1. Delayed Choice Quantum Eraser
Anita Kulkarni
April 17, 2018
Interesting experiment, application of Ph 125 1

2. Outline Motivation 1999 Realization of Delayed Choice Quantum Eraser
Complementarity and the double-slit experiment
Wheeler’s delayed choice experiments
1999 Realization of Delayed Choice Quantum Eraser
Popular but misguided interpretation
Solution with standard quantum mechanics
Discussion 2

3. Double-Slit Experiment
Image: 3

4. Original Interpretation of Double-Slit Experiment
Complementarity
Electrons (or photons, or anything), even individual ones, behave like waves when there is no detector to determine which slit they traveled through
The objects behave like particles when there is a detector present
Which-path knowledge means no interference patterns, and vice versa
Relationship between wave/particle behavior is like relationship between position and momentum in quantum mechanics
Wheeler’s question: When does the object decide to travel as a wave or as a particle? 4

5. Wheeler’s Thought Experiments
Wheeler posed several thought experiments to answer the question of wave/particle decision timing
One version will be described next
Variations of Wheeler’s thought experiments have been realized in the lab, including the famous experiment 5

6. Wheeler’s Interferometer Thought Experiment
If BS_output is absent, photons behave as particles and appear as single pulses at D1 and D2
If BS_output is present, photons behave as waves and interference patterns appear
Remove or introduce BS_output after photon has traveled through BS_input to see when wave/particle decision is made
1999 experiment is adapted from this thought experiment
Image source: Jacques, Vincent, et al. “Experimental Realization of Wheeler's Delayed-Choice Gedanken Experiment.” Science, vol. 315, no. 5814, 16 Feb. 2007, pp. 966–968., doi: /science 6

7. The Experiment
First delayed choice quantum eraser experiment, performed in 1999
Kim, Yoon-Ho, et al. “A Delayed Choice Quatum Eraser.” Physical Review Letters, vol. 84, no. 1, 3 Jan , pp. 1–5., doi: /physrevlett
Image source: Fankhauser, Johannes. “Taming the Delayed Choice Quantum Eraser.” 25 July 2017, arXiv: v2. 7

8. Single photon emerges from pump and enters double slit
Photon enters BBO crystal and gets destroyed
An entangled pair of photons with half of the original frequency is created in its place, traveling in opposite y directions
Amount of which-slit information (side of BBO crystal passed through) or interference pattern information obtained from both photons will be the same 8

9. Lens does not affect signal photon
One entangled photon, the signal photon, reaches detector D0 first, since path length of other photon is ns longer
Position of photon is measured at D0, much like in the classic double-slit experiment
Lens does not affect signal photon 9

10. Prism does not affect idler photon
Detectors D3 and D4 provide which-path information of idler photon if they click
For simplicity, assume the two beam splitters BS inside the red circle are actually mirrors that can be added or removed by experimenter
This way, experimenter can control whether which-path information is being obtained or not
Prism does not affect idler photon 10

11. If mirrors near D3 and D4 are not present, then idler photon goes to detectors D1 or D2
Beam splitter between D1 and D2 ensures that that there is a 50/50 chance of photon going to D1 or D2, regardless of the slit it came through
D1, D2, and BS form the quantum eraser: they “erase” which-path information 11

12. Summary/Naïve Predictions
Double Slit/BBO
Photon goes through double slit
Photon gets split into an entangled pair
Signal Photon Detection
Position of signal photon is detected at D0
We expect to see an interference pattern akin to classic double-slit experiment because no which-path information has been obtained yet
Idler Photon Detection
Which of 4 detectors detects idler photon should not affect pattern at D0
Joint detection patterns of D0/D1-D4 should all look identical 12

13. Actual Results
Signal Photon Detection
Position of signal photon is detected at D0
We expect to see an interference pattern akin to classic double-slit experiment because no which-path information has been obtained yet
Idler Photon Detection
Which of 4 detectors detects idler photon should not affect pattern at D0
Joint detection patterns of D0/D1-D4 should all look identical
Predictions are incorrect: D3/D4 do provide which-path information and D1/D2 act as a quantum eraser after D0 detection
Retrocausality?
Image source: Fankhauser, Johannes. “Taming the Delayed Choice Quantum Eraser.” 25 July , arXiv: v2; not the original paper data since Fankhauser’s schematic is clearer 13

14. Careful Analysis Photon just after leaving pump: 𝜓= 𝑒 𝑖 𝑘 𝑥 𝑥
Between double slits and BBO crystal:
𝜓= 𝜓 1 + 𝜓 2
𝜓 𝑖 = 𝑒 𝑖𝑘 𝑟 𝑖 𝑟 𝑖
BBO crystal creates entangled pair:
𝜓= 𝜓 1 ⊗ 𝜓 ′ 1 + 𝜓 2 ⊗ 𝜓 ′ 2
Primed component corresponds to idler photon, unprimed corresponds to signal photon
Each term in sum is orthogonal
Superposition of both slits still exists 14

15. Assume nothing has affected wavefunction since either photon left BBO
Overall probability distribution:
𝜓 2 = 𝜓 𝜓′ 𝜓 𝜓′ 𝜓 𝜓′ 𝜓 𝜓′ 2 2
Probability distribution of signal photon position looks something like this:
Image adapted from: 15

16. D3 and D4 measure in the 𝜓′ 1 / 𝜓′ 2 basis
Again assume that nothing has happened to the wavefunction after either photon left BBO: 𝜓= 𝜓 1 ⊗ 𝜓 ′ 1 + 𝜓 2 ⊗ 𝜓 ′ 2
D3 and D4 measure in the 𝜓′ 1 / 𝜓′ 2 basis
D3 clicks: wavefunction collapses to 𝜓 1 ⊗ 𝜓 ′ 1
D4 clicks: wavefunction collapses to 𝜓 2 ⊗ 𝜓 ′ 2 16

17. We have written the shifted wave functions in the D1/D2 detector basis
Yet again assume that nothing has happened to the wavefunction after either photon left BBO: 𝜓= 𝜓 1 ⊗ 𝜓 ′ 1 + 𝜓 2 ⊗ 𝜓 ′ 2
Each reflection at beam splitter or mirror introduces a 90-degree phase shift:
𝜓 1 ′ →𝑖 𝜓 𝐷1 − 𝜓 𝐷2
𝜓 2 ′ →𝑖 𝜓 𝐷2 − 𝜓 𝐷1
We have written the shifted wave functions in the D1/D2 detector basis
Plug these in and do a little algebra… 17

18. If D2 clicks, we get the same but with a phase shift:
If D1 clicks, wavefunction collapses and gives probability distribution as interference fringes:
𝜓 𝜓 −2Im( 𝜓 1 𝜓 2 )
If D2 clicks, we get the same but with a phase shift:
𝜓 𝜓 Im( 𝜓 1 𝜓 2 )
Since each detector is equally likely to click, on average we will see:
𝜓 𝜓 2 2
This exact phase difference will always be present even if another device is added since it will equally affect both paths to detectors 18

19. Putting the Pieces Together: D0 and D3/D4
Recall D0 detection: 𝜓 2 = 𝜓 𝜓 2 2
Cannot be sure which slit signal photon came from
A: more likely from bottom slit, or 𝜓 2
B: both 𝜓 1 and 𝜓 2 equally likely
C: more likely from top slit, or 𝜓 1
Technically this is a mixed state
𝜓 2
𝜓 1
Since D3 and D4 provide which-path information and measurement of 𝜓′ 1/2 directly corresponds to measurement of 𝜓 1/2 , there is consistency in D0/D3 and D0/D4 joint measurements:
If D0 measures A, then D4 is more likely to click
If D0 measures B, then D3 and D4 are equally likely to click
If D0 measures C, then D3 is more likely to click 19

20. Putting the Pieces Together: D0 and D1/D2
Recall D0 detection: 𝜓 2 = 𝜓 𝜓 2 2
Analogous to D3/D4 case, but we consider the basis of probability distributions corresponding to measurement of D1 or D2:
𝜓 𝐷0/𝐷1 2 = 𝜓 𝜓 −2Im( 𝜓 1 𝜓 2 )
𝜓 𝐷0/𝐷2 2 = 𝜓 𝜓 Im( 𝜓 1 𝜓 2 )
𝜓 𝐷0/𝐷 𝜓 𝐷0/𝐷2 2 = 𝜓 2 = 𝜓 𝜓 2 2
Examples of consistency:
If D0 measures J, then D1 is more likely to click
If D0 measures K, then D1 and D2 are equally likely to click
If D0 measures L, then D2 is more likely to click
No which-path information because both distributions are symmetric across 𝜓 1 and 𝜓 2 20

21. Summary
Only real difference between D3/D4 and D1/D2 measurement is the basis
D3/D4 measurements are in which-path basis
D1/D2 measurements are “orthogonal” to which-path basis
Measuring at D0 fixes likelihood of detection at D3/D4 or D1/D2 in a consistent way that is time/ordering-independent
𝜓 𝐷0/𝐷 𝜓 𝐷0/𝐷2 2 = 𝜓 2 = 𝜓 𝜓 = 𝜓 𝐷0/𝐷 𝜓 𝐷0/𝐷4 2 is crucial, otherwise D0 pattern before measuring D1- D4 would give away choice of D1-D4 measurement basis before D1-D4 measurement basis is decided by experimenter 21

22. DCQE is a Bell-Type Experiment
Alice and Bob have state ( 0 ⊗ ⊗ 1 )
Alice measures blue photon (idler), Bob measures red photon (signal)
If Alice measures in ( 0 , 1 ) basis (D3/D4/which-path), then Bob would get if Alice gets and Bob would get if Alice gets 1 ; ordering-independent
If Alice measures in ( + , − ) basis (D1/D2), where + = and − = − so that 0 = − and 1 = − − , then the original state is equivalent to ( + ⊗( )+ − ⊗( 0 − 1 ))
If Alice measures + , then Bob has (no which-path information) but nothing traveled back in time
If Alice measures − , then Bob has − (no which-path information) but nothing traveled back in time
Remember, we only care about the distribution of final outcomes, not the total amounts of different joint detections. 22

23. Final Thoughts
Entanglement is weird; it exists everywhere in the universe at the same time
Answer to Wheeler’s question: it is ill-defined
I think people jump to retrocausality explanation because entanglement is so counterintuitive
Exact mechanism of these results depends on favorite interpretation of quantum mechanics, but math is consistent 23

24. Acknowledgments and References
I would like to thank Ashmeet Singh for helping me understand the experiment.
Fankhauser, Johannes. “Taming the Delayed Choice Quantum Eraser.” 25 July 2017, arXiv: v2.
Jacques, Vincent, et al. “Experimental Realization of Wheeler's Delayed-Choice Gedanken Experiment.” Science, vol. 315, no , 16 Feb. 2007, pp. 966–968., doi: /science
Kim, Yoon-Ho, et al. “A Delayed Choice Quatum Eraser.” Physical Review Letters, vol. 84, no. 1, 3 Jan. 2000, pp. 1–5., doi: /physrevlett.84.1. 24