Quantum limit for laser interferometric gravitational wave - PowerPoint PPT Presentation

Quantum limit for laser interferometric gravitational wave detectors from optical dissipation Takuya Kawasaki September 28, 2018 @Ando lab. seminar

Contents • Background for the paper • Review of “Quantum limit for laser interferometric gravitational wave detectors from optical dissipation” • Discussion � 2

Quantum noise • Quantum noise is one of the major noise in gravitational- wave detectors. • Quantum noises arise from quantum fluctuations of the optical fields. Sensitivity of KAGRA � 3

Shot noise and radiation pressure noise • Quantum noise — Shot noise + Radiation pressure noise • Shot noise Phase fluctuation • Radiation pressure noise Phase fluctuation <- Amplitude fluctuation Suspended mirror (converter) • Trade-o ff between shot noise and radiation pressure noise Standard Quantum Limit (SQL) � 4

̂ ̂ Standard Quantum Limit • Standard quantum limit is a fundamental limit from quantum noise. • However, SQL can be surpassed by many schemes. • Because the Poisson distribution for photon number is assumed in derivation of SQL • In other words, photon numbers at two di ff erent times are independent. n ( t ) n ( t + Δ t ) classical laser beam commute • SQL is a kind of “classical” limit from quantum fluctuation. � 5

Beating SQL • one of the basic ideas to beat SQL is squeezing “squeez” fluctuation in the amplitude/phase quadrature plane ex.) Ponderomotive squeezing 
 (Optomechanical squeezing) Read a signal in the squeezed direction ex.) Homodyne detection � 6

Homodyne detection Output Phase 
 a 2 quadrature Signal Local Oscillator θ a 1 Amplitude 
 quadrature Signal • Homodyne detection measures the quadrature in θ -direction. 
 ( θ is the phase of local oscillator) � 7

Ponderomotive squeezing • Amplitude fluctuation make phase fluctuation through suspended mirrors. • Amplitude fluctuation and part of phase fluctuation have correlation. • Correlation means squeezing. Ponderomotive squeezing • Only in low frequency range � 8

input squeezing 
 & filter cavity • High frequency reflected by the filter cavity • Low frequency entering the filter cavity the quadrature rotates radiation pressure noise PRD 65.022002 Sensitivity shot noise Frequency � 9

Another quantum limit: 
 Quantum Cramer-Rao bound • SQL can be beaten What is the more fundamental limit? A. Quantum Cramer-Rao bound (PRL 119, 050801) • Enomomoto-san’s seminar on Feb. 9, 2017 � 10

Enomoto-san’s slide (last page) Feb 9 2017, Ando Lab Seminar 5. Discussion = Effect of loss = The quantum Cramér-Rao bound does not come from some trade-off. => the limit can ideally be infinitely small In reality, there are always losses everywhere. - perfect backaction evasion is impossible - squeezing (internal/external) degrades Incorporating the effect of losses will be important END 22 � 11

Review part: “Quantum limit for laser interferometric gravitational wave detectors from optical dissipation”

Introduction ↓ Fundamental quantum limit ↓ Derivation ↓ Application: Advanced LIGO

Introduction ↓ Fundamental quantum limit ↓ Derivation ↓ Application: Advanced LIGO

Summary for background • Standard quantum limit can be beaten by many means • Quantum Cramer-Rao bound (QCRB) fundamental idealistic in terms of optical loss • New limit presented the paper includes the e ff ect of optical loss � 15

Purpose of the paper • Giving a new quantum limit that highlights the role of optical losses • Useful to optimize the detector configuration in the presence of optical losses Previous research PRD 98.044044 Designing the detector configuration 
 ↓ 
 Numerical computation for the sensitivity Future Analytical calculation for the limit sensitivity 
 ← configuration- 
 independent ↓ 
 Designing the detector configuration � 16

Introduction ↓ Fundamental quantum limit ↓ Derivation ↓ Application: Advanced LIGO

New quantum limit 
 from optical dissipation • The result sensitivity consists of the quantum Cramer-Rao bound term and the loss term hh = S QCRB S min + S ϵ hh hh Loss term General QND scheme Quantum Cramer-Rao bound � 18

Review of QCRB ℏ 2 c 2 S QCRB ( Ω ) = hh 2 S PP ( Ω ) L 2 • is the spectral density of the power fluctuation in the S PP ( Ω ) interferometer arms • QCRB can be approached with optimal frequency-dependent homodyne detection. • According to number-phase uncertainty relation, a large uncertainty in the photon number is necessary to achieve accurate measurement of the phase. S QCRB ( Ω ) ∝ ( S PP ( Ω )) − 1 hh only if minimum uncertainty states are maintained � 19

Loss term (main result) ϵ arm + ( 1 + Ω 2 γ 2 ) ℏ c 2 T itm ϵ src S ϵ hh = + α T src ϵ ext 4 L 2 ω 0 P 4 • P: optical power inside each arm • ω 0 : laser frequency • ε arm , ε src : internal loss of the arm, the signal recycling cavity (10 -6 for 1ppm) • ε ext : external loss (including the quantum ine ffi ciency of the photodetector) • γ : bandwidth of the arm cavity • T itm : transmission of ITM • T src : e ff ective transmission of SRC • α = 1 if the internal squeezing is optimized 
 1/4 if the internal squeezing is negligible � 20

The arm cavity loss ϵ arm + ( 1 + Ω 2 γ 2 ) ℏ c 2 T itm ϵ src S ϵ hh = + α T src ϵ ext 4 L 2 ω 0 P 4 • The arm cavity loss sets a flat limit • Fluctuation is directly mixed with the GW signal inside the arm • The e ff ect is identical to GW signal � 21

The SRC loss ϵ arm + ( 1 + Ω 2 γ 2 ) ℏ c 2 T itm ϵ src S ϵ hh = + α T src ϵ ext 4 L 2 ω 0 P 4 • The SRC loss gives worse sensitivity at high frequency • because the GW signal inside the arm is suppressed when it exceed the arm cavity bandwidth. � 22

The external loss ϵ arm + ( 1 + Ω 2 γ 2 ) ℏ c 2 T itm ϵ src S ϵ hh = + α T src ϵ ext 4 L 2 ω 0 P 4 • The e ff ect of external loss depends on the transmission of the SRC • note: The transmission of the SRC can be frequency dependent current GW detectors configuration � 23

Introduction ↓ Fundamental quantum limit ↓ Derivation ↓ Application: Advanced LIGO

Derivation: 
 Model of GW detectors • Dark fringe Quantum noise is originated from vacuum entering from AS port. � 25

Derivation: 
 Simplification • signal recycling cavity -> an e ff ective mirror • Phase factors in the cavities are canceled by some active components. (Non-perfect cancelation will lead to worse sensitivity.*) M intra → M rot , M opt → M sqz *PRL 115, 211104 � 26


 
 ̂ Derivation: 
 Calculation • input-output relation 
 a out = M io ̂ T src ϵ int M c ̂ ϵ ext ̂ a in + n int + n ext + v h GW • where 
 M io ≡ − R src I + T src M c M rot M sqz M rot M c ≡ [ I − − 1 R src M rot M sqz M rot ] • Detector response v ≡ T src M c v 0 = ( 0, 2 ′ � ω 0 L 2 P / ( ℏ c 2 ) ) � 27

Derivation: 
 Approximation • From input-output relation, minimum fluctuation can be calculated complicated • Approximation T src << 1: to enhance signal recycling for large amplitude fluctuation in the arm (small QCRB) Θ << 1: to focus the frequency range smaller than FSR r << 1: for large amplitude fluctuation in the arm Amplitude fluctuation → Phase fluctuation � 28

Derivation: 
 Result expression hh ≈ S QCRB S min + S ϵ hh hh ℏ c 2 ( δ 2 − 4 r 2 ) 2 e − 2 r input T 2 src + 16 Θ 2 δ ≡ S QCRB = 16 L 2 ω 0 PT src [ δ 2 + 4 r 2 + 4 δ r sin ( θ + θ 0 ) ] hh θ 0 ≡ cot − 1 ( 4 Θ / T src ) • case1: optimal internal squeezing (r = δ /2) ℏ c 2 S QCRB S ϵ 4 L 2 ω 0 P ( ϵ int + T src ϵ ext ) = 0 hh = hh • case2: negligible internal squeezing (r = 0) = ℏ c 2 δ 2 e − 2 r input ℏ c 2 4 L 2 ω 0 P ( ϵ int + T sr 4 ϵ ext ) S QCRB S ϵ hh = hh 16 T src L 2 ω 0 P � 29

Introduction ↓ Fundamental quantum limit ↓ Derivation ↓ Application: Advanced LIGO

Application for Advanced LIGO: 
 Parameters ϵ arm + ( 1 + Ω 2 γ 2 ) ℏ c 2 T itm ϵ src S ϵ hh = + α T src ϵ ext 4 L 2 ω 0 P 4 • ε arm : 100 ppm • ε src : 1000 ppm (mainly contributed from the beam splitter) • ε ext : 0.1 (mainly contributed from OMC and photodiode quantum ine ffi ciency) • T itm : 0.014 default broadband detection mode • T src : 0.14 • Squeezing: 30 dB � 31

Application for Advanced LIGO: 
 Resulting sensitivity � 32

Application for Advanced LIGO: 
 Implication • Although QCRB can be suppressed by squeezing, optical losses will limit the sensitivity. uncompensated phase fixed read-out quadrature � 33

Discussion • Achieving the loss limit is a future prospect Optimized homodyne angle at each frequencies is necessary Designing the optical filters is the next step • How realistic this new limit is more realistic than QCRB How complicated is the filter cavity to achieve this limit? � 34

Summary • Quantum limit from optical dissipation is presented cannot be beaten (fundamental) realistic (including optical loss) ϵ arm + ( 1 + Ω 2 γ 2 ) ℏ c 2 T itm ϵ src S ϵ hh = + α T src ϵ ext 4 L 2 ω 0 P 4 • Sensitivities of future gravitational-wave detectors could be limited by optical losses minimization of optical loss is crucial � 35