1. Ψ WHITFIELD GROUP Ψ WHITFIELD GROUP
DARTMOUTH COLLEGE  PHYSICS AND ASTRONOMY Ψ
WHITFIELD GROUP
DARTMOUTH COLLEGE  PHYSICS AND ASTRONOMY Ψ

2. Boson Sampling and Vibronic Spectra
Steven Karson, James Whitfield

3. What are Bosons? Force carrier particles
Follow Bose-Einstein statistics
Have integer value spin
Multiple bosons can occupy the same quantum state, unlike fermions
Symmetric under interchange

4. Examples of Bosons
Photons
Gluons
W and Z bosons
Higgs Boson
Phonons

5. Boson Sampling Overview
Scatter n identical bosons distributed in m modes using a linear interferometer
At end of interferometer, detect photon distribution
Used to simulate quantum events
NOT a universal quantum computer, intended for use in specific cases
Modes can be thought of as types of qubits

6. Qubits
Two eigenstates

7. Qubits
A state is some linear combination of the two eigenstates

8. Bloch sphere

9. Boson Sampling: Create input state
Prepare an input state comprising n single photons in m modes

10. Optical Elements of Quantum System
Phase-shifters
Beam-splitters
Photodetectors

11. Phase-Shifter
Unitary operator
Changes phase

12. Beam Splitter
Unitary operator
Separates photons

13. Photodetectors Detects photons at end of interferometer
Guess how much the photon counter in the photo to the right costs?
$4755

14. Linear Optics Network Unitary Matrix
Evolve input state via passive linear optics network

15. Output State
The output state is a superposition of the different configurations of how the n photons could have arrived in the output modes
S is configuration, n is number of bosons in mode, γ is amplitude

16. Significance of Boson Sampling
Aaronson & Arkhipov argue that passive linear optics interferometer with Fock state inputs is unlikely to be classically simulated
Calculating an amplitude directly is O(2nn2)

17. Complications of Boson Sampling
Synchronization of pulses
Mode-matching
Quickly controllable delay lines
Tunable beam-splitters and phase-shifters
Single-photon sources
Accurate, fast, single photon detectors

18. Vibronic Spectra
Simultaneous changes in the vibrational and electronic energy states of a molecule
Word “vibronic” comes from “vibrational” and “electronic”

20. Model with Parabola
Can approximate Morse potential with simple harmonic oscillator

21. Born-Oppenheimer approximation
In this approximation, one can separate the wavefunction of a molecule into its electronic and nuclear components
In the context of molecular spectroscopy, we can treat the energy components separately:

22. Franck-Condon principle
Transitions are most likely to happen straight up and where there is overlap in the wavefunctions

23. Derivation of Franck-Condon factor
Begin by calculating molecular dipole operator μ
(ri is distance of electron, Ri is distance of nucleus

24. Derivation of Franck-Condon Factor (cont.)
Calculate probability amplitude of a transition between ψ and ψ’

25. Derivation of Franck-Condon Factor (cont.)
Use Born-Oppenheimer approximation to expand and simplify probability amplitude

26. Derivation of Franck-Condon Factor (cont.)
The Franck-Condon Profile is

27. Simulating Vibronic Spectra with Boson Sampling

29. Simulating Vibronic Spectra with Boson Sampling (Cont)
Overall operation is:

30. Final Apparatus
There are two proposed layouts for the boson sampling device

31. When can vibronic spectra be simulated classically
When can vibronic spectra be simulated classically? (No boson sampling needed)
At high temperature
At high mass

32. This Project’s Goal
Using “Gaussian” software, simulate the vibronic spectra of various molecules
Ultimately want to demonstrate that for high enough mass, vibronic spectra can be simulated classically

33. References
Bryan T. Gard,1 Keith R. Motes “An introduction to boson-sampling”
Joonsuk Huh*, Gian Giacomo Guerreschi, “Boson sampling for molecular vibronic spectra”

34. Ψ WHITFIELD GROUP Ψ WHITFIELD GROUP
DARTMOUTH COLLEGE  PHYSICS AND ASTRONOMY Ψ
WHITFIELD GROUP
DARTMOUTH COLLEGE  PHYSICS AND ASTRONOMY Ψ