We prove that if 0<\a<1 and f is in the H\"older class \L_\a(\R), then
for arbitrary self-adjoint operators A and B with bounded A−B, the
operator f(A)−f(B) is bounded and \|f(A)-f(B)\|\le\const\|A-B\|^\a. We
prove a similar result for functions f of the Zygmund class \L_1(\R):
\|f(A+K)-2f(A)+f(A-K)\|\le\const\|K\|, where A and K are self-adjoint
operators. Similar results also hold for all H\"older-Zygmund classes
\L_\a(\R), \a>0. We also study properties of the operators f(A)−f(B) for
f\in\L_\a(\R) and self-adjoint operators A and B such that A−B belongs
to the Schatten--von Neumann class \bS_p. We consider the same problem for
higher order differences. Similar results also hold for unitary operators and
for contractions.Comment: 6 page
A critical review of experimental studies of the so-called 'slow light'
arising due to anomalously high steepness of the refractive index dispersion
under conditions of electromagnetically induced transparency or coherent
population oscillations is presented. It is shown that a considerable amount of
experimental evidence for observation of the 'slow light' is not related to the
low group velocity of light and can be easily interpreted in terms of a
standard model of interaction of light with a saturable absorber.Comment: 17 pages, 8 figures, to be published in June issue of Phisics:
Uspekhi under the title "Notes on Slow Light
We develop a theory of spin noise spectroscopy of itinerant, noninteracting,
spin-carrying fermions in different regimes of temperature and disorder. We use
kinetic equations for the density matrix in spin variables. We find a general
result with a clear physical interpretation, and discuss its dependence on
temperature, the size of the system, and applied magnetic field. We consider
two classes of experimental probes: 1. electron-spin-resonance (ESR)-type
measurements, in which the probe response to a uniform magnetization increases
linearly with the volume sampled, and 2. optical Kerr/Faraday rotation-type
measurements, in which the probe response to a uniform magnetization increases
linearly with the length of the light propagation in the sample, but is
independent of the cross section of the light beam. Our theory provides a
framework for interpreting recent experiments on atomic gases and conduction
electrons in semiconductors and provides a baseline for identifying the effects
of interactions on spin noise spectroscopy
We consider the class of integral operators Q_\f on L2(R+) of the form
(Q_\f f)(x)=\int_0^\be\f (\max\{x,y\})f(y)dy. We discuss necessary and
sufficient conditions on ϕ to insure that Qϕ is bounded, compact,
or in the Schatten-von Neumann class \bS_p, 1<p<∞. We also give
necessary and sufficient conditions for Qϕ to be a finite rank
operator. However, there is a kind of cut-off at p=1, and for membership in
\bS_{p}, 0<p≤1, the situation is more complicated. Although we give
various necessary conditions and sufficient conditions relating to
Q_{\phi}\in\bS_{p} in that range, we do not have necessary and sufficient
conditions. In the most important case p=1, we have a necessary condition and
a sufficient condition, using L1 and L2 modulus of continuity,
respectively, with a rather small gap in between. A second cut-off occurs at
p=1/2: if \f is sufficiently smooth and decays reasonably fast, then \qf
belongs to the weak Schatten-von Neumann class \wS{1/2}, but never to
\bS_{1/2} unless \f=0.
We also obtain results for related families of operators acting on L2(R)
and ℓ2(Z).
We further study operations acting on bounded linear operators on
L2(R+) related to the class of operators Q_\f. In particular we
study Schur multipliers given by functions of the form ϕ(max{x,y}) and
we study properties of the averaging projection (Hilbert-Schmidt projection)
onto the operators of the form Q_\f.Comment: 87 page
This is a continuation of our paper \cite{AP2}. We prove that for functions
f in the H\"older class \L_\a(\R) and 1
, the operator f(A)−f(B)
belongs to \bS_{p/\a}, whenever A and B are self-adjoint operators with
A-B\in\bS_p. We also obtain sharp estimates for the Schatten--von Neumann
norms \big\|f(A)-f(B)\big\|_{\bS_{p/\a}} in terms of \|A-B\|_{\bS_p} and
establish similar results for other operator ideals. We also estimate
Schatten--von Neumann norms of higher order differences
\sum\limits_{j=0}^m(-1)^{m-j}(m\j)f\big(A+jK\big). We prove that analogous
results hold for functions on the unit circle and unitary operators and for
analytic functions in the unit disk and contractions. Then we find necessary
conditions on f for f(A)−f(B) to belong to \bS_q under the assumption
that A-B\in\bS_p. We also obtain Schatten--von Neumann estimates for
quasicommutators f(A)Q−Qf(B), and introduce a spectral shift function and
find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).Comment: 49 page
We describe an experimental study of spin-projection noise in a high
sensitivity alkali-metal magnetometer. We demonstrate a four-fold improvement
in the measurement bandwidth of the magnetometer using continuous quantum
non-demolition (QND) measurements. Operating in the scalar mode with a
measurement volume of 2 cm^3 we achieve magnetic field sensitivity of 22
fT/Hz^(1/2) and a bandwidth of 1.9 kHz with a spin polarization of only 1%. Our
experimental arrangement is naturally back-action evading and can be used to
realize sub-fT sensitivity with a highly polarized spin-squeezed atomic vapor.Comment: 4 page