3.3 How to calculate spectral densities of quantum noise in linear optical measurement?
In this section, we give a brief introduction to calculation of the power spectral densities of quantum
noise one usually encounters in linear optical measurement. In optomechanical sensors, as we have
discussed earlier, the outgoing light carries the information about the measured quantity (e.g., the
displacement due to GW tidal forces) in its phase and (sometimes) amplitude quadratures. The
general transformation from the input light characterized by a vector of quadrature amplitudes
to the readout quantity of a meter is linear and can be written in spectral form
as:
where
is the spectrum of the measured quantity,
are some complex-valued
functions of
that characterize how the light is transformed by the device. Quantum noise is
represented by the terms of the above expression not dependent on the measured quantity
, i.e.,
The measure of quantum noise is the power spectral density
that is defined by the following
expression:
Here
is the quantum state of the light wave.
In our review, we will encounter two types of quantum states that we have described above, i.e., vacuum
and squeezed vacuum
states. Let us show how to calculate the power (double-sided)
spectral density of a generic quantity
in a vacuum state. To do so, one should substitute Eq. (86)
into Eq. (87) and obtain that:
where we used the definition of the power spectral density matrix of light in a vacuum
state (65).
Similarly, one can calculate the spectral density of quantum noise if the light is in a squeezed state
, utilizing the definition of the squeezed state density matrix given in Eq. (83):
It might also be necessary to calculate also cross-correlation spectral density
of
with
some other quantity
with quantum noise defined as:
Using the definition of cross-spectral density
similar to (87):
one can get the following expressions for spectral densities in both cases of the vacuum state:
and the squeezed state:
Note that since the observables
and
that one calculates spectral densities
for are Hermitian, it is compulsory, as is well known, for any operator to represent a physical
quantity, then the following relation holds for their spectral coefficients
and
:
This leads to an interesting observation that the coefficients
and
should be real-valued
functions of variable
.
Now we can make further generalizations and consider multiple light and vacuum fields comprising the
quantity of interest:
where
stand for quadrature amplitude vectors of
independent electromagnetic fields, and
are the corresponding complex-valued coefficient functions indicating how these fields are
transmitted to the output. In reality, the readout observable of a GW detector is always a combination of
the input light field and vacuum fields that mix into the output optical train as a result of optical loss of
various origin. This statement can be exemplified by a single lossy mirror I/O-relations given by Eq. (34) of
Section 2.2.4.
Thus, to calculate the spectral density for such an observable, one needs to know the initial state of all
light fields under consideration. Since we assume
independent from each other, the initial state will
simply be a direct product of the initial states for each of the fields:
and the formula for the corresponding power (double-sided) spectral density reads:
with
standing for the i-th input field spectral density matrix. Hence, the total spectral
density is just a sum of spectral densities of each of the fields. The cross-spectral density for
two observables
and
can be built by analogy and we leave this task to the
reader.