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Cascaded Quantum Systems Introduction Cascaded Quantum System: Exciting one quantum system with another via a unidirectional coupling. Cascaded quantum networks (PRL 1997, Cirac, Zoller, Kimble, Mabuchi) In this paper, we study entanglement formation in driven few- and many-particle cascaded quantum networks as introduced by Gardiner [1] and Carmichael [2], where the unconventional coupling of multiple systems to a common unidirectional bath offers remarkable new opportunities for dissipative preparation of highly correlated states Original Context of [1]: A pair of two level systems where a coherently driven two level atom is used as the light source for a second two level system.Observation of anti-bunching and mutually anti-correlated: If photon measured from atom 1, it is not available to excite atom 2. Ref [2] Cascaded Quantum Network a cascaded quantum network consists of N systems coupled to a one-dimensional (1D) reservoir that has the unique feature that excitations can only propagate along a single direction, thereby driving successive systems in the network in a unidirectional way. [1] C. W. Gardiner, “Driving a Quantum System with the Output Field From Another Driven Quantum System”, Phys. Rev. Lett. 70 2269 (1993) [2]                       “Cascaded quantum systems: first system drives in a unidirectional coupling a second quantum system “ Open.png Fig. 1 What is a cascaded quantum system? What are the basic requirements/definitions. Need of an anisotropic coupling. Why are they interesting? Entangled state generation (develop in another section). History/Background In early exploration of the open quantum systems: driven quantum systems (ref), spectral properties of field leaking from quantum systems (ref). Early developments: Question: What happens when quantum field leaking from one system drives another quantum system. This is interesting subclass of open quantum systems because lack of back action (by treating the interactions between the subsystems in ansymmetric way) it allows simplification of the theoretical treatmant, allowing treatmant of more complex many body open systems. Problem with direct calculation: Infinite set of Heisenberg equations ->only solvable for simplest couplings Simple systems: Early development in this direction - C. Gardiner, two level system driven by squeezed light; Origin - Carmaichel (two cavities coupled together in unidirectional way, coupled unidirectionaly to the the reservoar) Summary of theoretical approaches: Quantum Trajectories (by H.C. what they precisely exploit); Quantum Langevin Equations (what they exactly exploit) Current interest: Quantum networks (J.I.Cirac et.al, PRL78:3221 (1997)) Promising approaches: Chiral nanophotonic fibers waveguide interface based on spin-orbit interaction of light Figure: Excitations between the systems can travel only in the single direction. System is coupled to common unidirectional bath. Nikola Methods of solution? Challenges? (Mathematical formulation) Master Equation (Araceli) NOTE: ref to the Fig 1, we are going to work on the framework of an unidirectional reservoir to obtain the master equation. Description of the Hamiltonian.

being the respective Hamiltonians for: systems 1 and 2 HR is the free Hamiltonian of a traveling-wave reservoir which couples the cavities in one direction only Hint is the interaction part which shows the unidirectional coupling via only right running modes Version 1 & Version 2 (Lindblad Form)? Example Systems ii) Two cavities connected by unidirectional coupling Thomas Experiment // Araceli and Jorge Example.png Quantum Network?? Ref. H.J. Kimble (2008 Nature) & wikipedia entry: Quantum network State preparation through unidirectional coupling Andrew said include G.Rempe 2014 experiment. (Ref P. Zoller)  Araceli and Jorge Chiral Quantum Networks: Reference “Quantum Spin-Dimers from Chiral Dissipation in Cold Atom Chains“ IVOR!! Rauschenbeutel et. al.: Science 2014 & arXiv 1406.2184