General linear measurements of the zero-point fluctuation

placement, we have

S_yy(ω)|_∆=ωm = κr

2 ¯G²₀[1 + 2Γ_AS(ω)]. (6.46) Similarly, by following the same line of thought, we get

Syy(ω)|_∆=−ωm = κr

2 ¯G²₀[1 + 2ΓS(ω)]. (6.47) The above two equations give identical results to Eq. (6.37). We therefore make a direct connection between the spectra of amplitude quadrature measured in the experiment and the photon emission spectra that is obtained if making photon-counting measurements.

6.3 General linear measurements of the zero-point fluctua-

are contributions from the free evolution of the initial Schr¨odingier operators (i.e., undisturbed by the environment), which decay over time, and get replaced by contributions from the environmental heat bath [integrals on the right-hand side of Eqs. (6.48a) and (6.48b)]. Note that for any oscillator with non-zero decay rate, it is essential to have bath operators entering over time, otherwise the commutation relation between position and momentum:

[ˆx_q(t),pˆ_q(t)] =i~ (6.50)

will not hold att >0 because of

[ˆxfree(t),pˆfree(t)] =i~e^−κmt. (6.51) This dictates that the heat bath must be such that the additional commutator from terms containing Fˆqexactly compensate for the decay in Eq. (6.51), which leads to the quantum fluctuation-dissipation theorem (see e.g., Ref. [37]).

It is interesting to note that this “replenishing” of commutators has a classical counterpart, since commutators are after all proportional to the classical Poisson Bracket. More specifically, for a classical oscillator with decay, we can write a similar relation for Poisson Brackets using the position and momentum of the oscillator, plus environmental degrees of freedom. The replenishing of the position-momentum Poisson Bracket by environmental variables, in classical mechanics, can also be viewed as a consequence of the conservation of phase-space volume, following the Liouville theorem.

A decaying oscillator’s phase-space volume will shrink, and violate the Liouville theorem — unless additional phase-space volume from the environmental degrees of freedom is introduced.

Nevertheless, the definitively quantum feature in our situation is a fundamental scale in the volume of phase space,~. Here we note that ifκmωm, when reaching the steady state with ˆxfree

and ˆqfreedecayed away, we have

∆x_q·∆p_q≈mω_m Z dω

2πS_xx^q (ω), (6.52)

whereS_xx^q ≡ |χ|²S_{F F}^q . AlthoughS_xx^q depends on the specific scenario, they are all constrained by a Heisenberg-like relation of,

S_xx^q (ω)≥2~Imχ(ω), (6.53)

which is a straightforward consequence of the commutation relation in Eq. (6.11). The equality is achieved at the ground state³. This enforces the same Heisenberg uncertainty relation:

∆xq·∆pq ≥~/2, (6.54)

3A generalization of this to thermal states will be the fluctuation-dissipation theorem [37].

as an ideal harmonic oscillator whose quantum fluctuations arise “on its own”, instead of having to be driven by the surrounding environment. Therefore, in the steady state, the zero-point fluctuation of the mechanical oscillator can be viewed as being imposed by the environment due to linearity of the dynamics.

6.3.2 Measuring the zero-point fluctuation

Having clarified the nature of quantum zero-point fluctuations of a mechanical oscillator in the steady state, let us argue that the effects seen in Ref. [24] are actually generic when one tries to probe such fluctuations, namely: the correlation between sensing and back-action noise can be at the level of the zero-point fluctuation itself.

Let us start our discussion here from Eq. (6.5), namely,

[ˆy(t),y(tˆ ⁰)] = 0, (6.55)

and the fact that ˆyconsists of sensing noise, back-action noise, and finally the zero-point fluctuation of the mechanical oscillator [cf. Eq. (6.8)]:

ˆ

y(t) = ˆz(t) α +α

Z t

−∞

χ(t−τ) ˆF_BA(τ)dτ+ ˆx_q(t). (6.56)

Here we have added a factorα, which labels the scaling of each term as the measurement strength which is proportional to the square root of the readout beam power. Let us assume that Eq. (6.55) continues to hold for the same set of ˆz and ˆF_BA, for a large set ofαandχ: basically the measuring device works for different mechanical oscillators with different measuring strengths.

Since Eq. (6.55) remains valid for all values of α, we extract terms with different powers of scaling, and obtain

[ˆz(t),z(tˆ ⁰)] =h

FˆBA(t),FˆBA(t⁰)i

= 0, (6.57)

and

Z t⁰

−∞

χ(t⁰−τ)h ˆ

z(t),FˆBA(τ)i dτ

− Z t

−∞

χ(t−τ)h ˆ

z(t⁰),FˆBA(τ)i dτ

+ [ˆx_q(t),xˆ_q(t⁰)] = 0, ∀t, t⁰. (6.58)

This becomes

Z +∞

0

χ(τ) [CzF(t−τ)−CzF(−t−τ)] dτ=−i~χ(t), (6.59)

for all values oft, where we have defined

CzF(t⁰−t)≡h ˆ

z(t),FˆBA(t⁰)i

. (6.60)

Here the dependence is only through t⁰ −t because the system is time-invariant. We also note that since ˆz is an out-going field,CzF must vanish when t >0 — otherwise any generalized force applied on ˆz can still affect ˆF at a later time [11, 35], which violates the causality. As proved in Appendix. 6.B, in order for Eq. (6.59) to be satisfied for all possible response functions of the oscillator, we must have

C_zF(t) =−i~δ₋(t), (6.61)

whereδ₋(t) is the Dirac delta function with support only fort <0. In other words,

[ˆz(t),FˆBA(t⁰)] =−i~δ₋(t⁰−t). (6.62)

Eq. (6.56), plus the commutation relations in Eqs. (6.57) and (6.62), then provide a general description of linear measuring devices which do not modify the dynamics of the mechanical oscillator

— simply from the requirement that the out-going field operators at different times must commute [cf. Eq. (6.55)]. In particular, the non-vanishing commutator [ˆxq(t),xˆq(t⁰)], which underlies the existence of the zero-point fluctuation, is canceled in a simple way by the non-vanishing commutator between the sensing noise and the back-action noise [cf. Eq. (6.62)].

Now turn to the noise content of the output ˆy(t):

Syy =Szz

α² + 2Re[χ^∗SzF] +α²S_{F F}^BA+S_xx^q . (6.63) Let us consider experiments with relatively low measurement strength, so that the first termS_zz/α² from the sensing noise dominates the output noise. The next-order terms contain: (i) correlation between the sensing noise and the back-action noise —SzF; and (ii) the mechanical fluctuation — S_xx^q . We assume nearly ground state for the mechanical oscillator

S_xx^q (ω)≈2~Imχ(ω), (6.64)

which, forκmωm, gives

Z dω

2πS^q_xx(ω)≈ ~ 2mωm

. (6.65)

IfS_zF(ω) does not change noticeably within the mechanical bandwidth, then Z dω

2π2Re[χ^∗(ω)SzF(ω)]≈ − 1 2mωm

ImSzF(ωm). (6.66)

Because of Eq. (6.62), the typical magnitude forSzF is naturally

|S_zF| ∼~. (6.67)

Therefore, contributions to the output noise from quantum correlation SzF and mechanical fluc- tuation S_xx^q can generically become comparable to each other when the mechanical oscillator is approaching the quantum ground state. The result presented in Ref. [24] therefore illustrates two typical cases of this generic behavior [cf. Eq. (6.20)].

6.3.3 Measuring an external classical force in presence of zero-point fluc- tuation

Finally, let us discuss the role of zero-point fluctuation in force measurement, when the mechanical oscillator is used as a probe of an external classical force not far away from the mechanical resonant frequency. Because of the commutation relations shown in Eqs. (6.57) and (6.62), a Heisenberg uncertainty relation exists between ˆz and ˆF_BA, which, in terms of the single-sided symmetrized spectrum density, can be written as

S_zz(ω)S^BA_{F F}(ω)−S_zF(ω)S_{F z}(ω)≥~². (6.68)

Note that because of Eq. (6.57), symmetrization is not necessary in the definition of Szz and S_{F F}^BA. When the two noises arenot correlated—SzF =SF z= 0, the above inequality represents a trade-off between sensing noise ˆzand back-action noise ˆFBA. Correspondingly, the force sensitivitySF, which is obtained by normalizing the displacement sensitivityS_yy with respect to the mechanical response functionχ, will have a lower bound:

SF(ω)≡ Syy(ω)

|χ(ω)|² = Szz(ω)

|χ(ω)|² +S_{F F}^BA(ω) +S_{F F}^q (ω)

≥ 2~

|χ(ω)|+ (4hni+ 2)~mκmωm. (6.69) If the mechanical oscillator is in its quantum ground state, namelyhni= 0, we obtain:

SF(ω)≥ 2~

|χ(ω)|+ 2~mκmωm≡S_F^Qtot. (6.70)

1 2

frequency range relevant to gravitational-wave

detectors table-top

experiments

Figure 6.3: (Color online.) Figure illustrating that total quantum limitation S_F^Qtot (red) for force sensitivity and contribution from zero-point fluctuation S_F^zp (blue). For clarity, we divide both by the SQL and use log-log scale.

The first term is the Standard quantum limit for force sensitivity with mechanical probes [11, 12]:

S^SQL_F ≡ 2~

|χ(ω)| = 2~mp

(ω²−ω_m²)²+κ²_mω². (6.71) The second term,

S_F^zp≡2~mκmωm, (6.72)

arising from the zero-point fluctuation due to mechanical quantization, also limits the sensitivity. As we can learn from Eqs. (6.68) and (6.69), the SQLS_F^SQL can be surpassed, in principle indefinitely, by building up quantum correlations between the sensing noise ˆz and the back-action noise ˆFBA

— although in practice the beating factor will be limited by the available optical power and the level of optical losses. However, the limit imposed by zero-point fluctuation cannot be surpassed — although can be mitigated by loweringκm.

Braginskyet al.[38] argued that mechanical quantization does not influence the force sensitivity when measuring a classical force with mechanical probes — one only needs to evaluate the quantum noise due to the readout field. But these authors had specifically pointed out that they were focusing on ideal mechanical probes with infinitely narrow bandwidth (κ_m→0) and observations outside of that frequency band. This corresponds to the situation of free-mass gravitational-wave detectors, in which the mechanical oscillator is the differential mode of four mirror-endowed test masses hung as pendulums with eigenfrequencies around 1 Hz and very high quality factor, while the detection band is above 10 Hz, well outside the pendulum resonance. Indeed, from Eqs. (6.71) and (6.72), we see that the effect of zero-point fluctuation is only significant not far away from resonance — which confirms Braginskyet al.’s result. More specifically, ifκmωm, we can write, for|ω−ωm| ωm,

S_F^SQL≈S_F^zp s

1 +

ω−ωm

κ_m/2 ²

. (6.73)

Figure 6.4: (Color online.) Figure illustrating effect of the mechanical decay rate (bandwidth)κm

on S_F^Qtot (solid) and S_F^zp (dashed) — the larger the mechanical bandwidth, the lower the force sensitivity (This plot is also in the log-log scale).

In particular, the limit imposed by zero-point fluctuation is equal to SQL on resonance, and becomes less important as |ω −ω_m| becomes comparable to or larger than the half bandwidth κ_m/2, as illustrated in Fig. 6.3.

Although the above relative comparison between the SQL and zero-point fluctuation indicates that the later plays a more important role near the oscillator’s resonant frequency, it is straightfor- ward to see from Eqs. (6.71) and (6.72) that on an absolute scale: (i) for a given oscillator at ground state,S_F^SQL(ω) is lower near mechanical resonance, whileS_F^zp(ω) is independent from frequency, and (ii) at any frequency, loweringκm, while fixingωmand keeping the oscillator at ground state, always results in lower noise, as illustrated in Fig. 6.4. Now suppose we are free to choose from ground-state mechanical oscillators with differentωm andκmas our probe, and that we are always able to reach the SQL at all frequencies, then: (i) if we know the frequency content of target signals, we should choose probes that are more resonant with the target, and (ii) regardless of signal frequency, we should always use probes with lowerκ_m, or equivalently, higher mechanical quality factor.