1. Some Ideas About a Vacuum Squeezer A.Giazotto INFN-Pisa

2. Some considerations on 2 very important quantum noises: Shot noise and Radiation pressure 1)Shot Noise: Uncertainty prin. ΔφΔN  1 The phase of a coherent light beam fluctuates as The phase produced by GW signal is The measurability condition is i.e. i.e shot noise decreases by increasing W 1/2 F FW W LASER 2W FW

3. 2)Radiation Pressure Noise The photon number fluctuations create a fluctuating momentum on the mirrors of the FP cavities : The spectral force on the mirrors is: For the measurability condition this force should be smaller than Riemann force: The measurability cond. for Shot noise and Radiation Pressure noise is:

4. The term WF 2 produces dramatic effects on the sensitivity; minimizing h with respect to W we obtain the Standard Quantum limit: This limit can only be reduced either by increasing L and mirror masses M or by using a Squeezed Vacuum Not to scale Radiation Pressure Shot Noise ĥ BS pow.10 3 W 10 4 W 10 5 W 10 6 W 10 7 W Hz 10 100 1000

5. Noise Budget

6. In Quantum Electrodynamics it is shown that Electric field E x and magnetic field H y of a z-propagating em wave do not commute and satisfy the following commutation relation: We may expand the vector potential and the Hamiltonian as a sum of creation and anihilation operators: Where s is the polarization From E&H commutation relations we obtain

7. In interferometric detectors, GW produce sidebands at frequency ω 0 ±Ω, where 2π.10<Ω<2π.10000 is the acoustical band; this process involves the emission of two correlated photons of frequency ω 0 ±Ω: ε cos Ω t ω 0 -Ω ω 0 ω 0 +Ω Consequently it is appropriate to deal with two correlated photon processes. The positive frequency of the fluctuating electric field is: Where a ± =a ω±Ω satisfy the following commutation relations: the prime means that a is evaluated at ω±Ω ’.

8. The interaction of EM field Vacuum Fluctuation with a real systems:The quadrature formalism Where We can then describe the evolution of the emf vacuum fluctuation propagating in a optical system by means of the QUADRATURE STATE: Intensity Fluct. Phase Fluct.

9. If we include the carrier amplitude α, the total field is defined: Where α=(2W/hν 0 ) 1/2 is the amplitude of the carrier in a coherent state and W is the laser power entering the cavity.See Fig.1. We may rewrite E in after reflection on a mirror moving by δX (keeping first order terms in a 0, π/2 only) From eq.1 it is evident the role of a 0,in and a π/2,in in creating radiation pressure and phase fluctuations respectively and also shows that reflection on a mirror moving by δX gives a phase shift contributing only to π/2 quadrature.. Eq. 1 Fig.1

10. Fabry-Perot Cavity L (T 1,R 1, B) (T 2,R 2,B) F+Q= Losses Vacuum Fluctuations Classical Field δL1δL1 δL2δL2 Time Decomposition of amplitudes in small+large (P+G) components

11. Decomposition of amplitudes in small+large (P+G) components Large Components L (T 1,R 1, B) (T 2,R 2,B) F+Q Contributions to small components

12. Small Components

13. F out and Q out fields The out Fields L (T 1,R 1, B) (T 2,R 2,B) F+Q

14. Matrix Inversion Rationalization of F out and Q out fields

15. Ponderomotive action Mirror displacement due to Rad. Press. Fluctuations L (T 1,R 1, B) (T 2,R 2,B) F+Q Intracavity Power Fluctuations

16. Phase shifts due to: Ext.+int. vacuums Pondero- motive action External Forces Classical Field

17. Ponderomotive action at Resonance Resonance condition

18. The Squeezed Out Vacuum State Squeezing factor Squeezed Out Vacuum State

19. FP Ponderomotive Cavity Output State Squeezing Factor Summary FP Ponderomotive Cavity Optical Matrix

20. Why is K the squeezing factor? Squeezed State Covariance Matrix The eigenvalues λ and the eigevectors V, in the large K approximation, are: tg Φ=K The squeezing factor is then. For having V - parallel to V π/2 out we must rotate the ellipse by an angle Φ. V+V+ V-V- Φ V π/2 out V 0 out The relevant parameters for measuring squeezing are: 1) the eigenvalue is giving the size of the ellipse minor axis i.e. the inverse of the squeezing factor. 2) the eigenvectors V  which gives the tangent of the rotation angle Φ needed to bring the ellipse minor axis parallel to π/2 quadrature

21. What happens if K 1  K 2 If the squeezed quadrature state is the overlap of two states having squeezing factors K 1 and K 2 respectively: The covariance matrix C is: The eigenvalues λ and the eigevectors V, in the large K approximation, are: Eq.s 7 V+V+ V-V- Φ V π/2 out V 0 out

22. Topology of the vacuum reflected by a resonant PC Our squeezed state is: V+V+ V-V- Φ V π/2 out V 0 out Losses are then responsible for squeezing degrade; infact the squeezing factor in presence of losses becomes and in the limit becomes

23. Rotation in Quadrature Space A Detuned FP cavity may produce rotations in Quadrature Space. The rotation operator is: By operating it Out of Resonance (L d =L RES +ΔL) on the Squeezed state we would like to obtain the optimally rotated state V+V+ V-V- Φ V π/2 out V 0 out

24. Detuning Evaluation The condition B=-AK, in the approximation 2ΩL RES /c<<1, gives: When the detuning is, the I.C. classical amplitude becomes half max.: 0 1-R 1 G0G0

25. Frequency behavior and finesse of a Detuned cavity It is remarkable that a detuned cavity produces rotations in the quadrature space having the same K frequency behavior i.e. 1/Ω 2. Had the ratio B/A not the same 1/Ω 2 functional character of K, a frequency independent optimal squeezing could not be obtained. Infact, by expanding B/A in series of 1/Ω 2 we obtain: From this equation it is possible to evaluate the detuned cavity finesse F d.

26. Radiation Pressure and Shot noise in a Michelson Interferometer Effects of the vacuum fluctuations entering the ports of the Beam Splitter of a FP Michelson interferometer FP1 L 1,R 1,δl 1,w 1 W 1,D 1 BS FP2 W 2, D 2 L 2,R 2,δl 2,w 2 U Laser W,ω 0 W 1,2 Power entering FP cavities L 1,2 FP cavity length D 1,2 FP-BS Length R 1,2 FP entrance mir. Transmitt. δl 1,2 FP mirrors displacement b, v Injected Vacuum fluct. w 1,, w 2 Internal Vacuum fluct. α, β Classical Fields

27. With reference to the previous diagram we obtain for the output field U: FP cavities lossesFP PonderomotiveGW signal

28. When the two FP cavities are equal i.e: D=D 1 =D 2 +nλ/2 L=L1=L2 +nλ/2 K=K 1 =K 2 Γ= Γ 1 = Γ 2 K=K 1 =K 2 we obtain: This Eq. clearly shows that the only contribution to the noise is given by the vacuum quadrature operators v 0,π/2 entering the BS from the dark fringe port; in this symmetric configuration there is no noise contribution from b 0,π/2.This is a fundamental result showing that we may SQL by injecting in the BS black port a squeezed vacuum. Sum of two incoherent vacuums

29. If we inject a K squeezed state with carrier βe iφ coherent with laser carrier α 0 we obtain (for the sake of simplicity we omit vacuums fields due to losses): Initial State Vacuum and and Classical field entering BS Laser port. Squeezed Vacuum and and Classical field entering BS Black port.

30. Let us inject this signal in a detuned cavity: In conclusion, if we inject in the BS black port a K squeezed state, for having a squeezing K the following inequalicies should hold:

31. A Vacuum Squeezer conceptual Diagram Ponderomotive 1 K 1 ITF-Squeezer relative phase alignment Φ LASER K2K2 Ponderomotive 2 (10 -3, 1) Detuned Recycling Mirror Detuning Phase Modulator WDWD 2W Polarization Rotators PC ωMωM L L+Δ Squeezed Vacuum output V π/2out V 0out All Locking amplifiers are low-pass filtered 10 Hz

32. Ponderomotive Cavities Equalization ψ PC Detuning (T 2 =60,R 2 =40) Variable BS (t 2 =1-r 2, r 2 ) Variable Attenuator (η) Polarizer Squeezed Vacuum LASER ITF-Squeezer relative phase alignment Arm 2 L +Δ Arm 1 L Squeezed vac. carrier ampl. ωMωM Φ K2K2 Ponderomotive 1 K 1 ωMωM ωMωM Detuned Ponderomotive 2 ωMωM PC ωMωM (T 2 =1- σ 2,σ 2 =10 -6 ) Polarization Rotators