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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 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/

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