The quantum eraser and delayed choice
Goal of the project: Work out mathematically and simulate numerically one of the most counterintuitive experiments in quantum mechanics. Show how which-path information about the particle destroys the interference and how correlations between events make it possible to “recover” it without violating causality.
The project is based on a modified double-slit experiment in which the coordinate \(x\) of the particle on the screen is entangled with its internal degree of freedom (for example, its spin).
1. Theory: entanglement and phase
Let the states of a particle that has passed through slit \(A\) or slit \(B\) be described by the functions \(\psi_A(x)\) and \(\psi_B(x)\). The state of the system in front of the screen is \[|\Psi(x)\rangle = \frac{1}{\sqrt{2}} \bigl( \psi_A(x)|\uparrow\rangle + \psi_B(x)|\downarrow\rangle \bigr).\]
Loss of interference Compute the probability density of detecting the particle on the screen, \[P(x) = \langle\Psi(x)|\Psi(x)\rangle.\] Show that the interference term vanishes because the spin states are orthogonal (\(\langle\uparrow|\downarrow\rangle = 0\)). Explain the physical meaning: why does information that in principle can be extracted from the spin destroy the wave pattern?
Change of basis (the quantum eraser) Suppose that we measure the spin in the diagonal basis \[|+\rangle = \frac{|\uparrow\rangle + |\downarrow\rangle}{\sqrt{2}}, \quad |-\rangle = \frac{|\uparrow\rangle - |\downarrow\rangle}{\sqrt{2}}.\] Rewrite the state \(|\Psi(x)\rangle\) in this basis and find the joint probability densities \(P(x,\pm)=|\langle\pm|\Psi(x)\rangle|^2\) for the subsamples of particles.
Recovering the pattern Show that the distributions \(P(x,+)\) and \(P(x,-)\) contain interference terms, shifted in phase by \(\pi\). Prove that \[P(x,+) + P(x,-) = P(x).\]
2. Practical part: a Monte Carlo simulation
An experimentalist needs to understand how the “magic” turns into statistics.
Generating the data Using
python, generate a sample of \(10^5\) events. For each event:draw the coordinate \(x\) from the distribution \(P(x)\);
for this \(x\), draw the result of the spin measurement (\(+\) or \(-\)) with probabilities \(P(x,+)/P(x)\) and \(P(x,-)/P(x)\).
Sorting the events Plot the histogram of all \(x\). Then, using the results of the spin measurements, plot two separate histograms for the “plus events” and the “minus events”. Show how two interference patterns emerge from a featureless distribution.
3. Discussion: causality and “erasing the past”
The delayed-choice paradox Suppose that the spin is measured after the particle has hit the screen. Explain why this is not a signal sent from the future into the past.
The no-signaling theorem Discuss whether an observer at the screen can learn instantly which basis a colleague at the other end of the laboratory has chosen for the spin measurement.
Interpretation Why is the phrase “we erased the which-path information and the particle changed its mind” methodologically harmful for a research physicist?
4. What to submit
An analytic calculation of the densities \(P(x,\pm)\).
A
pythonscript with plots: the total distribution (without interference) and the two correlated histograms (with interference).A short essay (half a page) resolving the causality paradox.
References
Y.-H. Kim, R. Yu, S.P. Kulik, Y. Shih, A Delayed Choice Quantum Eraser, Phys. Rev. Lett. 84, 1 (2000).
R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. 3 (Quantum Mechanics).