Quantum Physics of LightMatter Interactions Lecturer Claudiu Genes

Quantum Physics of Light-Matter Interactions Lecturer: Claudiu Genes Max Planck Institute for the Science of Light (Erlangen, Germany) -- Lecture 2 -- Genes Research Group Cooperative Quantum Phenomena

Topics covered in this lecture Lecture 2 q Quantization of the free electromagnetic field q Quantum states of light: number (Fock) basis, coherent states, thermal states q The minimal coupling Hamiltonian: the dipole approximation q The two-level system approximation q The simplified light-matter interaction in the two-level system approximation q Fundamental processes: stimulated absorption/emission and spontaneous emission q Preview of next class: derivation of an open system master equation in the case of spontaneous emission

Quantization of the free electromagnetic field in a fictitiuous box

The quantized electromagnetic field Quantization box q The box is used to simulate an infinite (when L is taken to infinity) space q No charges or currents present q Boundary conditions (periodic) on the walls of the box Take a fictitiuous box of dimensions L x L=V

The quantized electromagnetic field Maxwell’s equations

The quantized electromagnetic field Maxwell’s equations Vector and scalar potentials

The quantized electromagnetic field Maxwell’s equations Vector and scalar potentials q They are not unique so that one has gauge freedom q A gauge transformation involves adding a scalar field to the problem such that q The transformed potentials are

. . here one can use some useful formulas (from Jackson)

The quantized electromagnetic field Coulomb gauge q This is obtained by picking a scalar field such that

The quantized electromagnetic field Coulomb gauge q This is obtained by picking a scalar field such that q From Maxwell’s equations we have which implies that the scalar potential in the Coulomb gauge is constant and can be factored out

The quantized electromagnetic field Coulomb gauge q This is obtained by picking a scalar field such that q From Maxwell’s equations we have which implies that the scalar potential in the Coulomb gauge is constant and can be factored out The wave equation q Solutions can be separated into positive and negative frequency components

The quantized electromagnetic field Solving the spatial part q Solutions are of the form of plane waves

The quantized electromagnetic field Solving the spatial part q Solutions are of the form of plane waves q The normalization makes the solutions orthonormal q The index refers to a direction and amplitude of the allowed wavevectors and to one of the two transverse polarizations k-propagation direction 2 possible polarizations

The quantized electromagnetic field Solving the spatial part q Solutions are of the form of plane waves q From assuming PBC (periodic boundary conditions) we find the set of allowed wavevectors where the indexes range from minus to plus infinite integers

The quantized electromagnetic field Solving the spatial part q Solutions are of the form of plane waves q From assuming PBC (periodic boundary conditions) we find the set of allowed wavevectors where the indexes range from minus to plus infinite integers q The resulting expansion of the vector potential into plane waves

The quantized electromagnetic field Solving the spatial part q Solutions are of the form of plane waves q From assuming PBC (periodic boundary conditions) we find the set of allowed wavevectors where the indexes range from minus to plus infinite integers q The resulting expansion of the vector potential into plane waves q And the corresponding electric field

The quantized electromagnetic field Quantization q Replace c-numbers with non-commuting bosonic operators

The quantized electromagnetic field Quantization q Replace c-numbers with non-commuting bosonic operators q The electric field operator q Where the zero-point electric field amplitude is

The quantized electromagnetic field Quantization q Replace c-numbers with non-commuting bosonic operators q The electric field operator q Where the zero-point electric field amplitude is q One can compute q …showing that the total free field Hamiltonian is a sum over an infinite number of modes each characterized by a quantum harmonic oscillator operator

The quantized electromagnetic field A few aspects q The sum over the constant ground state energy diverges in the infinite volume limit – this gives rise to the Casimir-Polder force (for more details see https: //en. wikipedia. org/wiki/Casimir_effect). We will disregard this term in the following as it plays no role in the physics described in this course.

The quantized electromagnetic field A few aspects q The sum over the constant ground state energy diverges in the infinite volume limit – this gives rise to the Casimir-Polder force (for more details see https: //en. wikipedia. org/wiki/Casimir_effect). We will disregard this term in the following as it plays no role in the physics described in this course. q In the Heisenberg picture the operators acquire time dependence

The quantized electromagnetic field A few aspects q The sum over the constant ground state energy diverges in the infinite volume limit – this gives rise to the Casimir-Polder force (for more details see https: //en. wikipedia. org/wiki/Casimir_effect). We will disregard this term in the following as it plays no role in the physics described in this course. q In the Heisenberg picture the operators acquire time dependence q …so that the electric field operator expectation value can be computed from the initial state vector

The quantized electromagnetic field Quantum states of light q The number (Fock) basis

The quantized electromagnetic field Quantum states of light q The number (Fock) basis q How to construct the basis (for an specific mode)

The quantized electromagnetic field Quantum states of light q Thermal light *Covered in Exercises 1&2

The quantized electromagnetic field Quantum states of light q Thermal light q Coherent states *Covered in Exercises 1&2

The minimal coupling Hamiltonian: performing the dipole approximation

Light-matter Hamiltonian in the dipole approximation The minimal coupling Hamiltonian Coulomb gauge

Light-matter Hamiltonian in the dipole approximation The minimal coupling Hamiltonian Coulomb gauge q Eigenstates in the absence of the external field (a few examples of orbitals below) q Dimension of orbitals (at the level of Å)

Light-matter Hamiltonian in the dipole approximation The minimal coupling Hamiltonian Coulomb gauge q Eigenstates in the absence of the external field (a few examples of orbitals below) q Dimension of orbitals (at the level of Å) q Energy difference between orbitals corresponds to wavelengths of few hundred nm

Light-matter Hamiltonian in the dipole approximation The minimal coupling Hamiltonian Coulomb gauge q Eigenstates in the absence of the external field (a few examples of orbitals below) q Dimension of orbitals (at the level of Å) q Energy difference between orbitals corresponds to wavelengths of few hundred nm q !! The dipole approximation !!

Light-matter Hamiltonian in the dipole approximation The transformation to the length gauge

Light-matter Hamiltonian in the dipole approximation The transformation to the length gauge q Length gauge

Light-matter Hamiltonian in the dipole approximation The transformation to the length gauge q Length gauge q …leads to

Light-matter Hamiltonian in the dipole approximation The transformation to the length gauge q Length gauge q …leads to q And finally the simplified interaction Hamiltonian q …where the dipole moment is

The two level system (TLS) approximation

The two-level system The dipole operator q Only two levels coupled to the external fields

The two-level system The dipole operator q Only two levels coupled to the external fields q Define ladder operators q …with action

The two-level system The dipole operator q Only two levels coupled to the external fields q Define ladder operators q …with action q Notice that there is no transition dipole moment within the same orbital (use specific wavefunctions for Hydrogen atom to check this)

The two-level system The dipole operator q Only two levels coupled to the external fields q Define ladder operators q …with action q Notice that there is no transition dipole moment within the same orbital (use specific wavefunctions for Hydrogen atom to check this) q Write the dipole moment operator in the full 2 x 2 basis

The two-level system The full Hamiltonian q The free Hamiltonian q …based on the completeness of the basis

The two-level system The full Hamiltonian q The free Hamiltonian q …based on the completeness of the basis q A couple of simplifications q We then have the full Hamiltonian and substract the constant energy term

The two-level system The full Hamiltonian q The free Hamiltonian q …based on the completeness of the basis q A couple of simplifications q We then have the full Hamiltonian q …not yet done with approximations! and substract the constant energy term

The two-level system The rotating wave approximation q Assume a classical driving electric field q. . and transform the Hamiltonian into the Heisenberg picture (where it becomes time dependent)

The two-level system The rotating wave approximation q Assume a classical driving electric field q. . and transform the Hamiltonian into the Heisenberg picture (where it becomes time dependent) q Terms are quickly oscillating unless close to resonance condition is achieved where the laser frequency is very close to the atomic frequency splitting Rabi frequency

The fully quantum lightmatter Hamiltonian

The fully quantum light-matter Hamiltonian q Reintroduce the quantization box q Replace the electric field with operators

The fully quantum light-matter Hamiltonian q Reintroduce the quantization box q Replace the electric field with operators q The light-matter coupling strength per mode is q The interaction part can be read in terms of creation of a photon accompanying the transition of an electron from excited to ground state and viceversa

Fundamental processes: stimulated absorption/emission and spontaneous emission

Fundamental processes of light-matter interactions Initial excited state and a photon mode occupied

Fundamental processes of light-matter interactions Initial excited state and a photon mode occupied q Simply apply the interaction part of the Hamiltonian to the initial state

Fundamental processes of light-matter interactions Initial excited state and a photon mode occupied q Simply apply the interaction part of the Hamiltonian to the initial state q Notice the amplitude of given processes (if one starts with a coherent state, the stimulated emission will be amplified by the amplitude of the coherent state) Stimulated emission

Fundamental processes of light-matter interactions Initial excited state and a photon mode occupied q Simply apply the interaction part of the Hamiltonian to the initial state q Notice the amplitude of given processes (if one starts with a coherent state, the stimulated emission will be amplified by the amplitude of the coherent state) Stimulated emission Spontaneous emission

Fundamental processes of light-matter interactions Initial ground state and a photon mode occupied

Fundamental processes of light-matter interactions Initial ground state and a photon mode occupied q Simply apply the interaction part of the Hamiltonian to the initial state

Fundamental processes of light-matter interactions Initial ground state and a photon mode occupied q Simply apply the interaction part of the Hamiltonian to the initial state q The same argument as before applies here as well (if one starts with a coherent state, the stimulated absorption will be amplified by the amplitude of the coherent state) Stimulated absorption

Preview of Lecture 3

Open system dynamics - dissipation The need for a master equation q Dynamics in the quantization box is complicated to describe: owing to the infinite dimensional Hilbert space q One would desire to reduce the dynamics to the system of interest (the two level system – a two dimensional Hilbert space)

Open system dynamics - dissipation The need for a master equation q Dynamics in the quantization box is complicated to describe: owing to the infinite dimensional Hilbert space q One would desire to reduce the dynamics to the system of interest (the two level system – a two dimensional Hilbert space) q First step: dynamics can be followed at the level of the density operator (von. Neumann equation of motion)

Open system dynamics - dissipation The need for a master equation q Dynamics in the quantization box is complicated to describe: owing to the infinite dimensional Hilbert space q One would desire to reduce the dynamics to the system of interest (the two level system – a two dimensional Hilbert space) q First step: dynamics can be followed at the level of the density operator (von. Neumann equation of motion) q Some factorization assumption can be employed

Open system dynamics - dissipation The procedure… q Perform a formal integration (exact, there are no approximations) q Another formal time integration (still exact)

Open system dynamics - dissipation The procedure… q Perform a formal integration (exact, there are no approximations) q Another formal time integration (still exact)

Open system dynamics - dissipation The procedure… q Perform a formal integration (exact, there are no approximations) q Another formal time integration (still exact) q Keep doing this for a sequence of ordered times

Open system dynamics - dissipation The procedure… q Perform a formal integration (exact, there are no approximations) q Another formal time integration (still exact) q Keep doing this for a sequence of ordered times q Truncate (this is where the approximation comes in)

Open system dynamics - dissipation The important step… q Trace over the field states (equivalent to eliminating the box) q Next steps in the following lecture…