Introduction to decoherence

Fritz Haake

Fachbereich Physik,
Universität Essen, Essen, Germany

Quantum superpositions tend to decohere to mixtures, due to dissipative environmental influence. In particular, two superposed wave packets loose their relative coherence the faster the larger is their distance $d$. For sufficiently large $d$, the relative phase of packets is lost before any deformation of the shapes of the individual packets and any change of their distance become noticeable.

I shall illustrate the phenomenon by discussing recent efforts to take decoherence under experimental control (diffraction of Fullerenes in Vienna; superpositions of coherent states of microwave resonator modes in Paris; superpositions of wave packets of $Be$ ions in Paul traps in Boulder; superpositions of states with counterpropagating supercurrents in Delft and Stony Brook).

All of these experiments observe superpositions of packets whose distance $d$ is larger than the individual width $\lambda$, but the ratio $d/\lambda$ achieved is still so moderate that the environment imposed decoherence time $\tau_{{\rm dec}}$ is, while shorter than the time scale $\tau_{{\rm diss}}$ of dissipative changes of $d$, still longer than typical oscillation periods $\tau_{{\rm sys}}$ of the isolated system, i.e. $\tau_{{\rm sys}} < \tau_{{\rm dec}} < \tau_{{\rm diss}}$. The appropriate theoretical treatment is thus based on Fermi's Golden Rule or, equivalently, Markovian master equations.

I illustrate golden rule type decoherence for the damped harmonic oscillator, using the simple master equation familiar from quantum optics. The important prediction is $\tau_{{\rm dec}}/\tau_{{\rm diss}} = (\lambda/d)^2.$ The underlying perturbative treatment of the system-environment interaction requires the self-consistency condition $\tau_{{\rm sys}} \ll \tau_{{\rm dec}}$.

Emergence of classical behavior in the macroworld: meso- and macroscopic superpositions

Fritz Haake

Fachbereich Physik,
Universität Essen, Essen, Germany

The golden-rule prediction breaks down once the distance $d$ between the superposed wave packets is so large, compared to the quantum scale of reference $\lambda$, that $\tau_{{\rm dec}} \mathrel{\lower0.55ex\hbox{$\mathchar''3218$}\mkern-14mu \raise0.55ex\hbox{$<$}}\tau_{{\rm sys}}$. The golden rule can therefore not be invoked to explain the notorious absence of quantum interference effects from the macroworld. In the limit $\tau_{{\rm dec}} < \tau_{{\rm sys}}$ decoherence obviously no longer is a weak-damping phenomenon. A simple solution of the system $\oplus$ environment Schrödinger equation becomes possible when $\tau_{{\rm dec}} \ll \tau_{{\rm sys}}$, the limit of relevance for superpositions of macroscopically distinct wave packets. The simplicity of that limit rests on the fact that in the full Hamiltonian $H = H_{{\rm sys}} + H_{{\rm bath}} + H_{{\rm int}}$ the free-system part $H_{{\rm sys}}$ becomes an effectively small perturbation. The decoherence time scale is then found to obey the power law $\tau_{{\rm dec}} \propto \hbar^{\mu}/d^{\nu}$ with positive exponents $\mu, \nu$. That law is a universal one, independent of the character of the system and the environment. It is only based on the interaction Hamiltonian $H_{{\rm int}}$ additively involving a large number of degrees of freedom such that the central limit theorem holds for the reservoir means met with.

After treating the universal asymptotics of the limit $\tau_{{\rm dec}} \ll \tau_{{\rm sys}}$ I shall briefly discuss the crossover from that interaction dominated regime to golden-rule type decoherence. The crossover is system specific; I shall rely on an exactly solvable model system, an harmonic oscillator coupled to a reservoir which itself consists of harmonic oscillators.

Quantum measurement

Fritz Haake

Fachbereich Physik,
Universität Essen, Essen, Germany

Textbook wisdom has it that superpositions $\vert\psi\rangle = \sum\limits_i c_i\vert\psi_i\rangle$ of eigenstates $\psi_i$ of some observable of a microscopic object ``collapse'', under measurement of that observable, to a mixture, $\vert\varphi\vert \rangle \langle \varphi\vert \stackrel{\mbox{\rm {\tiny colla... ...arrow} \sum\limits_i\vert c_i\vert^2 \vert\varphi_i\rangle\langle\varphi_i\vert$, with probabilities $\vert c_i\vert^2$ as in the original pure state but all coherences $c^*_i\ c_j$ for $i \neq j$ gone. Such collapse can be understood as due to unitary time evolution of a tripartite system comprising, besides the micro-object, a macroscopic pointer (idealizable as having a single degree of freedom), and a many-freedom environment. The Hamiltonian must allow micro-object and pointer to become entangled as

$$\begin{array}{lll} \varphi^{\mbox{\rm\tiny {obj}}}(0) \psi^{... ... \psi^{\mbox{\rm\tiny {point}}}_i \hspace*{1.2cm}, \end{array}$$

with the various pointer states $\psi^{\mbox{\rm\tiny {point}}}_i$ corresponding to macroscopically distinct pointer displacements. Concomitantly, the many-freedom environment will decohere the superposition to the mixture $\sum_i\vert c_i\vert^2 \vert\varphi_i\ \rangle\langle \varphi_i\vert \otimes \v... ...\mbox{\rm\tiny {point}}}_i \rangle\langle \psi^{\mbox{\rm\tiny {point}}}_i\vert$; inasmuch as the pointer states $\vert\psi^{\mbox{\rm\tiny {point}}}_i\rangle$ are macroscopically distinct, the decay of the coherences $c^*_i\ c_j$ will occur with $\tau_{\mbox{\rm\tiny {dec}}} \ll \tau_{\mbox{\rm\tiny {syst}}}, \tau_{\mbox{\rm\tiny {diss}}}$, i.e. might appear as practically instantaneous. By disregarding all information about (i.e. tracing over) the pointer, one has the textbook-wisdom mixture for the micro-object.

I shall present exactly solvable models both for both processes involved, entanglement and decoherence.