Defense Intelligence Reference Document High-Frequency Gravitational Wave Communications

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where L1x is the position uncertainty, LJp is the momentum uncertainty, and fi is
Planck's reduced constant. Thus measuring x disturbs p, which in turn disturbs future
measurements of x
&(dt) = &(O) + dt[Llp(O)/m] (4)
where Llx(O) is the initial position uncertainty is, Llp(O) is the initial momentum
uncertainty, dt is the time of the future measurement, and m is the mass of the system
under measurement. E/c2 may be substituted for mass in an energy only system. This
is depicted in Figure 13.
To summarize, the quantum effects of measurements on future measurements is
quantum back action. Therefore the Standard Quantum Limit defines the lower
sensitivity limit for all measurement instruments, including gravitational-wave
detectors, according to the Heisenberg uncertainty principle. Detectors cannot avoid
quantum back action, however the use of higher energies in the detection process can
change the relative scale and impact of back action, and the use of squeezed states can
shift the relative distribution of back action into states not involved in measurement.
Tll\IE t = 0
Position x of
mass m being
mea_sure_d
(~'' I I /
~-
' '
'
_____.: ;.._ Ap(O) > fl/2
AX(O)
Measurement of x drives down A)((O)
wtiii;h drives up Ap(O)
Tll\IE t = dt
Quantum Back Action:
: x(dt) measurement affected
: by earlier x(O) measurement
''''''''''''''
,•·~
(( ~,
' ' - ~ '~ ~
•x(dt) ~ ax(dt) + dt[•p(O) I m]
Therefore Ap{O) drives up Ax(gt)
Figure 13. Quantum Back Action as a Mechanism for Creating the Standard Quantum Limit
Calculating the Standard Quantum Limit {SOL)
A method for calculating the Standard Quantum Limit (SQL) is introduced in this
section. The calculation of coherent versus stochastic SQL is compared and contrasted.
Important terms of the SQL calculation are described, including the impact of contained
energy levels within the detector on SQL, and the sources of Quality Factor and its
effect on SQL. Calculating the Standard Quantum Limit (SQL)
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