For concreteness, I present in Table 5.1 coherences of density matrices of the macroscopic atom-molecule superposition in a Bose condensate shown in Figures 4.3 and 4.4 of Section 4.3 at two instants of time: t1
(the end of phase I) and t2 (when the macroscopic superposition should have been formed). The parameter ξd describes the strength of the dom-inant interaction compared with the strength of decoherence. It appears that even for weak decoherence (scenario D), the system has lost about 20% of its coherence before instant t2. Thus, with probability ∼ 0.8 the measurement outcome is that the system is in a state that is a unitary
transformation of the initial pure state. With probability ∼ 0.2 the sys-tem has ended up to some other state, i.e., the macroscopic superposition in the system has collapsed with probability ∼0.2.
With this little example I will show in practice the use of the (invari-ant) coherence: by using it, it is possible to evaluate the probabilities to collapse of all systems that can be expressed in the density matrix for-malism. It is very handy tool in research of quantum computer, because the computational power of quantum computer is based on superposition states. For example, in the quantum computer with 400 qubits if each qubit loses ∼0.01units of coherence during the computation, the whole quantum computer has only ∼0.018 units of coherence left, i.e., it com-putes the desired calculation right with probability of ∼0.018. Thus, it is understandable why it is challenging to develop a quantum computer.
Chapter 6
About decoherence in a closed system
In this Chapter, I consider the decoherence of a closed system by mod-elling it with a simple spin 1/2 model. I will use density matrix formal-ism, since the decoherent histories approach is problematic (see Section 4.1), and in my opinion, the fundamental approach for describing quan-tum physical systems is a wave function (i.e., a state vector). Moreover, studying quantum physics by state vectors in the scale of the physical reality should be a valid approach and result in meaningful results. My purpose is to apply the definition of coherence and derive a phenomeno-logical concept of decoherence time τd from the dynamics of the realistic coherence Ξre(t). In Section 6.1 I present the spin model, and the sim-ulations and results are presented in Section 6.2. Finally, in Section 6.3 I consider the roles of decoherence, an environment and an observer in a closed system.
6.1 Spin 1/2 model
The Heisenberg spin model is a very suitable model for decoherence stud-ies, because it is simple enough to be solved, and yet complicated enough to simulate the properties of real quantum systems. Moreover, it is easily either simplified or expanded. A simplified version is good enough in re-searching the general properties of coherence in closed systems. Coupled spin systems are interesting from a quantum computational point of view, too.
My system, N interacting particles fixed in a space, has no environ-ment and, in that sense, the system forms a closed quantum universe. The particles are spin 1/2 particles, and the interaction between them is due
to their spin-z component (not necessary, but a simplification that makes the model analogous to the Ising model). When there is no coupling with the environment (i.e., no external magnetic field etc.), the spin states are degenerate and have the same energy, which is taken to be zero. Zurek [166], Omnès [19] and Schlosshauer [108, 109] have considered a similar, but simpler model in order to study the decay of off-diagonal elements (i.e., quantum correlations) of a reduced density matrix. They label one particle as the system and the others as the environment, and the parti-cles that form the environment do not interact with each other. I, on the contrary, am interested in studying the particle system as a whole.
The interaction Hamiltonian
describes the dynamics of the system. The interaction matrix G, where gij =gji, gives the interaction strength between particlesiandj. The in-teraction strength arises from the potentialV, but for formal calculations there is no need to know more about it, because particles are doomed to stay in one place. Fixing the positions of the particles is a justified assumption in decoherence studies since, in most cases, the decoherence time scale is the shortest time scale [104], at least shorter than the time scale of particle motion. In numerical simulations (Section 6.2), only the potentials of the type V =η/rǫ are considered.
Let me simplify the model a bit so that the initial state of simulations is a product state of single particle superposition states
|Ψ(0)i= gives the dynamics of the system, and with the given initial condition of Eq. (6.2) one gets the time dependence
|Ψ(t)i= exp The fate of the lth particle is solved by tracing over other particles,
ρl= Tr1,...,N6=lρ, (6.5)
where ρ = |Ψ(t)ihΨ(t)|. With Gibbsian coarse-graining [45, 173, 174] I make an effective theory of the particle system by tracing over the ”unin-teresting” particles (that form an effective environment to the particular particle in focus), as in the mean field approximation. The net effect of traced-out particles is described in a simpler form and with less degrees of freedom. Thus, It is interesting that the result of Eq. (6.6) is the same as in Ref. [166], if one drops off the indexl. The existence of indexlarises from the fact that in Ref. [166] only an interactionless environment has been considered, but my model counts all the interactions between particles.
Let us make our notation a bit lighter by denoting zl=alb∗l Thiszl (or its complex conjugate z∗l) describes the fate of the off-diagonal elements oflth particle. The eigenvalues of the reduced density matrixρl
are Now the single particle coherences are Ξl,single = Ξ(ρl), and the realistic coherence Ξre=N−1PNl=1Ξl,single of the closed system can be evaluated.
Inserting the maximum eigenvalues of Eq. (6.8) into the definition of coherence1 Eq. (5.5) results in
Ξre(t) = 1 whereN is the number of particles andM is the dimension of the reduced density matrix.
1See Chapter 5 for a detailed analysis of coherence.