LOG#195. Quantum theory of photon spin states.

This blog post is dedicated to my friend, Sergio Lukic. He wrote it as undergraduate student, and I got his permission to publish it here, on my blog.

This article exposes a treatment of photon polarization using the quantum features of photon states. Photons are massless particles with spin one in natural units. Using the formalism and framework of quantum statistics, we will derive known mathematical relationships.

In modern particle physics, we understand the electromagnetic field not as a continuum, but a set of discrete particles transmitting electromagnetic forces we name photons. This idea, in fact, can be generalized to any Yang-Mills field with semi-simple internal symmetry group. For instance, for SU(3) we build a Yang-Mills field whose mediator particle is the gluon. If you quantize the theory of quarks and gluons, you get the theory known as QCD (Quantum Chromodynamics), where in addition to gauge forces, you have also matter fields called quarks. Quantizing thesed fields, you get a field of quarks and gluons.

Going back to the electromagnetic field, this field of photons is described by a 4-vector potential A_\mu, when we define the target space as the usual 4D=3+1 space-time. This 4-vector is called the gauge potential, and the electromagnetic field is defined by the gauge field modulo gauge transformations A'=A+d\varphi. Then, we have 3 real degrees of freedom. For instance, choose A_0=0, then there are only 3 independent components for the electromagnetic field. Photon states have spin -1, 0, +1, apparently. We also know, as we said above, that photons are massless particles, and photons move at the speed of light. Therefore, we can delete ONE additional degree of freedom, and we have only 2 independent state spaces for its spin. In this way, the polarization state is fully described by a ray in some Hilbert state with dimension two. Usually, optical physicists call this space as Jones’ vector space. We are going to use the framework of quantum mechanics to describe it. This state space can be thought as the one having an orthogonal base, \vert +\rangle, \vert -\rangle, where, as usual, we denote \vert +\rangle as the  circular polarization state with positive helicity (left-handed), and \vert -\rangle as the circular polarization state with negative helicity (right-handed). Now, using the standard rules of Quantum Mechanics (QM), any pure state of polarization can be associated to a ray in Hilbert state given by the complex linear superposition

(1)   \begin{equation*} \boxed{\vert \Psi \rangle=z_1\vert +\rangle +z_2\vert -\rangle} \end{equation*}

These states are defined up to a constant by the quotient z_1/z_2. Mathematically, we have passed from the complex space \mathbb{C}^2 to the projective complex space \mathbb{CP}^1, using projective coordinates z_2/z_1. This projective plane is the extended complex plane \{\mathbb{C}\}+\infty, and it is generally represented by a sphere called Riemann sphere (alternatively, it has equivalent names), projecting stereographically the plane on that sphere, as the next figure shows:

This representation of spin states on the Riemann sphere was proposed by first time by the genius Ettore Majoranain 1932 (Atomi orientati in campo magnetico variabile, Nuovo Cimento, 9, 43-50). It is important to highlight the deep link of the Majorana description with the one given by Poincaré in 1892 (Theorie Mathematique de la Lumière, vol. 2., Paris, Georges Carre, 1892). In fact, different physical paradigms (or physmatics) often arrive at the same conclusions, in this case, the same geometrical construction. One of the wonderful aspects of the Majorana description is that if you write q=z_2/z_1, it represents the polarization state of a photon or the coherent photon ray (in superposition!). Moreover, p=\sqrt{q} specifies a vector on the Riemann sphere, so that the intersection of the perpendicular plane to that vector with the sphere gives us a circumference that projected onto the the complex plane defines the polarization ellipse, modulo a dilatation factor, as follows

The orientation is positive if p lives in the North hemisphere, and it is negative if it lives in the South hemisphere. Such a convention can be remembered with the conventional right-hand rule you know from electromagnetic courses. Now, introducing spherical coordinates 0\leq \theta\leq \pi and 0\leq \psi\leq 2\pi, on the Riemann sphere, we can parametrize the physical space of spin states of the photon in the next way:

(2)   \begin{equation*} \boxed{\vert \Psi\rangle=\cos (\theta/2)\vert +\rangle+e^{i\psi}\sin (\theta /2)\vert -\rangle} \end{equation*}

where we have normalized the state to 1, and we have fixed the relative phase \psi of any sate \vert \Psi\rangle in \mathbb{C}. Therefore, we can find the point (\theta,\psi) corresponding to any state. This comes with benefits. With a quantum description of the polarization state is that we can use the density matrices to describe ANY mixed state of polarization, i.e., we get a fully description of the polarization state of a photon ray even in incoherent superposition. You can compare this to the description given by the book Quantum Mechanics (3th ed., Wiley and Sons, E. Merzbacher, 1998), and to study the formalism of density matrices we are going to use here. As another example, we are going to calculate the density matrix in our space \vert +\rangle, \vert -\rangle, describing pure states. For any density matrix, using Dirac notation:

(3)   \begin{equation*} \boxed{\rho=\sum_o\vert \Psi_i\rangle p_i\langle \Psi_i\vert} \end{equation*}

and where tr(\rho)=\sum_i p_i=1. For the pure state given above, the sum is reduced to a single term, and the density matrix (check it yourself) is given by

(4)   \begin{equation*} \boxed{\rho=\rho (\theta, \psi)=\begin{pmatrix}\cos^2 (\theta/2) & \dfrac{1}{2}e^{-i\psi}\sin (\theta)\\ \dfrac{1}{2}e^{i\psi}\sin (\theta) & \sin^2 (\theta/2)\end{pmatrix}} \end{equation*}

The second example is to use this to get the polarization state of natural light. As any possible state is equally probable in the mixing, the sum of states becomes an integral onto the Riemann sphere of all the pure states of possible polarizations \rho (\theta, \psi), as given above. Using the usual measure on the sphere to perform the integral, you obtain:

(5)   \begin{equation*} \rho=\kappa \int_0^{2\pi}\int_0^\pi \begin{pmatrix}\cos^2 (\theta/2) & \dfrac{1}{2}e^{-i\psi}\sin (\theta)\\ \dfrac{1}{2}e^{i\psi}\sin (\theta) & \sin^2 (\theta/2)\end{pmatrix}\sin\theta d\theta d\psi=\kappa \begin{pmatrix}2\pi & 0\\ 0 & 2\pi\end{pmatrix} \end{equation*}

To normalize, you have only to impose tr (\rho)=1, so \kappa =1/4\pi, and finally:

(6)   \begin{equation*} \rho=\begin{pmatrix} 1/2 & 0\\ 0 & 1/2\end{pmatrix} \end{equation*}

So, the density matrix for natural polarization is proportional to the unit matrix. Moreover, this polarization state coincides with the experience and the maximal entropy expectation, since as it is the most probable, it has the biggest (maximum) entropy, i.e., S=-k_B tr(\rho\ln\rho)=k_B \ln 2, for an arbitray polarization state. It can be also derived from the condition:

    \[\dfrac{dS}{dp}=\dfrac{d}{dp}\left(-k_B\left[p\ln p+(1-p)\ln (1-p)\right]\right)=0\]

and where p and 1-p are the self-values of the density matrix \rho. Imposing this condition, the only value is that with p=1/2, and it is a maximum since S(1/2)=k_B\ln 2> S(1)=S(0)=0, as you can check yourself.

Observables in this density matrix formalism are polarization measurements, as \rho=P is an hermitian operator, P^+=P, associated to polarization measurements with eigenvectors

    \[\vert 1\rangle=\cos (\theta/2)\vert +\rangle+e^{i\psi}\sin (\theta/2)\vert -\rangle\]

    \[\vert 0\rangle=\sin (\theta/2)\vert +\rangle-e^{i\psi}\cos (\theta/2)\vert -\rangle\]

Polarization operatores, a.k.a. density matrices in this context, are projectors (P^2=P), and they evolve no unitarily the states unless they are eigenstates, since

    \[\vert \Psi\rangle \rightarrow P\vert\Psi\rangle\]

The evolution of any polarization state after passing through a polarizator P, will be

    \[\rho_f=P\rho_iP^{+}/tr(P\rho_i P^+)\]

and, as you can probe yourself

    \[tr(P\rho_iP^+)=\sum_i p_i\vert\langle \varphi_p\vert \Psi_i\rangle\vert^2\]

with P=\vert\varphi_p\rangle\langle\varphi_p\vert. The intensity will be

    \[I_f=I_itr(P\rho_i P^+)\]

and you can iterate the process any time you get a polarizator. After n polarizators, with tr(\rho_i)=1, the emergent ray will become a ray with

    \[\rho_f=P_nP_{n-1}\cdots P_1\rho_i P_1^+P_2^+\cdots P_n^+/tr(P_n\cdots P_1\rho_i P_1^+\cdots P_n^+)\]

This class of non unitary density matrix evolution is named as reduction of the state vector. We are interested, in principle, in linear polarizators. \theta=\pi/2. Then,

(7)   \begin{equation*} \rho=P=\begin{pmatrix} 1/2 & e^{-i\psi}/2\\ e^{i\psi}/2 & 1/2\end{pmatrix} \end{equation*}

Let me show how evolve a polarization state described by a monochromatic natural light when it passes a linear polarizator.

    \[\rho_i=\mathbf{1}/2\rightarrow \rho_f=P\rho_i P^+=P^2/2=P/2\]

    \[tr(P\rho_iP^+)=tr P^2=1/2\]

    \[I_f=I_i/2\]

Therefore, we get a state of linear polarization with a half of the intensity that the incident light-ray. This is, of course, a well-known result from the classical theory of light and its polarization. As practical examples, we will derive the Malus law for linearly polarizated states and we will study the effect of a quarter-wave plate on polarization states.

Example. Malus law. Let me begin with

(8)   \begin{equation*} \rho=\begin{pmatrix} 1/2 & e^{-i\psi}/2\\ e^{i\psi}/2 & 1/2\end{pmatrix} \end{equation*}

and a linear polarizator with observable

(9)   \begin{equation*} \rho=P=\begin{pmatrix} 1/2 & e^{-i\psi'}/2\\ e^{i\psi'}/2 & 1/2\end{pmatrix} \end{equation*}

The intensity after the light passes the polarizator is

    \[I_f=I_i tr(P\rho P^+)=\dfrac{1}{8}tr\left[ \begin{pmatrix} 1 & e^{-i\psi'}\\ e^{i\psi'} & 1\end{pmatrix}\begin{pmatrix} 1 & e^{-i\psi}\\ e^{i\psi} & 1\end{pmatrix}\begin{pmatrix} 1 & e^{-i\psi'}\\ e^{i\psi'} & 1\end{pmatrix}\right]\]

and then

    \[\boxed{I_f=\dfrac{I_i}{2}\left(1+\cos (\psi-\psi')\right)}\]

So this is the Malus law from density matrices! It is important to realize that \psi , \psi' have a very special relationship with the Riemann sphere picture. (z_2/z_1)^{1/2} is associated to the spatial representation of the polarization state that the complex number z_2/z_1 represents. In our case, we can see that p=\exp (i\psi /2) and p'=\exp (i\psi'/2) on the equator of the sphere, and denoting respectively a normal vector to the plane defining the linear polarization of \rho and a normal vector to the plane defining the polarization axis of our polarization device P. The relative angle \varphi between both axes, experimentally accessible, becomes \varphi=\phi/2-\psi'/2. Now, we can already rewrite the Malus law in its usual formulation, as follows:

    \[\boxed{I_f=\dfrac{I_i}{2}\left(1+\cos (2\varphi)\right)=I_i\cos^2\varphi}\]

Remark: I used polarizator as a terminator misnomer. The usual name is polarizer!

Example 2. Quarter-wave plate. Let me firstly review the \lambda/4 plate effect onto polarization states, based on experimental results:

  1. \lambda/4 plates have I_i=I_f, i.e., unit transmittance.
  2. \lambda/4 plates do not alter linear polarization states with parallel or perpendicular planes to the optical axis of the plate.
  3. \lambda/4 plates induce a phase \alpha to polarization states, typically \alpha=\pi/2. Then, linear polarization with +(-) \pi/4 plane with respect to the optical axis becomes circularly polarized with +(-) if \alpha=\pi/2 or -(+) if \alpha=-\pi/2.

We assign an evolution operator U to every \lambda/4 plate, so every state evolves

    \[\vert\Psi\rangle\rightarrow U\vert \Psi\rangle\]

    \[\rho\rightarrow U\rho U^+\]

for pure and mixed states, respectively. We have tr (\rho)=tr (U\rho u^+)\forall\rho. This holds if U\in SL(2,\mathbb{C}), and we can write

    \[U=\begin{pmatrix} a & b\\c & d\end{pmatrix}\]

with det (U)=ad-bc=1. For the state

    \[\vert \Psi\rangle=z_1\vert +\rangle+z_2\vert -\rangle\]

    \[U\vert \Psi\rangle=(az_1+bz_2)\vert +\rangle+(cz_1+dz_2)\vert -\rangle\sim \vert +\rangle+ f(q)\vert -\rangle\]

where q=z_2/z_1, and f(q)=(c+dq)/(a+bq). The conditions above imply that

    \[f(e^{i\psi'}=e^{i\psi'}\]

    \[f(-e^{i\psi'}=-e^{i\psi'}\]

    \[f(e^{i\psi'-i\pi/2})=0\]

and finally, for the \lambda/4 plate, you get that

    \[U=\dfrac{1}{\sqrt{2}}\begin{pmatrix} e^{i\psi'}& i\\i & e^{-i\psi'}\end{pmatrix}\]

and you can check that UU^+=1, so U is a unitary operator for SU(2). Using the representation on the Riemann sphere q\rightarrow f(q) is a rotation with angle \pi/2 defined by invariant states that are phases e^{i\psi'},-e^{i\psi'}. In summary, and generalizing all this stuff, \lambda/4 plates introduce phases \alpha onto the polarization states, they are associated to unitary operators U defined on the group SU(2), and they act on the Riemann sphere as rotations with angle \alpha! Remarkably, this is very different to the stochastic evolution of polarization observables at microscopic level. However, both operators are examples of the evolution of quantum states. The joined action of both ways of evolution can also happen. Imagine the following experiment: firstly, natural light passes through a linear polarizer, and then it passes a \lambda/4 plate, after it passes again another linear polarizer. The action can be fully defined by operators P_1, U, P_2. This experiment is usually done to get circularly polarized light, and the second polarizer acts an analyzer. Step by step, you get:

Step 1. \rho=\mathbf{1}/2, I=I_i. Natural light.

Step 2. Linear polarizer acts. \rho=P_1, I=I_i/2.

Step 3. \lambda/4 plate. \rho=MP_1M^+. I=I_itr(MP_1M^+)/2=I_i/2.

Step 4. \rho=P_2MP_1M^+P_2/tr(P_2MP_1M^+P_2). I=I_itr(P_2MP_1M^+P_2)/2.

Using spherical coordinates, the final result can be written as

    \[I_f=\left(1+\sin (\theta)\cos (2\varphi)\right)I'/2\]

You can get different final polarization states, depending on \theta, e.g., linear polarization (\theta=\pi/2), elliptical polarization (\theta=\pi/4), and circular polarization (\theta=0,\pi).

See you in the next blog post!

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