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 frequencyπ0, as illustrated in Fig. 4.2, can be described by the following Hamiltonian:
Λ
π»int=π~ππ (πΛπΛβ βπΛβ πΛ) +π~πΊ(πΛβ πΛβ βπΛπΛ), (4.1) with Λπ ,π ,Λ πΛ 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 ππ is the beam-splitter-type interaction strength between mode
Λ
π and Λπ, and πΊ describes the optomechanical interaction strength between mode Λπ and mode Λπ.
Considering the GW strain signal input β with coupling strength πΌ, as well as the coupling to the external bath Λπin with rate πΎ[32], we obtain the Heisenberg equations of motion for the three modes:
Β€Λ
π(π‘)=βππ πΛ(π‘) +π πΌ β(π‘),
Β€Λ
πβ (π‘)=πΊπΛ(π‘),
Β€Λ
π(π‘)=βπΎπΛ(π‘) +ππ πΛ(π‘) +πΊπΛβ (π‘) +p
2πΎπΛin(π‘),
(4.2)
with outgoing field given by Λπout(π‘)=βπΛin(π‘) +p
2πΎπΛ(π‘). The above equations can be solved in frequency domain via Fourier transform:
Λ
π(Ξ©) =F [πΛ(π‘)] β‘
β« β
ββ
π π‘ ππΞ©π‘πΛ(π‘), (4.3) where Ξ© is the sideband frequency in the rotating frame of frequency π0, and Λπ represents Λπ,π,Λ πΛ, or Λπin,out. The resulting solution for the outgoing field is:
Λ
πout(Ξ©) =πΞ©(πΎ +πΞ©) βπΊ2+π2
π
πΞ©(πΎβπΞ©) +πΊ2βπ2π Λ πin(Ξ©) +
πp
2πΎ ππ πΌ πΞ©(πΎβπΞ©) +πΊ2βπ2
π
β(Ξ©). (4.4)
Interestingly, regardless of the value ofπΊandππ , 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 Λπ = (πΛoutβ
93
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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=">AAACpXicdVHLahsxFJWnr9R9xGkXXWQjYkK7KPZMCLR0FdpNN4UEYidgT42kuR4LayQh3QmYYb6m2/aD+jfV2EOInfSC0OGc+77cKukxjv92okePnzx9tve8++Llq9f7vYM3Y29KJ2AkjDLumjMPSmoYoUQF19YBK7iCK7781uhXN+C8NPoSVxbSguVazqVgGKhZ7910wZCKj5vvZzXNWJ6Dq2e9fjyI10bvg6QFfdLa+eygczPNjCgL0CgU836SxBbTijmUQkHdnZYeLBNLlsMkQM0K8Gm1nqCmx4HJ6Ny48HRopGHvRlSs8H5V8OBZMFz4Xa0hH9ImJc4/p5XUtkTQYlNoXiqKhjbroJl0IFCtAmDCydArFQvmmMCwtK1MnOcOYKnSqqnCuVH1brLAZXTtRRXgbobbGY7pAtF+GQ4FMj0wLh/aZT5s1a0IK5t1/CdgI3bDpZLdu9wH45NBEvDFaf/sa3uzPXJIjsgHkpBP5Ix8J+dkRASpyS/ym/yJ3kc/ostovHGNOm3MW7Jl0ewfFsDSEw==</latexit> Λcβ
G
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Λb<latexit sha1_base64="cGc8wrnNZWhk9dzZbrYP2o24TUQ=">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</latexit><latexit sha1_base64="cGc8wrnNZWhk9dzZbrYP2o24TUQ=">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</latexit><latexit sha1_base64="cGc8wrnNZWhk9dzZbrYP2o24TUQ=">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</latexit><latexit sha1_base64="cGc8wrnNZWhk9dzZbrYP2o24TUQ=">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</latexit> β cΛβ signal
<latexit sha1_base64="ufZXSO0sBjZRmyW4q4RzA0hODAM=">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</latexit><latexit sha1_base64="ufZXSO0sBjZRmyW4q4RzA0hODAM=">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</latexit><latexit sha1_base64="ufZXSO0sBjZRmyW4q4RzA0hODAM=">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</latexit><latexit sha1_base64="ufZXSO0sBjZRmyW4q4RzA0hODAM=">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</latexit>
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=">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</latexit><latexit sha1_base64="+RsvLLT7iAJZ0YzZhS0lPFLJ2qo=">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</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=">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</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
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Λ a Λb
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β¦
!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 Λπ represents the differential mode of the arm cavity which couples to the GW strain β, Λπ represents the SRC (filter) mode, and Λπrepresents the mechanical mode of the suspended mirror. The coupling between Λπand Λπ( Λπand Λπ) is characterized byππ (πΊ). Mode Λπis coupled to the external ingoing field Λπin, which carries the vacuum noise, and the outgoing field Λπout, 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 Λπ, rather than the filter cavity mode Λπ. (c) The frequency-domain mode structure under single-mode (for both Λπand Λπ) and resolved sideband (for Λπ) approximations. The parametric interaction Λπβ πΛβ 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 π0, andΞ©is the sideband frequency with respect to it.
πΛβ
out)/(β
2π). The resulting signal-referred shot noise spectral density when mea- suring the phase quadrature is given by
πβ β(Ξ©)= Ξ©2πΎ2+ (πΊ2βπ2
π +Ξ©2)2 4πΎ π2
π πΌ2
. (4.5)
The sensitivity given different relations between πΊ and ππ is plotted in Fig. 4.3.
One interesting case is whenπΊ = ππ where the noise spectral density vanishes at DC as the signal response diverges:
πβ β(Ξ©) |πΊ=ππ = Ξ©2(Ξ©2+πΎ2) 4πΎ π2
π πΌ2
. (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
πΞ©(πΎ βπΞ©) +πΊ2βπ2
π =0. (4.7)