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 AMlight 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 phasemodulated 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 twodimensional 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 twodimensional 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 speedmeter interferometer. Left panel shows the plots of squeezing parameter and squeezing angle versus normalized sideband frequency (here is the interferometer halfbandwidth). 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 backaction noise, the meter readout observable, the actual probe’s displacement. 

Figure 22:
Sum quantum noise power (doublesided) 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 (doublesided) 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 (doublesided) 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 speedmeter scheme. 

Figure 27:
Real () and effective () coupling constants in the speedmeter 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 signalrecycled Fabry–Pérot–Michelson interferometer. 

Figure 31:
Effective model of the dualrecycled 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 dualrecycled Fabry–Pérot–Michelson interferometer, reduced to the single cavities using the scaling law model. 

Figure 33:
The differential mode of the dualrecycled Fabry–Pérot–Michelson interferometer in simplified notation (364). 

Figure 34:
Examples of the sum quantum noise spectral densities of the classicallyoptimized (, ) resonancetuned 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 (doublesided) spectral densities of the resonancetuned interferometers with frequencydependent 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 [prefiltering, Eq. (403)]: 10 dB squeezing, , frequencydependent squeezing angle. Dashes [postfiltering, Eq. (408)]: 10 dB squeezing, , frequencydependent homodyne angle. Dashdots [pre and postfiltering, Eq. (410)]: 10 dB squeezing, frequencydependent squeeze and homodyne angles. For all plots, and . 

Figure 36:
Plots of the locallyoptimized SQL beating factor (418) of the interferometer with crosscorrelated 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 prefiltering 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 postfiltering 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: Numericallyoptimized filtercavity parameters for a single cavity based pre and postfiltering schemes: halfbandwidth (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 frequencydependent squeeze and homodyne angle cases, see Eqs. (403) and (408). ‘Ordinary squeezing’: frequencyindependent 10 dB squeezing with . In all cases, , , and . 

Figure 39:
Examples of the sum quantum noise power (doublesided) spectral densities of the resonancetuned interferometers with the single filter cavity based pre and postfiltering. Left: prefiltering, see Figure 37 (left); dashes – 10 dB squeezing, , ideal frequencydependent squeezing angle (404); thin solid – 10 dB squeezing, , numericallyoptimized lossy prefiltering cavity with . Right: postfiltering, see Figure 37 (right); dashes: 10 dB squeezing, , ideal frequencydependent homodyne angle (407); thin solid – 10 dB squeezing, , numerically optimized lossy postfiltering 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 coupledcavities 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 (doublesided) spectral densities of the Sagnac speedmeter 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, , . “Postfiltering”: 10 dB squeezing, , ideal frequencydependent homodyne angle [see Eq. (408)]. For the Fabry–Pérot–Michelsonbased topologies, and . In the speedmeter case, and the bandwidth is set to provide the same highfrequency 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 (doublesided) spectral densities of the detuned interferometer. ‘Broadband’: double optimization of the Advanced LIGO interferometer for NSNS inspiraling and burst sources in presence of the classical noises [93] (, , , ). ‘Highfrequency’: lowpower configuration suitable for detection of the GW signals from the millisecond pulsars, similar to one planned for GEO HF [169] [, , , ]. ‘Secondorder pole’: the regime close to the secondorder 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 secondorder pole system (511); dots: the twopole 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). 
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Living Rev. Relativity 15, (2012), 5
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