Quantum trajectories for a system interacting with environment in a single-photon state: Counting and diffusive processes

We derived quantum trajectories for a system interacting with the environment prepared in a continuous mode single photon state as the limit of discrete filtering model with an environment defined as series of independent qubits prepared initially in the entangled state being an analogue of a continuous mode state. The environment qubits interact with the quantum system and they are subsequently measured. The initial correlation between the bath qubits is the source of the non-Markovianity. The conditional evolutions of the quantum system for limit of the continuous in time observations together with the formulas for the photon counting probabilities are given.

[1]  Nicholas I. M. Gould,et al.  SIAM Journal on Optimization , 2012 .

[2]  V. P. Belavkin,et al.  Measurements continuous in time and a posteriori states in quantum mechanics , 1991 .

[3]  G. J. Milburn,et al.  Phonon number measurements using single photon opto-mechanics , 2012, 1205.3240.

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

[5]  Hendra Ishwara Nurdin,et al.  Quantum filtering for systems driven by fields in single photon states and superposition of coherent states using non-Markovian embeddings , 2011, Quantum Information Processing.

[6]  Francesco Petruccione,et al.  The Theory of Open Quantum Systems , 2002 .

[7]  P. Meyer,et al.  Quantum Probability for Probabilists , 1993 .

[8]  G. J. Milburn.  Coherent control of single photon states , 2008 .

[9]  T. Ralph,et al.  Optimized generation of heralded Fock states using parametric down-conversion , 2009, 0909.4147.

[10]  G. Alber,et al.  Quantum Electrodynamics of One-Photon Wave Packets , 2010, 1002.3059.

[11]  Dla Polski,et al.  EURO , 2004 .

[12]  F Petruccione,et al.  Non-Markovian quantum repeated interactions and measurements , 2009, 0903.3859.

[13]  Thomas Hellman.  PHIL , 2018, Encantado.

[14]  Kindra M Kelly-Scumpia,et al.  51 , 2015, Tao te Ching.

[15]  Quantum Electrodynamics of One-Photon Wave Packets , 2010 .

[16]  R. Handel,et al.  Discrete approximation of quantum stochastic models , 2008, 0803.4383.

[17]  A. Brańczyk,et al.  N-photon wave packets interacting with an arbitrary quantum system , 2012, 1202.3430.

[18]  John Edward Gough,et al.  Stochastic Schrödinger Equations as Limit of Discrete Filtering , 2004, Open Syst. Inf. Dyn..

[19]  Cl'ement Pellegrini,et al.  Existence, uniqueness and approximation of a stochastic Schrödinger equation: The diffusive case , 2007, 0709.1703.

[20]  Luigi Accardi,et al.  Central limits of squeezing operators , 1989 .

[21]  JOHN GOUGH.  Holevo-Ordering and the Continuous-Time Limit for Open Floquet Dynamics , 2004 .

[22]  Matthew R. James,et al.  A Discrete Invitation to Quantum Filtering and Feedback Control , 2009, SIAM Rev..

[23]  Y. Pautrat,et al.  From Repeated to Continuous Quantum Interactions , 2003, math-ph/0311002.

[24]  Thomas Pellizzari,et al.  Photon-Wavepackets as Flying Quantum Bits , 1998 .

[25]  Gerd Leuchs,et al.  Perfect excitation of a matter qubit by a single photon in free space , 2008, 0808.1666.

[26]  Quantum random walks and their convergence to Evans-Hudson flows , 2008 .

[27]  B. M. Fulk.  MATH , 1992 .

[28]  Hendra Ishwara Nurdin,et al.  Quantum filtering for systems driven by fields in single-photon states or superposition of coherent states , 2012 .

[29]  A. Lvovsky,et al.  Quantum state reconstruction of the single-photon Fock state. , 2001, Physical Review Letters.

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

[31]  M. Wodzicki.  Lecture Notes in Math , 1984 .

[32]  Luc Bouten,et al.  Stochastic Schrödinger equations , 2003 .

[33]  Clément Pellegrini,et al.  Existence, uniqueness and approximation of the jump-type stochastic Schrödinger equation for two-level systems , 2010 .

[34]  Burkhard Kümmerer,et al.  Quantum Markov Processes and Applications in Physics , 2006 .

[35]  Nima Monshizadeh,et al.  2011 50TH IEEE CONFERENCE ON DECISION AND CONTROL AND EUROPEAN CONTROL CONFERENCE (CDC-ECC) , 2011, CDC 2011.

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

[37]  K. Banaszek,et al.  Pure-state single-photon wave-packet generation by parametric down-conversion in a distributed microcavity , 2005 .

[38]  A. Dąbrowska,et al.  Belavkin filtering with squeezed light sources , 2014, 1405.7795.

[39]  Lars V. Schäfer,et al.  ADVANCES IN QUANTUM CHEMISTRY, VOL 59 , 2010 .

[40]  M. James,et al.  Single photon quantum filtering using non-Markovian embeddings , 2012, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[41]  Yan Pautrat.  From Pauli Matrices to Quantum Itô Formula , 2005 .

[42]  Todd A. Brun,et al.  A simple model of quantum trajectories , 2002 .

[43]  Elsi-Mari Laine,et al.  Colloquium: Non-Markovian dynamics in open quantum systems , 2015, 1505.01385.

[44]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[45]  Guofeng Zhang,et al.  Continuous-Mode MultiPhoton Filtering , 2016, SIAM J. Control. Optim..

[46]  Christine Silberhorn,et al.  Spectral structure and decompositions of optical states, and their applications , 2006, quant-ph/0609004.

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

[48]  Ben Q. Baragiola,et al.  Quantum trajectories for propagating Fock states , 2017, 1704.00101.

[49]  Valerio Scarani,et al.  Efficient excitation of a two-level atom by a single photon in a propagating mode , 2010, 1010.4661.

[50]  M. Scully,et al.  The Quantum Theory of Light , 1974 .