Consulta en materia de delimitación territorial

So far only vacuum states have been considered as the input to a GW detector. However the input fields may be prepared in any minimum uncertainty state, such as a squeezed vacuum state, that modifies the noise statistics before they enter the interferometer. For squeezed state injection into an interferometer, the evolution of quantum field operators remains the same and the input state is replaced by an input field that has been modified by a squeeze operation

|in_i= ˆS(R,⇠)_|0ai (4.69)

where R is the squeeze factor of the prepared state and ⇠ is the squeeze angle. Evalu- ating the expectation of the strain referenced quantum noise gives the spectral density proportional to

hin|hnh†_n0|ini=h0a|Sˆ†(r,⇠)hnh_n†0Sˆ(r,⇠)|0ai=h0a|hnsh†_ns0|0ai, (4.70)

where the field evolution can once again be completely factored into the operator evolution. The transformed strain referenced noise for a squeezing injected interferometer is given by

hns= ˆS†(R,⇠)hnSˆ(R,⇠). (4.71)

8_{As an aside: it is important to note here that the initial of the test masses is not relevant for the}

characterisation of the quantum noise component of these interferometers. As Braginskyet al. argue in [116], the initial zero point quantum and thermal motion of the instrument appear only at the pendular frequency of the mirror suspensions and are removed entirely from the data by regular high-pass filtering. As samples of the output light all mutually commute, Fourier components at higher frequencies are not influenced by the initial zero-point energy encoded in the pendular motion.

9_{Under estimated losses a factor of 3.2 more power is was estimated by [18] to realise a desirable}

Here interferometer and input state preparation have been completely factored into operator evolution leaving pre-squeezer fields in a vacuum state. Only the diagonal elements of this vacuum, Equation 4.60, give spectral densities of Sa1(⌦) = Sa2(⌦) = 1

with no cross spectral density components. The input vacuum field quadratures may again be thought of as independent random processes propagated though the instrument through a set of transforms.

For the simple Michelson the strain equivalent noise operator for readout in quadrature ⇣

is

h⇣_n= hpSQL

2Ke

i _[ˆa

2+ ˆa1(cot⇣ K)2]. (4.72)

Where ˆa1,2 are the quadrature operators at the input of the interferometer. Applying a

squeeze transform for the case of injected vacuum squeezing, the strain equivalent noise becomes [18]: h⇣_ns = Sˆ†(R,⇠)h⇣_nSˆ(R,⇠) = hpSQL 2_K 1 sin h ˆ

a1(coshRcos sinhRcos[ 2( +⇠)])

ˆ

a2(coshRsin sinhRsin[ 2( +⇠)]) i

(4.73)

where the coupling factor K and readout angle⇣ are trigonometrically factored into the quantity

= arccot(_K cot⇣). (4.74) The resulting spectral density follows in a similar manner to the previous cases, where vacuum spectral densities are given by Equation 4.61, giving

S_hs⇣ = h 2 SQL 2_K 1 sin2 ⇥ e 2R+ sinh 2R(1 cos[2( +⇠)])⇤. (4.75) Special cases follow for specific choices of squeezing angle or readout quadrature.

Frequency-independent squeezing

In the very early quantum noise analysis carried out by Caves, frequency-independent squeezed state injection was envisioned to provide an improvement to quantum shot noise at the expense of amplitude quadrature radiation pressure shot noise [14]. Fixing the homodyne readout angle to be in quadrature with the generated GW sideband signal,

⇣ = ⇡/2, and applying a phase quadrature squeezed state, such that ⇠ = ⇡/2, Equation 4.75 reduces to, S_hs⇣=⇡/2= h 2 SQL 2 ✓ 1 Ke2R+Ke 2R ◆ (4.76)

Recalling that_K is proportional to laser power I0, the injection of a phase squeezed state

is exactly equivalent to increasing laser power. The quantum shot noise is reduced at the expense of increased radiation pressure noise. Squeezing injection can therefore be used to complement increases in laser power in cases where radiation pressure noise at low frequency is dominated by other technical noise sources.

Frequency-independent squeezed state injection was demonstrated in the GEO600 detector [64, 66], enhanced LIGO [123] and earlier on the prototype Caltech-40 m interferometer [124]. As these interferometers were not radiation pressure noise limited, they provided an overall improvement to quantum noise with demonstrated shot noise reduction of 3.5 dB and 2.1 dB respectively for the GEO600 and enhanced LIGO detectors [66, 123]. However, injection of non-classical states of this type does not reduce the strain sensitivity below the standard quantum limit,

hns=hSQL s 1 2 ✓ 1 Ke2R +Ke2R ◆ hSQL, (4.77)

as the frequency-dependent back-action involves the same essential compromise over the interferometer’s full frequency band. Figure 4.8 shows the strain referenced sensitivity of a simple arm cavity Michelson for three choices of injection squeeze angle. Phase squeezed states act with an e↵ect similar to increased power, similar to Figure 4.7(a), with amplitude quadrature squeezing acting to reduce the e↵ective power. Fixed quadrature squeezing o↵ers only a reduction in a specific frequency band for quantum noise, only directly counteracting the interferometer’s back-action at the point at which the strain sensitivity grazes the SQL. Fixed quadrature squeezing is therefore not a true broadband QND measurement strategy. 101 102 103 [Hz] 10-25 10-24 10-23 10-22 Frequency

Normalised strain sensitivity

Vacuum input Phase squeezed Amp. squeezed Sque. angle Freq. dep. squeeze Shot noise Radiation preasure

Figure 4.8: Strain referenced sensitivity plot of di↵erent injected squeezing regimes for di↵erent choices of fixed angle and frequency dependant squeezing.

Frequency-dependant squeezing

To achieve some broadband reduction of quantum noise below the standard quantum limit, injected squeezed states must counteract the frequency-dependent squeezed states inher- ently generated by the interferometer’s opto-mechanical coupling interactions. Recalling the squeeze picture of the arm cavities’ response introduced in _§4.5.3, the quantum noise

for the squeezed injection case is of the form ˆ b(SQZ Inject)_j = Sˆ†(R,⇠) ˆS†(r, ) ˆR†( ✓)ˆajei2 Rˆ( ✓) ˆS(r, ) | {z } Interferometer Transform ˆ S(R,⇠) (4.78)

where✓, andrare as defined earlier in Equation 4.53. When the injection squeeze angle is engineered to match the interferometer opto-mechanical response, the injected squeezing may exactly follow and cancel the interferometer’s anti-squeeze components. Setting the input squeezing angle to the frequency dependence ⇠(⌦) = (⌦) = arccot[K(⌦)] and commuting the outer squeeze operator pairs gives

ˆ_b(SQZ Inject)

j = Sˆ†(r, )e R/2Rˆ†( ✓)ˆajei2 Rˆ( ✓)e R/2Sˆ(r, ) (4.79)

Thus squeezing with matched frequency dependence to the interferometer gives an e↵ective squeeze to the input fields exactly in the quadrature of the interferometer’s back-action- induced anti-squeezing, reducing the total quantum noise impact of both the phase and amplitude quadrature. Equivalently, selecting a readout in quadrature with signal gener- ation, ⇣ = ⇡/2, and applying the above injection squeezing frequency dependence for ⇠, Equation 4.75 yields the strain referenced quantum noise

S_hs⇣= (⌦)= h 2 SQL 2 ✓ 1 K +K ◆ e 2R. (4.80)

Frequency-dependent squeezed states use the same e↵ective rotation as strategies for variational readout, except that it is applied to the states before they enter the inter- ferometer and not after. Similarly their frequency-dependent rotation may be achieved using filter cavities to shape frequency dependence on reflection [18, 22, 125]. This o↵ers significant advantages as the rotation operation acts on the quantum fields before they enter the interferometer, leaving the signal component of the field unchanged. This is a significant disadvantage in the variational readout strategy as frequency-dependent homodyne readout partially cancels the signal fields in order to measure along minimum noise quadrature of the output fields: ultimately reducing the available signal to noise of the detection.

Early work of Chelkowski et al. demonstrated the first use of a a detuned filter cavity to produce frequency dependent squeezing in the 12-18 MHz frequency band from a frequency-independent squeezing source (an optical parametric amplifier) [126]. Recent studies of requirements for realistic audio-band squeezing rotation for Advanced LIGO have analysed the possibility of a shorter 16 m detuned cavity [127, 128]. Recently a similar scheme was implemented to produce the first audio-band frequency-dependent squeezed states from a detuned cavity with a pole at 1.2 kHz, demonstrating output squeezing on the order of 5.4 dB and 2.6 dB at high and low frequencies respectively [129]. Continued development of frequency-dependent quadrature rotation combined with the development of stable frequency-independent squeezed sources, such as that described in Chapter 6, o↵ers one of the strongest paths toward broadband QND measurement for laser interferometric GW detectors.