Quantum mechanically [Unpolarized light](Unpolarized%20light.md) is light in which every [photon polarization state,](photon%20polarization.md) $|\phi_i\rangle$ in the stream of photons has equal probability of existing. And, as a result of this, the probability of measuring a photon in one polarization or another is independent of the [polarization state](photon%20polarization.md). Equivalently, this means that unpolarized light forms a [mixed state](mixed%20state.md) and it is in fact also [maximally mixed.](mixed%20state.md#Maximally%20mixed%20states) %%This should be obvious, but it could be more explicitly stated how mixed states are also uncorrelated states.%% # Unpolarized ensemble of photons from two sources Consider an [ensemble of circularly polarized photons](ensemble%20of%20circularly%20polarized%20photons.md) from two separate sources of polarized light that are combined into a single beam. This system is described by the following density matrix,  where for the case where the beam is unpolarized, $p_1=p_2=\frac{1}{2}$ and the density matrix is written as $\hat{\rho}_{\psi_1,\psi_2} = \frac{1}{2}|\psi_1\rangle\langle\psi_1|+\frac{1}{2}|\psi_2\rangle\langle\psi_2|=\frac{1}{2}\hat{\mathbb{1}}.$ We show [below](quantum%20mechanical%20description%20of%20monochromatic%20unpolarized%20light.md#Measuring%20an%20unpolarized%20ensemble%20of%20photons%20from%20two%20sources) that a measurement on this state is independent of the [state vectors.](State%20vector.md) In addition, the form of this density matrix is that of a [maximally mixed state](mixed%20state.md#Maximally%20mixed%20states) for a [two state ensemble.](ensemble%20of%20circularly%20polarized%20photons.md) %%Does being maximally mixd imply the statistical independence from the state vector? Check also that the left-right polarization basis works here. perhaps do it in a footnote.%% ## Measuring an unpolarized ensemble of photons from two sources We find the following probabilities to detect a right or left polarized photon respectively:   Generalized for an unknown polarization state $|\phi\rangle,$ the probability of detecting a photon with a polarization $|\phi\rangle$ is then  When the populations of states in the system are equal ($p_1=p_2=p$) then the resulting combined beam is unpolarized and the expression for probability reduces to $P_\phi = p|\langle \phi|\psi\rangle|^2$ which we may write in terms of the [completeness relation](State%20vector.md#State%20vector%20completeness%20relation) for the [polarization basis](polarization%20operator.md) such that $P_\phi = p\langle \psi| \Big(\sum_i |\phi_i\rangle\langle \phi_i|\Big)|\psi\rangle = p$ Choosing [left-right basis](photon%20polarization.md#circular%20polarization) $P_\phi=p(\langle\psi|( |L\rangle\langle L|+|R\rangle\langle R|)|\psi\rangle)=p$ ^585351 Thus we find that whatever measurement we make of the polarization on an unpolarized light beam is independent of the [polarization state vectors](photon%20polarization.md) in the ensemble. The quantum probability also matches the proportions of left and right polarized states in the ensemble. %%You need to show that this result generalizes for any number of sources. does it actually?%% # Determining whether a state is unpolarized Given a [measurement operator](Born%20rule.md#Role%20of%20projection%20operator), $\hat{P}_{\psi},$ and a density matrix $\rho,$ $\hat{\rho}$ is a [density matrix of an unpolarized beam](quantum%20mechanical%20description%20of%20monochromatic%20unpolarized%20light.md) if $\mathrm{tr}(\hat{\rho}\hat{P}_\psi)=\frac{1}{2}$ --- # Recommended Reading This page is constructed on the basis of a minimal model for unpolarized light that may be found in the following problem set: * [Schollwöck, U, J. Homework 2, Quantum Mechanics 1 (German) (2019,2020)](Schollwöck,%20U,%20J.%20Homework%202,%20Quantum%20Mechanics%201%20(German)%20(2019,2020).md) (Problems 1 and 2). Here we consider the simple model where photons may be vertically or horizontally polarized and these degrees of freedom in the quantum system are the only ones considered. A pedagogical description of unpolarized light is given in the following textbook: * [Baym, G., _Lectures on Quantum Mechanics_, Westview Press, 1990](Baym,%20G.,%20Lectures%20on%20Quantum%20Mechanics,%20Westview%20Press,%201990.md) pgs. 25-29. Here the purpose of this discussion is to introduce the notion of [mixed states.](mixed%20state.md) More general and complete descriptions of unpolarized light in terms of quantum mechanics are found here: * [Söderholm, J., Björk, G., Trifonov, Unpolarized light in quantum optics, arXivquant-ph0007099, 2000](Söderholm,%20J.,%20Björk,%20G.,%20Trifonov,%20Unpolarized%20light%20in%20quantum%20optics,%20arXivquant-ph0007099,%202000.md) * [Prakesh, H., Chandra, N., Density Operator of Unpolarized Radiation, Phys. Rev. A, vol. 4, No. 2 (1970)](Prakesh,%20H.,%20Chandra,%20N.,%20Density%20Operator%20of%20Unpolarized%20Radiation,%20Phys.%20Rev.%20A,%20vol.%204,%20No.%202%20(1970).md) %%There's a lot in this note that should be generalized for unpolarized beams!%% #QuantumMechanics/QuantumOptics #Electromagnetism/Optics/WaveOptics #PhysicalExamples/PhysicalExamplesInQuantumMechanics/PhysicalExamplesInQuantumOptics