quantum physics of light-matter interactions...quantum physics of light-matter interactions claudiu...

26 April 2019

Quantum Physics of Light-Matter Interactions

Claudiu Genes Max Planck Institute for the Science of Light (Erlangen, Germany)

Summer Semester 2019, FAU, Erlangen

Class details

Class structure

• Lectures: 12 each of two hours (classes cancelled on bridge days: May 31 and June 21)

Class details

Class structure


• Lecturer: Claudiu Genes – MPL Group Cooperative Quantum Phenomena (see website

where lecture notes are available)

• Exercises: 12 each for one hour (Michael Reitz)

https://www.mpl.mpg.de/research-at-mpl/research-groups-and-associated-groups/genes-research-group/

Class details

Class structure


• Lecturer: Claudiu Genes – MPL Group Cooperative Quantum Phenomena (where lecture

notes are available)


• First 6 lectures: fundamentals of light-matter interactions (learning how to write and solve

different models based on master equations, Langevin equations, Bloch

equations etc)


Class details

Class structure







equations etc)

• Last 6 lectures: The laser, Laser cooling, Optomechanics, Ion trap cooling and logic,

Vibronic coupling in molecules


Class details

Class structure







equations etc)

• Last 6 lectures: The laser, Laser cooling, Optomechanics, Ion trap cooling and logic,

Vibronic coupling in molecules

• Exam: 90 minutes written/10 minutes oral examination


Outline of lectures

Fundamentals of light-matter interactions

• The two-level system approximation

• Light-matter interaction – dipole and rotating wave approximations

• The master equation for spontaneous emission

• Solving the master equation – optical Bloch equations

• Fundamentals of cavity quantum electrodynamics – the Jaynes Cummings Hamiltonian and

quantum Langevin equations

Effects of light onto motion

• Collective decay of coupled quantum emitters

• subradiantly enhanced energy transfer

Superradiance/subradiance

• The full quantum optical force

• Doppler cooling, gradient cooling

• Ion traps, ion cooling, ion logic

• Cavity optomechanics

The laser

• Laser threshold

• Laser photon statistics

Molecules: vibronic coupling

• The Holstein Hamiltonian

• Emission and absorption spectra, energy transfer

Story in short

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Consider a quantum emitter: atom, molecule, quantum dot etc

Story in short

+

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How does it interact with electromagnetic waves?

Story in short

Simplified picture – two level system

Story in short

Light absorption

Story in short

Light emission

Story in short

There is also a quantum vacuum of electromagnetic modes (quantization in a big box)

Story in short

Effect – spontaneous emission – gives rise to finite lifetime (non-zero linewidth) for electronic transitions

• Master equation description of the dynamics

Story in short

Effect – spontaneous emission – gives rise to finite lifetime (non-zero linewidth) for electronic transitions

• Master equation description of the dynamics

• Optical Bloch equations

Story in short

Multiple resonances

Optical cavity

Story in short

Multiple resonances

Lorentzian profile – enhanced density of optical modes around resonances

Quasi-mode

frequency

Cavity mode linewidth

Optical cavity

Atom in cavity- Jaynes Cummings dynamics

The Jaynes-Cummings Hamiltonian

Story in short

Enhancement of photon-emitter scattering – cavity quantum electrodynamics

• Jaynes-Cummings Hamiltonian

• Quantum Langevin equation for the cavity field

• Input-output relations

Goals
















The laser

• Laser threshold





Motivation – laser physics

Source: youtube: How a laser works

Motivation – laser physics

Can you build a (theoretical) laser?

../Cool stuff/lasers_en.jar

Story in short

Enhancement of photon-emitter scattering – optical cavities

Pump leads to population inversion

Non-thermal light?

Story in short

Model 1

pump Lasing transition

Fast relaxation

Laser threshold

Story in short

Model 2

Laser photon statistics

after

Lasing transition

Outline of lectures
















The laser

• Laser threshold





Story in short

laser

Light carries momentum

Recoil momentum – recoil energy

Moving atom

Light can affect motion !

Laser cooling

laser

Possible processes:

• Stimulated absorption

Initial Final

• Stimulated absorption

Initial

Final

• Spontaneous emission

Initial

Final

recoil

recoil

recoil

Story in short

Effective Langevin equation

Spontaneous emission to other directions

Counterpropagating fields

Light can be used to control motion ! (for example cooling)

Laser cooling

Laser cooling

More complicated cooling schemes (Sysiphus cooling)

Story in short

Macroscopic mechanical resonator

Can it be brought close to the quantum ground state of vibrations?

Light can control motion even at the macroscale !

Cavity optomechanics

Free space

Single pass

Weak interaction

Single photon

Atoms

Fluorescent molecules

Ions

Quantum dotsSingle photon

Recoil momentum – recoil energy


Free space

Single pass

Weak interaction

Optical cavity

Multiple bounces (100000 or more)

Stronger interaction

Single photon

Single photon

Standing wave modes (like on a guitar string)

Atoms

Fluorescent molecules

Ions

Quantum dotsSingle photon


Nobel Prize 2017: Kip Thorne, Barry Barish, Rainer Weiss

Original motivation: improving precision in the detection of gravitation waves


kg

• LIGO, VIRGO

• Detection of gravitational

waves

Towards nano-optomechanics

Cavity Optomechanics, M. Aspelmeyer, T. Kippenberg, and F. Marquardt, Phys. Rev. Mod. (2014)

• Testing the classical-quantum

boundary at the large mass

scale

Basic science Technology

• Ultra-sensitive displacement

detection

• inertial sensors

Free space

Radiation pressure

force

Cavity enhancement

• Dynamical system – back action of motion onto

the cavity field

• Time-delayed equations

• Both amplification and cooling of mechanical

motion achievable

• Nonlinear dynamics – chaos, self-oscillations

Tiny effect

Goal: use light to control motion of massive mechanical resonators at the quantum level.


A macroscopic resonator in a thermal environment



Stochastic noise term

Induced damping




Induced damping

Noise correlationsThermal noise spectrum




Induced damping

Noise correlationsThermal noise spectrum

Equipartition!


Resolved-sideband cooling of a micromechanical oscillator, A. Schliesser, R. Rivière,

G. Anetsberger, O. Arcizet & T. J. Kippenberg, Nature Physics, 4, 415 (2008)

Cooling scheme analogous to sideband cooling of trapped ions!

https://www.nature.com/articles/nphys939#auth-1





Laser cooling

Frozen Motion, Ania Bleszynski Jayich, News and Views, Nature Physics 11, 710, 2015

Outline of lectures










• Subradiantly enhanced energy transfer






The laser

• Laser threshold





Story in short

+

-

Atomic, molecular system 1

+

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How do atoms decay when they see each other?

Story in short

+

-


+

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Answer: faster (superradiance) or slower (subradiance) than independent atoms

Outline of lectures










• Subradiantly enhanced energy transfer






The laser

• Laser threshold





More complex systems

+

-

Not always a good enough model

Molecules: fundamentally more complexAtoms: complicated fine and hyperfine structure

A simple model - diatomic molecules

A simple model for electronic to vibrations coupling in diatomic molecules

Story in short

Molecular systems – coupling to vibrations

Story in short

Molecular systems – coupling to vibrations

Holstein Hamiltonian

Story in short

Förster resonance energy transfer

Fig.1 The acceptor and donor fluorophores must be closer

than ~ 10 nm for the energy transfer to occur

(adopted from P. I.H. Bastiaens and R. Pepperkok, TIBS 25

(2000) 631).

An Introduction to Fluorescence Resonance

Energy Transfer (FRET) Technology and its

Application in Bioscience, Paul Held, Ph.D,

Applications Department, BioTek Instruments

(2005)

The two level system

Interaction with a classical light field

The Hydrogen atom

+

-

Simplifying assumptions:

• nucleus is heavy and therefore fixed in the origin

• non-relativistic description (Schrödinger equation suffices)

• spinless electron and proton

Radially symmetric potential (Coulomb) – exactly solvable model

The Hydrogen atom

+

-

In Dirac (ket) notation Hamiltonian is diagonalized

The Hydrogen atom

In the position representation

Solutions

Interaction with classical light

+

-

Propagating field


+

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Propagating field

Interaction Hamiltonian (in the dipole approximation)

Classical minimal coupling Hamiltonian

Vector potential Scalar potential


+

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Propagating field

Interaction Hamiltonian (in the dipole approximation)

+

-

Classical minimal coupling Hamiltonian

Dipole Hamiltonian

Length gauge transformation + quantization


A bit more on the Hamiltonian in the dipole approximation

+

-

Gauge transformation with


In more detail

+

-

Dipole Hamiltonian


In more detail

+

-

Dipole Hamiltonian

Expanding the dipole operator

Dipole matrix elements


Computing dipole matrix elements

unity unity


Computing dipole matrix elements

unity unity

Example: 1s to 2p dipole transitions

Only 1s to 2pz transitions allowed


Fulfilling the resonance condition

Field frequency should fit energetically the energy level difference – resonance condition


Putting it all together - Roadmap

1. Shine a laser field of a given frequency

(resonance condition)





2. Choose a given laser polarization (for

example linearly polarized in the z direction)

(selection rules for dipole-allowed transitions) only m=0 allowed





2. Choose a given laser polarization (for

example linearly polarized in the z direction)

(selection rules for dipole-allowed transitions) only m=0 allowed

Conclusion

Reduced dynamics in a two-state Hilbert space suffices!


+

-


Ladder operators

Projectors

Complete system – projectors add to unity


Ladder operators

Projectors

Complete system – projectors add to unity

Constant energy shift - ignore

Free Hamiltonian


Dipole operator


Dipole operator

+

-

Interaction in the dipole approximation


Dipole operator

+

-


Free operator evolution


Dipole operator

+

-


Free operator evolution

Fast rotating terms. Discard within the Rotating wave approximation (RWA)

Light matter interaction Hamiltonian

Result

Rabi frequency

• Imposing resonance and selection rules – reduction to a two-level system

• The dipole approximation (long wavelength approximation)

• Rotating wave approximation (discarding energy non-conserving terms)

Approximations made on the way

HAVE A NICE WEEKEND! (…after exercise class of course)

Light in a box – the dipolar coupling

Maxell’s equations (no sources)

Vector potential

Coulomb gauge

Wave equation

with

Light in a box – the dipolar coupling

Solutions

Wave equation

quantum physics of light-matter interactions...quantum physics of light-matter interactions claudiu...

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