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Transcript
Quantum Electro-Dynamical Time-Dependent
Density Functional Theory on a Lattice
Lattice-QED-TDDFT
Mehdi Farzanehpour, Ilya Tokatly
Nano-bio spectroscopy group, University of Basque Country
San Sebastian, Spain
eman ta zabal zazu
[email protected]
Universidad
del País Vasco
Euskal Herriko
Unibertsitatea
nano-bio
spectroscopy
group
What is Time dependent density functional theory?
TDDFT is a formulation of the quantum many-body problem based on
the 1:1 map from the time-dependent density
potential
to the driving external
.
A lattice example: Many electrons on an M-site lattice subject to an
external classical electric field.
What is Time dependent density functional theory?
TDDFT is a formulation of the quantum many-body problem based on
the 1:1 map from the time-dependent density
potential
to the driving external
.[1]
A lattice example: Many electrons on an M-lattice subject to external
classical electric field.
It can be shown that the time dependent state
of the system
and on-site potential
are unique functionals of the on-site density
provided that the density
and initial state
satisfy some well defined conditions.[2]
[1] E. Runge and E. K. U. Gross, PRL. 52, 997 (1984)
[2] M. Farzanehpour and I. V. Tokatly, PRB 86,125130 (2012).
Why QED-TDDFT?
●
Superconducting qubit using circuit quantum
electrodynamics
●
Atoms in optical cavity
●
Molecules interacting with cavity photons
H. Mabuchi and A. Doherty,Science 298,1372 (2002)
A. Blais etal. .PRA 69, 062320 (2004)
What is QED-TDDFT?
QED-TDDFT is a generalization of TDDFT for electronic systems
interacting with quantized electromagnetic field.
[1] I. V. Tokatly, PRL. 110, 233001
[2] M. Ruggenthaler etal PRA. 90, 012508
[3] Mehdi Farzanehpour, I. V. Tokatly, PRB 90, 195149
[4] C. Pellegrini etal, Arxiv 1412.4530
An example system
A two-level system coupled with a single cavity mode:
Schrödinger Equation:
A two-level system in a cavity
Electronic Hamiltonian:
Hopping On-site potential
A two-level system in a cavity
Photonic Hamiltonian:
A two-level system in a cavity
Interaction Hamiltonian:
Basic variables
Here we need to define a set of basic variables:
1. On-site density
2. Expectation value of the photonic coordinate:
Direct map
Given the Schrödinger equation and the initial state
unique map from the time dependent on-site potential
external current
to the on-site density
of the photonic coordinate
SE:
.
, there is a
and the
and expectation value
“Inverse” map
Statement of the theorem: The electron-photon wave function
unique functional of the electronic density
value of the photonic coordinate
and the density
and the expectation
, provided the initial state
satisfy a set of well defined conditions.
is a
How we show the “inverse” map?
From the equation for the second derivative of the density and
the field average:
(Driven harmonic oscillator)
(Divergence of the
force balance equation)
How we show the “inverse” map?
So we will have a set of equations which form a Non-linear SE
NLSE:
This is a set of semi-linear partial differential equations where Irving
Segal showed that it has a unique solution if
continuous in time.
[1] I. Segal, Annals of Mathematics, 78, 339(1963)
is Lipschitz and
What else?
1. This formalism can easily generalized to many electron on a lattice
coupled to many photonic modes. [1]
2. Not all densities on a lattice are v-representable.
3. However dynamics starting from the ground state density are
guaranteed to be v-representable.[1]
(v-representability: A time dependent density is v-representable if exists a time evolution
with a time-dependent potential which reproduces the density.)
[1] Mehdi Farzanehpour, I. V. Tokatly, PRB 90, 195149
Thank you
[email protected]