4.1 Quantum measurement of a classical force
4.1.1 Discrete position measurement
Let us consider a very simple measurement scheme, which, nevertheless, embodies all key features of a
general position measurement. In the scheme shown in Figure 20, a sequence of very short light pulses are
used to monitor the displacement of a probe body
. The position
of
is probed periodically
with time interval
. In order to make our model more realistic, we suppose that each pulse reflects from
the test mass
times, thus increasing the optomechanical coupling and thereby the information of
the measured quantity contained in each reflected pulse. We also assume mass
large enough to neglect
the displacement inflicted by the pulses radiation pressure in the course of the measurement
process.
Then each
-th pulse, when reflected, carries a phase shift proportional to the value of the test-mass
position
at the moment of reflection:
where
,
is the light frequency,
is the pulse number and
is the
initial (random) phase of the
-th pulse. We assume that the mean value of all these phases is equal to
zero,
, and their root mean square (RMS) uncertainty
is equal to
.
The reflected pulses are detected by a phase-sensitive device (the phase detector). The implementation
of an optical phase detector is considered in detail in Section 2.3.1. Here we suppose only that the phase
measurement error introduced by the detector is much smaller than the initial uncertainty of the
phases
. In this case, the initial uncertainty will be the only source of the position measurement error:
For convenience, we renormalize Eq. (96) as the equivalent test-mass displacement:
where
are the independent random values with the RMS uncertainties given by Eq. (97).
Upon reflection, each light pulse kicks the test mass, transferring to it a back-action momentum equal to
where
and
are the test-mass momentum values just before and just after the light pulse
reflection, and
is the energy of the
-th pulse. The major part of this perturbation is contributed by
classical radiation pressure:
with
the mean energy of the pulses. Therefore, one could neglect its effect, for it could be either
subtracted from the measurement result or compensated by an actuator. The random part, which cannot be
compensated, is proportional to the deviation of the pulse energy:
and its RMS uncertainly is equal to
with
the RMS uncertainty of the pulse energy.
The energy and the phase of each pulse are canonically conjugate observables and thus obey the
following uncertainty relation:
Therefore, it follows from Eqs. (97 and 103) that the position measurement error
and
the momentum perturbation
due to back action also satisfy the uncertainty relation:
This example represents a simple particular case of a linear measurement. This class of measurement
schemes can be fully described by two linear equations of the form (98) and (100), provided that both the
measurement uncertainty and the object back-action perturbation (
and
in this case) are
statistically independent of the test object initial quantum state and satisfy the same uncertainty relation as
the measured observable and its canonically conjugate counterpart (the object position and momentum in
this case).
4.1.2 From discrete to continuous measurement
Suppose the test mass to be heavy enough for a single pulse to either perturb its momentum noticeably, or
measure its position with the required precision (which is a perfectly realistic assumption for the
kilogram-scale test masses of GW interferometers). In this case, many pulses should be used to accumulate
the measurement precisions; at the same time, the test-mass momentum perturbation will be accumulated
as well. Choose now such a time interval
, which, on the one hand, is long enough to comprise a large
number of individual pulses:
and, on the other hand, is sufficiently short for the test-mass position
not to change considerably during
this time due to the test-mass self-evolution. Then one can use all the
measurement results to refine
the precision of the test-mass position
estimate, thus getting
times smaller uncertainty
At the same time, the accumulated random kicks the object received from each of the pulses random kicks,
see Eq. (102), result in random change of the object’s momentum similar to that of Brownian motion, and
thus increasing in the same diffusive manner:
If we now assume the interval between the measurements to be infinitesimally small (
), keeping
at the same time each single measurement strength infinitesimally weak:
then we get a continuous measurement of the test-mass position
as a result. We need more adequate
parameters to characterize its ‘strength’ than
and
. For continuous measurement we
introduce the following parameters instead:
with
This allows us to rewrite Eqs. (107) and (108) in a form that does not contain the time interval
:
To clarify the physical meaning of the quantities
and
let us rewrite Eq. (98) in the continuous
limit:
where
stands for measurement noise, proportional to the phase
of the light beam (in the continuous limit
the sequence of individual pulses transforms into a continuous beam). Then there is no difficulty in seeing
that
is a power (double-sided) spectral density of this noise, and
is a power double-sided spectral
density of
.
If we turn to Eq. (100), which describes the meter back action, and rewrite it in a continuous limit we
will get the following differential equation for the object momentum:
where
is a continuous Markovian random force, defined as a limiting case of the following discrete
Markov process:
with
the optical power,
its mean value, and ‘
’ here meaning all forces (if any), acting on
the object but having nothing to do with the meter (light, in our case). Double-sided power
spectral density of
is equal to
, and double-sided power spectral density of
is
.
We have just built a simple model of a continuous linear measurement, which nevertheless comprises the
main features of a more general theory, i.e., it contains equations for the calculation of measurement
noise (112) and also for back action (114). The precision of this measurement and the object back action in
this case are described by the spectral densities
and
of the two meter noise sources, which are
assumed to not be correlated in our simple model, and thus satisfy the following relation (cf. Eqs. (109)):
This relation (as well as its more general version to be discussed later) for continuous linear measurements
plays the same role as the uncertainty relation (105) for discrete measurements, establishing a universal
connection between the accuracy of the monitoring and the perturbation of the monitored
object.
Simple case: light in a coherent state.
Recall now that scheme of representing the quantized light wave as a
sequence of short statistically-independent pulses with duration
we referred to in Section 3.2. It is
the very concept we used here, and thus we can use it to calculate the spectral densities of the measurement
and back-action noise sources for our simple device featured in Figure 20 assuming the light to be in a
coherent state with classical amplitude
(we chose
thus making the mean
phase of light
). To do so we need to express phase
and energy
in the pulse in
terms of the quadrature amplitudes
. This can be done if we refer to Eq. (61) and make
use of the following definition of the mean electromagnetic energy of the light wave contained
in the volume
(here,
is the effective cross-sectional area of the light beam):
where
is the mean pulse energy, and
is a fluctuating part of the pulse energy.
We used here the definition of the mean pulse quadrature amplitude operators introduced in Eqs. (67). In
the same manner, one can define a phase for each pulse using Eqs. (14) and with the assumption of small
phase fluctuations (
) one can get:
Thus, since in a coherent state
the phase and energy uncertainties are equal to
and hence
Substituting these expressions into Eqs. (110, 109), we get the following expressions for the power
(double-sided) spectral densities of the measurement and back-action noise sources:
and
We should emphasize that this simple measurement model and the corresponding uncertainty
relation (116) are by no means general. We have made several rather strong assumptions in the course of
derivation, i.e., we assumed:
- energy and phase fluctuations in each of the light pulses uncorrelated:
;
- all pulses to have the same energy and phase uncertainties
and
, respectively;
- the pulses statistically independent from each other, particularly taking
with
.
These assumptions can be mapped to the following features of the fluctuations
and
in
the continuous case:
- these noise sources are mutually not correlated;
- they are stationary (invariant to the time shift) and, therefore, can be described by spectral
densities
and
;
- they are Markovian (white) with constant (frequency-independent) spectral densities.
The features 1 and 2, in turn, lead to characteristic fundamentally-looking sensitivity limitations, the
SQL. We will call linear quantum meters, which obey these limitations (that is, with mutually
non-correlated and stationary noises
and
), Simple Quantum Meters (SQM).