A (λ) = 1 d z f A (z) e λz λ z, π φ S (λ) = 1 d z f S (z) e λz λ z. (5.0) π Hence any of the three functions f N, f A and f S can be used to calculate a characteristic function with (5.0), which then reproduces the operator ˆF with (5.17). 1 C. Normal characteristic function and Q distribution After substituting the closure (4.5) in the definition of χ N in (5.9) in between the exponentials, we find that χ N (λ) = d z Q(z) e λ z+λz. (5.1) Here Q(z) = z ˆρ z /π is a normalized, positive, and real function over the complex z plane. Since Re z and Im z represent the two quadratures of the field, analogous to the position and momentum of a mechanical particle, Q(z) may be viewed as a distribution function over phase space. The density matrix is fully specified when Q(z) is known, as follows from (5.15) and (5.1). When an operator ˆF is expanded in the antinormal form of (5.17), we obtain an expression for its expectation value after substituting the closure (4.5) in between the annihilation and creation operators. This gives ˆF = Trˆρ ˆF = d z Q(z) f A (z). The expectation value takes the classical form of an integration over phase space of the product of a phase space distribution function Q(z), where now the function f A (z) represents the quantity ˆF. The function Q(z) is analytical as a function of the two complex quantities z and z, but not as a function of z alone. Therefore, it is often denoted as Q(z, z ) in the literature. The expression for χ N in terms of Q may be Fourier-inverted to give Q(z) = 1 π d λ χ N (λ) e λ z λz. D. Antinormal characteristic function and P distribution Suppose that the density matrix ˆρ can be represented as a diagonal expansion over coherent states, as ˆρ = d z z P (z) z. (5.) Substituting this expression in eq. (5.10) for χ A gives the Fourier relation and its inversion χ A (λ) = d z P (z) e λz λ z, P (z) = 1 π d λ χ A (λ) e λ z λz. The latter relation gives an expression for P (z) for any density matrix ˆρ. However, it is quite common for χ A (λ) to have a polynomial form. For instance, for a number state, when ˆρ = n n, χ N (λ) is a polynomial in λ of rank n. This implies that its Fourier transform P (z) contains higher derivatives of delta functions. In general, P is not analytical, but it is a distribution in the mathematical sense, which is well-defined under an integral. For a Hermitian and normalized density matrix ˆρ, P is real and normalized. It serves as a phase space distribution function for normally ordered operators. With (5.) and the normally ordered form of (5.17) one derives ˆF = TrρF = d z P (z) f N (z). (5.3) However, it cannot be viewed as a phase space distribution function in the classical sense, since it can attain negative values.

13 13 E. Symmetric characteristic function and Wigner distribution In analogy to the distributions Q(z) and P (z) in terms of χ N (λ) and χ A (λ), we define the distribution function W (z) relating to the symmetrized characteristic function as W (z) = 1 π d λ χ S (λ) e λ z λz, χ S (λ) = d z W (z) e λz λ z. (5.4) This is called the Wigner distribution function, which was introduced by Wigner for a mechanical particle rather than for a mode [7]. Again, W (z) is real and normalized for a Hermitian and normalized density matrix ˆρ. If we calculate the expectation value of ˆF by using the symmetrized from of both ˆρ (from (5.15)) and ˆF (from (5.17)), while using (5.16), we find ˆF = Tr ˆρ ˆF = 1 d λ χ S (λ)φ S ( λ) = d z W (z) f S (z). (5.5) π The attractive feature is that now the prescriptions for the functions f S (z) and W (z) in terms of the operators ˆF and ˆρ are identical (apart from a simple factor π). Just as P (z), W (z) can be negative in parts of phase space. Expressions for the Wigner distribution function in terms of the coordinate x and momentum y are usually defined as W (x, y) = 1 π dµ x 1 µ ˆρ x + 1 µ eiyµ = 1 4π dµ dν χ S (µ, ν)e iµy iνx, (5.6) where we separate z = 1 (x + iy), λ = 1 (µ + iν). When the density matrix ˆρ is expressed in momentum representation, the Wigner distribution takes the alternative form W (x, y) = 1 dν y 1 π ν ˆρ y + 1 ν e ixν. (5.7) The symmetric characteristic function is χ S (µ, ν) = = = dxdy W (x, y)e iµy+iνx dx e iνx x 1 µ ˆρ x + 1 µ dy e iµy y 1 ν ˆρ y + 1 ν = Tr ˆρ e iµŷ +iν ˆX. (5.8) Actually, this definition of the Wigner distribution function differs by a factor from the definition of W (z), since it obeys the normalization condition dxdy W (x, y) = 1, with dxdy = d z. The marginal integrals dx W (x, y) = y ˆρ y and dx W (x, y) = x ˆρ x are the momentum distribution and position distribution respectively. For a particle in three dimensions, the Wigner distribution function W ( r, p) is defined in complete analogy. VI. CLASSICAL AND QUANTUM BITS A. The concept of a qubit Classical information theory uses as a unit of information the bit. It corresponds to the information content of a single choice between two options, which are usually represented as 0 or 1. Hence a series of N bits can be represented as a series of N elements, each element being 0 or 1. Any piece of information, like the contents of a book, or the sequence of the nucleotides in a string of DNA, can be encoded in a string of classical bits. Such a string of length N may be viewed as a binary number of N digits, which represents one number out of N (0 to N 1). The natural quantum generalization of a classical bit is a two-state system (for instance two of the energy levels e and g of an atom), two number states of a radiation mode (for example the vacuum state 0 and the one-photon state 1 ), or two independent polarization states V (vertical) and H (horizontal) of a photon. In the context of quantum information theory, a two-state system is called a quantum bit, or qubit for short. To stress the analogy with a classical bit, we can denote the two basis states as 0 and 1 in all cases. In contrast to a classical bit, the

14 state of a qubit is a state vector in two-dimensional state space, of the general form ψ = α α 1 1. Using the normalization, and the fact that an overall phase factor has no physical significance, the full specification of the state requires knowledge of the complex number β = α 1 /α 0, which is equivalent to two real numbers. It is easy to verify that an arbitrary complex number β defines uniquely the normalized state vector ψ = (β )/ 1 + β, apart from an irrelevant overall phase factor. This suggests that a qubit contains an unlimited amount of classical information. On the other hand, each observable ˆQ of the qubit has two possible real eigenvalues, so that the information produced by a measurement performed on a qubit is just a single classical bit. This reminds us of the fact that an unknown state vector ψ cannot be determined by a single measurement. The act of measurement disturbs the state of the qubit, and subsequent measurements on the same qubit do not allow to reproduce with certainty the initial state prior to the first measurement. Moreover, the state of N qubits is a state vector in a Hilbert space of N dimensions. As basis of this state, we can take all possible states of length N, so that the basis vectors are enumerated precisely by all possible values of N classical bits. However, in contrast to the classical case, each linear combination of these basis states also is an allowed state of the N-qubit system, which implies that each of the classical states can in some sense be present in a single quantum state. 14 B. Spin model of a qubit Mathematically, all two-state systems are equivalent. A convenient picture of a physical realization is constituted by a spin with S = 1/, e.g. the spin of an electron or a proton. The three independent Hermitian spin operators Ŝ1, Ŝ and Ŝ3 are combined into a vector operator ˆ S. For convenience, and to avoid repeated occurrences of factors 1/, we introduce the vector of Pauli matrices ˆ σ = ˆ S. The eigenstates of ˆσ3 (or Ŝ3) are termed (spin up) and (spin down), which we shall use as an alternative notation to 0 and 1. On the basis of these states, the components of ˆ σ have the matrix form of the well-known Pauli matrices, so that ˆσ 1 = ( ), ˆσ = ( 0 i i 0 ), ˆσ 3 = ( ) 1 0. (6.1) 0 1 Each one of these operators has a vanishing trace, and two eigenvalues 1 and 1, which correspond to the two possible outcomes of a measurement. When we supplement these three Pauli matrices with the unit matrix ( ) 1 0 ˆσ 0 =, (6.) 0 1 we have a complete set of four Hermitian matrices ˆσ 1, ˆσ, ˆσ 3 and ˆσ 0. They obey the multiplication rule ˆσ iˆσ j = k iε ijkˆσ k + δ ij ˆσ 0, (6.3) for i, j, k = 1,, 3, with ε ijk the fully antisymmetric tensor of rank 3, with ε 13 = 1. Because of the completeness, we can expand any matrix in these four Pauli matrices. As a special case, we express the normalized density matrix ˆρ as with P obeying the identity ˆρ = 1 (ˆσ 0 + P ˆ σ), (6.4) ˆ σ = Tr ˆρ ˆ σ = P. (6.5) This shows that the real vector P is the expectation value of the Pauli vector, which is equal to ˆ S. C. Bloch sphere for spin states In order to check under what conditions the density matrix ˆρ represents a pure state, we evaluate the square ˆρ, and by using (6.3) we find ˆρ = 1 4 ˆσ 0(1 + P ) + 1 ˆ σ P. (6.6)

15 As mentioned before, the density matrix (6.4) represents a pure state if and only if Tr ˆρ = 1, which is the case if the vector P has the length 1. Since for a pure state ˆρ has one eigenvalue 1, and one eigenvalue 0, the operator u ˆ σ has the eigenvalues ±1 for all real unit vectors u. Since for a mixed state, the density matrix has two positive eigenvalues (that add up to 1), Eq. (6.4) represents a mixed state when the vector P has length P < 1. We conclude that the density matrix of a qubit can be uniquely represented by a point P in a sphere with radius 1. A mixed state corresponds to a vector P with P < 1, which is represented by a point inside the sphere. The state is pure when P = u is a unit vector, specified by a point on the surface of the unit sphere. This sphere is called the Bloch sphere when the qubit is a spin 1/. A pure-state vector is represented by a point on the surface of the Bloch sphere, apart from an overall phase factor. Now we consider a unit vector specified by the spherical angles θ and φ, so that u(θ, φ) = (cos φ sin θ, sin φ sin θ, cos θ), with 0 φ π, 0 θ π. We have seen that the operator u ˆ σ has an eigenstate with eigenvalue 1, for which the expectation value of the Pauli vector (or the spin vector) is directed parallel to u, and an eigenstate with eigenvalue 1, corresponding to a spin vector that is antiparallel to u. These eigenstates follow from the eigenstates and after a rotation in spin space ˆR(θ, φ) = exp( iφˆσ 3 /) exp( iθˆσ /) exp(iφˆσ 3 /). (6.7) The operator ˆR(θ, φ) consists of a rotation over an angle φ about the 3-axis, then a rotation over an angle θ about the -axis, and finally a rotation over an angle φ about the 3-axis. This rotation transforms the positive 3-direction into the direction u. The matrix form of this rotations follows from the matrices for the rotations about the axes exp( iθˆσ /) = ( cos(θ/) sin(θ/) sin(θ/) cos(θ/) The eigenstate of u ˆ σ with eigenvalue 1 is then ), exp( iφˆσ 3 /) = ( e iφ/ 0 0 e iφ/ 15 ). (6.8) ˆR(θ, φ) = cos θ + sin θ eiφ. (6.9) This is the pure state vector that is represented by the point u on the surface of the Bloch sphere. The opposite point u represents the state vector ˆR(θ, φ) = sin θ e iφ + cos θ, (6.10) that is orthogonal to the state vector (6.9). The North pole of the Bloch sphere represents the state, and the state is represented by the South pole. A point on the Equator (θ = π/) with azimuthal angle φ indicates the eigenstate of the Pauli-vector component ˆσ 1 cos φ/ + ˆσ sin φ/ with eigenvalue 1. VII. POINCARÉ SPHERE AND SCHWINGER REPRESENTATION FOR TWO MODES A. Poincaré sphere for polarization states Another important realization of a qubit is provided by the two-dimensional polarization degree of freedom of single photons. As discussed before, a mode of the field is a complex vector function in space. For plane wave modes with a wave vector k, the mode function is a product of a spatial mode and a polarization. In this language, each spatial mode still has two possible polarization modes, for which we take the circular polarizations e ±. When we arbitrarily choose the z-axis parallel to k, these two orthonormal polarization vectors are e ± = 1 ( e x ± i e y ), (7.1) with e x and e y the unit vectors in the x- and the y-direction. The corresponding modes are specified by the mode functions R ±, and photons in these two modes are created by the operators â ±. Any normalized polarization vector e can be written as a unitary linear combination of these two basis vectors. These basis vectors e + and e can be mapped on the two orthogonal spin states and, so that an arbitrary linear combination of the basis polarization vectors is mapped on the same linear combination of the spin states. In

16 this way, the two-dimensional space of polarization vectors is represented as points on the surface of the unit sphere. The sphere representing polarization vectors is termed the Poincaré sphere [8]. Then the point on the Poincaré sphere with the spherical angles θ and φ represents the polarization vector e up (θ, φ) = e + cos θ + e sin θ eiφ, (7.) in analogy to the spin state (6.9). An alternative expression for the same polarization vector is obtained by separating (7.) as e up (θ, φ) = exp(iφ/) ( e R (θ, φ) + i e I (θ, φ)), with e R (θ, φ) = 1 ( cos θ + sin θ ) ( e x cos φ + e y sin φ ), e I (θ, φ) = 1 ( cos θ sin θ ) ( e x sin φ + e y cos φ ). (7.3) Since e R and e I are orthogonal, it is easy to recognize the shape of the polarization ellipse. Since e I is smaller than e R (except at the poles), the direction of e R indicates the long axis of the ellipse. The North pole of the Poincaré sphere represents right circular polarization e +, the South pole represents left circular polarization e. The point on the Equator (θ = π/) with azimuthal angle φ specifies linear polarization at an angle φ/ with the x-axis. In between the poles and the Equator, the polarization is elliptical. Opposite points on the sphere represent orthogonal polarizations. The polarization corresponding to the opposite spin state (6.10) is equal to e down (θ, φ) = e + sin θ e iφ + e cos θ. (7.4) We consider now the two modes with mode functions R ±, which have the same spatial behavior (ideally a plane wave with wave vector k = k e z ), and opposite circular polarization e ±. The two-dimensional one-photon state space is spanned by the basis set â ± 0, 0, with 0, 0 the two-mode vacuum state. The one-photon state with the polarization (7.) is then obtained as the corresponding linear combination of these basis states, and the same is true for the one-photon state with the opposite polarization (7.4). These states result when the creation operators 16 â (θ, φ) = â + cos θ + â sin θ eiφ, ˆb (θ, φ) = â + sin θ e iφ + â cos θ (7.5) act on the vacuum state. Hence, the operators (7.5) create a photon with polarization e up (θ, φ) or e down (θ, φ). Each one-photon state in these two modes is uniquely represented by a point on the surface of the unit sphere. A mixed state, represented as a density matrix on these basis states is represented by a real vector P inside the Poincaré sphere, in full analogy to the Bloch sphere representing density matrices of a spin 1/. B. Stokes operators In the case of a spin 1/, the three directions 1, and 3 and the points on the Bloch sphere indicate the components of the spin vector along the x, y and z axes in real space. In the case of polarization, the space of the Pauli operators and the points on the Poincaré sphere refer to a fictitious space, that is defined in mere analogy to the spin case. The three components of the unit vector u = (u 1, u, u 3 ) correspond respectively to the degree of linear polarization along the x and the y axis (u 1 ), the degree of linear polarization in the directions under 45 with the x- and the y-axis (u ), and the degree of circular polarization (u 3 ). This can be checked from the significance of the vector u as the expectation value of the Pauli vector ˆ σ. In classical optics these quantities are known as the Stokes parameters, that together fully specify the polarization vector [8]. From the Bloch-Poincaré analogy we know that for an arbitrary one-photon state â (θ, φ) 0, 0, the expectation value of the Pauli vector is ˆ σ = u, where the Pauli operators have the form of the Pauli matrices on the basis of the states â ± 0, 0. Similarly, when a density matrix ˆρ on this two-dimensional state space of one-photon states takes the form (6.4), the expectation value is ˆ σ = P, and the density matrix can be uniquely represented by the point P inside the Poincaré sphere. Within the two-dimensional state space of one-photon states the action of the Pauli operators ˆσ 1, ˆσ and ˆσ 3 coincides with the action of the operators ˆ Σ, defined by the three components ˆΣ 1 = â â + + â +â,

17 17 ˆΣ = iâ â + iâ +â, ˆΣ 3 = â +â + â â, (7.6) which can be summarized in an elegant fashion by the notation ( ) ˆ Σ = (â + â â+ )ˆ σ â (7.7) The operators ˆΣ 1 /, ˆΣ / and ˆΣ 3 / obey the commutation rule of angular momentum operators, just as the spin operators Ŝ1 = ˆσ 1 /, Ŝ = ˆσ / and Ŝ3 = ˆσ 3 /. On the other hand, these operators obviously are defined on arbitrary states of the system consisting of the two modes R ±, not just the one-photon states. These operators play the role of the Stokes operators, which may be regarded as a quantum version of the classical Stokes vector that specifies the polarization state of a beam of light [8]. The operators ˆ Σ conserve the number of photons, and thereby commute with the total photon number ˆΣ0 = ˆN + + ˆN = â +â + + â â. The quantum version of the Stokes vector is the vector P, defined by its components P i = ˆΣ i ˆΣ 0, (7.8) with i = 1,, 3. For the case of one-photon states, the denominator is always equal to 1, so that this definition coincides with the earlier definition in this special case. Only then does the vector P determine the density matrix completely. In the N +1-dimensional subspace of N photons, specification of the full density matrix requires N +N parameters, with the vector P, defined by (7.8) specifying 3 of them. C. Schwinger representation of two modes We have noticed that the three operators ˆ Σ/ behave as the components of an angular momentum. This is the basis of the Schwinger representation, which builds on the equivalence of the J + 1-dimensional state space of an angular momentum J with the states of two boson modes with total boson number N = J [9]. The eigenstate JM of Ĵ3 with eigenvalue M corresponds to the eigenstate n +, n = n +, N n + of ˆΣ 3 / with M = (n + n )/. It is interesting to notice that the spin angular momentum of the photon state is equal to (n + n ) = M. This reminds us of the fact that, in contrast to the Bloch sphere, the Poincaré sphere does not generally specify the angular momentum vector of the state of the two modes. The components of the angular-momentum operator are Ĵi = ˆΣ i /. The representation of the rotation group SU() with dimension J + 1 = N + 1 is generated by the N-boson states. The rotation corresponding to Eq. (6.7) in the two-mode space takes the form ˆR(θ, φ) = exp( iφˆσ 3 /) exp( iθ ˆΣ /) exp(iφˆσ 3 /) = exp( iφĵ3) exp( iθĵ) exp(iφĵ3). (7.9) This rotation operator acting on one-photon states with polarizations e ± transforms these into the polarizations e up (θ, φ) and e down (θ, φ). This corresponds to the rotation transformations of the creation operators â ± into the operators (7.5), as expressed by ˆR(θ, φ)â + ˆR (θ, φ) = â (θ, φ), ˆR(θ, φ)â ˆR (θ, φ) = ˆb (θ, φ). (7.10) The same rotation operator transforms the circularly polarized N-photon state N, 0 into an N-photon state with pure polarization at the point u(θ, φ) on the Poincaré sphere: ˆR(θ, φ) N, 0 = 1 N! ˆR(θ, φ)(â +) N 0, 0 = 1 N! (â (θ, φ)) N 0, 0 (7.11) In the angular-momentum language, this is the state vector with maximal angular momentum in the direction u(θ, φ). In analogy to the coherent states of a mode of the radiation field, it is commonly termed a spin-coherent state [10]. Just as a coherent state (4.6) is a displaced version of the vacuum state, the spin coherent state is a rotated version of the state with maximal angular momentum along the 3-axis. This analogy is particularly clear when we rewrite the rotation operator (7.9) as exp( iφĵ3) exp( iθĵ) exp(iφĵ3) = exp[ iθ(ĵ cos φ Ĵ1 sin φ)] = exp(zĵ z Ĵ + ), (7.1) with z = (θ/) exp(iφ). Here we denoted as usual Ĵ ± = Ĵ1 ± iĵ. Note the similarity between the rotation operator (7.1) and the displacement operator (4.7), with Ĵ+ playing the part of â, and Ĵ of â.

18 18 VIII. ANGULAR MOMENTUM OF MONOCHROMATIC PARAXIAL BEAMS A. Paraxial approximation The paraxial approximation for the radiation field applies when the wave vectors of the field fall within a narrow cone with a small opening angle. This is the case for light beams, as they are produced by lasers. In this approximation the electric field of a light beam with frequency ω that propagates in vacuum in the positive z-direction can be expressed as the product of a plane wave and a slowly-varying envelope. The components of E in the transverse (x, y)-plane can be expressed as E t ( r, t) = u(ρ, z)e i(kz ωt) + c.c., (8.1) with ω = ck. Here ρ = (x, y) is the D transverse position vector and r = (ρ, z) is the position vector in three dimensions. The propagation equation for u follows from the Helmholtz equation E = k E for the electric field. The paraxial approximation is justified when u/ ρ /(ku) 1. In that case the transverse profile of u varies only slowly with z, so that the second derivative with respect to z can be ignored. Then the propagation of the light beam is well described by the paraxial wave equation [11, 1] ( ρ + ik ) u(ρ, z) = 0, (8.) z where ρ is the gradient operator in the transverse direction. The vector field u lies in the transverse (xy) plane. The paraxial approximation can be viewed as a lowest-order term of an expansion in the small paraxial parameter δ = 1/(kγ 0 ), with γ 0 the beam waist [11]. The magnetic field in the transverse plane is B t ( r, t) = 1 c e z u(ρ, z)e i(kz ωt) + c.c., (8.3) Equation (8.3) shows that the components of the magnetic field in the transverse plane has the same pattern as the electric field, with a polarization that is equal to the electric polarization vector rotated over an angle π/ in the positive (anti-clockwise) direction. The z-components of the fields E and B are non-vanishing in higher order. Since both fields are divergence-free, their first-order terms are proportional to the transverse divergence of E t and B t, and we find E z = i k ρ u e i(kz ωt) + c.c., B z = i k ρ ( e z u) e i(kz ωt) + c.c.. (8.4) B. Angular momentum of monochromatic beam The momentum density has a leading term ɛ 0 Et B t, which points in the z-direction. After using the expressions (8.1) and (8.3), and eliminating the rapidly oscillating terms by averaging over a few optical cycles, the zeroth-order contribution to the momentum density is found as p z ( R, z) = ɛ 0 c u u. (8.5) It is easy to verify that the leading term in the Poynting vector S = E H is equal to its z-component S z = c p z = cw, with w( R, z) = 1 ( ) ɛ 0 E t + c B t = ɛ 0 u u (8.6) the energy density of the beam. When we use the photon energy ω as an energy quantum, the photon density is n = w/( ω), and the momentum density (8.5) amounts to n k, which corresponds to k per photon. The energy per unit length is denoted as W = d ρ w(ρ, z) = ɛ 0 d ρ u u. (8.7)

19 However, we are not interested in the angular momentum arising from this photon momentum along the axis, but in the component j z of the angular-momentum density in the propagation direction. Since j z = ρ p t, (8.8) this z-component arises from the components of the momentum density in the transverse (xy) plane. To first order in δ, the transverse component of the momentum density is [ p t = ɛ 0 E z ( e z B t ) + ( E ] t e z )B z. (8.9) After substituting the expressions (8.1), (8.3) and (8.4), and averaging over an optical cycle, one finds that j z can be separated into the sum j z = l + s, where l and s are given by the expressions in cylindrical coordinates l(ρ, z) = ɛ 0 iω u φ u + c.c., s(ρ, z) = ɛ 0 iω ρ ρ ( u u). (8.10) The contribution l is determined by the phase gradient of the two components of u in the azimuthal direction. This expression has the flavor of a density of orbital angular momentum, as is obvious when we compare it to the expression for the z-component of the orbital angular momentum of a particle in elementary quantum mechanics. The separation of j z in l and s holds exactly for the contributions to the density of angular momentum. The quantity s arises from the gradient in the radial direction of the cross product ( u u) /i of the transverse mode amplitude. We recall that for an arbitrary radiation field, the separation (3.) of J into L and S could only be made for the total angular momenta, integrated over the entire space. It is remarkable that for a paraxial beam the separation of j z as l + s arises for the densities, in each point of space separately. Even so, the expression (8.10) for l is identical to the z-component of the integrand in the expression (3.3) for L, when A and E are represented by their monochromatic paraxial expressions. The spin per unit beam length is given by the integral Σ dρ s(ρ, z), and the orbital angular momentum per unit length is equal to Λ dρ l(ρ, z). We use partial integration with respect to φ for Λ, and with respect to ρ for Σ, and we obtain Λ = ɛ 0 ω d ρ u 1 i φ u, Σ = ɛ 0 d ρ ( u u) /i. (8.11) ω It is easy to show that both Σ and Λ do not vary with the propagation coordinate z under free propagation [13]. One also easily verifies that the integrand of this expression for Σ coincides with the integrand in (3.3) for the z-component of S. Again, this is remarkable, since the integration in (8.11) runs only over the transverse plane, not over the entire volume. When we separate the complex vector field u(ρ, z) as u = u e, with e the complex normalized local polarization vector, and u = u the local field strength, we arrive at the identity ɛ 0 ( u u) /i = σw, where the cross product σ = ( e e) /i is the local helicity of the beam. The helicity σ is a real number that is zero for linear polarization, and it takes the value ±1 for circular polarization e ± = ( e x ± i e y )/. The spin density in (8.10) is found to be localized in the region of the radial gradient of the product σw. However, equation (8.11) for Σ may be read as an integration of n σ, which is the product of the photon density n = w/( ω) and the spin σ per photon, where both the photon density and the helicity may depend on the transverse position ρ. 19 C. Uniform orbital and spin angular momentum The expressions (8.10) and (8.11) generalize the results for a monochromatic beam with uniform polarization. In that case, we can write u(ρ, z) = eu(ρ, z), where the polarization vector e is independent of position. Then the helicity σ is uniform over the cross section of the beam, and we recover from Eqs. (8.10) the known expressions [14] l(ρ, z) = ɛ 0 σ iω u u + c.c., s(ρ, z) = φ ω ρ w(ρ, z). (8.1) ρ This shows that the spin density is determined by the radial derivative of the energy density. The integrated spin momentum obeys the relation Σ = σw/ω, which corresponds to σ per photon, as expected [15]. However, the spin is localized in the region of the gradient of energy density, so that it vanishes in the region of uniform intensity. On the other hand, when a fraction of the light is absorbed by a particle, or when it is cut out by an aperture, the relation Σ = σw/ω also applies for this fraction. In this sense it is justified to say that light with a uniform helicity σ carries a spin σ per photon [16].

20 0 Of special interest are mode profiles of the form u(ρ, φ, z) = F m (ρ, z) exp(imφ), (8.13) where the φ-dependence is given by the factor exp(imφ). In order that the mode is continuous, m must be an integer. Then the density of orbital angular momentum is equal to l = mw/ω = n m, and the orbital angular momentum per photon is m. They are eigenmodes of the differential operator / φ. However, it would be confusing to state that they are eigenmodes of orbital angular momentum. In the classical context we are discussing here, orbital angular momentum is just a classical quantity, not an operator. For any classical beam the amount of orbital angular momentum has a well-defined specific value, and the same is true for the spin. What is special about these modes is that the density of orbital angular momentum is proportional to the energy density. In this sense, the orbital angular momentum can be said to be uniform over the beam profile. The modes (8.13) have an orbital angular momentum that can be quantified as m per photon. Since the paraxial wave equation (8.) is isotropic, this φ-dependence is conserved during free propagation. The radial mode function F m obeys the radial paraxial wave equation ( ρ + 1 ρ ρ m ρ + ik ) F m (ρ, z) = 0. (8.14) z A well-known example is provided by the Laguerre-Gaussian modes [14, 17]. For these modes the radial mode functions are denoted as F mp (ρ, z), where p is the radial mode number. The real function F mp is the product of a Gaussian function, a factor ρ m, and an associated Laguerre polynomial that depends only on the absolute value m [17]. These mode functions have the special property that their radial shape is invariant during free propagation, apart from a scaling factor. The z-dependent scaling factor is the width of the radial profile. Around the beam axis, the profiles of these beams are proportional to ρ m exp(imφ) = (x ± iy) m, depending on the sign of m. This shows that the beams have a phase singularity which corresponds to a vortex of charge m. D. Non-uniform polarization When a beam with non-uniform polarization passes a polarizer, the mode profile of the outgoing beam depends on the setting of the polarizer. This means that the mode function u does not factorize into the form eu(ρ, z), with a fixed polarization vector. On the quantum level, this means that for each photon in the beam its polarization and its translational degrees of freedom are entangled. At present, light beams with a non-uniform linear polarization and axial symmetry are widely studied. They can be generated by spatially varying dielectric gratings [18, 19]. As an example, we consider the superposition of two Laguerre-Gaussian light beams with opposite azimuthal mode number ±m, and with opposite circular polarizations. We consider a monochromatic beam characterized by the mode pattern u(ρ, φ, z) = F mp (ρ, z) [ e + e imφ + e e imφ] /. (8.15) The mode function (8.15) is real everywhere, and it is the superposition of two components with orbital angular momentum per photon m, and spin ± per photon. The vector multiplying F mp in Eq. (8.15) is a φ-dependent linear polarization vector e(φ) = e x cos(mφ) + e y sin(mφ). This polarization vector is in the x direction for φ = 0, and along a circle around the beam axis the polarization direction makes m full rotations in the positive direction. The directions of linear polarization as a function of φ are indicated by the black arrows in Figure 3. The linearly polarized field oscillates in phase everywhere along such a circle. For negative values of m, the polarization direction rotates in the negative direction along the circle. In the special case that m = 1, the number of rotations is 1, and the pattern is rotationally invariant. Then the density of angular momentum j z = l + s is zero, and the beam is invariant for rotation around the axis. The polarization direction is always in the radial direction. When we replace φ by φ φ 0 in the right-hand side of (8.15), the pattern is still isotropic, and the polarization direction makes an angle φ 0 with the radial direction. The density of orbital and spin angular momentum of the mode (8.15) can be evaluated with equation (8.10), and are both found to be zero. In fact, this mode is a superposition of two terms with orbital angular momentum equal to m, and spin ± per photon. The energy density is w(ρ, z) = ɛ 0 F mp (ρ, z). (8.16) Accordingly, near the axis, the pattern of phase and polarization is described by the expression u(x, y) ( e x + i e y )(x iy) m + ( e x + i e y )(x + iy) m. (8.17)

21 1 m=1 m= m=-1 m=- FIG. 3: Sketch of the position-dependent linear polarization for a mode as described by Eq. (8.15). The arrows indicate the direction of the linear polarization. This describes a singularity in phase and polarization with a mixed charge. An interesting generalization is the case of a similar superposition of modes with opposite circular polarization, and φ-dependent phase terms with two arbitrary m-values. This gives a transverse mode function [ u(ρ, φ) = F (ρ) e + e im φ + e e imφ] /, (8.18) prepared in a single transverse plane, where now the azimuthal mode numbers m and m are arbitrary integer numbers. We omitted the z-dependence of the mode, since in the general case, the two terms will undergo different diffraction, so that for different transverse planes the radial mode functions will no longer be identical. When we extract a phase factor exp(i(m + m )φ/), the remaining polarization vector is e(φ) = e x cos((m m )φ/) + e y sin((m m )φ/). The number of rotations of the polarization vector along a circle around the beam axis is now (m m )/. This is a half-integer value when m m is odd. The polarization pattern is illustrated in Figure 4 for the cases that m m = ±1. The overall phase factor exp(i(m + m )φ/) indicates that the phase of the polarized field varies along the circle. m - m = 1 m - m = -1 FIG. 4: Sketch of the position-dependent linear polarization for a mode as described by Eq. (8.18). The arrows indicate the direction of the linear polarization. IX. MODE PROFILES AND STATE VECTORS OF QUANTUM HARMONIC OSCILLATORS From here on, we disregard the polarization of the light beams, and the transverse mode profiles are indicated as scalar functions u( r) = u(ρ, z)

22 A. Paraxial beams and quantum harmonic oscillators It is well-known that the paraxial wave equation has a complete set of solutions in the form of a Gaussian function multiplied by a Hermite polynomial [17]. These Hermite-Gaussian mode functions have an intensity pattern that is invariant under propagation, apart from a scaling factor. They closely resemble the eigenfunctions of the twodimensional quantum harmonic oscillator (HO). In fact, the Hermite-Gaussian mode functions can be written as [0] u nxn y ( r) = 1 γ ψ n x ( x γ ) ψ ny ( y γ ) exp ( ) ikρ R iχ(n x + n y + 1). (9.1) These mode functions are determined by three mode parameters that depend on the propagation coordinate z. The width γ determines the spot size, R is the radius of curvature of the wave fronts, and χ is the Gouy phase, that determines the phase delay over the beam focus. When the transverse plane z = 0 coincides with the focal plane, the z-dependence of these parameters is determined by the equalities 1 γ(z) ik R(z) = k b + iz, tan χ(z) = z b. (9.) Here b is the diffraction length (or Rayleigh range) of the beam. The Gouy phase increases by an amount π from z = to. The mode functions (9.1) are exact normalized solutions of the paraxial wave equation (8.). The spot size at focus is γ 0 = γ(0) = b/k. Here the functions ψ n (ξ) for n = 0, 1,... are the real normalized energy eigenfunctions of the one-dimensional quantum harmonic oscillator in dimensionless form. Hence they are eigenfunctions of the Hamiltonian Ĥ ξ = 1 ) ( ξ + ξ (9.3) with eigenvalue n + 1/. The explicit form of the normalized eigenfunctions is ψ n (ξ) = 1 n n! π e ξ / H n (ξ), (9.4) in terms of the Hermite polynomials H n, and the eigenvalues are n + 1/. The Gouy phase term in the Hermite-Gaussian modes (9.1) is proportional to the eigenvalue n x + n y + 1 of the Hamiltonian of the two-dimensional quantum harmonic oscillator. This allows us to express an arbitrary solution of the paraxial wave equation in terms of an arbitrary time-dependent solution of the Schrödinger equation of the harmonic oscillator, provided that we replace time by the Gouy phase χ. In dimensionless notation, this equation takes the form ) (Ĥξ χ Ψ(ξ, η, χ) = i + Ĥη Ψ(ξ, η, χ). (9.5) Obviously, wave functions ψ nx (ξ)ψ ny (η) exp( i(n x + n y + 1)χ) are solutions of this Schrödinger equation, and by taking linear combination of these one obtains the most general solution Ψ(ξ, η, χ). On the other hand, an arbitrary solution of the paraxial wave equation is a linear combination of the Hermite-Gaussian modes (9.1). We conclude that an arbitrary solution Ψ(ξ, η, χ) of (9.5) gives an arbitrary solution u( r) of (8.), by the identification [1] u( r) = 1 ( ) ikρ γ Ψ(ξ, η, χ) exp, (9.6) R(z) with ξ = x/γ, η = y/γ, and where the z-dependent parameters γ, R and χ are specified by Eq. (9.) as functions of z. This identification (9.6) is exact, and it works both ways: there is a one-to-one correspondence between a timedependent state of the two-dimensional harmonic oscillator and a monochromatic paraxial beam of light. For a given harmonic oscillator wave function, we find a mode function after choosing as free parameter the diffraction length b, which is a measure of the size of the focal region. Moreover, a solution of the quantum harmonic oscillator remains a solution under a shift of time. If we substitute in (9.6) Ψ(ξ, η, χ χ 0 ) for Ψ(ξ, η, χ), we find a different paraxial beam in general. For an arbitrary mode, a phase shift χ 0 = π/ leads to an interchange of the mode pattern in focus and in the far field. Since the Gouy phase increases by an amount π, the mode function u from z = to corresponds to half a cycle of the oscillator. A stationary state of the harmonic oscillator is a linear superposition of products ψ nx (ξ)ψ ny (η) with n x + n y = n constant. Then Ψ(ξ, η, χ) is proportional to exp( i(n x + n y + χ)), and the corresponding paraxial beam retains its shape during propagation, apart from scaling by the width γ(z) and the z-dependent radius of curvature R(z).

23 3 B. Dirac notation of a paraxial beams As is well-known, the paraxial wave equation (8.) for a monochromatic beam is mathematically equivalent to the Schrödinger equation of a free quantum particle in two dimensions, where the propagation coordinate z replaces time. This analogy suggests to denote the mode function u as a function of ρ in a single transverse plane as a state vector u(z) in Dirac notation [], so that u(ρ, z) = ρ u(z). The scalar product of two state vectors v u = d ρ v (ρ) u(ρ) (9.7) involves an integration over the transverse coordinates. The relation between quantum and classical mechanics is analogous to the relation between paraxial wave optics and ray optics. A ray that passes a transverse plane is completely determined by its position ρ, and the -vector θ = (θ x, θ y ) of direction angles. The direction angles are the ratio between the transverse momentum and the longitudinal momentum k, and the operators for the direction angles therefore take the form ˆθ = (ˆθ x, ˆθ y ) = (i/k) ρ. The paraxial wave equation (8.) can be written as z u(z) = ik ˆθ u(z). (9.8) The effect on a state vector of free propagation over a distance L is then described by the operator ( Û prop (L) = exp ik ) ˆθ L. (9.9) The position operators ˆρ and the direction operators ˆθ are both Hermitian, and they obey the commutation rules where the indices a and b run over the x and y components. It is convenient to work with mode functions that are normalized to 1, so that [ˆρ a, kˆθ b ] = iδ ab, (9.10) u u = Ψ Ψ = 1. (9.11) Then the orbital angular momentum per photon along the beam axis of a monochromatic beam as given in (8.11) is ωλ W = k u ˆρ ˆθ u = u i φ u = Ψ Ψ, (9.1) i φ with Λ and W the orbital angular momentum and the energy per unit length. This expression (9.1) resembles that for the orbital angular momentum of a quantum particle in two dimensions. We emphasize that although the operator notation is borrowed from quantum mechanics, the description is applied here to classical paraxial optics. C. Raising and lowering operators for a paraxial beam As is well-known from textbook quantum mechanics, the eigenstates of the quantum harmonic oscillator are linked by ladder operators that raise or lower the quantum number n. In dimensionless notation, the lowering operator for the wave function ψ n (ξ) takes the form â = (ξ + / ξ)/. This formalism can be carried over to the Hermite-Gaussian modes u nx n y (z), so that these modes are linked to the fundamental mode u 00 (z) by the standard relation u nx n y (z) = 1 (â x (z) ) n x (â y (z) ) n y u00 (z). (9.13) nx!n y! The Hermite-Gaussian modes factorize into a product of a function of x and a function of y. They do not have vortex structures, but only contain line dislocations, with opposite phases on opposite sides of the lines. In terms of the canonical operators ˆρ and ˆθ the ladder operators take the form [0] â x (z) = k ) (ˆx + i(b + iz)ˆθ x, â y (z) = b k ) (ŷ + i(b + iz)ˆθ y. (9.14) b

24 These ladder operators obey the standard bosonic commutation rules [â x (z), â x(z)] = 1, etc. (9.15) The z-dependence of the ladder operators obeys the transformation rule â x (z) = Û(z)â x(0)û (z) and a similar identity for â y (z), which ensures that the mode functions solve the paraxial wave equation. Moreover, the lowering operators â x and â y give zero when acting on the fundamental mode u 00. In order to understand the relation between Hermite-Gaussian and Laguerre-Gaussian modes, it is attractive to combine the raising operators â x and â y into raising operators with a circular flavor, just as linear polarization vectors e x and e y are combined into circular polarization vectors. Therefore we write in analogy to (7.1) â +(z) = 1 (â x(z) + iâ y(z)), â (z) = 1 (â x(z) iâ y(z)). (9.16) In analogy to the Hermite-Gaussian modes (9.13), we can introduce the circular modes 1 ( ) nx+ ( u n+ n (z) = â +(z) â n (z)) u00 (z). (9.17) n+!n! From the expressions 9.14) and (9.16) for the operators, one directly checks that the operator ˆM = â +(z)â + (z) â (z)â (z) = k[ˆxˆθ y ŷˆθ x ] = i / φ (9.18) for all values of z. The mode (9.17) is eigenmode of the operator ˆM with eigenvalue m = n+ n. Therefore, according to Eq. (9.1), these modes carry an orbital angular momentum equal to m per photon. The total mode number n = n + + n of the modes (9.17) is eigenvalue of the operator ˆN = â +(z)â + (z) + â (z)â (z) (9.19) The mode (9.17) is the Laguerre-Gaussian mode L mp with azimuthal mode index m = n + n, and radial mode index p = min(n +, n ). This notation also allows us to obtain an expansion of the Hermite-Gaussian mode of a given total mode order n = n x + n y in the Laguerre-Gaussian modes of the same total order p + m = n + + n = n. Just as in Sec. VII C, the two ladder operators â ± (z) generate Schwinger-type SU() operators with the commutation rules of angular-momentum operators, and the corresponding rotation operators. Now these operators act on the space of classical modes, rather than on the state space of photon number state of two modes. 4 D. The Hermite-Laguerre sphere In analogy to Eq. (7.5), for each point on the sphere we can introduce the raising operators â (z; θ, φ) = â +(z) cos θ + â (z) sin θ eiφ, ˆb (z; θ, φ) = â +(z) sin θ e iφ + â (z) cos θ. (9.0) For each point on a sphere, these operators define a basis of modes 1 u na n b (z) = (â (z) ) n (ˆb ) nb a (z) u00 (z). (9.1) na!n b! Again, these modes are linear combinations of the Gaussian modes of the total order n, with n = n a + n b. The normalized mode u na n b (z) with n a = n and n b = 0 can be written as ( â (z; θ, φ) ) n u00 (z) / n!. This is the analogue of a spin coherent state, since it is just the rotated version of the state u n+n (z) with n + = n and n = 0, ( n which is the same as â +(z)) u00 (z) / n!. The rotation operator ˆR(θ, φ) is identical in form as in Eqs. (7.9) or (7.1) in terms of the ladder operators â ±. The mode basis { u nan b (z) } is defined by the two ladder operators â(z; θ, φ) and ˆb(z; θ, φ). These two operators are represented by two opposite points on the unit sphere, and together they generate a complete basis of modes. The operators represented by the North pole and the South pole (θ = 0) generate the Laguerre-Gaussian modes. Likewise, the operators â x and â y, which generate the Hermite-Gaussian modes, are represented by the points on the Equator and the 1-axis. The operators represented by the points on the Equator (θ = π/) with azimuthal angle φ and φ + π produce Hermite-Gaussian modes that are rotated over an angle φ/. This sphere is naturally called the Hermite-Laguerre sphere. Generalized basis sets of modes arise for points on the sphere between the poles and the equator. They have an elliptical vortex on the axis. For the lowest-order modes, the intensity pattern is illustrated in Figure 5.

25 5 FIG. 5: Intermediate modes with θ = π/3, φ = 0. a. Intensity distribution. b. Phase distribution. The phase is indicated by discrete grey tones, with increasing phase with brighter tones. FIG. 6: Unfolding an optical resonator into an equivalent periodic lens-guide; the mirrors are replaced by lenses with the same focal lengths and the reference plane is indicated by the dashed line. X. OPERATOR DESCRIPTION OF RESONATOR MODES We want to find explicit expressions for the modes of an astigmatic two-mirror optical resonator. We shall show that these modes can be expressed in terms of the ladder operators connecting them. These ladder operators in turn are specified by the eigenvectors of the ray matrix that describes the transformation of a ray during a round-trip through the resonator. A. Two-mirror cavity represented as lens-guide We consider an optical resonator consisting of two ideal mirrors facing each other. Resonant modes arise when light is bouncing back and forth between the mirrors. Such a system can be unfolded into an equivalent periodic lens-guide, as indicated in Figure 6. The mirrors are replaced by lenses with the same focal lengths. One period of the lens-guide is equivalent to a single round-trip through the resonator. The propagation coordinate z is measured in the lens-guide. Resonant modes are found by the requirement that the beam profile u(ρ, z) is periodic apart from an overall phase factor. In the lens-guide, regions of free propagation with the cavity length L are separated by thin lenses, that impose a phase profile. A spherical mirror, just as the equivalent lens, is characterized by its focal length f, and the imposed phase factor is exp( ikρ /f ). We want to allow the mirrors to be astigmatic, which means that they have different focal lengths along two orthogonal axes. For instance, the effect of a thin astigmatic and lossless lens that is aligned along the x and y axes can be expressed by the input-output relation u(ρ, z+ ) = exp{ ik(x /fx + y /fy )/}u(ρ, z ). (10.1) In ray optics, a ray that passes a transverse plane is completely determined by the -vectors ρ and θ. These can be combined into a 4-vector, which we indicate by the symbol µ. When the position and direction are replaced by the corresponding wave operators acting on the Hilbert space of transverse modes, the 4-vector µ turns into the 4-dimensional ray operator µ. These two definitions can be summarized in the form µ µ ρ ρ µ=, µ =. (10.) θ θ The ray travelling through the lens-guide is then described by the z-dependent vector µ(z). In paraxial ray optics with Gaussian lenses, the transformation of the ray is linear, and it can be described by the relation µ(z) = M (z)µ(0).

26 The transformation for free propagation through free space over the cavity length L can be expressed as ( ) 1 L1 M prop (L) =. (10.3) 0 1 Here 0 and 1 denote the zero and unit matrices respectively. The transformation matrix for an astigmatic mirror (or lens) can be written as ( ) 1 0 M mirror (F) = F 1 (10.4) 1 where F is a real and symmetric matrix. The eigenvalues of F are the focal lengths of the mirror and the corresponding mutually orthogonal real eigenvectors fix the orientation of the mirror in the transverse plane. The ray matrix that describes the transformation of a round-trip through an astigmatic resonator can be obtained by unfolding the resonator into the corresponding lens-guide and multiplying the matrices that represent the transformations of the different elements in the correct order M rt = M mirror (F 1 )M prop (L)M mirror (F )M prop (L). (10.5) Here F 1, are the matrices that specify the focal lengths and the orientation of the mirrors. Likewise, we can define the ray matrix M(z) for any distance z from the reference plane at z = 0. When we describe the mode profile by the vector u(z) in mode space, the effect of free propagation over the cavity length L is specified in Eq. (9.9), and the effect of a mirror (or a lens) with focal matrix F is ( ) ik Û rt (F) = exp ρf 1 ρ. (10.6) The propagation operator for a round-trip through the resonator (or a period in the lens-guide) is then in analogy to (10.5) Û = Ûmirror(F 1 )Ûprop(L)Ûmirror(F )Ûprop(L). (10.7) Likewise, we can define the propagation operator Û(z) for any distance z from the reference plane at z = 0, so that u(z) = Û(z) u(0). The unitary propagation operator is always an exponential of a quadratic form in the components of the transverse coordinate operator ˆρ and the operator ˆθ for the direction angles. Such operators form a group (the metaplectic group M p(d)). 6 B. Relation between ray and wave optics When the modes are normalized as u(z) u(z) = 1, the expectation values u(z) ˆρ u(z) and u(z) ˆθ u(z) have the significance of the average transverse position and the average propagation direction of the field. These -dimensional expectation values can be combined into the 4-dimensional vector u(z) ˆµ u(z) = u(0) Û (z)ˆµû(z) u(0). These four expectation values specify the ray of light in the transverse plane z. The quadratic exponential form of the propagation operators, the Heisenberg transformation Û (z)ˆµû(z) of the ray operator is indeed linear, so that we can write Û (z)ˆµû(z) = M(z)ˆµ, (10.8) where M(z) the ray matrix. As opposed to the unitary transformation Û, which acts upon the operator nature of the ray operator ˆµ, the matrix M acts upon its vector nature. The canonical commutation rules (9.10) are preserved under unitary transformation. It follows that M is real and obeys the identity ( ) M T 0 1 (z)gm(z) = G with G =, (10.9) 1 0 with M T the transpose of M. Both M and G are 4 4 matrices. Physically speaking, the identity (10.8) expresses that the operator expectation values u(z) ˆµ u(z) of the transverse position and propagation direction correpond to the path of a ray. It shows how paraxial ray optics emerges from paraxial wave optics and, as such, it may be viewed as an optical analogue of the Ehrenfest theorem in quantum mechanics. All real linear transformations M that obey the relation (10.9), or, equivalently, preserve the canonical commutation rules (9.10), are ray matrices. The product of two ray matrices is again a ray matrix, so that the ray matrices form a group, which is called the symplectic group Sp(d, R).

27 7 C. Ladder operators for astigmatic resonator We shall introduce ladder operators â 1 (z) and â (z) that connect the modes of the astigmatic resonator of different order. Defining properties of these ladder operators are: (i) They must obey the bosonic commutation rules of the type (9.15) for all values of z, in order to ensure that they produce a complete basis set of modes. (ii) They must obey the transformation rule â i (z)û(z) = Û(z)â i(0) (10.10) to ensure that a ladder operator acting on a solution of the paraxial wave equation produces another solution. (iii) They must be periodic in z, so that â i (z + L) is equal to â i (z), apart from a phase factor, in order to guarantee the periodicity. Since the bosonic commutation rules must follow from the canonical commutation rules (9.10), it is natural to assume that the ladder operators are linear in the operators ˆρ and ˆθ. It turns out that the ladder operators can be expressed in terms of the eigenvectors µ i of the round-trip ray matrix M rt. From the general property (10.9) of the transfer matrix we can derive some important properties of the eigenvalues and eigenvectors. The eigenvalue relation is generally written as M rt µ i = m i µ i (10.11) where µ i are the four eigenvectors and m i are the corresponding eigenvalues. By taking matrix elements of the matrix identity (10.9) between two eigenvectors, we find µ i Gµ j = m i m j µ i Gµ j. (10.1) The matrix element µ i Gµ i vanishes, so this relation gives no information on the eigenvalue for i = j. For different eigenvectors µ i µ j, we conclude that either m i m j = 1, or µ i Gµ j = 0. Since M rt is real, when an eigenvalue m i is complex, the same is true for the eigenvector µ i. Moreover, µ i is an eigenvector of M with eigenvalue m i. Provided that the matrix element µ i Gµ i 0, the eigenvalue must then obey the relation m i m i = 1, so that the complex eigenvalue m i has absolute value 1. The cavity is stable only when all eigenvalues have absolute value 1. Apart from accidental degeneracies, we conclude that a stable astigmatic resonator has two complex conjugate pairs of eigenvectors µ 1, µ 1, and µ, µ with unitary eigenvalues m 1, m 1, and m, m, so that m 1 = e iχ 1, m = e iχ. (10.13) Hence the eigenvalues now specify two different round-trip Gouy phase angles. The complex eigenvectors obey the identities µ 1 Gµ = 0, and µ 1Gµ = 0 (10.14) On the other hand, the matrix elements µ 1Gµ 1 and µ Gµ are usually nonzero. These matrix elements are imaginary, and without loss of generality we may assume that they are equal to the imaginary unit i times a positive real number. This can always be realized, when needed by interchanging µ 1 and µ 1 (or µ and µ ), which is equivalent to a sign change of the matrix element. The eigenvectors are uniquely defined when we impose the normalization condition µ 1Gµ 1 = µ Gµ = i. (10.15) These relations (10.14) and (10.15) can be viewed as symplectic orthonormality relations. They can be applied to obtain the expansion of an arbitrary ray µ in the eigenrays, with the result µ = 1 i (µ i Gµ µ i µ i Gµ µ i ). (10.16) i=1, These eigenvectors µ i refer to the reference plane at z = 0. The z-dependence of the vectors µ i is imposed according to the relation µ i (z) = M(z)µ p, with M(z) the ray matrix that specifies the transformation of a ray from the reference plane to the transverse plane z. We can now specify the ladder operators in the reference plane by the expressions [3] â i (z) = k µ igˆµ = k ( ) ρ i (z)ˆθ θ i (z)ˆρ, â k i (z) = µ i Gˆµ = k ( ) ρ i (z)ˆθ θi (z)ˆρ. (10.17)

28 where we separate the vectors µ i (z) in the two-dimensional subvectors ρ i (z) and θ i (z). The bosonic commutation rules are ensured by the relations (10.14) and (10.15). The z-dependence of these ladder operators indeed obeys the relation (10.10), as can be checked by using the Ehrenfest transformation (10.8). Finally, the ladder operators have the same periodicity as the eigenrays, so that â i (z + L) = exp(iχ i )â(z). The expressions for the ladder operators in terms of the ray operator ˆµ can be directly inverted by using the general expansion (10.16). When we substitute the operator ˆµ for µ, we obtain ˆµ = 1 i k This expression is valid for all transverse planes z. 8 ) (µ i â i µ i â i. (10.18) i=1, D. Structure of the modes and resonance spectrum The full set of modes for all values of z is now determined by the relations u nm (z) = 1 ( ) n ) m â 1 (â (z) (z) u00 (z), (10.19) n!m! in terms of the fundamental mode u 00 (z). An explicit analytical expression for the normalized fundamental mode can be given when we introduce the matrices R(z) = (ρ 1 (z) ρ (z)) and T(z) = (θ 1 (z) θ (z)), (10.0) in terms of the two-dimensional column vectors ρ i and θ i. The symplectic orthonormality properties (10.14) and (10.15) of the vectors µ 1, can be expressed as R T T R = i1, and R T T T T R = 0 (10.1) which hold for all values of z. An explicit analytical expression of the normalized mode function as it propagates through the lens guide is given by k u 00 (ρ, z) = exp ( kρs(z)ρ/), (10.) π det R(z) where S = itr 1 is a matrix. Because of the definitions of R and T in terms of the eigenvectors of the round-trip matrix M, the fundamental mode returns to itself after a round trip, as expressed by u 00 (z + L) = u 00 (z) exp( i(χ 1 + χ )/). (10.3) It is directly checked that acting upon u 00 with the lowering operators â 1 (z) and â (z) gives zero, and that it obeys the paraxial wave equation both in the sections of free propagation [4], and across the lenses [3]. Moreover, the second relation (10.1) guarantees that S is symmetric. This is obvious when we multiply the relation from the left with (R T ) 1, and from the right with R 1. The real and imaginary parts S r and S i of S respectively characterize the astigmatism of the intensity and phase patterns. The real part can be written as S r = ( itr 1 + i(r ) 1 T )/. With the first relation (10.1) this shows that RS r R = 1. This leads to the identity RR = S 1 r (10.4) This shows that S r is positive definite, so that the fundamental mode is square-integrable. The curves of constant intensity are ellipses. Depending on the sign of det S i the curves of constant phase are ellipses, hyperbolas or parallel straight lines. Under free propagation, S is a slowly varying smooth function of z. Optical elements, on the other hand, may instantaneously modify the astigmatism. The astigmatism of both the intensity and the phase patterns is characterized by two widths in mutually perpendicular directions and one angle that specifies the orientation of the curves of constant intensity or phase. As opposed to R and T, the matrix S is a matrix in real space ρ = (x, y) and transforms accordingly under coordinate transformations.

29 9 FIG. 7: Intensity pattern on a fixed mirror for a resonator where the other mirror has varying angles between the axes of two identical mirrors. The focal lengths are f ξ = L in the (horizontal) ξ-direction, and f η = 10L in the (vertical) η-direction. Top, from left to right: rotation angle is 0, π/6, π/3. Bottom, from left to right: rotation angle is π/, π/3, 5π/6. The orientation of the other mirror is indicated by the white crossing lines. Because of the periodicity properties of the fundamental mode u 00 (z) and of the ladder operators, the higher-order modes transform over a round trip as ( u nm (L) = exp i(n + 1 )χ 1 i(m + 1 ) )χ u nm (0). (10.5) The field exp(ikz)u nm (R, z) must repeat itself after a round trip, which gives the resonance condition for the wavenumber kl (n + 1 )χ 1 (m + 1 )χ = πq. (10.6) Thus the frequencies of the modes are specified by the transverse mode indices n and m, and the longitudinal mode index q is ω = c ( πq + (n + 1 L )χ 1 + (m + 1 ) )χ. (10.7) This shows that the general astigmatism does not show up in the frequency spectrum of the resonator. All that can be seen is the presence of two different round-trip Gouy phases. There are two different ways in which the corresponding frequency spectrum can be degenerate. For a resonator that has cylinder symmetry the two eigenvalue spectrum of the transfer matrix is degenerate (i.e. if χ 1 = χ ) and its modes are frequency degenerate in the total mode number n + m. As a result any linear combination of eigenmodes with the same total mode number is an eigenmode too. The second kind of degeneracy arises when one of the Gouy phases is a rational fraction of π. Then the combs of modes at different values of q overlap so that many different modes appear at the same frequency. The intensity pattern of the modes can be directly calculated from the algebraic expressions, once the eigenrays are found. These are illustrated in Figure 7. It is remarkable that the wave-optical mode pattern is determined in terms of the eigenvectors of the ray matrix of the resonator. Moreover, the concepts of the ladder operators that link ray optics to wave optics have the flavor of quantum operators, even though the field is described in a fully classical way. Finally, it is worthwhile to notice that the results of this section remain valid when the number of (transverse) dimensions is different. In particular, the same method gives explicit expressions for complete orthogonal sets of time-dependent wave functions that solve the Schrödinger equation of a free particle in three-dimensional space. E. Orbital angular momentum The twisted modes in the case of two non-aligned astigmatic mirrors consist of a wave that is travelling back and forth, and which can carry a non-vanishing orbital angular momentum. The angular momentum per photon in the lens-guide mode u nm can be expressed as L nm = k u nm ˆρ ˆθ u nm (10.8)