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Tel.: +7(095)135-14-49
E-mail: pechen@mi.ras.ru
Degree: Master degree in Physics (2001)
Thesis: Calculation of the higher order corrections to the stochastic limit (Diploma with Honor)
Thesis advisor: Professor I.V. Volovich
Postgraduate study: Steklov Mathematical Institute ( Department of Mathematical Physics) 2001–2004
Field of Study: Mathematical Physics
PhD Thesis: Method of stochastic asymptotic in quantum dynamics
Thesis advisor: Professor I.V. Volovich
A. Pechen works on study of the long time dynamics of open quantum systems in the weak coupling and low density limits. In the weak coupling limit one considers a quantum system interacting with a reservoir in the case the interaction between the system and reservoir is small, whereas in the low density limit the density of particles of reservoir (Bose gas) is small. L.Accardi, Y.G.Lu and I.V.Volovich developed a powerful approach – the stochastic limit method (the white noise approach) to study the dynamics in the weak coupling limit.
In joint publications of A.Pechen with L.Accardi and I.Volovich the nonperturbative white noise approach to the low density limit was developed [1, 3]. It is shown that the dynamics of the compound system in this regime is described by the unitary solution (adapted process) of a quantum stochastic differential equation, driven by a quantum Poisson noise. Using the developed white noise approach, in these papers a non-perturbative derivation, starting from exact microscopic Schrodinger equation, of the quantum stochastic equation for the limiting evolution operator is given. As a consequence, the quantum Langevin and master equation for dynamics of the test particle are derived. At a very first time the Stochastic Golden Rule (a set of simple rules for derivation of the limiting equations) is found in the low density limit. This approach uses the Fock-antiFock (or Gel’fand-Naimark-Segal) representation for the CCR algebra of the gas, determined by its quasifree state.
In [5] A.Pechen developed a white noise approach to derivation of the quantum stochastic equations directly in terms of the correlation functions, without use of the Fock-antiFock representation. This approach simplifies the derivation of the limiting quantum stochastic equations and allows to express the intensity of the quantum Poisson process directly in terms of the one-particle S-matrix, describing scattering of the test particle on one particle of the gas. The notions of a causal state and causal time-energy quantum white noise, which are very relevant to the low density limit, are introduced.
Another part of A.Pechen’s research is devoted to study the higher order corrections to the weak coupling limit [2]. The dynamics in the weak coupling limit is described by the limiting evolution operator, which is the solution of a quantum stochastic equation driven by a quantum Brownian motion. In joint work of A.Pechen and I.Volovich it is shown that a new object describes the higher order corrections. That is quantum multipole noise operators, an operator-valued distribution which acts in a pseudo–Hilbert space and extends essentially the white noise analysis.
A.Pechen also works on study of the role of spatial dependence of correlators in Bell's inequality. I.Volovich proposed that the spatial dependence of wave functions could play crucial role in experiments aimed to test Bell's inequalities. This dependence, although of a very general nature, requires careful analysis for all concrete experiments. In [4] A.Pechen jointly with A.Baranov and I.Volovich considered the spatial dependence in Franson-type experiments. It is shown that in this case it is impossible to violate experimentally Bell’s inequalities for large distance between detectors.
2. A.N. Pechen, I.V. Volovich, Quantum Multipole Noise and Generalized Quantum Stochastic Equations, Infinite Dimensional Analysis, Quantum Probability and Related Topics 5 N4 (2002) 441-464; http://arxiv.org/abstracts/math-ph/0202046 abstract in IDAQP
3. L. Accardi, A.N. Pechen, I.V. Volovich, A Stochastic Golden Rule and Quantum Langevin Equation for the Low Density Limit, Infinite Dimensional Analysis, Quantum Probability and Related Topics 6 N3 (2003) 431-453; http://arxiv.org/abstracts/math-ph/0206032 abstract in IDAQP
4. A. Baranov, A.N. Pechen, I.V. Volovich, Space Dependence of Entangled States and Franson-type EPR Experiments, http://arxiv.org/abstracts/quant-ph/0203152
5. A.N. Pechen, Quantum Stochastic Equation for a Test Particle Interacting with a Dilute Bose Gas, J. Math. Phys. 45 (2004) 400-417, http://arxiv.org/abstracts/math-ph/0303020 JMP
2. Invited talk White noise approach to the low density limit, given by A.Pechen at the international conference "Classical and Quantum Levy Processes: Theory and Applications", 27 Sept.–3 Oct. 2003, Levico Terme (Trento), Italy.
2. "Quantum multipole noise and higher order corrections to the weak coupling limit", Steklov Mathematical Institute of Russian Academy of Sciences, Sept. 2002
3. "Quantum multipole noise and dynamics of quantum open systems", Lebedev Physical Institute of Russian Academy of Sciences, Oct. 2002
4. "Quantum multipole noise in pseudo-Hilbert spaces", Moscow State University, Faculty of Mathematics, Oct. 2002
5. "A stochastic golden rule for the low density limit", Bari University (Italy), Jan. 2003
6. "The white noise approach to the low density limit", Bari University (Italy), Jan. 2001
7. "Quantum multipole noise and dynamics of quantum open systems", Palermo Univ. (Italy), Jan. 2003
2. Quantum Interacting Particle Systems, 23–29 September 2000, Levico Terme (Trento), Italy.
3. School on Quantum Markov Chains and their Applications in Physics and Quantum Information, 14–20 December 2001, Levico Terme (Trento), Italy.
4. Open Quantum Systems, 15 June – 05 July 2003, Institut J.Fourier, Grenoble, France.
5. Les Houches summer school "Quantum Entanglement and Information Processing", 01–25 July 2003, Les Houches, France.