PT-symmetric amplifier .1 Theoretical model.1Theoretical model

PT-SYMMETRIC AMPLIFIER: BROADBAND SENSITIVITY IMPROVEMENT VIA COHERENT QUANTUM FEEDBACK

3.2 PT-symmetric amplifier .1 Theoretical model.1Theoretical model

Suppose the mode of a weak-signal sensor has annihilation/creation operators (𝑎,ˆ 𝑎ˆ^†), which can be reorganized into quadrature operators ˆ𝑎₁ = (𝑎ˆ + 𝑎ˆ^†)/√

2, ˆ

𝑎₂= (𝑎ˆ−𝑎ˆ^†)/(√

2𝑖), and the detector is coupled to a signalℎvia ˆ𝑉_𝑠 =−𝛼𝑎ˆ₁ℎ. For a GW detector, ˆ𝑎is the cavity mode,ℎis GW strain, while𝛼is achieved by strong carrier field in the cavity (ignoring the radiation-pressure effects [26]).

As shown in Fig. 3.2, ˆ𝑎and ˆ𝑏are coupled with rate𝜅, while ˆ𝑏and ˆ𝑐are amplified by a non-degenerate amplifier with rate𝜒, the Hamiltonian reads (Sec. 2 of Ref. [43]):

𝑉 =𝑖~𝜅(𝑎ˆ𝑏ˆ^†−𝑎ˆ^†𝑏ˆ) +𝑖~𝜒(𝑏ˆ^†𝑐ˆ^†−𝑏ˆ𝑐ˆ). (3.2) The mode ˆ𝑏 is read out from ˆ𝑣, which in turn gives input quantum noise ˆ𝑢. The resulting Heisenberg equations reads:

¤ˆ

𝑎 =𝑖 𝛼 ℎ−𝜅𝑏,ˆ 𝑐¤ˆ^†= 𝜒𝑏,ˆ 𝑏¤ˆ =−𝛾_𝑅𝑏ˆ +𝜅𝑎ˆ+𝜒𝑐ˆ^†+p

2𝛾_𝑅𝑢,ˆ (3.3) and ˆ𝑣=𝑢ˆ−p

2𝛾_𝑅𝑏ˆ. The combination

𝜒𝑎ˆ+𝜅𝑐ˆ^† (3.4)

forms an internal modethat responds purely to the signal, i.e., 𝑑(𝜒𝑎ˆ +𝜅𝑐ˆ^†)/𝑑 𝑡 = 𝑖 𝜒𝛼 ℎ, with the vacuum noise coherently canceled out. When 𝜒 = 𝜅, this internal mode coincides with theextraction mode

𝜅𝑎ˆ+ 𝜒𝑐ˆ^† (3.5)

that directly couples to mode ˆ𝑏, as shown in last equation in Eqs. (3.2). A similar system was analyzed in Ref. [36] as an amplifier to manipulate the injected signal,

a,a^†

−αa₁h

c,c^†

b,b^† ab^† b^†c^† signal

sWLC readout (with PT symmetry)

g G

u x,p

−μ α(a+a^†)x

X,P μ

α(c+c^†)X u vuWLC readout (no PT symmetry)

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

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unstable

stable

sWLC uWLC

stable

unstable

fixed pole Ω = 0 χ = 0

χ =κ

χ = 0 χ =κ

χ = 0 χ = 0

χ =κ χ =κ χ =κ

χ = 0

unstable

stable

sWLC uWLC

stable

unstable

fixed pole Ω = 0 χ = 0

χ =κ

χ = 0 χ =κ

χ = 0 χ = 0

χ =κ χ =κ χ =κ

χ = 0

Figure 3.2: Mode interaction and stability analysis. a, A weak signal sensor [mode (𝑎,ˆ 𝑎ˆ^†), plus (𝑥 , 𝑝) for the test mass in case of GW detectors] plus coherent quantum feedback with modes(𝑏,ˆ 𝑏ˆ^†),(𝑐,ˆ 𝑐ˆ^†). For the “sWLC readout”, the mode ˆ𝑎 couples to ˆ𝑏(via ˆ𝑎^†𝑏ˆ), which in turn couples to ˆ𝑐(via ˆ𝑏^†𝑐ˆ^†). The mode ˆ𝑏is coupled to an external continuum. The system isPT-symmetric when𝜅 = 𝜒. Mode ˆ𝑐may also couple to (𝑋 , 𝑃) with negative mass to evade the back-action noise caused by

𝑎’s coupling to(𝑥 , 𝑝)(see Fig. 3.6). The uWLC scheme (in gray) is recovered when ˆ

𝑎 directly couples to continuum; further setting𝜅→ 0 reduces it to a conventional detector. b, Poles trajectories of the two WLC dynamics in the complex plane, where arrows show the directions for increasing𝜒with𝜅fixed. Left: the sWLC has a fixed pole atΩ =0, invisible to the input-output relation in Eq. (3.6); the other two poles move up and down along the imaginary axis after meeting each other, until one reaches 0 at 𝜒=𝜅. For 𝜒 > 𝜅, the system becomes unstable. Right: the uWLC is unstable when 𝜒 ≠0.

Dalam dokumen for Detector Improvement and High-Precision Modeling of Binary Black Hole Ringdown Waveform (Halaman 79-84)