We report that the true intrinsic degree of freedom of the photon is neither the polarization nor the spin. It describes a local property in momentum space and is represented in the local representation by the Pauli matrices. This result is achieved by treating the transversality condition on the vector wavefunction as a nonholonomic quantum constraint. We find that the quantum constraint makes it possible to generalize the Stokes parameters to characterize the polarization of a general state. Unexpectedly, the generalized Stokes parameters are specified in a momentum-space local reference system that is fixed by another degree of freedom, called Stratton vector. Only constant Stokes parameters in one particular local reference system can convey the intrinsic degree of freedom of the photon. We show that the optical rotation is one of such processes that change the Stratton vector with the intrinsic quantum number remaining fixed. Changing the Stratton vector of the eigenstate of the helicity will give rise to a Berry's phase.
We report that the true intrinsic degree of freedom of the photon is neither the polarization nor the spin. It describes a local property in momentum space and is represented in the local representation by the Pauli matrices. This result is achieved by treating the transversality condition on the vector wavefunction as a nonholonomic quantum constraint. We find that the quantum constraint makes it possible to generalize the Stokes parameters to characterize the polarization of a general state. Unexpectedly, the generalized Stokes parameters are specified in a momentum-space local reference system that is fixed by another degree of freedom, called Stratton vector. Only constant Stokes parameters in one particular local reference system can convey the intrinsic degree of freedom of the photon. We show that the optical rotation is one of such processes that change the Stratton vector with the intrinsic quantum number remaining fixed. Changing the Stratton vector of the eigenstate of the helicity will give rise to a Berry's phase.
It has been accepted that the polarization of the photon in vector beams is entangled with its momentum. Here a quantum description is advanced for the polarization that shows entanglement with the momentum. This is done by showing that the Jones vector at each value of the momentum plays the role of the polarization wavefunction in the sense that the Pauli matrices represent the Cartesian components of the polarization in the local reference system with respect to which the Jones vector is defined. The unit vector that the constraint of transversality condition requires to specify the local reference system turns out to be a gauge degree of freedom that determines the entanglement of the polarization with the momentum and has observable effects.
Whether the total angular momentum of the photon can be separated into spin and orbital parts has been a long-standing problem due to the constraint of transversality condition on its vector wavefunction. A careful analysis shows that the situation arises from a misuse of the constraint in quantum mechanics. To use the constraint properly, we convert the vector representation into a two-component representation in which the wavefunction is not constrained by any conditions. Upon doing so, we not only separate the spin conceptually from the orbital angular momentum but also identify the Berry gauge in which the two-component wavefunction can be canonically quantized. A corollary is that only in one particular Berry gauge can a radiation field be canonically quantized in terms of the plane waves. The degree of freedom to fix the Berry gauge, which reflects a symmetry of the constraint, turns out to have observable quantum effects.
We comprehensively review the quantum theory of the polarization properties of light. In classical optics, these traits are characterized by the Stokes parameters, which can be geometrically interpreted using the Poincaré sphere. Remarkably, these Stokes parameters can also be applied to the quantum world, but then important differences emerge: now, because fluctuations in the number of photons are unavoidable, one is forced to work in the three-dimensional Poincaré space that can be regarded as a set of nested spheres. Additionally, higher-order moments of the Stokes variables might play a substantial role for quantum states, which is not the case for most classical Gaussian states. This brings about important differences between these two worlds that we review in detail. In particular, the classical degree of polarization produces unsatisfactory results in the quantum domain. We compare alternative quantum degrees and put forth that they order various states differently. Finally, intrinsically nonclassical states are explored, and their potential applications in quantum technologies are discussed.
In the context of the two-dimensional (2D) polarization states of light, the degree of polarization P2 is equal to the maximum value of the degree of coherence over all possible bases. Therefore, P2 can be referred to as the intrinsic degree of coherence of a 2D state. In addition to (i) the maximum degree of coherence interpretation, P2 also has the following interpretations: (ii) it is the Frobenius distance between the state and the maximally incoherent identity state, (iii) it is the norm of the Bloch vector representing the state, (iv) it is the distance to the center of mass in a configuration of point masses with magnitudes equal to the eigenvalues of the state, (v) it is the visibility in a polarization interference experiment, and (vi) it is the weightage of the pure part of the state. Among these six interpretations of P2, the Bloch vector norm, Frobenius distance, and center-of-mass interpretations have previously been generalized to derive an analogous basis-independent measure PN for N-dimensional (ND) states. In this paper, by extending the concepts of visibility, degree of coherence, and weightage of the pure part to ND spaces, we show that these three remaining interpretations of P2 also generalize to the same quantity PN, establishing PN as the intrinsic degree of coherence of ND states. We then extend PN to the N→∞ limit to quantify the intrinsic degree of coherence P∞ of infinite-dimensional states in the orbital angular momentum, photon number, and position-momentum degrees of freedom.
We show that the quantum Stokes parameters aptly describe the nonclassical two-polarization-mode Fock states and the transformation of their polarization states. In this description, the two-polarization-mode Fock states are the simultaneous eigenstates of the first and second Stokes parameters. Transformations of the polarization states, which are caused, for example, by optically active media, are described by the rotations in the Stokes vector space, which are given by unitary operators generated from the Hermitian Stokes parameters. The SU(2) symmetry of the Stokes parameters provides two independent degrees of freedom of polarimetric measurements that can exceed the standard quantum limit.
The laws of Arago and Fresnel relate the polarization of light to its capacity for interference. We investigate their quantum counterpart at the level of the fundamental helicity states of a single photon. To this end, we introduce displaced photon helicity states (DPHS), constructed by coherently displacing single-photon helicity eigenstates. The displacement bridges the microscopic spin degree of freedom and macroscopic polarization observables, enabling interference effects that would otherwise be inaccessible to isolated photons. We derive analytical expressions for the Stokes parameters associated with the interference between DPHS and coherent optical fields and show that opposite helicities generate distinct parameter dependences in the resulting fringe structure. The interference constitutes a quantum realization of Arago-Fresnel interference and permits the helicity of an unknown photon to be inferred from the Stokes response of the combined field. Beyond this application, DPHS generalize the framework of displaced nonclassical states from photon-number excitations to the helicity degree of freedom of light.