List of Figures

View Image Figure 1:
Scheme of a simple weak force measurement: an external signal force G pulls the mirror from its equilibrium position x = 0, causing displacement δx. The signal displacement is measured by monitoring the phase shift of the light beam, reflected from the mirror.
View Image Figure 2:
Scheme of a Michelson interferometer. When the end mirrors of the interferometer arms Mn,e are at rest the length of the arms L 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 δL, the photodetector at the dark port should measure light intensity I δL Idark(δL) = -02 (1 − cos4π λ-).
View Image Figure 3:
Action of the GW on a Michelson interferometer: (a) h+-polarized GW periodically stretch and squeeze the interferometer arms in the x- and y-directions, (b) h ×-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 ∘ 45 with respect to the x and y directions of the frame. The lower pictures feature field lines of the corresponding tidal acceleration fields ∝ ¨h+,×.
View Image 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 ω 0 around the origin of the complex plane frame. Sideband fields are depicted as blue vectors. The lower (ω0 − Ω) 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 Ω. The upper (ω0 + Ω) 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 Ω. 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 Ω. 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 π ∕2 constant phase shift with respect to the carrier field (note factor i in front of the corresponding terms in Eq. (22View Equation); 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 Ω. 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.
View Image Figure 5:
Scheme of light reflection off the coated mirror.
View Image Figure 6:
Scheme of a beamsplitter.
View Image Figure 7:
Model of lossy mirror.
View Image Figure 8:
Reflection of light from the movable mirror.
View Image Figure 9:
Schematic view of light modulation by perfectly reflecting mirror motion. An initially monochromatic laser field Ein(t) with frequency ω = 2 πc∕λ 0 0 gets reflected from the mirror that commits slow (compared to optical oscillations) motion x(t) (blue line) under the action of external force G. Reflected the light wave phase is modulated by the mechanical motion so that the spectrum of the outgoing field Eout(ω) 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).
View Image 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 ω 0, two smaller peaks on either side of the carrier represent the signal sidebands with the shape defined by the mechanical motion spectrum x(Ω ); the noisy background represents laser noise.
View Image Figure 11:
Schematic view of homodyne readout (left panel) and DC readout (right panel) principle implemented by a simple Michelson interferometer.
View Image Figure 12:
Schematic view of heterodyne readout principle implemented by a simple Michelson interferometer. Green lines represent modulation sidebands at radio frequency ΩRF and blue dotted lines feature signal sidebands
View Image Figure 13:
Light field in a vacuum quantum state |vac⟩. 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 |vac⟩ that encircles the area of single standard deviation for a two-dimensional random vector ˆa 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 𝕍vac eigenvalues, i.e., to √ -- 1∕ 2.
View Image Figure 14:
Wigner function W |vac⟩(X 𝜀,Y 𝜀) of a ground state of harmonic oscillator (left panel) and its representation in terms of the noise ellipse (right panel).
View Image Figure 15:
Light field in a coherent quantum state |α (ω )⟩. 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 ⟨Eˆ(t)⟩. The red arrow in the right panel features the vector of the mean values of quadrature amplitudes, i.e., A, while the red dashed circle displays the error ellipse for the state |α(ω )⟩ that encircles the area of single standard deviation for a two-dimensional random vector ˆa 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 𝕍 coh, i.e., to √ -- 1∕ 2.
View Image Figure 16:
Schematic plot of a vacuum state transformation under the action of the squeezing operator Sˆ[r,ϕ ]. Eqs. (74View Equation) demonstrate the equivalence of the general squeezing operator ˆS[r,ϕ] to a sequence of phase plane counterclockwise rotation by an angle ϕ (transition from a) to b)), phase plane squeezing and stretching by a factor r e (transition from b) to c)) and rotation back by the same angle ϕ (transition from c) to d)). Point P ′ tracks how transformations change the initial state marked with point P.
View Image Figure 17:
Left panel: Wigner function of a squeezed vacuum state with squeeze parameter r = 0.5 (5 dB) and rotation angle ϕ = − π∕4. Right panel: Error ellipse corresponding to that Wigner function.
View Image Figure 18:
Light field in a squeezed state |sqzα(r,ϕ )⟩. Upper row features time dependence of the electric field strength E (t) in three different squeezed states (10 dB squeezing assumed for all): a) squeezed vacuum state with squeezing angle ϕ = π ∕4; b) displaced squeezed state with classical amplitude Ac = 5 (mean field strength oscillations ⟨ ˆE (t)⟩sqz are given by red dashed line) and amplitude squeezing (ϕ = π∕2); c) displaced squeezed state with classical amplitude Ac = 5 and phase squeezing (ϕ = 0). Lower row features error ellipses (red dashed lines) for the corresponding plots in the upper row.
View Image 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 r (Ω ) dB and squeezing angle ϕ (Ω ) versus normalized sideband frequency Ω∕ γ (here γ is the interferometer half-bandwidth). Right panel features a family of error ellipses for different sideband frequencies Ω that illustrates the squeezed state defined by rdB(Ω) and ϕ(Ω) drawn in the left panel.
View Image Figure 20:
Toy example of a linear optical position measurement.
View Image Figure 21:
General scheme of the continuous linear measurement with G standing for measured classical force, ˆ𝒳 the measurement noise, ˆℱ the back-action noise, Oˆ the meter readout observable, ˆx the actual probe’s displacement.
View Image Figure 22:
Sum quantum noise power (double-sided) spectral densities of the Simple Quantum Meter for different values of measurement strength Ωq (red ) < Ωq (green) < Ωq(blue). Thin black line: SQL. Left: free mass. Right: harmonic oscillator
View Image Figure 23:
Sum quantum noise power (double-sided) spectral densities of the Simple Quantum Meter in the h-normalization for different values of measurement strength: Ωq (red ) < Ωq(green) < Ωq (blue ). Thin black line: SQL. Left: free mass. Right: harmonic oscillator.
View Image 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: Ωq (red ) < Ωq(green) < Ωq (blue ). Thin black line: SQL.
View Image Figure 25:
Toy example of a linear optical position measurement.
View Image Figure 26:
Toy example of the quantum speed-meter scheme.
View Image Figure 27:
Real (β(t)) and effective (α(t)) coupling constants in the speed-meter scheme.
View Image Figure 28:
Scheme of light reflection off the single movable mirror of mass M pulled by an external force G.
View Image Figure 29:
Fabry–Pérot cavity
View Image Figure 30:
Power- and signal-recycled Fabry–Pérot–Michelson interferometer.
View Image 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.
View Image 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.
View Image Figure 33:
The differential mode of the dual-recycled Fabry–Pérot–Michelson interferometer in simplified notation (364View Equation).
View Image Figure 34:
Examples of the sum quantum noise spectral densities of the classically-optimized (ϕLO = π∕2, 𝜃 = 0) resonance-tuned interferometer. ‘Ordinary’: J = JaLIGO, no squeezing. ‘Increased power’: J = 10JaLIGO, no squeezing. ‘Squeezed’: J = JaLIGO, 10 dB squeezing. For all plots, γ = 2π × 500 s−1 and ηd = 0.95.
View Image 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, ηd = 0.95. ‘Ordinary’: no squeezing, ϕLO = π ∕2. ‘Squeezed’: 10 dB squeezing, 𝜃 = 0, ϕ = π ∕2 LO (these two plots are provided for comparison). Dots [pre-filtering, Eq. (403View Equation)]: 10 dB squeezing, ϕLO = π ∕2, frequency-dependent squeezing angle. Dashes [post-filtering, Eq. (408View Equation)]: 10 dB squeezing, 𝜃 = 0, frequency-dependent homodyne angle. Dash-dots [pre- and post-filtering, Eq. (410View Equation)]: 10 dB squeezing, frequency-dependent squeeze and homodyne angles. For all plots, J = J aLIGO and γ = 2π × 500 s− 1.
View Image Figure 36:
Plots of the locally-optimized SQL beating factor ξ(Ω ) (418View Equation) of the interferometer with cross-correlated noises for the “bad cavity” case Ω0 ≪ γ, for several different values of the optimization frequency Ω0 within the range √ --- 0.1 × Ωq ≤ Ω0 ≤ 10 × Ωq. Thick solid lines: the common envelopes of these plots; see Eq. (420View Equation). Left column: ηd = 1; right column: ηd = 0.95. Top row: no squeezing, r = 0; bottom row: 10 dB squeezing, e2r = 10.
View Image 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 ϕLO.
View Image Figure 38:
Left: Numerically-optimized filter-cavity parameters for a single cavity based pre- and post-filtering schemes: half-bandwidth γf1 (solid lines) and detuning δf (dashed lines), normalized by γf0 [see Eq. (442View Equation)], as functions of the filter cavity specific losses Af ∕Lf. Right: the corresponding optimal SNRs, normalized by the SNR for the ordinary interferometer [see Eq. (455View Equation)]. Dashed lines: the normalized SNRs for the ideal frequency-dependent squeeze and homodyne angle cases, see Eqs. (403View Equation) and (408View Equation). ‘Ordinary squeezing’: frequency-independent 10 dB squeezing with 𝜃 = 0. In all cases, J = JaLIGO, γ = 2π × 500 s−1, and ηd = 0.95.
View Image 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, ϕLO = π ∕2, ideal frequency-dependent squeezing angle (404View Equation); thin solid – 10 dB squeezing, ϕ = π∕2 LO, numerically-optimized lossy pre-filtering cavity with −9 − 7 −6.5 − 6 Af∕Lf = 10 , 10 10 and 10. Right: post-filtering, see Figure 37 (right); dashes: 10 dB squeezing, 𝜃 = 0, ideal frequency-dependent homodyne angle (407View Equation); thin solid – 10 dB squeezing, 𝜃 = 0, numerically optimized lossy post-filtering cavity with A ∕L = 10−9, 10−8 and 10− 7 f f. In both panes (for the comparison): ‘Ordinary’ – no squeezing, ϕLO = π∕2; ‘Squeezed’: 10 dB squeezing, 𝜃 = 0, ϕLO = π∕2
View Image 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].
View Image 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 λ ∕4-plates [85, 50].
View Image 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, ηd = 0.95, the losses part of the bandwidth − 1 γ2 = 1.875 s (which corresponds to the losses −4 Aarm = 10 per bounce in the 4 km length arms). “Ordinary”: no squeezing, ϕLO = π ∕2. “Squeezed”: 10 dB squeezing, 𝜃 = 0, ϕLO = π ∕2. “Post-filtering”: 10 dB squeezing, 𝜃 = 0, ideal frequency-dependent homodyne angle [see Eq. (408View Equation)]. For the Fabry–Pérot–Michelson-based topologies, J = J aLIGO and γ = 2π × 500 s− 1. In the speed-meter case, J = 2J aLIGO and the bandwidth is set to provide the same high-frequency noise as in the other plots (− 1 γ = 2π × 385 s in the lossless case and −1 γ = 2π × 360 s in the lossy one).
View Image Figure 43:
Plots of the SQL beating factor (485View Equation) of the detuned interferometer, for different values of the normalized detuning: 0 ≤ β ≡ arctan(δ∕γ ) < π ∕2, and for unified quantum efficiency η = 0.95. Thick solid line: the common envelope of these plots. Dashed lines: the common envelopes (420View Equation) of the SQL beating factors for the virtual rigidity case, without squeezing, r = 0, and with 10 dB squeezing, 2r e = 10 (for comparison).
View Image Figure 44:
Roots of the characteristic equation (494View Equation) as functions of the optical power, for γ ∕δ = 0.03. Solid lines: numerical solution. Dashed lines: approximate solution, see Eqs. (498View Equation)
View Image 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] (J = JaLIGO ≡ (2π × 100)3 s−3, Γ = 3100 s−1, β = 0.80, ϕLO = π ∕2 − 0.44). ‘High-frequency’: low-power configuration suitable for detection of the GW signals from the millisecond pulsars, similar to one planned for GEO HF [169] [J = 0.1JaLIGO, −1 Γ = 2π × 1000 s, β = π∕2 − 0.01, ϕLO = 0]. ‘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 [Stech = 0.1SSQL, J = JaLIGO, Γ = 1050 s− 1, β = π∕2 − 0.040, ϕLO = 0.91]. In all cases, ηd = 0.95 and the losses part of the bandwidth − 1 γ2 = 1.875 s (which corresponds to the losses − 4 Aarm = 10 per bounce in the 4 km long arms).
View Image Figure 46:
Left panel: the SQL beating factors ξ2 for Ωq ∕Ω0 = 0.1. Thick solid: the second-order pole system (511View Equation); dots: the two-pole system with optimal separation between the poles (528View Equation), (529View Equation); dashes: the harmonic oscillator (170View Equation); thin solid – SQL of the harmonic oscillator (171View Equation). Right: the normalized SNR (526View Equation). Solid line: analytical optimization, Eq. (530View Equation); pluses: numerical optimization of the spectral density (514View Equation) in the lossless case (η = 1); diamonds: the same for the interferometer with J = (2π × 100)3 s−3, ηd = 0.95 and the losses part of the bandwidth γ2 = 1.875 s−1 (which corresponds to the losses Aarm = 10− 4 per bounce in the 4 km long arms).