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arXiv:quant-ph/0602116 · Physical Review A

Open-quantum-system formulation of particle decay

Reinhold A. Bertlmann, Walter Grimus, Beatrix C. Hiesmayr

Published May 1, 2006• 95 citations
arXivHTMLPDF

Abstract

We consider an open quantum system which contains unstable states. The time evolution of the system can be described by an effective non-Hermitian Hamiltonian ${H}_{\mathrm{eff}}$, in accord with the Wigner-Weisskopf approximation, and an additional term of the Lindblad form, the so-called dissipator. We show that, after enlarging the original Hilbert space by states which represent the decay products of the unstable states, the non-Hermitian part of ${H}_{\mathrm{eff}}$---the ``particle decay''---can be incorporated into the dissipator of the enlarged space via a specific Lindblad operator. Thus the formulation of the time evolution on the enlarged space has a Hermitian Hamiltonian and is probability conserving. The equivalence of the formulation with the original one demonstrates that the time evolution which is governed by a non-Hermitian Hamiltonian and a dissipator of the Lindblad form is nevertheless completely positive, just as systems with Hermitian Hamiltonians.

DOI
10.1103/physreva.73.054101

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We consider an open quantum system which contains unstable states. The time evolution of the system can be described by an effective non-Hermitian Hamiltonian Heff{H}_{\mathrm{eff}}Heff​, in accord with the Wigner-Weisskopf approximation, and an additional term of the Lindblad form, the so-called dissipator. We show that, after enlarging the original Hilbert space by states which represent the decay products of the unstable states, the non-Hermitian part of Heff{H}_{\mathrm{eff}}Heff​---the ``particle decay''---can be incorporated into the dissipator of the enlarged space via a specific Lindblad operator. Thus the formulation of the time evolution on the enlarged space has a Hermitian Hamiltonian and is probability conserving. The equivalence of the formulation with the original one demonstrates that the time evolution which is governed by a non-Hermitian Hamiltonian and a dissipator of the Lindblad form is nevertheless completely positive, just as systems with Hermitian Hamiltonians.
Utilizing the GKLS master equation we show that the decay property of a particle can be straightforwardly incorporated. In standard particle physics the decay is often described by an efficient non-hermitian Hamiltonian, in accord with the seminal Wigner-Weisskopf approximation. We show that by enlarging the Hilbert space and defining specific GKLS operators we have attained a formalism with a hermitian Hamiltonian and probability conserving states. This proves that the Wigner-Weisskopf approximation is Markovian and completely positive. In addition, this formalism allows a straightforward generalization to many-particle decays. Last, but not least, some impacts of the GKLS master equation onto systems at high energies are reported, such as for neutral meson, neutrino and hyperon systems.
The Lindblad equation is commonly used to model dissipation and decoherence in open quantum systems. Yet many of the properties of Lindblad dynamics are still little understood. This thesis explores various aspects of the Lindblad equation and its connections with non-Hermitian quantum theories. The evolution speed of Lindblad dynamics is investigated and an explicit expression for the speed is derived. The radial component, connected to the rate at which the purity of the state changes, is shown to be determined by the modified skew information of the Lindblad operators. The semiclassical time evolution of Gaussian states under Lindblad dynamics is then considered and a new type of phase-space dynamics is derived for the centre of a Gaussian Wigner function. By viewing the phase-space Lindblad equation as a Schroedinger equation with a non-Hermitian Hamiltonian, a further set of semiclassical equations are derived. These can describe the interference terms in Wigner functions. Further, a PT-symmetric two-level system driven through two consecutive exceptional points at finite speed is considered and an analytical expression for the probability of a non-adiabatic transition is derived. It is shown that the transition through the exceptional points can be experimentally addressed in a PT-symmetric lattice using Bloch oscillations. Finally, a quantum implementation of this PT-symmetric system using ultracold atoms in an optical lattice with single-particle losses is investigated.
The Wigner–Weyl transform and phase space formulation of a density matrix approach are applied to a non-Hermitian model which is quadratic in positions and momenta. We show that in the presence of a quantum environment or reservoir, mean lifetime and decay constants of quantum systems do not necessarily take arbitrary values, but could become functions of energy eigenvalues and have a discrete spectrum. It is demonstrated also that a constraint upon mean lifetime and energy appears, which is used to derive the resonance conditions at which long-lived states occur. The latter indicate that quantum dissipative effects do not always lead to decay but, under certain conditions, can support stability of a system.
We present the probability-preserving description of the decaying particle within the framework of quantum mechanics of open systems, taking into account the superselection rule prohibiting the superposition of the particle and vacuum. In our approach the evolution of the system is given by a family of completely positive trace-preserving maps forming a one-parameter dynamical semigroup. We give the Kraus representation for the general evolution of such systems, which allows one to write the evolution for systems with two or more particles. Moreover, we show that the decay of the particle can be regarded as a Markov process by finding explicitly the master equation in the Lindblad form. We also show that there are remarkable restrictions on the possible strength of decoherence.
A semiclassical approximation for an evolving density operator, driven by a 'closed' Hamiltonian operator and 'open' Markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of Hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra 'open' term is added to the double Hamiltonian by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. The particular case of generic Hamiltonian operators, but linear dissipative Lindblad operators, is studied in more detail. A Liouville-type equivariance still holds for the corresponding classical evolution in double phase space, but the centre subspace, which supports the Wigner function, is compressed, along with expansion of its conjugate subspace, which supports the chord function. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by a propagator in a mixed representation, so that a further 'small-chord' approximation leads to a simple generalization of the quadratic theory for evolving Wigner functions.
Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Królikowski–Rzewuski (KR) equation. KR equation results from the Schrödinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for n-particle, two-particle and for one-particle subspaces. In a general case these effective Hamiltonians depend on time t. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one-particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when t → ∞ whereas the imaginary part of this effective Hamiltonian tends to zero as t → ∞.