WHITFIELD GROUP
WHITFIELD GROUP
DARTMOUTH COLLEGE
PHYSICS AND ASTRONOMY
Ψ
DARTMOUTH COLLEGE
PHYSICS AND ASTRONOMY
Ψ
Boson Sampling and Vibronic Spectra
Steven Karson, James Whitfield
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
Examples of Bosons
• Photons
• Gluons
• W and Z bosons
• Higgs Boson
• Phonons
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
Qubits
• Two eigenstates
Qubits
• A state is some linear combination of the two eigenstates
Bloch sphere
Boson Sampling: Create input state
• Prepare an input state comprising n single photons in m modes
Optical Elements of Quantum System
• Phase-shifters
• Beam-splitters
• Photodetectors
Phase-Shifter
• Unitary operator
• Changes phase
Beam Splitter
• Unitary operator
• Separates photons
Photodetectors
• Detects photons at end of
interferometer
• Guess how much the photon
counter in the photo to the right
costs?
• $4755
Linear Optics Network Unitary Matrix
• Evolve input state via passive linear optics network
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
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)
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
Vibronic Spectra
• Simultaneous changes in the vibrational and electronic energy states
of a molecule
• Word “vibronic” comes from “vibrational” and “electronic”
Model with Parabola
• Can approximate Morse potential
with simple harmonic oscillator
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:
Franck-Condon principle
• Transitions are most likely to
happen straight up and where
there is overlap in the
wavefunctions
Derivation of Franck-Condon factor
• Begin by calculating molecular dipole operator μ
• (ri is distance of electron, Ri is distance of nucleus
Derivation of Franck-Condon Factor (cont.)
• Calculate probability amplitude of a transition between ψ and ψ’
Derivation of Franck-Condon Factor (cont.)
• Use Born-Oppenheimer approximation to expand and simplify
probability amplitude
Derivation of Franck-Condon Factor (cont.)
• The Franck-Condon Profile is
Simulating Vibronic Spectra with Boson
Sampling
Simulating Vibronic Spectra with Boson
Sampling (Cont)
• Overall operation is:
Final Apparatus
• There are two proposed
layouts for the boson
sampling device
When can vibronic spectra be simulated
classically? (No boson sampling needed)
• At high temperature
• At high mass
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
References
• Bryan T. Gard,1 Keith R. Motes “An introduction to boson-sampling”
• Joonsuk Huh*, Gian Giacomo Guerreschi, “Boson sampling for
molecular vibronic spectra”
• https://en.wikipedia.org/
WHITFIELD GROUP
WHITFIELD GROUP
DARTMOUTH COLLEGE
PHYSICS AND ASTRONOMY
Ψ
DARTMOUTH COLLEGE
PHYSICS AND ASTRONOMY
Ψ