ESTADO DE SITUACIÓN ACTUAL DE LAS PYMES HOSTELERAS EN EL

In this section we calculate the radiation pressure contribution ˆqrp contained in the mechanical signal ˆq. Naturally one obtains the mechanical signal from the matrix

˜

M[Ω], where ˆq[Ω] is described in the (2N + 1)th _{row. (We remind ourselves that}

˜

M[Ω] is the extended description of the optomechanical system, which includes the cavity modes as well as the mechanical oscillator.) Without further specifying ˜M[Ω] we can write

ˆ

q[Ω] = ˜M2N+1·δ ~XM[Ω]≡qˆrp[Ω] +χef f(Ω) ˆFthe[Ω]. (7.54)

Here the radiation pressure part contains all the parts that are proportional to noise operators acting on the optical field quadratures and the rest describes the response to external forces. We point out that the effective susceptibility χef f, which is

also defined by ˜M[Ω], includes damping or heating effects that are also a response to the optical field. Yet its influence on the mechanical motion is not considered a radiation pressure signal here and it is thus not contained in ˆqrp_{, because it originates}

from the classical and noise-free portion of the radiation pressure and is as such a deterministic (i.e., time reversible) contribution.

Here the effective susceptibility play a minor role and its knowledge is not required for most of the calculations. It is sufficient to know that χef f(Ω) ˆFthe is describes

the displacement of the mechanics. In a simplifying way one could say that the last chapter was about χef f(Ω) and its integral, here we are interested in the light

fluctuations that limit the resolution of the mechanical signal.

More importantly one could get the impression that in any case the full optomechan- ical system needs to be solved to obtain the expression for ˜M2N+1 that describes the

7.3 Multi-mode cavity optomechanics 135

our time with the initial calculations that involve only the (much simpler) matrix ˜

S[Ω] for the cavity modes. However strikingly we find that the first 2N entries of ˜

M2N+1 – exactly the ones that describe the radiation pressure contribution – can

be expressed by the matrix ˜S[Ω] and the effective susceptibility χef f(Ω) only.

˜ M2N+1,1, . . . ,M˜2N+1,2N =χef f(Ω) N X j=1 gmS˜2j−1 =gmχef f(Ω)T~0|·S[Ω]˜ (7.55)

From Equation 7.55 we obtain an expression for the cavity length fluctuations due to radiation pressure and, as the effective susceptibility appears, we can directly infer the expression for the radiation pressure force fluctuations.

ˆ qrp[Ω] = gmχef f(Ω)T~0|·S[Ω]˜ ·δ ~X[Ω] ⇒δFˆrp[Ω] = ~gm x0 ~ T₀|·S[Ω]˜ ·δ ~X[Ω] (7.56) = ~gm x0 ~ T₀|·S[Ω]˜ ·√κ ~T0δ ~Xsum[Ω] + √ κ ~Tπ/2δ ~Ysum[Ω] (7.57)

Notably the expression T~₀|·S[Ω]˜ ·δ ~X is the sum over the amplitude fluctuations, a result, which one could have guessed as well. Moreover, in the last line, we do not differentiate between input noise and vacuum fluctuations,7_{which is reasonable when}

the dissipative heat baths have the same temperature. Again the expression is suited for direct evaluation with the Mathematica code in appendix E, where we provide examples for the three resonance transducer and the single mode optomechanical scheme with arbitrary detuning ∆.

Now we have obtained all the terms that contribute to the output fluctuations and we summarize the results the results.

ˆ X_θout[Ω] = Xˆx,θ[Ω] + ˆXF,θ[Ω] + ˆXm,θ[Ω] (7.58) ˆ Xx,θ[Ω] = √ ηcκ ~T_Θ| ·S[Ω]˜ ·δ ~X[Ω]−Xˆθin[Ω] (7.59) ˆ XF,θ[Ω] = λθ(Ω)χef f(Ω) ˆFrp[Ω] (7.60) ˆ Xm,θ[Ω] = λθ(Ω)χef f(Ω) ˆFthe[Ω] (7.61)

In the following we will use these results to calculate the output spectrum, as well as the scaled imprecision spectrum that reflects the readout limitations of the trans- ducer.

136 7. Multi cavity mode transducers

Calculating the imprecision spectrum

In this paragraph we introduce the imprecision spectral density S_θimp(Ω), which has the unit of a displacement sensitivity and is a function of the noise acting on the optical modes. The displacement imprecision is related to the output fluctuations via the absolute value square of the transduction function.

S_θθout(Ω) =|λθ(Ω)| 2 S_θimp(Ω) +|λθ(Ω)| 2 |χef f(Ω)| 2 S_{F F}the(Ω) (7.62)

Then we can calculate S_xx,θimp(Ω) from the scaled optical contributions of the output fluctuation spectrum. δ(Ω + Ω0) 2π S imp θ (Ω) = D ˆ Xx,θ[Ω] + ˆXF,θ[Ω] Xˆx,θ[−Ω0] + ˆXF,θ[−Ω0] E |λθ(Ω)|2 . (7.63) Computing the expectation values, one obtains three terms, associated with different combinations of ˆIx,Θ[Ω] and ˆIF,Θ[Ω].

S_xx,θimp(Ω) = Sxx,θ(Ω) +|χef f(Ω)|2SF F(Ω) (7.64)

+χef f(Ω)SxF,θ(Ω) +χ∗ef f(Ω)SF x,θ(Ω) (7.65)

Here the photo shot noise spectral densitySxx,θ(Ω) is the inverse of the susceptibility |λΘ(Ω)|2 and SF F(Ω) is the quantum backaction force spectral density.

Sxx,θ(Ω) =

1

2|λθ(Ω)|

−2

(7.66) In the the analysis of experimental data the equivalent shot noise spectral density is typically evaluated at Ω = Ωm to scale the whole (flat) noise background. Plot-

ting Sxx,θ(Ω) as a function of Ω also reflects the cavity transmission and gives the

impression of a result that is not measured in the experiment. Moreover, when a calibration signal detuned from the mechanical resonance is applied, one needs ver- ify whether is experiences a different sensitivity.

The quantum backaction contribution, which is more precisely a radiation pressure noise backaction, is explicitly given by the expression

SF F(Ω) = 1 2 ˆ R dΩ0 2π D ˆ Frp[Ω] ˆFrp[−Ω]0E. (7.67)

In contrast to the photon shot noise the imprecision coming from radiation pressure is restricted to the bandwidth defined by the mechanical oscillation. In fact it leads to a real world mechanical displacement and it is this false signal that limits the sensitivity. However there is a number of experiments that aim at the observation of radiation pressure and up to now it is typically the thermal noise of the mechanics that overwhelms the radiation pressure effect [239, 177, 240]. To this end we have proposed the triple mode transducer scheme in section 7.1, where the radiation pressure effect can be largely enhanced.