3.1 Quantization of light: Two-photon formalism

From the point of view of quantum field theory, a freely propagating electromagnetic wave can be characterized in each spatial point with location vector r = (x, y,z) at time t by a Heisenberg operator of an electric field strength Eˆ(r,t).5 The electric field Heisenberg operator of a light wave traveling along the positive direction of the z-axis can be written as a sum of a positive- and negative-frequency parts:
{ (+) (−) } Eˆ(x,y,z;t) = u(x,y, z) Eˆ (t) + Eˆ (t) , (48 )
where u(x,y,z) is the spatial mode shape, slowly changing on a wavelength λ scale, and
āˆ˜ ------ ∫ ∞ d ω 2πā„ ω [ ]† ˆE (+ )(t) = --- -----ˆaωe −iωt, and Eˆ(−)(t) = Eˆ(+)(t) . (49 ) 0 2π š’œc
Here, š’œ is the effective cross-section area of the light beam, and ˆaω (ˆa† ω) is the single photon annihilation (creation) operator in the mode of the field with frequency ω. The meaning of Eq. (49View Equation) is that the travelling light wave can be represented by an expansion over the continuum of harmonic oscillators – modes of the electromagnetic field, – that are, essentially, independent degrees of freedom. The latter implies the commutation relations for the operators ˆaω and ˆa †ω:
[ † ] ′ [ † †] ˆa ω,ˆaω′ = 2πδ(ω − ω ), and [ˆaω, ˆaω′] = ˆaω,aˆω′ = 0. (50 )

In GW detectors, one deals normally with a close to monochromatic laser light with carrier frequency ω0, and a pair of modulation sidebands created by a GW signal around its frequency in the course of parametric modulation of the interferometer arm lengths. The light field coming out of the interferometer cannot be considered as the continuum of independent modes anymore. The very fact that sidebands appear in pairs implies the two-photon nature of the processes taking place in the GW interferometers, which means the modes of light at frequencies ω1,2 = ω0 ± Ω have correlated complex amplitudes and thus the new two-photon operators and related formalism is necessary to describe quantum light field transformations in GW interferometers. This was realized in the early 1980s by Caves and Schumaker who developed the two-photon formalism [39Jump To The Next Citation Point, 40Jump To The Next Citation Point], which is widely used in GW detectors as well as in quantum optics and optomechanics.

One starts by defining modulation sideband amplitudes as

ˆa+ = ˆaω0+Ω, ˆa− = aˆω0− Ω,
and factoring out the oscillation at carrier frequency ω0 in Eqs. (48View Equation), which yields:
š’ž e− iω0t∫ ∞ d Ω š’ž ∫ ∞ dΩ Eˆ(+)(t) = -0√----- ---λ+ (Ω )ˆa+e −iΩt ā‰ƒ √-0e−iω0t ---λ+(Ω )ˆa+e− iΩt, 2 −ω0 2π 2 −∞ 2π (−) š’ž0eiω0t ∫ ω0 dΩ † −iΩt š’ž0 iω t∫ ∞ dΩ † − iΩt ˆE (t) = --√---- ---λ− (Ω )ˆa− e ā‰ƒ √--e 0 ---λ− (Ω )ˆa− e , (51 ) 2 −∞ 2π 2 −∞ 2π
where we denote āˆ˜ ----- š’ž0 ≡ 4πā„ω0 š’œc and define functions λ± (Ω) following [39Jump To The Next Citation Point] as
āˆ˜ -------- ω0-±--Ω λ± (Ω ) = ω0 ,

and use the fact that ω0 ā‰« ΩGW enables us to expand the limits of integrals to ω0 → ∞. The operator expressions in front of ±iω0t e in the foregoing Eqs. (51View Equation) are quantum analogues to the complex amplitude ā„° and its complex conjugate ā„°∗ defined in Eqs. (14View Equation):

∫ ∞ ∫ ∞ ˆ -š’ž0- š’ž0-- dΩ- −iΩt ˆ† š’ž0--† š’ž0-- dΩ- † −iΩt ā„° (t) = √ 2 ˆa(t) ≡ √ 2 −∞ 2π λ+ (Ω)ˆa+e , and ā„° (t) = √2 ˆa (t) ≡ √ 2 −∞ 2π λ− (Ω)ˆa− e .

Again using Eqs. (14View Equation), we can define two-photon quadrature amplitudes as:

ˆā„°(t) + ā„°ˆ†(t) š’ž0 ∫ ∞ dΩ λ+ ˆa+ + λ− ˆa† š’ž0 ∫ ∞ dΩ ˆā„°c(t) = ----√-------= √--- --------√------−-e−iΩt ≡ √--- ---ˆac(Ω )e− iΩt 2 2 − ∞ 2π 2 2 − ∞ 2π ˆā„°(t) − ˆā„°†(t) š’ž0 ∫ ∞ dΩ λ+ˆa+ − λ− ˆa†− −iΩt š’ž0 ∫ ∞ d Ω −iΩt ˆā„°s(t) = ----√-------= √--- --- -----√-------e ≡ √--- ---ˆas(Ω )e . (52 ) i 2 2 −∞ 2 π i 2 2 −∞ 2π
Note that so-introduced operators of two-photon quadrature amplitudes ˆac,s(t) are Hermitian and thus their frequency domain counterparts satisfy the relations for the spectra of Hermitian operator:
ˆa†c,s(t) = ˆac,s(t) =⇒ ˆa†c,s(Ω ) = ˆac,s(− Ω ).

Now we are able to write down commutation relations for the quadrature operators, which can be derived from Eq. (50View Equation):

[ ] [ ] ˆac(Ω ),ˆa†c(Ω′) = ˆas(Ω),ˆa†s(Ω ′) = 2π Ω-δ(Ω − Ω′), (53 ) [ ] [ ] ω0 ˆac(Ω ),ˆa†(Ω′) = ˆa†(Ω),ˆas(Ω ′) = 2πiδ(Ω − Ω′), (54 ) s c

The commutation relations represented by Eqs. (53View Equation) indicate that quadrature amplitudes do not commute at different times, i.e., [ˆac(t),ˆac(t′)] = [ˆas(t),ˆas(t′)] ā„= 0, which imply they could not be considered for proper output observables of the detector, for a nonzero commutator, as we would see later, means an additional quantum noise inevitably contributes to the final measurement result. The detailed explanation of why it is so can be found in many works devoted to continuous linear quantum measurement theory, in particular, in Chapter 6 of [22Jump To The Next Citation Point], Appendix 2.7 of [43Jump To The Next Citation Point] or in [19Jump To The Next Citation Point]. Where GW detection is concerned, all the authors are agreed on the point that the values of GW frequencies Ω (1 Hz ≤ Ωāˆ•2 π ≤ 103 Hz ), being much smaller than optical frequencies ω0āˆ•2π ∼ 1015 Hz, allow one to neglect such weak commutators as those of Eqs. (53View Equation) in all calculations related to GW detectors output quantum noise. This statement has gotten an additional ground in the calculation conducted in Appendix 2.7 of [43Jump To The Next Citation Point] where the value of the additional quantum noise arising due to the nonzero value of commutators (53View Equation) has been derived and its extreme minuteness compared to other quantum noise sources has been proven. Braginsky et al. argued in [19Jump To The Next Citation Point] that the two-photon quadrature amplitudes defined by Eqs. (52View Equation) are not the real measured observables at the output of the interferometer, since the photodetectors actually measure not the energy flux

∫ ∫ ˆ ∞ dωdω-′ √ ---′† i(ω−ω′)t ā„ (t) = (2π)2 ā„ ωω ˆaω ˆaω′e (55 ) 0
but rather the photon number flux:
∫ ∫ ∞ dωd ω′ † i(ω− ω′)t š’©ˆ(t) = ----2-ˆaωˆaω ′e . (56 ) 0 (2π )
The former does not commute with itself: [ˆā„(t), ˆā„(t′)] ā„= 0, while the latter apparently does [š’©ˆ(t), ˆš’© (t′)] = 0 and therefore is the right observable for a self-consistent quantum description of the GW interferometer output signal.

In the course of our review, we shall adhere to the approximate quadrature amplitude operators that can be obtained from the exact ones given by Eqs. (52View Equation) by setting λ (Ω ) → 1 ±, i.e.,

∫ ∞ dΩ ˆa + ˆa† ˆa + ˆa † ˆac(t) = ----+√---−-e−iΩt ⇐ ⇒ ˆac(Ω ) = -+√----−, −∞ 2π 2 2 ∫ ∞ dΩ ˆa+ − ˆa† ˆa+ − aˆ† ˆas(t) = ------√--−-e−iΩt ⇐ ⇒ ˆas(Ω ) = ---√---−. (57 ) −∞ 2π i 2 i 2

The new approximate two-photon quadrature operators satisfy the following commutation relations in the frequency domain:

′ ′ ′ ′ [ˆac(Ω),ˆas(Ω )] = 2πiδ(Ω + Ω ), and [ˆac(Ω ),ˆac(Ω )] = [ˆas(Ω),ˆas(Ω )] = 0, (58 )
and in the time domain:
[ˆa (t),aˆ (t′)] = iδ(t − t′), and [ˆa (t),ˆa (t′)] = [ˆa (t),ˆa (t′)] = 0 (59 ) c s c c s s

Then the electric field strength operator (48View Equation) can be rewritten in terms of the two-photon quadrature operators as:

ˆE (x, y,z;t) = u(x,y,z)š’ž0 [ˆac(t)cos ω0t + ˆas(t)sin ω0t]. (60 )
Hereafter, we will omit the spatial mode factor u(x,y,z ) since it does not influence the final result for quantum noise spectral densities. Moreover, in order to comply with the already introduced division of the optical field into classical carrier field and to the 1st order corrections to it comprising of laser noise and signal induced sidebands, we adopt the same division for the quantum fields, i.e., we detach the mean values of the corresponding quadrature operators via the following redefinition ˆaold→ A + ˆanew cs c,s c,s with old Ac,s ≡ āŸØˆac,sāŸ©. Here, by old āŸØˆac,sāŸ© we denote an ensemble average over the quantum state |ψāŸ© of the light wave: āŸØAˆāŸ© ≡ āŸØψ| ˆA |ψ āŸ©. Thus, the electric field strength operator for a plain electromagnetic wave will have the following form:
Eˆ(x,y;t) = š’ž0[(Ac + ˆac(t))cosω0t + (As + ˆas(t))sinω0t]. (61 )
Further, we combine the two-photon quadratures into vectors in the same manner as we used to do for classical fields:
[Ac ] [ ˆac] A ≡ , and aˆ≡ . As aˆs

Now, when we have defined a quantum Heisenberg operator of the electric field of a light wave, and introduced quantum operators of two-photon quadratures, the last obstacle on our way towards the description of quantum noise in GW interferometers is that we do not know the quantum state the light field finds itself in. Since it is the quantum state that defines the magnitude and mutual correlations of the amplitude and phase fluctuations of the outgoing light, and through it the total level of quantum noise setting the limit on the future GW detectors’ sensitivity. In what follows, we shall consider vacuum and coherent states of the light, and also squeezed states, for they comprise the vast majority of possible states one could encounter in GW interferometers.


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