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
• ptical fields.

3

Sensitivity of KAGRA

Shot noise and radiation pressure noise

• Quantum noise — Shot noise + Radiation pressure noise
• Shot noise

Phase fluctuation

• Radiation pressure noise

Phase fluctuation <- Amplitude fluctuation

• Trade-off between shot noise and radiation pressure noise

Standard Quantum Limit (SQL)

4

Suspended mirror (converter)

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 different times are

independent. classical laser beam

• SQL is a kind of “classical” limit from quantum fluctuation.

5

̂ n(t) ̂ n(t + Δt)

commute

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

• Homodyne detection measures the quadrature in θ-direction.


(θ is the phase of local oscillator)

7

Signal Local Oscillator Output

a1 a2

Amplitude
 quadrature Phase
 quadrature Signal

θ

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

9

Sensitivity Frequency radiation pressure noise shot noise

PRD 65.022002

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)

11

Feb 9 2017, Ando Lab Seminar 22

• 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

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 effect 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
• ptical losses

16

Previous research

Designing the detector configuration
 ↓
 Numerical computation for the sensitivity

Future

Analytical calculation for the limit sensitivity
 ↓
 Designing the detector configuration ←configuration-
 independent

PRD 98.044044

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

18

Smin

hh = SQCRB hh

+ Sϵ

hh

Loss term Quantum Cramer-Rao bound

General QND scheme

Review of QCRB

• is the spectral density of the power fluctuation in the

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.

• nly if minimum uncertainty states are maintained

19

SQCRB

hh

(Ω) = ℏ2c2 2SPP(Ω)L2

SQCRB

hh

(Ω) ∝ (SPP(Ω))−1 SPP(Ω)

Loss term (main result)

• P: optical power inside each arm
• ω0: laser frequency
• εarm, εsrc: internal loss of the arm, the signal recycling cavity (10-6 for 1ppm)
• εext: external loss (including the quantum inefficiency of the photodetector)
• γ: bandwidth of the arm cavity
• Titm: transmission of ITM
• Tsrc: effective transmission of SRC
• α = 1 if the internal squeezing is optimized


1/4 if the internal squeezing is negligible

20

Sϵ

hh =

ℏc2 4L2ω0P ϵarm + (1 + Ω2 γ2 ) Titmϵsrc 4 + αTsrcϵext

The arm cavity loss

• The arm cavity loss sets a flat limit
• Fluctuation is directly mixed with the GW signal inside the

arm

• The effect is identical to GW signal

21

Sϵ

hh =

ℏc2 4L2ω0P ϵarm + (1 + Ω2 γ2 ) Titmϵsrc 4 + αTsrcϵext

The SRC loss

• 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

Sϵ

hh =

ℏc2 4L2ω0P ϵarm + (1 + Ω2 γ2 ) Titmϵsrc 4 + αTsrcϵext

The external loss

• The effect of external loss depends on the transmission of

the SRC

• note: The transmission of the SRC can be frequency

dependent

23

Sϵ

hh =

ℏc2 4L2ω0P ϵarm + (1 + Ω2 γ2 ) Titmϵsrc 4 + αTsrcϵext

current GW detectors configuration

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 effective mirror
• Phase factors in the cavities are canceled by some active
• components. (Non-perfect cancelation will lead to worse

sensitivity.*) Mintra → Mrot, Mopt → Msqz

26

*PRL 115, 211104

Derivation:
 Calculation

• input-output relation

• where


• Detector response

27

̂ aout = Mio ̂ ain + TsrcϵintMc ̂ nint + ϵext ̂ next + vhGW Mio ≡ − RsrcI + TsrcMcMrotMsqzMrot Mc ≡ [I − RsrcMrotMsqzMrot]

−1

v ≡ TsrcMcv0

= (0, 2 ω0L2P/(ℏc2))

′

Derivation:
 Approximation

• From input-output relation, minimum fluctuation can be

calculated complicated

• Approximation

Tsrc << 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

28

Amplitude fluctuation → Phase fluctuation

Derivation:
 Result expression

• case1: optimal internal squeezing (r = δ/2)
• case2: negligible internal squeezing (r = 0)

29

Smin

hh ≈ SQCRB hh

+ Sϵ

hh

SQCRB

hh

= ℏc2 (δ2 − 4r2)

2 e−2rinput

16L2ω0PTsrc [δ2 + 4r2 + 4δr sin (θ + θ0)] SQCRB

hh

= 0 Sϵ

hh =

ℏc2 4L2ω0P (ϵint + Tsrcϵext) SQCRB

hh

= ℏc2δ2e−2rinput 16TsrcL2ω0P Sϵ

hh =

ℏc2 4L2ω0P (ϵint + Tsr 4 ϵext) δ ≡ T2

src + 16Θ2

θ0 ≡ cot−1 (4Θ/Tsrc)

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

Application for Advanced LIGO:
 Parameters

31

• εarm: 100 ppm
• εsrc: 1000 ppm (mainly contributed from the beam splitter)
• εext: 0.1 (mainly contributed from OMC and photodiode

quantum inefficiency)

• Titm: 0.014
• Tsrc: 0.14
• Squeezing: 30 dB

Sϵ

hh =

ℏc2 4L2ω0P ϵarm + (1 + Ω2 γ2 ) Titmϵsrc 4 + αTsrcϵext

default broadband detection mode

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.

33

fixed read-out quadrature uncompensated phase

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)

• Sensitivities of future gravitational-wave detectors could

be limited by optical losses minimization of optical loss is crucial

35

Sϵ

hh =

ℏc2 4L2ω0P ϵarm + (1 + Ω2 γ2 ) Titmϵsrc 4 + αTsrcϵext