Astronomical quantum opticsHow rapid a variability can be detected?
Increasing the temporal resolution to microseconds, one should encounter
successively more rapid events, on timescales such as those theoretically
expected for magnetic instabilities in accretion systems, or for non-radial
oscillations is neutron stars. However, there do not yet appear to exist
any predicted macroscopic processes in astrophysical sources that
would be observable in the nanosecond domain. Such resolutions, however,
lead into the microscopic realm of quantum optics, and the quantum-mechanical
statistics of photon counts. To understand what phenomena exist at such
timescales, and what information they carry, we have to examine the physical
properties of light.
Nanoseconds and quantum optics
Classical physics merges all radiation of a certain wavelength into the
quantity "intensity". When instead treating radiation as a three-dimensional
photon gas, other effects also become significant, e.g. higher-order coherence
and the temporal correlation between photons. The best-known non-classical
property of light is the bunching of photons, first measured by Hanbury
Brown and Twiss in those experiments that led to the astronomical intensity
interferometer (Hanbury Brown 1974).
Although its function was perhaps not fully appreciated at the time it
was developed, it is now realized that such an instrument works correctly
only for sources whose emitted light is in a maximum entropy, thermodynamic
equilibrium state. Different physical processes in the generation of light
may cause quantum-statistical differences (different degrees of photon
bunching in time) between light with otherwise identical spectrum, polarization,
intensity, etc., and studies of such non-classical properties of light
are actively pursued in laboratory optics. Although its observability in
celestial sources is not yet established, quantum optics could ultimately
offer a fundamentally novel information channel also for astronomy.
These quantum correlation effects are fully developed over timescales equal
to the inverse bandwidth of light. For example, the use of a 1 nm bandpass
optical filter gives a frequency bandwidth of 1012
Hz, and the effects are then fully developed on timescales of 10-12
seconds. Instrumentation with continuous data processing facilities with
such resolutions is not yet available, but it is possible to detect the
effects, albeit with a decreased amplitude, also at the more manageable
nanosecond timescales.
Beyond imaging, photometry and spectroscopy
Conventional optical instruments, like photometers, spectrometers, polarimeters
or interferometers, are capable of measuring properties of light such as
its intensity, spectrum, polarization or coherence. However, such properties
are generally insufficient to determine the physical conditions under which
light has been created. Thus it is not possible, not even in principle,
to distinguish between e.g. spontaneously emitted light reaching the observer
directly from the source; similar light that has undergone scattering on
its way to the observer; or light predominantly created through stimulated
emission, provided these types of light have the same intensity, polarization
and coherence as function of wavelength. The deduction of the processes
of light emission is therefore made indirectly via theoretical models.
Yet, such types of light may have quantum-statistical differences regarding
collective multi-photon properties in the photon gas. Such properties are
known for light from laboratory sources and, might ultimately become experimentally
measurable also for astronomical sources.
To understand the "parameter domains" in "knowledge space" that are accessed
by e.g. photometers or spectrometers, we need to understand their working
principles on a very fundamental level, i.e. not superficial specifications
such as field-of-view or spectral resolution, but rather their workings
concerning the fundamental physical observables measured.
One-photon experiments
We describe light as an electromagnetic wave of one linear polarization
component whose electric field E contains terms of the type exp(-iwt) for
angular frequencies w. All classical optical instruments measure properties
of light that can be deduced from the first-order correlation function
of light, G(1), for two coordinates in space r and
time t (Glauber, 1970). The different classes are collected in a Figure,
where < > denotes time average, and * complex conjugate. For example,
a bolometer measures <E*(0,0) E(0,0)>, yielding the classical field
intensity irrespective of the spectrum or geometry of the source. For the
case r1 = r2
but t1 =/ t2,
G(1) becomes the autocorrelation function with respect
to time, <E*(0,0) E(0,t)>, whose Fourier transform yields the power
density as function of electromagnetic frequency. That is the spectrum
of light which is measured by spectrometers. The function is explicitly
sampled by Fourier transfer spectrometers while e.g. gratings "perform"
the transform to the spectrum through diffractive interference. For the
case r1 =/ r2
but t1 = t2
we have the spatial autocorrelation function <E*(0,0) E(r,0)>, which
is measured by interferometers and yields the angular distribution of the
source power density. The need for accurate timekeeping at both sites
r1 and r2
originates from the requirement t1 = t2.
In the absence of absolute flux calibrations, G(1)
is usually normalized to the first-order coherence g(1).
Figure: Fundamental quantities measured in one-photon experiments. All such measurements can be ascribed to quantities of type E*E, corresponding to intensity I, which in the quantum limit means observations of individual photons or of statistical one-photon properties. To this category belong all direct and interferometric imagers, spectrometers, and photometers, i.e. all ordinary instruments used in astronomy. Time average is denoted by < > while * marks complex conjugate.
Two- and multi-photon properties of light
Thus, classical measurements do not distinguish light sources with identical
G(1). All such measurements can be ascribed to quantities
of type E*E, corresponding to intensity I, which in the quantum limit means
observations of individual photons or of statistical one-photon properties.
Thus possible multi-photon phenomena in the photon stream reaching the
observer are not identified, not even in principle.
The description of collective multi-photon phenomena in a photon gas, in
general requires a quantum-mechanical treatment since photons have integer
spin (S = 1), and therefore constitute a boson fluid with properties different
from a fluid of classical distinguishable particles. The first treatment
of the quantum theory of coherence in a photon gas was by Glauber (1963a,
1963b), although some properties were inferred earlier from classical treatments,
notably the bunching of photons in chaotic (thermal) light, first observed
by Hanbury Brown and Twiss. Glauber showed that an arbitrary state of light
can be specified with a series of coherence functions essentially describing
one-, two-, three-, etc. -photon-correlations. A simplified expression
for the second-order correlation function is given in the next figure.
It describes the correlation of intensity between two coordinates in space
and time.
Since a detection of a photon (measurement of I) enters twice, G(2)
describes two-photon properties of light. G(2) is
often normalized to the second-order coherence of light, g(2).
Although its strict definition involves quantum-mechanical operators, a
simplified expression can be given in terms of intensities: g(2)
= <I(r1,t1)
I(r2,t2)>/<I(r1,t1)>
<I(r2,t2)>.
If the distribution of photons is chaotic, i.e. the photon gas is in a
maximum entropy state, the second-order coherence g(2)
can be deduced as g(2) = [g(1)]2
+ 1 (e.g. Loudon, 1983). This property can be used to determine |g(1)|
from measurements of g(2). In the intensity interferometer
this is measured for r1 =/ r2
but t1 = t2:
<I(0,0) I(r,0)>, thus deducing angular sizes of stars, reminiscent of
a classical interferometer. For r1 = r2
but t1 =/ t2
we instead have an intensity-correlation spectrometer, which measures <I
(0,0) I (0,t)>, determining the spectral width of e.g. scattered laser
light.
Figure: Fundamental quantities measured in two-photon experiments. All such measurements can be ascribed to quantities of type I*I, i.e. intensity multiplied by itself, which in the quantum limit means observations of pairs of photons or of statistical two-photon properties. The intensity interferometer was the first astronomical instrument in this category.
In thermodynamic equilibrium, the [chaotic] distribution of photons corresponds
to the value g(2) = 2 for first-order coherent [g(1)
= 1] light. Such photons follow a Bose-Einstein distribution, analogous
to a Maxwellian one for classical particles. However, away from equilibrium,
photons may deviate from Bose-Einstein distributions (just as classical
particles can be non-Maxwellian).
For example, light created by stimulated emission in the limiting case
of a stable wave without any intensity fluctuations has g(2)
= 1, corresponding to analogous states in other boson fluids, e.g. superfluidity
in liquid helium. Chaotic light scattered against a Gaussian frequency-redistributing
medium has g(2) = 4. In the laboratory, one can observe
how the physical nature of the photon gas gradually changes from chaotic
(g(2) = 2) to ordered [g(2)=
1] when a laser is "turned on" and the emission gradually changes from
spontaneous to stimulated. Measuring g(2) and knowing
the laser parameters involved, it is possible to deduce the atomic energy
level populations, which is an example of an astrophysically important
parameter (non-LTE departure coefficient) which cannot be directly observed
with classical measurements of one-photon properties. Just as it is not
possible to determine whether one individual helium atom is superfluid
or not, it is not possible to determine whether one individual photon is
due to spontaneous or stimulated emission: both cases require studies of
statistical properties of the respective boson fluid.
For a source with g(2) =/ 2, neither an intensity
interferometer nor an intensity-correlation spectrometer will yield correct
results. E.g. a monochromatic point source emitting a monochromatic stable
wave whose g(2) = 1 everywhere, would appear to be
spatially resolved by an intensity interferometer at any spatial baseline
and spectrally resolved by an intensity-correlation spectrometer at any
temporal baseline and hence give the false impression of an arbitrarily
large source emitting white light. This example merely indicates that additional
measurements are required to fully extract the information content of light.
Many different quantum states of optical fields exist, not only those mentioned
above (which can be given classical analogs) but also e.g. photon antibunching
which with g(2) = 0 is a purely quantum-mechanical
state. This implies that neighboring photons "avoid" one another in space
and time. While such properties are normal for fermions (e.g. electrons),
which obey the Pauli exclusion principle, ensembles of bosons (e.g. photons)
show such properties only in special situations. An antibunching tendency
implies that the detection of a photon at a given time is followed by a
decreased probability to detect another immediately afterward.
Experimentally, this is seen through sub-Poissonian statistics, i.e. narrower
distributions of recorded photon counts than would be expected in a "random"
situation. For an introduction to the theory of such quantum optical phenomena,
see e.g. Loudon (1980; 1983), Meystre & Sargent (1990), or Mandel &
Wolf (1995). Experimental procedures for studying photon statistics are
in Saleh (1978).
........
Figure: Properties of light, measurable in multi-photon experiments. Such measurements can be ascribed to quantities of type In, i.e. intensity multiplied n times by itself, which in the quantum limit means observations of groups of n photons or of statistical n-photon properties. The information contained in such higher-order photon correlations may include thermodynamic information of how the light was created or how it has been redistributed (scattered) since its creation. Although such problems are studied in theoretical astrophysics, they are not yet accessible to direct observational tests.
Astrophysical applications of quantum optics
One can envision applications of nanosecond resolution optical observations
to give insight in the physical processes of radiative deexcitation of
astrophysical plasmas, fields of study which presently are the exclusive
realm of theoreticians.
The physics of emission processes
What is the quantum nature of the light emitted from a volume with departures
from thermodynamic equilibrium of the atomic energy level populations?
Will a spontaneously emitted photon stimulate others, so that the path
where the photon train has passed becomes temporarily deexcited and remains
so for perhaps a microsecond until collisions and other effects have restored
the balance? Does then light in a spectral line perhaps consist of short
photon showers with one spontaneously emitted photon leading a trail of
others emitted by stimulated emission? One could search for such amplified
spontaneous emission ("laser action") in atomic emission lines from extended
stellar envelopes or solar active regions. Such [partial] laser effects
have been predicted for celestial sources, for example in mass-losing high-temperature
stars, where the rapidly recombining plasma in the stellar envelope can
act as an amplifying medium (e.g. Lavrinovich & Letokhov 1974; Varshni
& Lam 1976; Varshni & Nasser 1986). Analogous effects could exist
in accretion disks (Fang 1981). In the infrared, there are several cases
where laser action is predicted for specific atomic lines (e.g. Ferland
1993; Greenhouse et al. 1993; Peng & Pradhan 1994). Somewhat analogous
situations (corresponding to a laser below threshold) have been studied
in the laboratory. The radiation structure from "free" clouds (i.e. without
any laser resonance cavity) of excited gas with population inversion can
be analyzed. One natural mode of radiative deexcitation indeed appears
to be the emission of "photon showers" triggered by one spontaneously emitted
photon which is stimulating others along its flight vector out from the
volume.
In principle, quantum statistics of photons should permit to determine
whether the Doppler broadening of a spectral line has been caused by motions
of those atoms that emitted the photons or by those intervening atoms that
have scattered the already existing photons. Thus, for such scattered light,
its degree of partial redistribution in frequency might be directly measurable.
Although the existence in principle of such effects may be clear,
their practical observability is not yet known. At first sight, it might
even appear that light from a star should be nearly chaotic because of
the very large number of independent radiation sources in the stellar atmosphere,
which would randomize the photon statistics. However, since the time constants
involved in the maintenance of atomic energy level overpopulations (e.g.
by collisions) may be longer than those of their depopulation by stimulated
emission (speed of light), there may exist, in a given solid angle, only
a limited number of radiation modes reaching the observer in a given time
interval (each microsecond, say) and the resulting photon statistics might
well be non-chaotic. Proposed mechanisms for pulsar emission include stimulated
synchrotron and curvature radiation ("free-electron laser") with suggested
timescales of nanoseconds, over which the quantum statistics of light would
be non-chaotic. In general, photon statistics for the radiation from any
kind of energetic source could convey something about the processes where
the radiation was liberated. For example, the presence of photon "bubbles"
in photohydrodynamic turbulence in very hot stars has been suggested. The
bubbles would be filled with light and the photon-gas pressure inside would
balance the surrounding gas pressure but due to buoyancy the bubbles would
rise through the stellar surface, giving off photon bursts (Prendergast
& Spiegel 1973; Spiegel 1976). Obviously, the list of potential astrophysical
targets could be made longer.
Interpreting observed photon statistics
The theoretical problem of light scattering in a [macroscopic] turbulent
medium is reasonably well studied. In particular, the equations of transfer
for I2 and higher-order moments of intensity
have been formulated and solved (e.g. Uscinski 1977). A result that is
familiar to many people implies that stars twinkle more with [moderately]
increasing atmospheric turbulence. The value of I, i.e. the total number
of photons transmitted may well be constant, but I2
increases with greater fluctuations in the medium. The quantum mechanical
problem of scattering of light against atoms is somewhat related, except
that the timescales involved are now those of the coherence time of light.
However, theoretical treatments of astrophysical radiative transfer have
so far concentrated on the first-order quantities of intensity, spectrum
and polarization, and not on the transfer of I2
and higher-order terms. There are notable exceptions, however, like the
analytical solution of the higher-order moment equation relevant for radio
scintillations in the interstellar medium (Lee and Jokipii 1975; Lerche
1979a; 1979b) and attempts to formulate the quantum mechanical description
of the transfer of radiation, including non-Markovian effects (i.e. such
referring to more than one photon at a time) in a photon gas (Machacek
1978; 1979), the transfer equation for the density matrix of phase space
cell occupation number states (Sapar 1978; Ojaste & Sapar 1979), or
the need to introduce concepts from non-linear optics (Wu 1993). Still,
there do not yet appear to exist any theoretical predictions for specific
astronomical sources of any spectral line profiles of higher-order than
one (i.e. the ordinary intensity versus wavelength). Until the availability
of such theoretical predictions (of e.g. the second-order coherence versus
wavelength), this work will continue to have an exploratory character.
Updated JD 2,451,700
