Generalized Schrieffer-Wolff formalism for dissipative systems

We present a formalized perturbation theory for Markovian open systems in the language of a generalized Schrieffer-Wolff (SW) transformation. A non-unitary rotation decouples the unper- turbed steady states from all fast degrees of freedom, in order to obtain an effective Liouvillian, that reproduces the exact low excitation spectrum of the system. The transformation is derived in a constructive way, yielding a perturbative expansion of the effective Liouville operator. The presented formalism realizes an adiabatic elimination of fast degrees of freedom to arbitrary orders in the perturbation. We exemplarily employ the SW formalism to two generic open systems and discuss general properties of the different orders of the perturbation.

[1]  Fernando Pastawski,et al.  Quantum memories based on engineered dissipation , 2010, 1010.2901.

[2]  W. Marsden.  I and J , 2012 .

[3]  H. Carmichael.  Analytical and numerical results for the steady state in cooperative resonance fluorescence , 1980 .

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

[5]  B. Uchoa,et al.  Kondo quantum criticality of magnetic adatoms in graphene. , 2010, Physical Review Letters.

[6]  Khaled Karrai,et al.  Quantum-Dot Spin-State Preparation with Near-Unity Fidelity , 2006, Science.

[7]  Alex C. Hewson,et al.  The Kondo Problem to Heavy Fermions , 1993 .

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

[9]  D. Klein.  Degenerate perturbation theory , 1974 .

[10]  Raymond G. Beausoleil,et al.  Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond , 2008 .

[11]  D. DiVincenzo,et al.  Schrieffer-Wolff transformation for quantum many-body systems , 2011, 1105.0675.

[12]  N. Kampen,et al.  Elimination of fast variables , 1985 .

[13]  D. DiVincenzo,et al.  Quantum simulation of many-body Hamiltonians using perturbation theory with bounded-strength interactions. , 2008, Physical review letters.

[14]  Florentin Reiter,et al.  Effective operator formalism for open quantum systems , 2011, 1112.2806.

[15]  L. Davidovich,et al.  General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology , 2011, 1201.1693.

[16]  L. T. Redmon,et al.  Quasidegenerate perturbation theories. A canonical van Vleck formalism and its relationship to other approaches , 1980 .

[17]  Blatt,et al.  Laser cooling of trapped ions in a standing wave. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[18]  A. Hewson,et al.  The Kondo Problem to Heavy Fermions by Alexander Cyril Hewson , 1993 .

[19]  G. Giedke,et al.  Nuclear spin cooling using Overhauser-field selective coherent population trapping. , 2010, Physical review letters.

[20]  F. Verstraete,et al.  Quantum computation and quantum-state engineering driven by dissipation , 2009 .

[21]  M. Vojta,et al.  Fermi-liquid instabilities at magnetic quantum phase transitions , 2006, cond-mat/0606317.

[22]  Proposed Realization of the Dicke-Model Quantum Phase Transition in an Optical Cavity QED System , 2006, quant-ph/0607115.

[23]  G. Lindblad.  On the generators of quantum dynamical semigroups , 1976 .

[24]  Julia Kempe,et al.  The Complexity of the Local Hamiltonian Problem , 2004, FSTTCS.

[25]  J. Schrieffer,et al.  Relation between the Anderson and Kondo Hamiltonians , 1966 .

[26]  Christine Guerlin,et al.  Dicke quantum phase transition with a superfluid gas in an optical cavity , 2009, Nature.

[27]  Optical superradiance from nuclear spin environment of single-photon emitters. , 2010, Physical review letters.

[28]  J. Cirac,et al.  Improvement of frequency standards with quantum entanglement , 1997, quant-ph/9707014.

[29]  Melvin Lax,et al.  FORMAL THEORY OF QUANTUM FLUCTUATIONS FROM A DRIVEN STATE , 1963 .

[30]  J. Cirac,et al.  Dissipative phase transition in a central spin system , 2012, 1205.3341.

[31]  D. T. Pegg,et al.  JOURNAL OF PHYSICS B. ATOMIC AND MOLECULAR PHYSICS. LETTERS TO THE EDITOR , 1970 .

[32]  A. Parkins,et al.  Dissipation-driven quantum phase transitions in collective spin systems , 2008, 0805.1256.

[33]  E. Farhi,et al.  Perturbative gadgets at arbitrary orders , 2008, 0802.1874.

[34]  K. Flensberg,et al.  Vibrational sidebands and the Kondo effect in molecular transistors. , 2004, Physical review letters.