The angular momentum of light plays an important role in many areas, from optical trapping to quantum information. In the usual three-dimensional setting, the angular momentum quantum numbers of the photon are integers, in units of the Planck constant ħ. We show that, in reduced dimensions, photons can have a half-integer total angular momentum. We identify a new form of total angular momentum, carried by beams of light, comprising an unequal mixture of spin and orbital contributions. We demonstrate the half-integer quantization of this total angular momentum using noise measurements. We conclude that for light, as is known for electrons, reduced dimensionality allows new forms of quantization.

Keywords

Here we show, in analogy to the theory of fractional spin particles ( 14 ), that an unexpected half-integer total angular momentum can arise for light. To do this, we note that the form J z = L z + S z for the total angular momentum of light follows from the rotational symmetry of Maxwell’s equations ( 17 , 18 ). However, experiments involve beams of light propagating in a particular direction; thus, this full rotational symmetry is not present. The only potential symmetries, which determine the form of the angular momentum operators according to Noether’s theorem, are rotations of the two-dimensional cross section of the beam around the propagation direction. We will show that this restricted symmetry leads to a new form of total angular momentum, which has a half-integer, that is, fermionic, spectrum. We will experimentally demonstrate this quantization by showing that the noise in the total angular momentum current corresponds to the fractional quantum ħ/2.

However, a general feature of two-dimensional systems is that angular momentum need not be quantized in the usual way. The orbital angular momentum of an electron orbiting in two dimensions around a magnetic flux need not be an integer, but can include an arbitrary fractional offset ( 14 ). The same mechanism introduces a phase factor in the exchange of particle-flux composites, implying that such particles have generalized or fractional statistics ( 15 ) as well as fractional spin. These concepts have played an important role in understanding the quantum Hall effect, where the low-lying quasiparticles have fractional statistics that are related to their fractional charge ( 16 ).

Central to these developments is the quantization of the angular momenta of the photon, which forms a discrete state space ( 12 ). The relevant quantum numbers are the eigenvalues of the spin and orbital angular momentum operators, S z and L z , in units of the reduced Planck constant ħ. The spin quantum number describes the circular polarization of light and takes values of ±1. The orbital quantum number appears in twisted beams, with phase-winding factors e ilθ , where θ is the azimuthal angle, and takes integer values l ( 13 ). Thus, the quantum numbers for the total angular momentum, J z = L z + S z , are the integers.

Effects due to the angular momentum of light have been studied since the first measurements of the torques exerted on wave plates ( 1 ). Versions of these optomechanical effects now appear in experiments on optical trapping and manipulation ( 2 ) and enable the remote detection of rotation ( 3 ). Angular momentum effects are also emerging in the radio-frequency domain, for applications in astronomy and communications ( 4 ). Fundamental interest focuses on optical angular momentum in the quantum regime ( 5 ). The angular momentum of single photons has been measured ( 6 ), and entanglement ( 7 ) and Einstein, Podolsky and Rosen correlations ( 8 ) have been studied. This unique degree of freedom provides a basis for quantum information applications, with high-dimensional entanglement ( 9 ), quantum dense coding ( 10 ), and efficient object identification ( 11 ) recently demonstrated.

RESULTS

Forms of total angular momentum We begin by establishing the possible forms of total angular momentum operator for photons in a beam of light. We consider the case of a paraxial beam, which is well approximated in experiments, and where the separation of spin and orbital angular momenta is well established (19). Such a beam is specified by a two-component complex vector field E (Jones vector), whose components give the amplitudes of each polarization across the beam. We take circularly polarized states as the basis and use polar coordinates (r,θ) across the beam. The angular momentum operators are the generators of rotations that act on this field. In the paraxial limit, they include the third Pauli matrix S z = ħσ 3 , which rotates the polarization direction homogeneously across the beam, and the usual orbital form L z = −iħ(d/dθ), which rotates the beam profile (image) but leaves polarization unchanged (20). The eigenstates of S z correspond to uniform, circularly polarized beams, and those of L z correspond to uniformly polarized beams, such as Gauss-Laguerre beams, where the amplitude varies as eilθ. For a general three-dimensional field, S z and L z are not valid as independent angular momenta because they do not preserve the transversality of the electromagnetic field (17, 21). They appear in the usual combination J z = L z + S z for the total angular momentum, which is uniquely determined by the rotational invariance of Maxwell’s equations in three space dimensions (18, 21). However, in a beam of light, both polarization and image rotations, around the beam axis, keep the fields transverse, such that both spin and orbital angular momenta are valid and independent (13, 22). Thus, we may consider the possibility of a total angular momentum that is a general linear combination, J z,γ = L z + γS z . The operator J z,γ generates simultaneous rotations of the polarization and image, in general, through different angles. As discussed below, it is related to measurements using devices that couple the spin and orbital degrees of freedom and can be defined for arbitrary γ. However, it corresponds to an angular momentum of the photon only if the field can be expanded in terms of its eigenfunctions. Thus, we seek those eigenfunctions, that is, the beams that are invariant, up to a phase factor, under the associated rotation. The solutions of the eigenvalue equation (1)are of the form (2)where e R/L are right and left circularly polarized basis vectors, and the irrelevant radial dependence of the eigenmode is omitted. These modes are superpositions of two states with definite spin and orbital angular momenta. The quantum number, j γ , of the conserved total angular momentum is given by Equation 1 holds for any values of l 1 and l 2 . However, the field should be unchanged by a complete rotation, implying that l 1 and l 2 must be integers. Beams with fractional l have been considered, but are not angular momentum eigenstates because they contain discontinuities that destroy rotational symmetry (23, 24). On demanding integer l, we find that γ and j γ are either both integers or both half-integers. Thus, we find two families of angular momentum operators. One family includes the existing forms L, S, and L + S, among others, where we now drop the z subscript. These have the expected bosonic spectrum with integer eigenvalues. The other family, typified by L + S/2, however, has a fermionic spectrum, comprising half-integer eigenvalues. This half-integer total angular momentum is a quantized property of the photon. To see this, we note that the quantum theory is constructed by expanding the field in a complete set of transverse modes, with circularly polarized Laguerre-Gauss modes being the natural choice in the context of optical angular momentum (25). The eigenfunctions of J γ , however, lead to other representations. For each γ, we find that there is an associated second-quantized angular momentum operator, which is the sum over modes of the number of photons in each, multiplied by the eigenvalue The mechanism behind this unexpected spectrum is analogous to that of an electron orbiting a fractional quantum of magnetic flux. For the electron, there is a fractional offset in the spectrum arising from the Aharonov-Bohm phase accumulated over a complete orbit around the flux line (14). For photons, a similar offset can be generated by choosing a non–uniformly polarized basis. The Berry phase (26) associated with the variation of polarization around a closed orbit then provides a synthetic gauge field, which shifts the angular momentum spectrum.

Measurement of generalized total angular momentum These new forms of total angular momentum differ from the standard one, but nonetheless have the physical properties we expect. The established method for measuring an optical angular momentum, be it L, S, or J 1 = L + S, involves rotating the beams traversing a Mach-Zehnder interferometer (6). This measurement exploits the fact that eigenstates pick up a phase factor eijφ when rotated, where j is the quantum number of the measured angular momentum and φ is the rotation angle. We can generalize this technique to measure J 1/2 , as shown in Fig. 1, by choosing wave plates and prisms, such that the image rotates by twice as much as the polarization, that is, by implementing the rotation corresponding to J 1/2 . Fig. 1 The generalized total angular momentum of light. (A) Experimental arrangement to study the generalized total angular momentum of light J 1/2 . Photons in a variable superposition of two angular momentum eigenstates |j = ±1/2〉 can be generated from the Gaussian input beam using a linear polarizer (LP), a quarter–wave plate (QWP), and a biaxial crystal (BC). The angular momentum currents can then be measured using an interferometer, introducing rotations in the optical paths to sort the beam according to angular momentum. Measuring J 1/2 entails rotating the image and polarization by different angles, in this case using two polarization-preserving Dove prisms (DP1 and DP2) to rotate the image by 180° and two half–wave plates (HWP1 and HWP2) to rotate the polarization by 90°. BS1 and BS2 are beam splitters used to separate and recombine the optical paths; DP1 and DP2 are at 90° to one another, and HWP1 and HWP2 are at 45°. The piezo delay is tuned such that each eigenstate interferes constructively at one output and destructively at the other. (B) Calculated intensity (grayscale) and polarization (red arrows) for the |j = 1/2〉 component of the input beam. We argue that the operator J 1/2 is an angular momentum because it is a generator of rotations and because it can be measured by interferometric techniques analogous to those previously used (6). It also has the required mechanical effects, as we now show. As with spin and orbital angular momenta, the torque exerted on an object depends on how it couples to the field. A half–wave plate, for example, reverses the sign of the spin quantum number σ = ±1 but leaves orbital angular momentum unchanged, and hence experiences a torque 2ħσ per photon (1). The inversion of orbital angular momentum, which is achieved by an ideal polarization-preserving Dove prism, implies a torque 2ħl per photon (27). For the total angular momentum J γ , the quantum number is reversed on transmission through a polarization-preserving Dove prism followed by two half–wave plates, one with a constant fast axis and the other with a fast axis at an angle γθ where the azimuthal angle is θ [that is, a q plate (28) with charge q = γ]. We calculate that the torque exerted on such an element by a beam with quantum number j γ is 2ħj γ per photon. Thus, the change in the eigenvalue of J γ is related to a torque as it should be, confirming on mechanical grounds that it represents a form of angular momentum. Because j 1/2 has a half-integer spectrum, the minimum torque exerted when this quantum number reverses is ħ, whereas for the standard quantum numbers l, σ, and j, the corresponding value is 2ħ. We have used this interferometer to measure the angular momentum J 1/2 of photons in beams formed from two of its eigenstates, |j 1/2 = ±1/2〉. These beams are generated by the conical refraction (29) of the light from a helium-neon laser. As shown in Fig. 1, the light is first elliptically polarized before passing through a biaxial crystal, leading to a superposition of angular momentum eigenstates. The amplitudes in the superposition are controlled by the angle of the QWP, θ qwp , with the beam varying from purely |j 1/2 = 1/2〉 to purely |j 1/2 = −1/2〉 as the QWP rotates by 90°. This beam then enters a Mach-Zehnder interferometer, where two polarization-preserving Dove prisms at 90° and two half–wave plates at 45° rotate the beam to impart a relative phase of π between the components |j 1/2 = ±1/2〉. The path lengths are tuned such that each component interferes constructively at one output port and destructively at the other, and the signal is detected with a photodiode. The angular momentum current is thus related to the rates of photon arrivals P 1 , P 2 at the two outputs and the corresponding photocurrents I 1 , I 2 by (3)[the quantum efficiency of the detector will be irrelevant for the following because we have Poissonian intensity statistics (30), and so is taken as one]. The average angular momentum per photon is obtained by dividing by the total flux or photocurrent. The result is shown in Fig. 2A and confirms that the average angular momentum per photon varies between +ħ/2 and −ħ/2. Fig. 2 Experimental results. (A) Average of the total angular momentum J 1/2 per photon, in beams comprising a variable superposition of its eigenstates |j = 1/2〉 and |j = −1/2〉. The solid line is the predicted result, corrected for the measured visibility of the interferometer. (B) Measured fluctuations in the angular momentum current quantified by its Fano factor. At the minima, the noise is predominantly shot noise reflecting the discreteness of angular momentum. The corresponding charge goes below ħ and approaches the expected value of ħ/2, showing that the quantized angular momentum of the photon is a fraction of ħ. The solid lines show the predicted result in the ideal case, with full visibility and no classical noise.

Noise in angular momentum currents To establish the quantization of angular momentum, we have studied its fluctuations, in particular the noise in the angular momentum current. Electrical current noise is known to reveal the discreteness of charge and has been used particularly to demonstrate the fractional charge of quasiparticles in quantum Hall states (31–33). This suggests that angular momentum current noise could, analogously, reveal the discreteness of optical angular momentum. To establish the possibilities of such diagnostics, we first calculate the noise properties of the angular momentum currents. We consider measurements involving a finite response time T, such that the operator for the angular momentum current is , and use wave packet quantization to calculate the moments of . For a single photon in an eigenstate of J 1/2 , we obtain , as expected; we also find that the variance of the current operator , , is zero. This is consistent with the assertion that each photon carries an exact amount of this total angular momentum. In contrast, for this state, we obtain a nonzero variance for the orbital and spin current operators, and . Physically, a measurement of orbital or spin angular momentum projects onto the components of the superposition in Eq. 2, introducing quantum noise in these currents. Alternatively, we can consider the angular momentum currents in the semiclassical limit, that is, for the coherent states originating from the laser in our experiment. In this limit, in general, quantization of current carriers appears as shot noise (30), with power spectral density 2qI, where q is the charge and I is the current. The discrete charge q can thus be obtained from the Fano factor, that is, the ratio between current noise and current. To elucidate this, we calculate the mean and variance of the angular momentum currents for coherent states. For a spatial mode that is an eigenstate of J γ , we find which is the expected result for shot noise with quantized charge ħj γ . To calculate the noise for a general beam, we note that any beam is a superposition of Laguerre-Gauss modes with amplitudes c l,σ and write â = ∑ l,σ c l,σ â l,σ . In this way, we find that the Fano factor in a general coherent state is (4) Thus, the minimum noise of an angular momentum current in the semiclassical limit corresponds to the quantum ħ/2 and is achieved in measurements of J 1/2 on beams that are eigenstates thereof. This noise is solely the shot noise associated with the discreteness of angular momentum. Note that for any angular momentum operator, Eq. 4 implies that the minimum noise is achieved in the corresponding eigenstate, specifically that with the lowest-magnitude nonzero eigenvalue. This shot noise limit is, thus, minimized for operators with half-integer spectra. Moreover, we see that for beams that are not eigenstates, the noise is larger, as a result of the uncertainty in the angular momentum of each photon.