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arXiv:1301.5661 · Journal of Physics A Mathematical and Theoretical

Initial states of qubit–environment models leading to conserved quantities

Bartłomiej Gardas, Jerzy Dajka

Published May 21, 2013• 3 citations
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Abstract

It is possible to prepare a composite qubit–environment system so that its time evolution will guarantee the conservation of a preselected qubit's observable. In general, this observable is not associated with a symmetry. The latter may not even be present in the subsystem. The initial states which lead to such a quantity conserved dynamics form a subspace of the qubit–environment space of states. General construction of this subspace is presented and illustrated by two examples. The first one is the exactly solvable Jaynes–Cummings model and the second is the multi-photon Rabi model.

DOI
10.1088/1751-8113/46/23/235301

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It is possible to prepare a composite qubit–environment system so that its time evolution will guarantee the conservation of a preselected qubit's observable. In general, this observable is not associated with a symmetry. The latter may not even be present in the subsystem. The initial states which lead to such a quantity conserved dynamics form a subspace of the qubit–environment space of states. General construction of this subspace is presented and illustrated by two examples. The first one is the exactly solvable Jaynes–Cummings model and the second is the multi-photon Rabi model.
Recently, Gardas and Dajka (2013 J. Phys. A: Math. Theor. 46 235301) proposed a theoretical scheme on how to prepare a proper initial state of a composite qubit–boson system which could assure no energy exchange between the two subsystems. For illustration, they applied the proposed scheme to both the Jaynes–Cummings model and the multi-photon Rabi model. However, it has already been rigorously proven that the k -photon Rabi model for k > 2 does not have eigenstates in the Hilbert space spanned by the eigenstates of the corresponding k -photon Jaynes–Cummings model and that the two-photon Rabi model has a discrete eigenenergy spectrum with normalizable eigenstates only if the coupling strength is smaller than a critical value. Hence, the proposed theoretical scheme is not applicable to the multi-photon Rabi model.
Dissipative properties of a composite system interacting with an environment are studied in order to obtain insight into relaxation and decoherence phenomena. As a typical system with internal degree of freedom, a Jaynes–Cummings (JC) model is considered where a spin (qubit) interacts with a single bosonic mode. When the whole system is initially in a state of at most single excitation, the model becomes solvable for excitation number conserving Hamiltonians, even if the JC system interacts with its environment composed of harmonic oscillators. Effects of initial correlations between the system and the environment are also taken into account. Basic equations for the density matrix are solved exactly to determine dynamics of the reduced systems. Mutually related problems of initial correlations, entanglement and non-Markovian property are studied explicitly by examining the time evolution of the reduced density matrices, the concurrence and the trace distance.
We consider a particularly simple exactly solvable model for a qubit coupled to sequentially nested environments. The purpose is to exemplify the coherence conserving effect of a central system, that has been reported as a result of increasing the coupling between near and far environment. The paradigmatic example is the Jaynes–Cummings Hamiltonian, which we introduce into a Kossakowski–Lindblad master equation using alternatively the lowering operator of the oscillator or its number operator as Lindblad operators. The harmonic oscillator is regarded as the near environment of the qubit, while effects of a far environment are accounted for by the two options for the dissipative part of the master equation. The exact solution allows us to cover the entire range of coupling strength from the perturbative regime to strong coupling analytically. The coherence conserving effect of the coupling to the far environment is confirmed throughout.
In this paper we return to the problem of reduced-state dynamics in the presence of an interacting environment. The question we investigate is how to appropriately model a particular system evolution given some knowledge of the system-environment interaction. When the experimenter takes into account certain known features of the interaction such as its invariant subspaces or its non-local content, it may not be possible to consistently model the system evolution over a certain time interval using a standard Stinespring dilation, which assumes the system and environment to be initially uncorrelated. Simple examples demonstrating how restrictions can emerge are presented below. When the system and environment are qubits, we completely characterize the set of unitaries that always generate reduced dynamics capable of being modeled using a consistent Stinespring dilation. Finally, we show how any initial correlations between the system and environment can be certified by observing the system transformation alone during certain joint evolutions.
For a bi-partite quantum system defined in a finite dimensional Hilbert space we investigate in what sense entanglement change and interactions imply each other. For this purpose we introduce an entanglement operator, which is then shown to represent a non-conserved property for any bi-partite system and any type of interaction. This general relation does not exclude the existence of special initial product states, for which the entanglement remains small over some period of time, despite interactions. For this case we derive an approximation to the full Schroedinger equation, which allows the treatment of the composite systems in terms of product states. The induced error is estimated. In this factorization-approximation one subsystem appears as an effective potential for the other. A pertinent example is the Jaynes-Cummings model, which then reduces to the semi-classical rotating wave approximation.
Abstract This article is a response to the continued assumption, cited even in reports and reviews of recent experimental breakthroughs and advances in theoretical methods, that the antiJaynes-Cummings (AJC)interaction is an intractable energy non-conserving component of the quantum Rabi model (QRM). We present three key features of QRM dynamics : (a) the AJC interaction component has a conserved excitation number operator and is exactly solvable (b) QRM dynamical space consists of a rotating frame (RF) dominated by an exactly solved Jaynes-Cummings (JC) interaction specied by a conserved JC excitation number operator which generates the U(1) symmetry of RF and a correlated counter-rotating frame (CRF) dominated by an exactly solved antiJaynes-Cummings (AJC) interaction specied by a conserved AJC excitation number operator which generates the U(1) symmetry of CRF (c) for QRM dynamical evolution in RF, the initial atom-eld state je0i is an eigenstate of the effective AJC Hamiltonian HAJC, while the effective JC Hamiltonian HJC drives this initial state je0i into a time evolving entangled state, and, in a corresponding process for QRM dynamical evolution in CRF, the initial atom-eld state jg0i is an eigenstate of the effective JC Hamiltonian, while the effective AJC Hamiltonian drives this initial state jg0i into a time evolving entangled state, thus addressing one of the long-standing challenges of theoretical and experimental QRM dynamics; consistent generalizations of the initial states je0i , jg0i to corresponding n 0 entangled eigenstates j+en i , j 􀀀g ni of the AJC in RF and JC in CRF, respectively, provides general dynamical evolution of QRM characterized by collapses and revivals in the time evolution of the atomic, eld mode, JC and AJC excitation numbers for large initial photon numbers ; the JC and AJC excitation numbers are conserved in the respective frames RF, CRF, but each evolves with time in the alternate frame.