Quantum noise and the Standard Quantum Limit

Arising from the Heisenberg Uncertainty Principle, the quantum noise present within a classical interferometer2_{limits its sensitivity.}

Classical laser light in the coherent state, approximating what a standard laser will out- put, contains equal fractional amplitude and phase uncertainties. The phase fluctuations appearing at the sensor used to measure the output of the interferometer and amplitude fluctuations interacting with the interferometer’s test masses create noise at the measure- ment ports3_{. A fundamental limit to the sensitivity of classical interferometers arises from} the combination of these two effects; this is described in more detail in section 2.2.3.3. The following subsections summarise results from ref. [40].

2.2.3.1 Quantum shot noise

As described in section 2.2.1, open ports in the interferometer allow vacuum noise to enter, and when this noise is measured by a photodetector it appears as quantum shot noise. The phase fluctuations upon the light produce a varying photocurrent due to the stochastic arrival of photons at the sensor. The displacement-equivalent power spectral density of shot noise in an interferometer is

̃

𝑥2_shot = ℏ𝑐

2

𝑃 𝜔₀, (2.17)

in units of m2_{/Hz, for power 𝑃 and laser angular frequency 𝜔}

0. As this noise arises from spontaneous creation and annihilation of photons in space, it is a statistical random pro- cess and so the spectral density has equal power at all frequencies; it is white. The strain- equivalent power spectral density is equation (2.17) normalised to the arm length 𝐿:

̃ℎ2

shot=

ℏ𝑐2

𝑃 𝜔₀𝐿2. (2.18)

Since it scales with input power, the detrimental effect on the sensitivity due to phase un- certainty is mitigated by an increase in the classical light power injected into the interfer- ometer.

2.2.3.2 Quantum radiation pressure noise

Despite being massless, photons impart momentum to mirrors upon reflection inversely proportional to their wavelength. The strongest effect this has on an interferometer is 2_{Note the misnomer: a classical interferometer can still be limited by quantum noise. The name refers to}

the readout technique, namely the measurement of classical light intensity to determine displacement.

via dc radiation pressure, which arises from the classical light power circulating within the interferometer. In a suspended interferometer this radiation pressure effect extends the microscopic arm cavity length, with the equilibrium point being defined by the equivalence of the radiation pressure force to the suspension’s restoring force.

Quantumradiation pressure, on the other hand, arises from the fluctuating momentum im- parted onto the test masses by fluctuations in the number of photons present within the interferometer from the laser and loss points. As with quantum shot noise this effect is re- lated to the input power of the interferometer, but in this case fluctuations in the number of input photons creates amplitude noise that is transformed into equivalent strain noise via the dynamics of the mirror. Amplitude fluctuations upon the light beat with the classical field, creating a force noise. This fluctuating force changes the position of the mirror mi- croscopically via its mechanical susceptibility and this appears as phase noise at the output port. As the spectrum of noise from virtual photons is white the energy imparted to the mirror is the same at all frequencies. The mechanical susceptibility of a suspended mirror follows an inverse square law in frequency above the resonant frequency, and so in terms of strain this noise source is most important at low frequencies. The radiation pressure noise power spectral density is given in this case by

̃

𝑥2_rp= 𝑃 ℏ𝜔0

𝑐2_𝑚2_𝜔4, (2.19)

with reduced mirror mass 𝑚 and angular frequency of mirror oscillation 𝜔 = 2𝜋𝑓. The reduced mirror mass is the effective mass of the mechanical mode, given in the case of a Fabry-Perot Michelson interferometer as

𝑚= 𝑚1𝑚2

𝑚₁+ 𝑚₂, (2.20)

where 𝑚1and 𝑚2denote the individual cavity test masses. The strain-equivalent power spectral density is

̃ℎ2

rp =

𝑃 ℏ𝜔₀

𝑐2_𝑚2_𝜔4_𝐿2. (2.21)

The strain amplitude noise, ̃ℎ_rp, is proportional to 1

𝜔2 as expected from a free mass.

2.2.3.3 The Standard Quantum Limit

Note that equation (2.19) is proportional to power while equation (2.17) is inversely propor- tional to power. This implies the existence of a lower bound on the achievable sensitivity at a given observation frequency 𝑓 in the case of uncorrelated shot and radiation pressure

quantum noise sources. This bound, known as the standard quantum limit (SQL) [41], is a direct consequence of the Heisenberg Uncertainty Principle in a continuous measurement of a test mass.

The SQL is the point at which the sum power spectral density of shot and radiation pressure noise is minimised, and this occurs when the individual components are equal. For each laser power there exists a single frequency at which the SQL can be reached. The SQL forms a sensitivity limit with amplitude spectral density proportional to 1

𝑓 which can only be surpassed with special, sub-SQL techniques. The presence of cavities in the arms of a Michelson interferometer (formed by placing an additional, partially reflecting mirror in each arm) can enhance the power available to be able to reach the SQL. In terms of the strain-equivalent power spectral density, the SQL is specified for two free test masses separated by a distance 𝐿 by [42] ̃ℎ2 𝑆𝑄𝐿 = 8ℏ 𝑚𝜔2_𝐿2, (2.22) with units of Hz−1_.

The strain-equivalent power spectral density noise for a Michelson interferometer with arm cavities can be written with respect to the SQL [43] as

𝑆ℎ 𝑀 𝐼 = ̃ℎ2 𝑆𝑄𝐿 2 (₁ 𝜅 + 𝜅 ) , (2.23)

where the SQL is reached only at a single frequency. The term 𝜅 is the (dimensionless) opto-mechanical coupling factor [43]:

𝜅 = 𝑃0

𝑃_𝑆𝑄𝐿

2𝛾4

𝜔2(_𝛾2_{+ 𝜔}2), (2.24)

with 𝑃0the laser power at the test masses, 𝑃𝑆𝑄𝐿the laser power required to reach the SQL at the cavity pole frequency and 𝛾 the arm cavity half-bandwidth. 𝑃𝑆𝑄𝐿is given as [43]

𝑃_𝑆𝑄𝐿 = 𝑚𝐿

2_𝛾4

4𝜔₀ . (2.25)

The effect of 𝜅 is described in more detail in section 4.1.1.

The SQL is a locus defined at all frequencies, while the spectral density of a quantum noise limited interferometer touches the SQL at only one frequency. By injecting more photons into the interferometer to carry more information regarding the motion of the mirrors, we see a smaller shot noise spectral density while we see a larger radiation pressure noise spectral density [44]. This situation is illustrated in figure 2.3 for different input powers.

101 102 103 104 Frequency (Hz) 10−25 10−24 10−23 10−22 10−21 10−20 10−19 10−18 Strain ( 1 √ Hz ) 1 W 1 kW 1 MW SQL

Figure 2.3: The SQL for a Michelson interferometer with arm cavities of length 1 km, mirrors with reduced mass 50 kg and optimal frequency 100 Hz, along with quantum noise limited sensitivity curves for three different intracavity powers. The effect of the cavity pole frequency is visible in the case of the blue curve. The higher the intracavity power, the higher the strain sensitivity can be pushed, but at the expense of higher radiation pressure noise and thus higher optimal frequency for a given interferometer configuration. Quantum non-demolition techniques can be used to surpass the SQL (see section 2.4).

An important distinction to make here is that the SQL is defined for uncorrelated shot and radiation pressure noise. Techniques exist in theory and practice to reduce overall noise by introducing correlations between the two noise components with so-called quantum non- demolitioninterferometry, and this is discussed in greater detail in section 2.4 and chapter 4.