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From single-mode approximation to full analysis

OPTOMECHANICAL REALIZATION OF PT-SYMMETRIC INTERFEROMETER

4.2 From single-mode approximation to full analysis

In this section, we will first recap the idealized Hamiltonian dynamics under the single-mode approximation as analyzed in Refs. [1, 31] for the sWLC, and also

revisit the stability requirement. We will then introduce the analyzing framework considering the realistic setup, by abandoning the approximations applied in the previous treatment. The detailed analysis in the frequency and time domain will be carried in Sec. 4.3 and 4.4, respectively.

The idealized sWLC mode interaction in the rotating frame of frequencyπœ”0, as illustrated in Fig. 4.2, can be described by the following Hamiltonian:

Λ†

𝐻int=𝑖~πœ”π‘ (π‘ŽΛ†π‘Λ†β€ βˆ’π‘ŽΛ†β€ π‘Λ†) +𝑖~𝐺(π‘Λ†β€ π‘Λ†β€ βˆ’π‘Λ†π‘Λ†), (4.1) with Λ†π‘Ž ,𝑏 ,Λ† 𝑐ˆ being the quantum operators of the differential optical mode of the arm cavity, the SRC (i.e., the filter cavity) optical mode, and the mechanical mode, respectively. Here πœ”π‘  is the beam-splitter-type interaction strength between mode

Λ†

π‘Ž and ˆ𝑏, and 𝐺 describes the optomechanical interaction strength between mode ˆ𝑏 and mode ˆ𝑐.

Considering the GW strain signal input β„Ž with coupling strength 𝛼, as well as the coupling to the external bath ˆ𝑏in with rate 𝛾[32], we obtain the Heisenberg equations of motion for the three modes:

Β€Λ†

π‘Ž(𝑑)=βˆ’πœ”π‘ π‘Λ†(𝑑) +𝑖 𝛼 β„Ž(𝑑),

Β€Λ†

𝑐†(𝑑)=𝐺𝑏ˆ(𝑑),

Β€Λ†

𝑏(𝑑)=βˆ’π›Ύπ‘Λ†(𝑑) +πœ”π‘ π‘ŽΛ†(𝑑) +𝐺𝑐ˆ†(𝑑) +p

2𝛾𝑏ˆin(𝑑),

(4.2)

with outgoing field given by ˆ𝑏out(𝑑)=βˆ’π‘Λ†in(𝑑) +p

2𝛾𝑏ˆ(𝑑). The above equations can be solved in frequency domain via Fourier transform:

Λ†

π‘œ(Ξ©) =F [π‘œΛ†(𝑑)] ≑

∫ ∞

βˆ’βˆž

𝑑 𝑑 π‘’π‘–Ξ©π‘‘π‘œΛ†(𝑑), (4.3) where Ξ© is the sideband frequency in the rotating frame of frequency πœ”0, and Λ†π‘œ represents Λ†π‘Ž,𝑏,Λ† 𝑐ˆ, or ˆ𝑏in,out. The resulting solution for the outgoing field is:

Λ†

𝑏out(Ξ©) =𝑖Ω(𝛾 +𝑖Ω) βˆ’πΊ2+πœ”2

𝑠

𝑖Ω(π›Ύβˆ’π‘–Ξ©) +𝐺2βˆ’πœ”2𝑠 Λ† 𝑏in(Ξ©) +

𝑖p

2𝛾 πœ”π‘ π›Ό 𝑖Ω(π›Ύβˆ’π‘–Ξ©) +𝐺2βˆ’πœ”2

𝑠

β„Ž(Ξ©). (4.4)

Interestingly, regardless of the value of𝐺andπœ”π‘ , the outgoing field is not squeezed, as the modulus of input-to-output transfer function remains equal to unity. The signal is contained entirely in the phase quadrature [33, 34] defined as Λ†π‘Œ = (𝑏ˆoutβˆ’

93

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Λ† a,<latexit sha1_base64="4AnPCbZrEKQxwgz0Hd8xgEW9mRg=">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</latexit><latexit sha1_base64="4AnPCbZrEKQxwgz0Hd8xgEW9mRg=">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</latexit><latexit sha1_base64="4AnPCbZrEKQxwgz0Hd8xgEW9mRg=">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</latexit><latexit sha1_base64="4AnPCbZrEKQxwgz0Hd8xgEW9mRg=">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</latexit> aˆ†

Λ†b,<latexit sha1_base64="5zX0Pqok8cKCxTwLIYdpY3Tgl9E=">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</latexit><latexit sha1_base64="5zX0Pqok8cKCxTwLIYdpY3Tgl9E=">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</latexit><latexit sha1_base64="5zX0Pqok8cKCxTwLIYdpY3Tgl9E=">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</latexit><latexit sha1_base64="5zX0Pqok8cKCxTwLIYdpY3Tgl9E=">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</latexit> Λ†b†

Λ† c,<latexit sha1_base64="2ql0QdTzBnjYTKcWb8GWekv8Tt8=">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</latexit><latexit sha1_base64="2ql0QdTzBnjYTKcWb8GWekv8Tt8=">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</latexit><latexit sha1_base64="2ql0QdTzBnjYTKcWb8GWekv8Tt8=">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</latexit><latexit sha1_base64="2ql0QdTzBnjYTKcWb8GWekv8Tt8=">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</latexit> Λ†c†

G

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readout

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h

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Λ† aout

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Λ† ain

<latexit sha1_base64="+RsvLLT7iAJZ0YzZhS0lPFLJ2qo=">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</latexit><latexit sha1_base64="+RsvLLT7iAJZ0YzZhS0lPFLJ2qo=">AAACnXicdVFNixNBEO2MH7vGj83q0YONYcFTMiOCsqdFETyIrGA2C8kQqjuVSZOe7qa7ZiEM+SVe9Uf5b+xJBjFZLWh4vFevqrpKOK0CpemvTnLn7r37R8cPug8fPX5y0jt9ehVs5SWOpNXWXwsIqJXBESnSeO08Qik0jsXqQ6OPb9AHZc03WjvMSyiMWigJFKlZ72S6BOIwq6e+5MpsZr1+Oki3wW+DrAV91sbl7LRzM51bWZVoSGoIYZKljvIaPCmpcdOdVgEdyBUUOInQQIkhr7eTb/hZZOZ8YX18hviW/dtRQxnCuhQxswRahkOtIf+lTSpavMtrZVxFaOSu0aLSnCxv1sDnyqMkvY4ApFdxVi6X4EFSXNZeJSEKj7jSed10EcLqzWGxyM35NotrpMMKf/5wxpdE7nw4lARmYH0xdKti2Kp7DqeadfzHsBO78VLZ4V1ug6vXgyzir2/6F+/bmx2z5+wle8Uy9pZdsE/sko2YZBX7zn6wn8mL5GPyOfmyS006recZ24tk/BuGQc7m</latexit><latexit sha1_base64="+RsvLLT7iAJZ0YzZhS0lPFLJ2qo=">AAACnXicdVFNixNBEO2MH7vGj83q0YONYcFTMiOCsqdFETyIrGA2C8kQqjuVSZOe7qa7ZiEM+SVe9Uf5b+xJBjFZLWh4vFevqrpKOK0CpemvTnLn7r37R8cPug8fPX5y0jt9ehVs5SWOpNXWXwsIqJXBESnSeO08Qik0jsXqQ6OPb9AHZc03WjvMSyiMWigJFKlZ72S6BOIwq6e+5MpsZr1+Oki3wW+DrAV91sbl7LRzM51bWZVoSGoIYZKljvIaPCmpcdOdVgEdyBUUOInQQIkhr7eTb/hZZOZ8YX18hviW/dtRQxnCuhQxswRahkOtIf+lTSpavMtrZVxFaOSu0aLSnCxv1sDnyqMkvY4ApFdxVi6X4EFSXNZeJSEKj7jSed10EcLqzWGxyM35NotrpMMKf/5wxpdE7nw4lARmYH0xdKti2Kp7DqeadfzHsBO78VLZ4V1ug6vXgyzir2/6F+/bmx2z5+wle8Uy9pZdsE/sko2YZBX7zn6wn8mL5GPyOfmyS006recZ24tk/BuGQc7m</latexit><latexit sha1_base64="+RsvLLT7iAJZ0YzZhS0lPFLJ2qo=">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</latexit>

uWLC scheme<latexit sha1_base64="6WZK88f6T1UDjdxbUWFIVUSlLBs=">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</latexit><latexit sha1_base64="6WZK88f6T1UDjdxbUWFIVUSlLBs=">AAACnnicdVHbahsxEJU3vaTuzUkf+yJqAn2yd0uhpU8hgdKHlqZQxwZ7MZI8XgvrhjRrMIv/JK/NP+VvqrWXUjvtgOBwzpyZ0Qx3SgZM07tWcvTg4aPHx0/aT589f/Gyc3J6HWzpBQyEVdaPOAugpIEBSlQwch6Y5gqGfHlZ68MV+CCt+YlrB7lmhZFzKRhGatrpTLym5fDr5YQGsQAN00437aXboPdB1oAuaeJqetJaTWZWlBoMCsVCGGepw7xiHqVQsGlPygCOiSUrYByhYRpCXm1H39CzyMzo3Pr4DNIt+7ejYjqEteYxUzNchEOtJv+ljUucf8wraVyJYMSu0bxUFC2t90Bn0oNAtY6ACS/jrFQsmGcC47b2KnFeeIClyqu6C+dWbQ6LRW5Gt1lUAR5W+POHM7pAdJ/6fYHM9Kwv+m5Z9Bt1z+FkvY7/GHZiO14qO7zLfXD9rpdF/ON99/yiudkxeU3ekLckIx/IOflCrsiACLIiN+QXuU1o8jn5lnzfpSatxvOK7EUy+g23dc70</latexit><latexit sha1_base64="6WZK88f6T1UDjdxbUWFIVUSlLBs=">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</latexit><latexit sha1_base64="6WZK88f6T1UDjdxbUWFIVUSlLBs=">AAACnnicdVHbahsxEJU3vaTuzUkf+yJqAn2yd0uhpU8hgdKHlqZQxwZ7MZI8XgvrhjRrMIv/JK/NP+VvqrWXUjvtgOBwzpyZ0Qx3SgZM07tWcvTg4aPHx0/aT589f/Gyc3J6HWzpBQyEVdaPOAugpIEBSlQwch6Y5gqGfHlZ68MV+CCt+YlrB7lmhZFzKRhGatrpTLym5fDr5YQGsQAN00437aXboPdB1oAuaeJqetJaTWZWlBoMCsVCGGepw7xiHqVQsGlPygCOiSUrYByhYRpCXm1H39CzyMzo3Pr4DNIt+7ejYjqEteYxUzNchEOtJv+ljUucf8wraVyJYMSu0bxUFC2t90Bn0oNAtY6ACS/jrFQsmGcC47b2KnFeeIClyqu6C+dWbQ6LRW5Gt1lUAR5W+POHM7pAdJ/6fYHM9Kwv+m5Z9Bt1z+FkvY7/GHZiO14qO7zLfXD9rpdF/ON99/yiudkxeU3ekLckIx/IOflCrsiACLIiN+QXuU1o8jn5lnzfpSatxvOK7EUy+g23dc70</latexit>

sWLC scheme

<latexit sha1_base64="YGpwc39ezLvqPftKik6cR0//ecw=">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</latexit><latexit sha1_base64="YGpwc39ezLvqPftKik6cR0//ecw=">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</latexit><latexit sha1_base64="YGpwc39ezLvqPftKik6cR0//ecw=">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</latexit><latexit sha1_base64="YGpwc39ezLvqPftKik6cR0//ecw=">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</latexit>

Λ† a Λ†b

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⌦

!0 !0+!m

(a)

(b)

(c)

Figure 4.2: Idealized mode interaction structure of the optomechanical system illustrated in Fig. 4.1. (a) The sWLC scheme. Here Λ†π‘Ž represents the differential mode of the arm cavity which couples to the GW strain β„Ž, ˆ𝑏 represents the SRC (filter) mode, and ˆ𝑐represents the mechanical mode of the suspended mirror. The coupling between Λ†π‘Žand ˆ𝑏( ˆ𝑏and ˆ𝑐) is characterized byπœ”π‘  (𝐺). Mode ˆ𝑏is coupled to the external ingoing field ˆ𝑏in, which carries the vacuum noise, and the outgoing field ˆ𝑏out, which carries the signal and will be read out. (b) The uWLC scheme for comparison. In the intracavity readout scheme presented in Ref. [30], the external fields couple to the arm cavity mode Λ†π‘Ž, rather than the filter cavity mode ˆ𝑏. (c) The frequency-domain mode structure under single-mode (for both Λ†π‘Žand ˆ𝑏) and resolved sideband (for ˆ𝑏) approximations. The parametric interaction ˆ𝑏†𝑐ˆ†is realized by the optomechanical coupling under blue-detuned pumping by the mechanical resonant frequency. Note that all mode operators are defined in the rotating frame of frequency πœ”0, andΞ©is the sideband frequency with respect to it.

𝑏ˆ†

out)/(√

2𝑖). The resulting signal-referred shot noise spectral density when mea- suring the phase quadrature is given by

π‘†β„Ž β„Ž(Ξ©)= Ξ©2𝛾2+ (𝐺2βˆ’πœ”2

𝑠 +Ξ©2)2 4𝛾 πœ”2

𝑠𝛼2

. (4.5)

The sensitivity given different relations between 𝐺 and πœ”π‘  is plotted in Fig. 4.3.

One interesting case is when𝐺 = πœ”π‘  where the noise spectral density vanishes at DC as the signal response diverges:

π‘†β„Ž β„Ž(Ξ©) |𝐺=πœ”π‘  = Ξ©2(Ξ©2+𝛾2) 4𝛾 πœ”2

𝑠𝛼2

. (4.6)

The system stability, as analyzed in Ref. [1], is determined by the poles of the transfer function in Eq. (4.4), i.e., the roots of

𝑖Ω(𝛾 βˆ’π‘–Ξ©) +𝐺2βˆ’πœ”2

𝑠 =0. (4.7)