 |
Figure 1:
Scheme of a simple weak force measurement: an external signal force pulls the
mirror from its equilibrium position , causing displacement . The signal displacement is
measured by monitoring the phase shift of the light beam, reflected from the mirror. |
 |
Figure 2:
Scheme of a Michelson interferometer. When the end mirrors of the interferometer arms
are at rest the length of the arms is such that the light from the laser gets reflected
back entirely (bright port), while at the dark port the reflected waves suffer destructive interference
keeping it really dark. If, due to some reason, e.g., because of GWs, the lengths of the arms changed
in such a way that their difference was , the photodetector at the dark port should measure
light intensity . |
 |
Figure 3:
Action of the GW on a Michelson interferometer: (a) -polarized GW periodically
stretch and squeeze the interferometer arms in the - and -directions, (b) -polarized GW
though have no impact on the interferometer, yet produce stretching and squeezing of the imaginary
test particle ring, but along the directions, rotated by with respect to the and directions
of the frame. The lower pictures feature field lines of the corresponding tidal acceleration fields
. |
 |
Figure 4:
Phasor diagrams for amplitude (Left panel) and phase (Right panel) modulated light.
Carrier field is given by a brown vector rotating clockwise with the rate around the origin of the
complex plane frame. Sideband fields are depicted as blue vectors. The lower ( ) sideband
vector origin rotates with the tip of the carrier vector, while its own tip also rotates with respect to
its origin counterclockwise with the rate . The upper ( ) sideband vector origin rotates
with the tip of the upper sideband vector, while its own tip also rotates with respect to its origin
counterclockwise with the rate . Modulated oscillation is a sum of these three vectors an is given
by the red vector. In the case of amplitude modulation (AM), the modulated oscillation vector is
always in phase with the carrier field while its length oscillates with the modulation frequency .
The time dependence of its projection onto the real axis that gives the AM-light electric field strength
is drawn to the right of the corresponding phasor diagram. In the case of phase modulation (PM),
sideband fields have a constant phase shift with respect to the carrier field (note factor in
front of the corresponding terms in Eq. (22); therefore its sum is always orthogonal to the carrier field
vector, and the resulting modulated oscillation vector (red arrow) has approximately the same length
as the carrier field vector but outruns or lags behind the latter periodically with the modulation
frequency . The resulting oscillation of the PM light electric field strength is drawn to the right
of the PM phasor diagram and is the projection of the PM oscillation vector on the real axis of the
complex plane. |
 |
Figure 5:
Scheme of light reflection off the coated mirror. |
 |
Figure 6:
Scheme of a beamsplitter. |
 |
Figure 7:
Model of lossy mirror. |
 |
Figure 8:
Reflection of light from the movable mirror. |
 |
Figure 9:
Schematic view of light modulation by perfectly reflecting mirror motion. An initially
monochromatic laser field with frequency gets reflected from the mirror
that commits slow (compared to optical oscillations) motion (blue line) under the action of
external force . Reflected the light wave phase is modulated by the mechanical motion so that the
spectrum of the outgoing field contains two sidebands carrying all the information about
the mirror motion. The left panel shows the spectral representation of the initial monochromatic
incident light wave (upper plot), the mirror mechanical motion amplitude spectrum (middle plot)
and the spectrum of the phase-modulated by the mirror motion, reflected light wave (lower plot). |
 |
Figure 10:
The typical spectrum (amplitude spectral density) of the light leaving the interferometer
with movable mirrors. The central peak corresponds to the carrier light with frequency , two
smaller peaks on either side of the carrier represent the signal sidebands with the shape defined by
the mechanical motion spectrum ; the noisy background represents laser noise. |
 |
Figure 11:
Schematic view of homodyne readout (left panel) and DC readout (right panel) principle
implemented by a simple Michelson interferometer. |
 |
Figure 12:
Schematic view of heterodyne readout principle implemented by a simple Michelson
interferometer. Green lines represent modulation sidebands at radio frequency and blue dotted
lines feature signal sidebands |
 |
Figure 13:
Light field in a vacuum quantum state . Left panel (a) features a typical result one
could get measuring the (normalized) electric field strength of the light wave in a vacuum state as a
function of time. Right panel (b) represents a phase space picture of the results of measurement. A red
dashed circle displays the error ellipse for the state that encircles the area of single standard
deviation for a two-dimensional random vector of measured light quadrature amplitudes. The
principal radii of the error ellipse (equal in vacuum state case) are equal to square roots of the
covariance matrix eigenvalues, i.e., to . |
 |
Figure 14:
Wigner function of a ground state of harmonic oscillator (left panel) and
its representation in terms of the noise ellipse (right panel). |
 |
Figure 15:
Light field in a coherent quantum state . Left panel a) features a typical result
one could get measuring the (normalized) electric field strength of the light wave in a coherent state
as a function of time. Right panel b) represents a phase space picture of the results of measurement.
The red dashed line in the left panel marks the mean value . The red arrow in the right panel
features the vector of the mean values of quadrature amplitudes, i.e., , while the red dashed circle
displays the error ellipse for the state that encircles the area of single standard deviation
for a two-dimensional random vector of quadrature amplitudes. The principle radii of the error
ellipse (equal in the coherent state case) are equal to square roots of the covariance matrix ,
i.e., to . |
 |
Figure 16:
Schematic plot of a vacuum state transformation under the action of the squeezing
operator . Eqs. (74) demonstrate the equivalence of the general squeezing operator
to a sequence of phase plane counterclockwise rotation by an angle (transition from a) to b)),
phase plane squeezing and stretching by a factor (transition from b) to c)) and rotation back
by the same angle (transition from c) to d)). Point tracks how transformations change the
initial state marked with point . |
 |
Figure 17:
Left panel: Wigner function of a squeezed vacuum state with squeeze parameter
(5 dB) and rotation angle . Right panel: Error ellipse corresponding to that Wigner
function. |
 |
Figure 18:
Light field in a squeezed state . Upper row features time dependence of the
electric field strength in three different squeezed states (10 dB squeezing assumed for all): a)
squeezed vacuum state with squeezing angle ; b) displaced squeezed state with classical
amplitude (mean field strength oscillations are given by red dashed line) and
amplitude squeezing ( ); c) displaced squeezed state with classical amplitude and
phase squeezing ( ). Lower row features error ellipses (red dashed lines) for the corresponding
plots in the upper row. |
 |
Figure 19:
Example of a squeezed state of the continuum of modes: output state of a speed-meter
interferometer. Left panel shows the plots of squeezing parameter and squeezing angle
versus normalized sideband frequency (here is the interferometer half-bandwidth).
Right panel features a family of error ellipses for different sideband frequencies that illustrates
the squeezed state defined by and drawn in the left panel. |
 |
Figure 20:
Toy example of a linear optical position measurement. |
 |
Figure 21:
General scheme of the continuous linear measurement with standing for measured
classical force, the measurement noise, the back-action noise, the meter readout
observable, the actual probe’s displacement. |
 |
Figure 22:
Sum quantum noise power (double-sided) spectral densities of the Simple Quantum
Meter for different values of measurement strength . Thin black
line: SQL. Left: free mass. Right: harmonic oscillator |
 |
Figure 23:
Sum quantum noise power (double-sided) spectral densities of the Simple
Quantum Meter in the -normalization for different values of measurement strength:
. Thin black line: SQL. Left: free mass. Right: harmonic oscillator. |
 |
Figure 24:
Sum quantum noise power (double-sided) spectral densities of Simple Quantum Meter
and harmonic oscillator in displacement normalization for different values of measurement strength:
. Thin black line: SQL. |
 |
Figure 25:
Toy example of a linear optical position measurement. |
 |
Figure 26:
Toy example of the quantum speed-meter scheme. |
 |
Figure 27:
Real ( ) and effective ( ) coupling constants in the speed-meter scheme. |
 |
Figure 28:
Scheme of light reflection off the single movable mirror of mass pulled by an external
force . |
 |
Figure 29:
Fabry–Pérot cavity |
 |
Figure 30:
Power- and signal-recycled Fabry–Pérot–Michelson interferometer. |
 |
Figure 31:
Effective model of the dual-recycled Fabry–Pérot–Michelson interferometer, consisting
of the common (a) and the differential (b) modes, coupled only through the mirrors displacements. |
 |
Figure 32:
Common (top) and differential
(bottom) modes of the dual-recycled Fabry–Pérot–Michelson interferometer, reduced to the single
cavities using the scaling law model. |
 |
Figure 33:
The differential mode of the dual-recycled Fabry–Pérot–Michelson interferometer in
simplified notation (364). |
 |
Figure 34:
Examples of the sum quantum noise spectral densities of the classically-optimized
( , ) resonance-tuned interferometer. ‘Ordinary’: , no squeezing.
‘Increased power’: , no squeezing. ‘Squeezed’: , 10 dB squeezing. For all
plots, and . |
 |
Figure 35:
Examples of the sum quantum noise power (double-sided) spectral densities of the
resonance-tuned interferometers with frequency-dependent squeezing and/or homodyne angles. Left:
no optical losses, right: with optical losses, . ‘Ordinary’: no squeezing, .
‘Squeezed’: 10 dB squeezing, , (these two plots are provided for comparison).
Dots [pre-filtering, Eq. (403)]: 10 dB squeezing, , frequency-dependent squeezing angle.
Dashes [post-filtering, Eq. (408)]: 10 dB squeezing, , frequency-dependent homodyne angle.
Dash-dots [pre- and post-filtering, Eq. (410)]: 10 dB squeezing, frequency-dependent squeeze and
homodyne angles. For all plots, and . |
 |
Figure 36:
Plots of the locally-optimized SQL beating factor (418) of the interferometer
with cross-correlated noises for the “bad cavity” case , for several different values of the
optimization frequency within the range . Thick solid lines: the
common envelopes of these plots; see Eq. (420). Left column: ; right column: .
Top row: no squeezing, ; bottom row: 10 dB squeezing, . |
 |
Figure 37:
Schemes of interferometer with the single filter cavity. Left: In the pre-filtering scheme,
squeezed vacuum from the squeezor is injected into the signal port of the interferometer after the
reflection from the filter cavity; right: in the post-filtering scheme, a squeezed vacuum first passes
through the interferometer and, coming out, gets reflected from the filter cavity. In both cases the
readout is performed by an ordinary homodyne detector with frequency independent homodyne angle
. |
 |
Figure 38:
Left: Numerically-optimized filter-cavity parameters for a single cavity based pre-
and post-filtering schemes: half-bandwidth (solid lines) and detuning (dashed lines),
normalized by [see Eq. (442)], as functions of the filter cavity specific losses . Right: the
corresponding optimal SNRs, normalized by the SNR for the ordinary interferometer [see Eq. (455)].
Dashed lines: the normalized SNRs for the ideal frequency-dependent squeeze and homodyne angle
cases, see Eqs. (403) and (408). ‘Ordinary squeezing’: frequency-independent 10 dB squeezing with
. In all cases, , , and . |
 |
Figure 39:
Examples of the sum quantum noise power (double-sided) spectral densities of the
resonance-tuned interferometers with the single filter cavity based pre- and post-filtering. Left:
pre-filtering, see Figure 37 (left); dashes – 10 dB squeezing, , ideal frequency-dependent
squeezing angle (404); thin solid – 10 dB squeezing, , numerically-optimized
lossy pre-filtering cavity with . Right: post-filtering, see
Figure 37 (right); dashes: 10 dB squeezing, , ideal frequency-dependent homodyne
angle (407); thin solid – 10 dB squeezing, , numerically optimized lossy post-filtering cavity
with . In both panes (for the comparison): ‘Ordinary’ – no squeezing,
; ‘Squeezed’: 10 dB squeezing, ,  |
 |
Figure 40:
Left: schematic diagram of the microwave speed meter on coupled cavities as given
in [21]. Right: optical version of coupled-cavities speed meter proposed in [127]. |
 |
Figure 41:
Two possible optical realizations of zero area Sagnac speed meter. Left panel: The ring
cavities can be used to spatially separate the ingoing fields from the outgoing ones, in order to redirect
output light from one arm to another [42]. Right panel: The same goal can be achieved using an
optical circulator consisting of the polarization beamsplitter (PBS) and two -plates [85, 50]. |
 |
Figure 42:
Examples of the sum quantum noise power (double-sided) spectral densities of the
Sagnac speed-meter interferometer (thick solid line) in comparison with the Fabry–Pérot–Michelson
based topologies considerd above (dashed lines). Left: no optical losses, right: with optical losses,
, the losses part of the bandwidth (which corresponds to the losses
per bounce in the 4 km length arms). “Ordinary”: no squeezing, .
“Squeezed”: 10 dB squeezing, , . “Post-filtering”: 10 dB squeezing, ,
ideal frequency-dependent homodyne angle [see Eq. (408)]. For the Fabry–Pérot–Michelson-based
topologies, and . In the speed-meter case, and the
bandwidth is set to provide the same high-frequency noise as in the other plots (
in the lossless case and in the lossy one). |
 |
Figure 43:
Plots of the SQL beating factor (485) of the detuned interferometer, for different
values of the normalized detuning: , and for unified quantum efficiency
. Thick solid line: the common envelope of these plots. Dashed lines: the common
envelopes (420) of the SQL beating factors for the virtual rigidity case, without squeezing, ,
and with 10 dB squeezing, (for comparison). |
 |
Figure 44:
Roots of the characteristic equation (494) as functions of the optical power, for
. Solid lines: numerical solution. Dashed lines: approximate solution, see Eqs. (498) |
 |
Figure 45:
Examples of the sum noise power (double-sided) spectral densities of the detuned
interferometer. ‘Broadband’: double optimization of the Advanced LIGO interferometer for NS-NS
inspiraling and burst sources in presence of the classical noises [93] ( ,
, , ). ‘High-frequency’: low-power configuration
suitable for detection of the GW signals from the millisecond pulsars, similar to one planned for
GEO HF [169] [ , , , ]. ‘Second-order
pole’: the regime close to the second-order pole one, which provides a maximum of the SNR
for the GW burst sources given that technical noise is smaller than the SQL [ ,
, , , ]. In all cases, and the
losses part of the bandwidth (which corresponds to the losses per
bounce in the 4 km long arms). |
 |
Figure 46:
Left panel: the SQL beating factors for . Thick solid: the second-order
pole system (511); dots: the two-pole system with optimal separation between the poles (528),
(529); dashes: the harmonic oscillator (170); thin solid – SQL of the harmonic oscillator (171).
Right: the normalized SNR (526). Solid line: analytical optimization, Eq. (530); pluses: numerical
optimization of the spectral density (514) in the lossless case ( ); diamonds: the same for
the interferometer with , and the losses part of the bandwidth
(which corresponds to the losses per bounce in the 4 km long arms). |