Time-continuous Bell measurements.

We combine the concept of Bell measurements, in which two systems are projected into a maximally entangled state, with the concept of continuous measurements, which concerns the evolution of a continuously monitored quantum system. For such time-continuous Bell measurements we derive the corresponding stochastic Schrödinger equations, as well as the unconditional feedback master equations. Our results apply to a wide range of physical systems, and are easily adapted to describe an arbitrary number of systems and measurements. Time-continuous Bell measurements therefore provide a versatile tool for the control of complex quantum systems and networks. As examples we show that (i) two two-level systems can be deterministically entangled via homodyne detection, tolerating photon loss up to 50%, and (ii) a quantum state of light can be continuously teleported to a mechanical oscillator, which works under the same conditions as are required for optomechanical ground-state cooling.

[1]  J. Cirac,et al.  Discrete entanglement distribution with squeezed light. , 2003, Physical review letters.

[2]  Milburn,et al.  Quantum theory of optical feedback via homodyne detection. , 1993, Physical review letters.

[3]  H. Weinfurter,et al.  Experimental quantum teleportation , 1997, Nature.

[4]  V. P. Belavkin,et al.  Measurement, filtering and control in quantum open dynamical systems , 1999 .

[5]  J. Raimond,et al.  Manipulating quantum entanglement with atoms and photons in a cavity , 2001 .

[6]  Alberto Barchielli,et al.  Quantum Trajectories and Measurements in Continuous Time: The Diffusive Case , 2009 .

[7]  G. Rempe,et al.  Feedback cooling of a single neutral atom , 2011, 2011 Conference on Lasers and Electro-Optics Europe and 12th European Quantum Electronics Conference (CLEO EUROPE/EQEC).

[8]  J. Raimond,et al.  Exploring the Quantum , 2006 .

[9]  Daniel Sigg,et al.  Optical dilution and feedback cooling of a gram-scale oscillator to 6.9 mK. , 2007, Physical review letters.

[10]  J. Dowling.  Exploring the Quantum: Atoms, Cavities, and Photons. , 2014 .

[11]  H. Kimble,et al.  Efficient engineering of multiatom entanglement through single-photon detections. , 2003, Physical review letters.

[12]  A. N. Didenko,et al.  Radio Eng Electron Phys , 1972 .

[13]  Milburn,et al.  Quantum theory of field-quadrature measurements. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[14]  H. Kimble,et al.  Demonstration of a state-insensitive, compensated nanofiber trap. , 2012, Physical review letters.

[15]  M. Aspelmeyer,et al.  Quantum entanglement and teleportation in pulsed cavity optomechanics , 2011, 1108.2586.

[16]  P. Zoller,et al.  Preparation of entangled states by quantum Markov processes , 2008, 0803.1463.

[17]  A. Silberfarb,et al.  Efficient quantum-state estimation by continuous weak measurement and dynamical control. , 2006, Physical review letters.

[18]  E. Knill,et al.  Deterministic quantum teleportation of atomic qubits , 2004, Nature.

[19]  Christine A Muschik,et al.  Entanglement distillation by dissipation and continuous quantum repeaters. , 2010, Physical review letters.

[20]  M. Aspelmeyer,et al.  Laser cooling of a nanomechanical oscillator into its quantum ground state , 2011, Nature.

[21]  P. Knight,et al.  Proposal for teleportation of an atomic state via cavity decay , 1999, quant-ph/9908004.

[22]  Florian Marquardt,et al.  Quantum theory of cavity-assisted sideband cooling of mechanical motion. , 2007, Physical review letters.

[23]  R. Chaves,et al.  Quantum computing with incoherent resources and quantum jumps. , 2011, Physical review letters.

[24]  M. Wolf,et al.  Quantum capacities of bosonic channels. , 2006, Physical review letters.

[25]  Photon-by-photon feedback control of a single-atom trajectory , 2009, Nature.

[26]  Kurt Jacobs,et al.  A straightforward introduction to continuous quantum measurement , 2006, quant-ph/0611067.

[27]  A. D. Boozer,et al.  Trapped atoms in cavity QED: coupling quantized light and matter , 2005 .

[28]  G. Milburn,et al.  Quantum Measurement and Control , 2009 .

[29]  Matthew R. James,et al.  An Introduction to Quantum Filtering , 2006, SIAM Journal of Control and Optimization.

[30]  C. Simon,et al.  Robust long-distance entanglement and a loophole-free bell test with ions and photons. , 2003, Physical review letters.

[31]  Collett,et al.  Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation. , 1985, Physical review. A, General physics.

[32]  C. Regal,et al.  Observation of Radiation Pressure Shot Noise on a Macroscopic Object , 2012, Science.

[33]  K. Parthasarathy.  An Introduction to Quantum Stochastic Calculus , 1992 .

[34]  H. J. Kimble,et al.  Position-momentum Einstein-Podolsky-Rosen state of distantly separated trapped atoms , 2000 .

[35]  A. N. Korotkov,et al.  Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback , 2012, Nature.

[36]  46 , 2015, Slow Burn.

[37]  D. Stamper-Kurn,et al.  Observation of quantum-measurement backaction with an ultracold atomic gas , 2007, 0706.1005.

[38]  J. Cirac,et al.  Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network , 1996, quant-ph/9611017.

[39]  R J Schoelkopf,et al.  Circuit QED and engineering charge-based superconducting qubits , 2009, 0912.3902.

[40]  D. McClelland,et al.  Cooling of a Gram-Scale Cantilever Flexure to 70 mK with a Servo-Modified Optical Spring. , 2008, Physical review letters.

[41]  K. Jacobs,et al.  FEEDBACK CONTROL OF QUANTUM SYSTEMS USING CONTINUOUS STATE ESTIMATION , 1999 .

[42]  J. Cirac,et al.  Quantum simulations based on measurements and feedback control , 2008, 0811.1844.

[43]  Yanbei Chen,et al.  Macroscopic quantum mechanics: theory and experimental concepts of optomechanics , 2013, 1302.1924.

[44]  B. Muzykantskii,et al.  ON QUANTUM NOISE , 1995 .

[45]  M. Aspelmeyer,et al.  Squeezed light from a silicon micromechanical resonator , 2013, Nature.

[46]  A. Silberfarb,et al.  Quantum control of the hyperfine spin of a Cs atom ensemble. , 2007, Physical review letters.

[47]  Kurt Jacobs,et al.  Feedback cooling of a nanomechanical resonator , 2003 .

[48]  Graham,et al.  Linear stochastic wave equations for continuously measured quantum systems. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[49]  Robin L. Hudson,et al.  Quantum Ito's formula and stochastic evolutions , 1984 .

[50]  M. B. Plenio,et al.  Robust creation of entanglement between ions in spatially separate cavities. , 2003 .

[51]  F. Schmidt-Kaler,et al.  Deterministic quantum teleportation with atoms , 2004, Nature.

[52]  L. DiCarlo,et al.  Deterministic entanglement of superconducting qubits by parity measurement and feedback , 2013, Nature.

[53]  M Pinard,et al.  High-sensitivity optical monitoring of a micromechanical resonator with a quantum-limited optomechanical sensor. , 2006, Physical review letters.

[54]  Stefano Mancini,et al.  Engineering an interaction and entanglement between distant atoms , 2004 .

[55]  Mazyar Mirrahimi,et al.  Real-time quantum feedback prepares and stabilizes photon number states , 2011, Nature.

[56]  Craig A. Tracy,et al.  Communications in Mathematical Physics The Pearcey Process , 2006 .

[57]  Thierry Botter,et al.  Non-classical light generated by quantum-noise-driven cavity optomechanics , 2012, Nature.

[58]  T. Pellizzari,et al.  Quantum Networking with Optical Fibres , 1997 .

[59]  T. Nieuwenhuizen.  What are quantum fluctuations , 2007 .

[60]  J. Cirac,et al.  IDEAL QUANTUM COMMUNICATION OVER NOISY CHANNELS : A QUANTUM OPTICAL IMPLEMENTATION , 1997, quant-ph/9702036.

[61]  Karsten Danzmann,et al.  Entanglement of macroscopic test masses and the standard quantum limit in laser interferometry. , 2007, Physical review letters.

[62]  T. R. Gerholm.  Reports on progress in physics: Volume XIX (1956) (The Physical Society, London 1956, 367 pages price to non-fellows £ 2 10 s., to fellows, one copy, £ 1 7 s 6 d). , 1957 .

[63]  T J Kippenberg,et al.  Theory of ground state cooling of a mechanical oscillator using dynamical backaction. , 2007, Physical review letters.

[64]  S. Dawkins,et al.  Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber. , 2009, Physical review letters.

[65]  Samuel L. Braunstein,et al.  Quantum-state transfer from light to macroscopic oscillators , 2003 .

[66]  Christoph Becher,et al.  Feedback cooling of a single trapped ion. , 2006, Physical review letters.

[67]  B. Englert,et al.  Cavity quantum electrodynamics , 2006 .

[68]  J. Ignacio Cirac,et al.  Optically Levitating Dielectrics in the Quantum Regime: Theory and Protocols , 2010, 1010.3109.

[69]  Gardiner,et al.  Wave-function quantum stochastic differential equations and quantum-jump simulation methods. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[70]  H. Carmichael.  An open systems approach to quantum optics , 1993 .

[71]  Thomas G. Walker,et al.  Quantum information with Rydberg atoms , 2009, 0909.4777.

[72]  Christine A Muschik,et al.  Entanglement generated by dissipation and steady state entanglement of two macroscopic objects. , 2010, Physical review letters.

[73]  Stefano Mancini,et al.  Scheme for teleportation of quantum states onto a mechanical resonator. , 2003, Physical review letters.

[74]  Wiseman,et al.  Quantum theory of continuous feedback. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[75]  Hideo Mabuchi,et al.  Quantum feedback control and classical control theory , 1999, quant-ph/9912107.

[76]  Simon C Benjamin,et al.  Measurement-based entanglement under conditions of extreme photon loss. , 2007, Physical review letters.

[77]  M. Gu,et al.  Unconditional preparation of entanglement between atoms in cascaded optical cavities. , 2003, Physical review letters.

[78]  P. Corkum,et al.  Journal of Physics B: atomic, molecular and optical physics , 2015 .

[79]  J. Teufel,et al.  Sideband cooling of micromechanical motion to the quantum ground state , 2011, Nature.

[80]  G. Vidal,et al.  Computable measure of entanglement , 2001, quant-ph/0102117.

[81]  T. Hayler,et al.  Observation of a kilogram-scale oscillator near its quantum ground state , 2009 .

[82]  Jacob F. Sherson,et al.  Quantum teleportation between light and matter , 2006, Nature.

[83]  A. Sørensen,et al.  Quantum interface between light and atomic ensembles , 2008, 0807.3358.