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Open quantum systems
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Open quantum systems are usually modelled in a stochastic way
to account for the unpredictability of the measurement process.
(Note that a measurement is any non-negligible interaction with the
environment, whether or not it is observed by something deserving
the name 'detector' or 'observer').
In the simplest setting in which states can be assumed to
be pure and measurements occur at definite, a priori known times
and have a negligible duration, an open quantum system is a discrete
stochastic process with values psi(t) in the Hilbert space of state
vectors, normalized to norm 1. Between two consecutive measurements,
the system is assumed to be closed.
Thus between two consecutive measurements at times t' and t''>t',
the normalized state psi(t) evolves according to the Schroedinger
equation
i hbar psidot = H psi,
so that
psi(t''-0)= P psi(t'+0), P = exp (i/hbar (t'-t'')H). (1)
(In the interaction picture, H=0 and psi remains constant between
measurements.)
A measurement at time t is assumed to happen in infinitesimal time
and replaces psi(t-0) independent of other measurements with
probability p_s by
psi(t+0)= P_s psi(t-0)/p_s if p_s>0, (2)
where the P_s are linear operators determined by the experimental
arrangement, satisfying the relation
sum_s P_s^*P_s = 1, (3)
and
p_s=|P_s\psi(t-0)|^2 (4)
guarantees that psi(t+0) remains normalized. Clearly the p_s are
nonnegative and by (3), they sum up to 1 (since psi(t-0) is normalized).
(For measurements with more than countably many possible outcomes,
one must replace the probabilities by probability densities and the
sums by integrals.)
Thus this is a well-defined stochastic process.
A von-Neumann measurement of a self-adjoint linear operator A
corresponds to the special case where P_s is an orthogonal projector
to the eigenspace corresponding to the eigenvalue a_s of A
(respective to the set of eigenvalues corresponding to the s-th
interval in a partition of the continuous spectrum of A.)
If the measurement at different times has the same (or different)
nature, the P_s at these times are the same (or different).
It is possible to introduce 'empty measurements' at arbitrary
intermediate times with a trivial sum over a singleton s, where P_s=1.
For continuous measurements (where the open system cannot be considered
closed at all but a discrete number of times), one needs to take
a continuum limit of the above description. Depending how one takes
the limit, one gets quantum diffusion processes or quantum jump
processes. In this case, the density matrix for the associated
deterministic expectation evolves according to a Lindblad dynamics.
Realistic measurements (i.e. those taking into account the unavoidable
uncertainty) are not modelled by von-Neumann measurements, but rather
by positive operator valued measures, short POVMs. These are well
explained in
http://en.wikipedia.org/wiki/POVM
For more on real measurement processes (as opposed to the
von-Neumann measurement caricature treated in typical textbooks
of quantum mechanics), see, e.g.,
V.B. Braginsky and F.Ya. Khalili,
Quantum measurement,
Cambridge Univ. Press, Cambridge 1992