Polarizing beam-splitters: measuring photon polarization
Polarizing beam-splitters are optical devices used to measure the polarization of photons. They work by directing photons into different output channels depending on their polarization state. Each output channel is equipped with a detector, allowing for the registration of individual photons. For a single photon measurement, only one detector will register a “click”, indicating either transmission (let’s say outcome +1) or reflection (outcome -1).
The quantum observable describing this measurement, denoted as \mathbf A, has eigenstates |\mathbf 1_x\rangle and |\mathbf 1_y\rangle corresponding to horizontal and vertical polarizations, with eigenvalues +1 and -1 respectively. Mathematically, this observable is expressed as:
\mathbf A = | \mathbf 1_x \rangle \langle \mathbf 1_x | - | \mathbf 1_y \rangle \langle \mathbf 1_y |
In the basis defined by (|\mathbf 1_x\rangle, |\mathbf 1_y\rangle), the observable \mathbf A is represented by the matrix:
\mathbf A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}
Consider a photon polarized at an angle \theta relative to the x-axis, described by the state |\mathbf 1_\theta\rangle = \cos \left(\theta\right) |\mathbf 1_x\rangle + \sin \left(\theta\right) |\mathbf 1_y\rangle. To determine the probabilities of measurement outcomes when using an x-aligned polarizing beam-splitter, I project |\mathbf 1_\theta \rangle onto the eigenstates of the measurement operator.
The probability of transmission, corresponding to the +1 outcome, is given by the squared modulus of the projection of |\mathbf 1_\theta\rangle onto |\mathbf 1_x\rangle:
P(+1) = |\langle \mathbf 1_x | \mathbf 1_\theta \rangle|^2 = \cos^2 \left(\theta\right)
Similarly, the probability of reflection, corresponding to the -1 outcome, is given by the squared modulus of the projection of |\mathbf 1_\theta\rangle onto |\mathbf 1_y\rangle:
P(-1) = |\langle \mathbf 1_y | \mathbf 1_\theta \rangle|^2 = \sin^2 \left(\theta\right)
These probabilities, \cos^2 \left(\theta\right) and \sin^2 \left(\theta\right), align with expected results for polarizing beam-splitter measurements. A single measurement yields a binary result, providing only one bit of information, insufficient to fully determine the polarization angle \theta of a single photon. To estimate \theta, one must perform repeated measurements on identically prepared photons to determine the outcome probabilities.
It’s important to note that the no-cloning theorem in quantum mechanics prevents the perfect copying of arbitrary quantum states. Therefore, amplifying a single photon to create multiple identical copies for polarization measurement is fundamentally impossible. This quantum limit restricts the ability to directly obtain multiple copies of a single photon for measurement purposes.
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