Quantum Physics of LightMatter Interactions So Se 20

Quantum Physics of Light-Matter Interactions So. Se 20 --- April 24 th to July 24 th 2020 -- Lecturer: Claudiu Genes Max Planck Institute for the Science of Light (Erlangen, Germany) -- Lecture 3 --- 15. 05. 2020 -- Genes Research Group Cooperative Quantum Phenomena

From the last class… Modes of light inside the quantization box k-propagation direction 2 possible polarizations

From the last class… Modes of light inside the quantization box q Replace c-numbers with non-commuting bosonic operators q The electric field operator q Where the zero-point electric field amplitude is k-propagation direction 2 possible polarizations

From the last class… Light-matter interactions q The fully quantum light-matter Hamiltonian q The light-matter coupling strength per mode is

From the last class… Light-matter interactions q The fully quantum light-matter Hamiltonian 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

Spontaneous emission of an initially excited TLS Initial excited state and no photon mode occupied q Only one possible process (destroy excitation in the TLS and excite a photon mode)

Spontaneous emission of an initially excited TLS Initial excited state and no photon mode occupied q Only one possible process (destroy excitation in the TLS and excite a photon mode) q Notice that there an infinite number of channels for decay each with a small probability related to the TLS-photon coupling

Spontaneous emission of an initially excited TLS Initial excited state and no photon mode occupied q Only one possible process (destroy excitation in the TLS and excite a photon mode) q Notice that there an infinite number of channels for decay each with a small probability related to the TLS-photon coupling q Dynamics is purely unitary in an infinitely dimensional Hilbert space with basis states ,

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 at some initial time t

Open system dynamics - dissipation The procedure… q Start with the full quantum light-matter Hamiltonian

Open system dynamics - dissipation The procedure… q Start with the full quantum light-matter Hamiltonian q Transformation to an interaction picture with unitary evolution q Resulting full Hamiltonian in the interaction picture

Open system dynamics - dissipation The procedure… q Start with the full quantum light-matter Hamiltonian q Transformation to an interaction picture with unitary evolution q Resulting full Hamiltonian in the interaction picture q Notice that the two operators above are only referring to the field states

Open system dynamics - dissipation The procedure… q Perform a formal integration (exact, there are no approximations) t t 1 Δt time

Open system dynamics - dissipation The procedure… q Perform a formal integration (exact, there are no approximations) q Another formal time integration (still exact) t t 2 t 1 Δt time

Open system dynamics - dissipation The procedure… q Perform a formal integration (exact, there are no approximations) q Another formal time integration (still exact) q …and another one… t t 3 t 2 t 1 Δt time

Open system dynamics - dissipation The procedure… q One can keep doing this for a sequence of ordered time t t 3 t 2 t 1 Δt time

Open system dynamics - dissipation The procedure… q One can keep doing this for a sequence of ordered time q However, we truncate assuming weak photon-electron interactions t t 3 t 2 t 1 Δt time

Open system dynamics - dissipation The procedure… q One can keep doing this for a sequence of ordered time q However, we truncate assuming weak photon-electron interactions q …and perform the trace over the field degree of freedom t t 3 t 2 t 1 Δt time

Open system dynamics - dissipation The unavoidable mathematical steps… q Tracing procedure q …where the compound index means

Open system dynamics - dissipation The unavoidable mathematical steps… q Tracing procedure q …where the compound index means q For example the trace over the density matrix of the vacuum leads to

Open system dynamics - dissipation The unavoidable mathematical steps… q Tracing procedure q …where the compound index means q For example the trace over the density matrix of the vacuum q The first order contribution leads to

Open system dynamics - dissipation The unavoidable mathematical steps… q Tracing procedure q …where the compound index means q For example the trace over the density matrix of the vacuum q The first order contribution leads to

Open system dynamics - dissipation The unavoidable mathematical steps… q Tracing procedure q …where the compound index means q For example the trace over the density matrix of the vacuum q The first order contribution q We conclude leads to

Open system dynamics - dissipation The unavoidable mathematical steps… q Second order traces (4 terms)

Open system dynamics - dissipation The unavoidable mathematical steps… q Second order traces (4 terms) q …following steps as described before (see Lecture notes for more details) one gets

Open system dynamics - dissipation The unavoidable mathematical steps… q Second order traces (4 terms) q …following steps as described before (see Lecture notes for more details) one gets Only matter part Only field part

Open system dynamics - dissipation Partial result q A difference equation for the matter part only Remember the starting point

Open system dynamics - dissipation Partial result q A difference equation for the matter part only q …where the coefficients stem from time integration and the correlations of the time dependent operators referring to the field Remember the starting point

Open system dynamics - dissipation Partial result q A difference equation for the matter part only q …where the coefficients stem from time integration and the correlations of the time dependent operators referring to the field q To ruin the surprise, of course we wish to get the terms above proportional to we can turn the equation above into a differential equation Remember the starting point in which case

Open system dynamics - dissipation …a few more tedious steps q Starting with

Open system dynamics - dissipation …a few more tedious steps q Starting with q We find that the trace has a simple form . . using

Open system dynamics - dissipation …a few more tedious steps q Starting with q We find that the trace has a simple form . . using q This brings us to the task ahead which is to estimate

Open system dynamics - dissipation Evaluating the coefficients q To evaluate q First the sum can be turned into an integral (with the density of optical states )

Open system dynamics - dissipation Evaluating the coefficients q To evaluate q First the sum can be turned into an integral (with the density of optical states q With a few more steps (see Lecture notes for details) we learn that )

Open system dynamics - dissipation Evaluating the coefficients q To evaluate q First the sum can be turned into an integral (with the density of optical states ) q With a few more steps (see Lecture notes for details) we learn that q The time integral is a bit complicated but detailed in the Lecture notes so I’ll state the result

Open system dynamics - dissipation The master equation for spontaneous emission q We turn the difference equation into a differential equation via

Open system dynamics - dissipation The master equation for spontaneous emission q We turn the difference equation into a differential equation via q Then we end up with the following equation

Open system dynamics - dissipation The master equation for spontaneous emission q We turn the difference equation into a differential equation via q Then we end up with the following equation

Open system dynamics - dissipation The master equation for spontaneous emission q We turn the difference equation into a differential equation via q Then we end up with the following equation Linblad form

Open system dynamics - dissipation Effect of the Linblad term on population and coherence q Let us explicitly write the effect of the Lindblad superoperator onto the evolution of the system’s density matrix. We do this in matrix form by the following identifications q …and for the matrices

Open system dynamics - dissipation Effect of the Linblad term on population and coherence q Let us explicitly write the effect of the Lindblad superoperator onto the evolution of the system’s density matrix. We do this in matrix form by the following identifications q …and for the matrices q One can then write the free evolution and Linblad terms in matrix form

Open system dynamics - dissipation Effect of the Linblad term on population and coherence q Let us explicitly write the effect of the Lindblad superoperator onto the evolution of the system’s density matrix. We do this in matrix form by the following identifications q …and for the matrices q One can then write the free evolution and Linblad terms in matrix form q Finally we derive

Open system dynamics - dissipation Effect of the Linblad term on population and coherence q Starting with this q Evolution of excited state population (decay at twice the decay rate) q Evolution of the coherence (half of the decay rate of the population and rotation at the TLS frequency splitting) q Ground state accumulation of population

Bloch equations Time dynamics of the driven-dissipative TLS q Let us now add a laser drive such that the Hamiltonian is

Bloch equations Time dynamics of the driven-dissipative TLS q Let us now add a laser drive such that the Hamiltonian is q The master equation is q Let’s sandwich it with bras and kets from the left and write and compute the evolution of the coherence

Bloch equations Time dynamics of the driven-dissipative TLS q Let us now add a laser drive such that the Hamiltonian is q The master equation is q Let’s sandwich it with bras and kets from the left and write and compute the evolution of the coherence q …and after a few simple calculations we get

Bloch equations Time dynamics of the driven-dissipative TLS q Let us remove the time dependence in the Bloch equations q. . by the following changes of variables

Bloch equations Time dynamics of the driven-dissipative TLS q Let us remove the time dependence in the Bloch equations q. . by the following changes of variables q We end up with time-independent coupled linear differential equations q where the detuning is defined as

…in the next class Time dynamics of the driven-dissipative TLS q Distinction between transient versus steady state regime q Steady state response of a TLS to classical drive: absorption profile, dispersive response

…in the next class Time dynamics of the driven-dissipative TLS q Distinction between transient versus steady state regime q Steady state response of a TLS to classical drive: absorption profile, dispersive response q Transient dynamics: pi and pi/2 pulses (for quantum gate operations)

…in the next class Time dynamics of the driven-dissipative TLS q Distinction between transient versus steady state regime q Steady state response of a TLS to classical drive: absorption profile, dispersive response q Transient dynamics: pi and pi/2 pulses (for quantum gate operations) Connection between the macroscopic picture and microscopic processes q We will consider a simple model of a dielectric q From the response of the individual TLS coherence we derive the macroscopic polarization

…in the next class Time dynamics of the driven-dissipative TLS q Distinction between transient versus steady state regime q Steady state response of a TLS to classical drive: absorption profile, dispersive response q Transient dynamics: pi and pi/2 pulses (for quantum gate operations) Connection between the macroscopic picture and microscopic processes q We will consider a simple model of a dielectric q From the response of the individual TLS coherence we derive the macroscopic polarization q We identify the linear and nonlinear parts of the dielectric susceptibility