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Open Quantum Systems
Basics of Cavity QED
There are two competing rates: the atom in the excited state coherently emitting a photon
into the cavity and the atom emitting incoherently in
free space
Basics of Cavity QED
To Detector Free Space κ
Cavity (Photons) Atom There are two competing rates: the atom in the excited state coherently emitting a photon
into the cavity and the atom emitting incoherently in
free space
Open Quantum Systems
system
• 
Reservoir or Bath In reality, the system is not isolated but coupled to a reservoir (characterized by
many degrees of freedom or many particles or modes)
Open Quantum Systems
system
• 
Reservoir The system, the reservoir and their (little) interaction can all be described by
interaction energies or Hamiltonians
Open Quantum Systems
system
Reservoir •  So solve the larger system!
But that is much harder
and anyways we are usually interested only in the system behavior.
Open Quantum Systems
•  Density Matrix:
In general:
system
For pure states:
Operator expectation
value:
•  Try to get an effective equation for the system
Reservoir Open Quantum Systems
•  Density Matrix:
Evolution:
system
System and Environment:
System:
•  Try to get an effective equation for the system
Reservoir Master Equation
system
Reservoir •  To take the trace on the R.H.S one needs to approximate the reservoir to a
point that it becomes a mere spectator:
Born approximation:
The reservoir is so large that it remains unaffected by the system
Markov approximation:
The reservoir has no memory of the past.
Master Equation
system
For atoms:
Reservoir For cavities:
Master Equation: Criticism
system
Reservoir •  All degrees of freedom of the reservoir are integrated out. No information is
left in the reservoir.
•  It is hard to develop physical intuition with density matrix!
•  Computationally costly: N^2-1 elements as opposed to n elements
“Atom-Photon Interactions” –Cohen-Tanoudjii
Stochastic Wave-function approach
system
Reservoir Jump Operator that changes the state
• 
Develop an effective wave-function approach from the master equation
Carmichael: “An open systems approach to Quantum Optics” and Lecture Notes, 1991.
Dalibard, Castin, Molmer :
Stochastic Wave-function approach
system
Reservoir Lesson: As long as the quantum jump probabilities are small, one can
effectively use a non-Hermitian Hamiltonian with a wave-function to describe
the dynamics.
Atoms in cavity: Vacuum Rabi Splitting
Infact, the primary aim of experimental design for cavity QED is also to satisfy
this condition: to observe coherent atom-photon dynamics
Strong coupling condition
Atoms in cavity
Adiabatically following the
ground state manifold
κ
Eliminating the excited state (EIT)
Eliminating the excited state (EIT)
The “Dark State” is completely decoupled from the excited state: No spontaneously sca.ered photons Atoms in cavity
Adiabatically following the
ground state manifold
Either choose a vary good cavity
(Harcoche et. al.)
Or terminate the process, when a
photon is detected: Quantum Feedback
κ
Feedback
!me The process is terminated at the instance a photon is detected out of the cavity
Atoms in cavity: Vacuum Rabi Splitting
Diagonalize in the manifold:
Line-width of the splitting
and
The energy spectrum is split and
the splitting persist even for no photons!
When can one see this? Strong coupling condition
Strong condition:
Weaker condition:
Time Larmor Precession of atoms in cavity
B Frequency Phys. Rev. Le-.,103,043601(2009). . Strong coupling: Single atom co-operativity
Single atom cooperativity:
Universality: Independent of any details of the atoms:
Physical Systems
Optical cavity
Two mirrors with high reflectivity placed a distance L apart
Photo Source: G. Rempe group, Max planck, Germany
How a cavity looks like
•  Laser cooled atoms
Atoms in cavity
Video: Thanks to Kimble Lab. Photo Source: Nature 424, 839, 2003
Systems
•  Trapped ions
•  Quantum dots
•  Defect centers
•  Super-conducting
Qubits
•  And several others
Applications
“Exploring the Quantum: Atoms, Cavities and Photons” – Haroche and Raimond
Collapse and Revival
Phys. Rev. Le-.,94,0104001(2009). . SuperconducSng qubit C
Josephson tunnel juncSons LJ
100 µm Quantum jumps Quantum jumps due to spontaneous decay
R. Vijay, D.H. Slichter, and I. Siddiqi (Phys. Rev. Le[. 106, 110502 (2011)) Transferring a photon from one atom to another
via a cavity
•  Measured visibility of 88 %
Nature, 449, 438 (2007).
Nature, 499, 443 (2007).
Nature Physics, 3, 765, (2007).
Seeing Photons
ChrisSne Guerlin, Julien Bernu, Samuel Dele´glise, Cle´ment Sayrin, Se´basSen Gleyzes, Stefan Kuhr, Michel Brune, Jean-­‐Michel Raimond and Serge Haroche, Nature, 448, 889 (2007) Physics Nobel -­‐ 2012 Serge Haroche College de France, Paris Dave Wineland NIST and JILA, Boulder, Colorado The Nobel Prize in Physics 2012 was awarded jointly to Serge Haroche and David J. Wineland "for ground-­‐breaking experimental methods that enable measuring and manipula!on of individual quantum systems“ Informa!on: h-p://www.nobelprize.org/nobel_prizes/physics/laureates/2012/ Quantum Network
•  Material systems form nodes
•  Single photon channels connect the nodes
•  Motivation : Quantum cryptography, computation and
simulations
Quantum Network
Agenda:
•  Store and retrieve single photons
•  Entangled nodes
•  Unitary operations
Cavity QED: a quantum node