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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