How Can We Interpret Polarisation and Frequency When Dealing With One Photon

I gave this answer on Physics Stack Exchange to the following question:

If polarization is interpreted as a pattern/direction of the electric-field in an electromagnetic wave and the frequency as the frequency of oscillation, how can we interpret polarization and frequency when we are dealing with one single photon?

and the answer, slightly edited, is as follows:

Maxwell’s equations exactly define the propagation of a lone photon in free space. The state of a photon can be defined by a vector valued state in Hilbert space and this vector valued state is a precise mathematical analogy of the $\vec{E}$ and $\vec{H}$ fields of a macroscopic, classical field. That’s not to say that, for a single photon, the $\vec{E}$ and $\vec{H}$ are to be interpreted as electric and magnetic field: the vector valued $\vec{E}$ and $\vec{H}$ state is the unitarily evolving quantum state before any measurement is made. But:

There is a one-to-one, onto correspondence between every classical electromagnetic field for a given system and a one photon quantum state for a photon propagating in that system.

This is the first quantized description of the photon. To understand what measurements a photon state implies, one has to shift to a second quantized description where we have electic and magnetic field observables, whose measurements behave more and more like classical measurements as the number of photons gets bigger. A classical state is a coherent state of the second quantised field. But, given a photon can be described by a vector valued quantum state, it should be clear that polarization and all like “classical” attributes are meaningful for a lone photon.

In particular, a photon can be a quantum superposition of eigenstates, so:

One photon can be spread over a range of frequencies and wavelengths (i.e. it can be in a superposition of energy eigenstates), with possibly different polarisation for all components of the superposition.

One can even broaden this concept to propagation through dielectric mediums: the light becomes a quantum superposition of free photons and excited matter states, and the lone, first quantized quasiparticle that results from this superposition (strictly speaking a “polariton” rather than a true, fundamental, photon) has a quantum state which evolves following Maxwell’s equations solved for the medium. Thus, for example, we talk about lone photons propagating in the bound modes of optical fibres.

Another take on the one photon state is given in the first chapter of Scully and Zubairy “Quantum Optics”. The one photon state $\psi$ can be defined by the ensemble mean values of the second quantized electric and magnetic field observables:

$$\vec{E} = \left(\begin{array}{c}\left<\psi | \hat{E}_x | \psi\right>\\\left<\psi | \hat{E}_y | \psi\right>\\\left<\psi | \hat{E}_z | \psi\right>\end{array}\right);\quad\quad
\vec{B} = \left(\begin{array}{c}\left<\psi | \hat{B}_x | \psi\right>\\\left<\psi | \hat{B}_y | \psi\right>\\\left<\psi | \hat{B}_z | \psi\right>\end{array}\right)$$

where $\hat{E}_j$ is the $j^{th}$ component of the vector valued electric field observable and $\hat{B}_j$ that of the magnetic induction observables. ($[\hat{E}_j, \hat{B}_j]=0$ for $j\neq k$ and, in the right units, $[\hat{E}_j, \hat{B}_j]=i\,\hbar\,I$). For a one-photon state $\psi$, these means:

  1. Propagate exactly following Maxwell’s equations;
  2. Unquely define the light field’s quantum state for a one-photon state, even though they are not the state. This is in the same way that the mean of the classical Poisson probability distribution uniquely defines the distribution (even though it is a lone number, not a distribution).

Things are much more complicated for general, $N$ photon states so we need much more information than simple means to fully define the quantum state particularly with entangled states. Going back to our classical probability distribution analogy, the normal distribution needs two independent parameters, mean and variance, to wholly specify it, so it’s a more complicated thing than the Poisson distribution, which is defined by only its mean (which equals the variance). So quantum fields are hugely more complicated things than classical ones. But a coherent state of any photon is again uniquely defined by the mean values of the field observables, which means again propagate following the same Maxwell equations as the one-photon means: hence the one-to-one, onto correspondence between classical and one-photon states I spoke of – I like to call this the one photon correspondence principle (“OpCoP”). Why our macroscopic EM fields seem to behave like coherent quantum states rather than hugely more general, entangled ones (unless one goes to considerable experimental effort to observe entanglement) is still an open question. It is interesting to note, though, that the class of coherent states is the unique class of quantum harmonic oscillator states that achieve the lower bound of the Heisenberg uncertainty inequality.

Also see my answers to:

  1. If photons carry 1 spin unit, why does visible light seem to have no angular momentum? and
  2. Electromagnetic radiation and quanta.

Incidentally, even though general, entangled light states are hugely more complicated than one-photon (and, equivalently through the OpCoP, classical) light states, in principle we can still decompose them into a quantum superposition of tensor products of coherent states and so represent a general state by a set of field observable means. This was one of the contributions of 2005 Nobel Laureate Roy Glauber, who showed the above in 1963 in:

R. Glauber, “Coherent and Incoherent States of the Radiation Field”, Phys. Rev. 131, 2766–2788 (1963)

The coherent state tensor products are, however, overcomplete so the decomposition of a general quantum state into coherent states is highly not unique. Nonetheless, such a decomposition allows classical-like techniques to be brought to bear on entangled quantum states (in principle – in practice it is still complicated!).

If you google Iwo Bialynicki-Birula and his work on the photon wave function, he has a great deal more to say about the one-photon wave function. He defines the photon wave function as the positive frequency part of left and right circularly polarized eigenfunctions $\vec{F}_\pm = \sqrt{\epsilon} \vec{E} \pm i \sqrt{\mu} \vec{H}$. Iwo Bialynicki-Birula’s personal website is http://cft.edu.pl/~birula and all his publications are downloadable therefrom. $|\vec{F}_+|^2 + |\vec{F}_-|^2$ is the electromagnetic energy density. He defines the pair $(\vec{F}_+, \vec{F}_-)$, normalised so that $|\vec{F}_+|^2 + |\vec{F}_-|^2$ becomes a probability density to absorb the photon at a particular point, to be a first quantized photon wave function (without a position observable). There is special, nonlocal inner product to define the Hilbert space and in such a formalism the general Hamiltonian observable is $\hbar\, c\, \mathrm{diag}\left(\nabla\wedge, -\nabla\wedge\right)$. Please also see Arnold Neumaier’s pithy summary (here) of a key result in section 7 of Bialynicki-Birula’s “Photon wave function” in Progress in Optics, 36 V (1996), pp. 245-294 also downloadable from arXiv:quant-ph/0508202. The Hilbert space of Riemann Silberstein vector pairs that Bialynicki-Birula defines is acted on by an irreducible unitary representation, defined by Bialynicki-Birula’s observables $\hat{H}$, $\hat{\mathbf{P}}$, $\hat{\mathbf{K}}$ and $\hat{\mathbf{J}}$, of the full Poincaré group presented in the paper.