5.1 Movable mirror

The scheme of the mirror is drawn in Figure 28View Image. It is illuminated from both sides by the two independent laser sources with frequency ωp, and mean power values ℐ1 and ℐ2. In terms of the general linear measurement theory of Section 4.2 we have two meters represented by these two incident light waves. The two arbitrary quadratures of the reflected waves are deemed as measured quantities ˆ O1 and ˆ O2. Measurement can be performed, e.g., by means of two independent homodyne detectors. Let us analyze quantum noise in such a model keeping to the scheme given by Eqs. (133View Equation).
View Image

Figure 28: Scheme of light reflection off the single movable mirror of mass M pulled by an external force G.

5.1.1 Optical transfer matrix of the movable mirror

The first one of Eqs. (133View Equation) in our simple scheme is represented by the input-output coupling equations (30View Equation, 36View Equation) of light on a movable mirror derived in Section 2.2.4. We choose the transfer matrix of the mirror to be real:

[ √ --√ --] 𝕄real = −√ -R √ T- (243 ) T R
according to Eq. (29View Equation). Then we can write down the coupling equations, substituting electric field strength amplitudes β„°β„°β„°1,2 and ˆe1,2(Ω ) by their dimensionless counterparts as introduced by Eq. (61View Equation) of Section 3.1:
[ ˆ ] [ˆ ] [ˆ ] [ ] [ ] B1(Ω ) = 𝕄real ⋅ A1 (Ω) , and b1(Ω) = 𝕄real ⋅ ˆa1(Ω ) + R1 xˆ(Ω), (244 ) ˆB2(Ω ) ˆA2 (Ω) ˆb2(Ω) ˆa2(Ω ) R2
where
√ --ω [ ] √--ω [ ] R1 = 2 R --p A1s , and R2 = 2 R --p A2s . (245 ) c − A1c c − A2c
Without any loss of generality one can choose the phase of the light incident from the left to be such that A1s = 0 and ∘ ---------- A1c = 2ℐ1βˆ•(ℏωp). Then factoring in the constant phase difference between the left and the right beams equal to Φ0, one would obtain for the left light ∘ ---------- {A2c, A2s} = 2 ℐ2βˆ•(ℏωp){cos Φ0,sinΦ0 }.

The two output measured quantities will then be given by the two homodyne photocurrents:

T Oˆ1 (Ω) = H [Ο•1]ˆb1 = ˆb1c(Ω )cosΟ•1 + ˆb1s(Ω) sin Ο•1, (246 ) Oˆ2 (Ω) = HT [Ο•2]ˆb2 = ˆb2c(Ω )cosΟ•2 + ˆb2s(Ω) sin Ο•2, (247 )
where vector H [Ο• ] was first introduced in Section 2.3.2 after Eq. (47View Equation) as:
[ ] cos Ο• H [Ο•] ≡ sin Ο• . (248 )
Then, after substitution of (243View Equation) into (244View Equation), and then into (246View Equation), one gets
(0) T √ -- √ -- Oˆ1 (Ω ) = H [Ο•1](− R aˆ1 + T ˆa2), (249 ) Oˆ(0)(Ω ) = HT [Ο• ](√T-aˆ + √R--aˆ), (250 ) 2 2 1 2
and
2∘2R--ℏ-ω-ℐ-- χO1F (Ω) = HT [Ο•1 ]R1 = − ---------p-1 sin Ο•1, (251 ) ∘ -ℏc------ T 2 2Rℏ ωpℐ2 χO2F (Ω) = H [Ο•2 ]R2 = − -----ℏc----- sin(Ο•2 − Φ0 ). (252 )

5.1.2 Probe’s dynamics: radiation pressure force and ponderomotive rigidity

Now we can write down the equation of motion for the mirror assuming it is pulled by a GW tidal force G:

m ¨ˆx(t) = Fˆ (t) + G(t) =⇒ − M Ω2 ˆx(Ω) = Fˆ (Ω ) + G (Ω ), (253 ) r.p. r.p.
that gives us the probe’s dynamics equation (the third one) in (133View Equation):
1 ˆx(Ω) = χxx (Ω)[ ˆFb.a.(Ω) + G (Ω)] = −----2[F ˆr.p.(Ω ) + G(Ω )], (254 ) M Ω
where the free mirror mechanical susceptibility 2 χxx(Ω ) = − 1βˆ•(M Ω ). The term ˆFb.a.(t) stands for the radiation pressure force imposed by the light that can be calculated as
ˆβ„ (Ω) + ˆβ„ (Ω ) − ˆβ„ (Ω) − ˆβ„ (Ω) Fˆr.p.(Ω ) = F0 + ˆFb.a.(Ω ) = -a1-------b1-------a2-------b2----≃ [ c ] ℏkp- T T T T T Tˆ T Tˆ 2 (A1 A1 + B 1B1 − A 2A2 − B2 B2) + 2(A 1 ˆa1(t) + B 1b1(t) − A 2aˆ2 (t) − B 2b2(t)) (255 )
where kp ≡ ωpβˆ•c
√ ----∘ ----- F = ℏkp(AT A + BT B − AT A − BT B ) = 2R-(ℐ − ℐ ) − 4--RT-- ℐ ℐ cosΦ , (256 ) 0 2 1 1 1 1 2 2 2 2 c 1 2 c 1 2 0
is the regular part of the radiation pressure force10. It is constant and thus can be compensated by applying a fixed restoring force of the same magnitude but with opposite direction, which can be done either by employing an actuator, or by turning the mirror into a low-frequency pendulum with ωm β‰ͺ ΩGW by suspending it on thin fibers, as is the case for the GW interferometers, that provides a necessary gravity restoring force in a natural way. However, it does not change the quantum noise and thus can be omitted from further consideration. The latter term represents a quantum correction to the former one
T T ˆ T T ˆ T T Fˆb.a.(Ω ) ≃ ℏkp(A 1 ˆa1(Ω ) + B1 b1(Ω) − A 2 ˆa2(Ω ) − B 2 b2(Ω )) = F1 ˆa1(Ω ) + F 2 ˆa2(Ω) − K ˆx(Ω()257 )
where ˆF(0)≡ F Tˆa1(Ω ) + FT ˆa2(Ω) b.a. 1 2 is the random part of the radiation pressure that depends on the input light quantum fluctuations described by quantum quadrature amplitudes vectors ˆa1(Ω ) and ˆa2 (Ω ) with coefficients given by vectors:
∘ -------[ √ ---- √ ---- ] ∘ -------[√ ---- √ ---- ] F = 2--2-ℏωpR- Rℐ1√−---T ℐ2cos Φ0 , and F = − 2---2ℏωpR- Tℐ1√ +---R ℐ2 cosΦ0 , 1 c − Tℐ2 sin Φ0 2 c R ℐ2sin Φ0

and the term − K ˆx(Ω) represents the dynamical back action with

√ -------- 8ωp--RT--ℐ1ℐ2sin-Φ0- K = c2 (258 )
being a ponderomotive rigidity that arises in the potential created by the two counter propagating light waves. Eq. (257View Equation) gives us the second of Eqs. (133View Equation). Here χFF (Ω) = − K.

5.1.3 Spectral densities

We can reduce both our readout quantities to the units of the signal force G according to Eq. (140View Equation):

ˆF 𝒳ˆ1-(Ω)- ˆ ˆ F -ˆπ’³2(Ω-) ˆ O1 = χexffx(Ω ) + β„±(Ω ) + G, O2 = χeffxx(Ω ) + β„± (Ω) + G (259 )
and define the two effective measurement noise sources as
Oˆ(0) ˆO (0) 𝒳ˆ1 (Ω ) = ----1---, and ˆπ’³2(Ω ) = ----2---- (260 ) χO1F (Ω) χO2F(Ω )
and an effective force noise as
(0) β„±ˆ(Ω ) = ˆFb.a., (261 )
absorbing optical rigidity K into the effective mechanical susceptibility:
χeff(Ω ) = ---χxx(Ω)----= ----1-----. (262 ) xx 1 + K χxx(Ω) K − M Ω2
One can then easily calculate their power (double-sided) spectral densities according to Eq. (144View Equation):
S (Ω ) [S (Ω)] SF11 = --𝒳1𝒳1--- + S β„±β„±(Ω ) + 2Re -𝒳1β„±----- , |χexffx(Ω )|2 [ χexffx(Ω )] F S 𝒳2𝒳2(Ω ) S𝒳2β„± (Ω) S22 = --eff----2 + S β„±β„±(Ω ) + 2Re --eff----- , |χxx(Ω )| [ χxx(Ω ) ] F F∗ S-𝒳1𝒳2(Ω-) S𝒳1β„±-(Ω)- S-∗𝒳2β„±(Ω)- S 12 = S 21 = |χeff(Ω )|2 + S β„±β„±(Ω ) + χeff(Ω) + χeff∗(Ω ) , (263 ) xx xx xx
where, if both lights are in coherent states that implies
† ⟨ˆai(Ω ) ∘ ˆaj(Ω ′)⟩ = 2ππ•Švacδijδ (Ω − Ω′), (i,j) = (1,2)
, one can get:
2 2 S (Ω) = ------ℏc--------, S (Ω ) = ---------ℏc-----------, S (Ω) = 0, 𝒳1𝒳1 16ωp ℐ1R sin2Ο•1 𝒳2𝒳2 16ωp ℐ2R sin2(Ο•2 − Φ0 ) 𝒳1𝒳2 4ℏω R(ℐ + ℐ ) ℏ ℏ S β„±β„±(Ω ) = ----p---1----2-, S𝒳1β„± (Ω ) = --cotΟ•1, S𝒳2β„± (Ω) = --cot(Ο•2 − Φ0 ). (264 ) c2 2 2
Comparison of these expressions with the Eqs. (195View Equation) shows that we have obtained results similar to that of the toy example in Section 4.4. If we switched one of the pumping carriers off, say the right one, the resulting spectral densities for ˆF O 1 (Ω ) would be exactly the same as in the simple case of Eqs. (195View Equation), except for the substitution of 2 Π³ ℐ0 → R ℐ1 and Ο•1 → Ο•LO.

5.1.4 Full transfer matrix approach to the calculation of quantum noise spectral densities

It was easy to calculate the above spectral densities by parts, distinguishing the effective measurement and back-action noise sources and making separate calculations for them. Had we considered a bit more complicated situation with the incident fields in the squeezed states with arbitrary squeezing angles, the calculation of all six of the above individual spectral densities (264View Equation) and subsequent substitution to the sum spectral densities expressions (263View Equation) would be more difficult. Thus, it would be beneficial to have a tool to do all these operations at once numerically.

It is achievable if we build a full transfer matrix of our system. To do so, let us first consider the readout observables separately. We start with Oˆ1 and rewrite it as follows:

( ) Oˆ = HT [Ο• ] − √R-ˆa + χeffR F Taˆ + √T-ˆa + χeff R F T ˆa + χeffHT [Ο• ]R G 1 1 1 xx 1 1 1 2 xx 1 2 2 xx 1 1 [ˆa1] = HT [Ο•1] ⋅ 𝕄1 ⋅ + χeffxxHT [Ο•1]R1G, (265 ) ˆa2
where we omitted the frequency dependence of the constituents for the sake of brevity and introduced a 2 × 4 full transfer matrix 𝕄 1 for the first readout observable defined as
[ √ -- eff T √ -- eff T] 𝕄1 = − R 𝕀 + χ xxR1F 1 T 𝕀 + χ xxR1F 2 (266 )
with outer product of two arbitrary vectors α T αα = {α1,α2} and β T ββ = { β1,β2} written in short notation as:
[ ] αααβββT ≡ α1 β1 α1β2 . (267 ) α2 β1 α2β2
In a similar manner, the full transfer matrix for the second readout can be defined as:
[√ -- eff T √ -- eff T ] 𝕄2 = T𝕀 + χxxR2F 1 R 𝕀 + χ xxR2F 2 . (268 )
Having accomplished this, one is prepared to calculate all the spectral densities (263View Equation) at once using the following matrix formulas:
HT [Ο• ] ⋅ 𝕄 π•Š 𝕄 †⋅ H [Ο• ] + HT [Ο• ] ⋅ 𝕄 ∗π•Š 𝕄T ⋅ H [Ο• ] SFij(Ω ) = ---1---------i----i-in--j------j---------i----j-in--i------j-, (i,j) = (1, 2) (269 ) 2|χexffx|2 HT [Ο•i]RiR †jH [Ο•j ]
where ∗ † T 𝕄 ≡ (𝕄 ) and π•Šin is the 4 × 4-matrix of spectral densities for the two input fields:
[ ] π•Šin = π•Šsqz[r1,πœƒ1] 0 (270 ) 0 π•Šsqz[r2,πœƒ2]
with π•Šsqz[ri,πœƒi] defined by Eq. (83View Equation).

Thus, we obtain the formula that can be (and, actually, is) used for the calculation of quantum noise spectral densities of any, however complicated, interferometer given the full transfer matrix of this interferometer.

5.1.5 Losses in a readout train

Thus far we have assumed that there is no dissipation in the transition from the outgoing light to the readout photocurrent of the homodyne detector. This is, unfortunately, not the case since any real photodetector has the finite quantum efficiency η < 1 d that indicates how many photons absorbed by the detector give birth to photoelectrons, i.e., it is the measure of the probability for the photon to be transformed into the photoelectron. As with any other dissipation, this loss of photons gives rise to an additional noise according to the FDT that we should factor in. We have shown in Section 2.2.4 that this kind of loss can be taken into account by means of a virtual asymmetric beamsplitter with transmission coefficients √ --- ηd and √ ------ 1 − ηd for the signal light and for the additional noise, respectively. This beamsplitter is set into the readout optical train as shown in Figure 8View Image and the i-th readout quantity needs to be modified in the following way:

√ --- ∘ ------ Oˆloiss(Ω ) = ηd ˆOi(Ω ) + 1 − ηdˆni(Ω ), (271 )
where
T ˆni(Ω ) = H [Ο•i]nˆi (Ω) = ˆni,c(Ω )cosΟ•i + ˆni,s(Ω )sin Ο•i
is the additional noise that is assumed to be in a vacuum state. Since the overall factor in front of the readout quantity does not matter for the final noise spectral density, one can redefine Oˆloss(Ω) i in the following way:
∘ -1----- ˆOloiss(Ω ) = ˆOi(Ω) + πœ–dˆni(Ω), where πœ–d ≡ ---− 1. (272 ) ηd

The influence of this loss on the final sum spectral densities (269View Equation) is straightforward to calculate if one assumes the additional noise sources in different readout trains to be uncorrelated. If it is so, then Eq. (269View Equation) modifies as follows:

{ T † T ∗ T F,loss ---1--- H---[Ο•i] ⋅ 𝕄i-π•Šin𝕄-j-⋅ H-[Ο•j] +-H-[Ο•i] ⋅ 𝕄-jπ•Šin𝕄-i-⋅ H-[Ο•j] Sij (Ω ) = 2|χexffx|2 HT [Ο•i]RiR †H [Ο•j] + j } πœ–2dδij ---T--------†----- , (i,j) = (1,2). (273 ) H [Ο•i]RiR iH [Ο•i]

Now, when we have considered all the stages of the quantum noise spectral densities calculation on a simple example of a single movable mirror, we are ready to consider more complicated systems. Our next target is a Fabry–Pérot cavity.


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