Squeezing and Dual Recycling in Laser Interferometric Gravitational Wave Detectors

We calculate the response of an ideal Michelson interferometer incorporating both dual recycling and squeezed light to gravitational waves. The photon counting noise has contributions from the light which is sent in through the input ports as well as the vacuum modes at sideband frequencies generated by the gravitational waves. The minimum detectable gravity wave amplitude depends on the frequency of the wave as well as the squeezing and recycling parameters. Both squeezing and the broadband operation of dual recycling reduce the photon counting noise and hence the two techniques can be used together to make more accurate phase measurements. The variance of photon number is found to be time-dependent, oscillating at the gravity wave frequency but of much lower order than the constant part. email: [email protected] email: [email protected] 1 Laser-interferometric gravitational wave detectors [1] operate by sensing the difference in phase shifts imposed on the laser light in the two orthogonal arms of a Michelson type of interferometer by a gravitational wave. This phase shift manifests itself in the observed intensity change of the interference pattern. The sensitivity of the detector is determined by two fundamental sources of quantum mechanical noise : the photon counting error and the error originating due to fluctuations in radiation pressure on the mirrors. At the present level of the laser power available, the smallest detectable signal is limited by the photon counting statistics and various efforts have been made to increase the sensitivity level of these interferometers. Caves[2] first realized that the photon number fluctuations at the output could be understood due to the interference of the vacuum fluctuations of light which enters through the unused input port of the beam splitter with the ingoing laser light. He suggested that instead a squeezed photon state could be injected through this port to reduce the photon counting noise. For a Michelson interferometer operating on a dark fringe, most of the light escapes towards the laser source. Therefore, it had been suggested [3] that this light can be recycled by putting a mirror in front of the source to enhance the sensitivity of the interferometer. This technique is known as Power recycling. Brillet et al[4] argued that squeezing and power recycling are compatible with each other and that both can be used together to improve the signal-to-noise ratio. Gravitational waves modulate the phase of the laser light, thus generating sidebands which, travel towards the photodetector in an interferometer operating at the dark fringe [5]. These sidebands comprise the signal which can also be recycled by another mirror placed in front of the photodetector. The above method used in conjunction with power recycling is known as dual recycling[6]. It is, therefore, important to attempt an analysis of the quantum mechanical noise present in a dual recycling interferometer that also uses squeezed light and to investigate how well these two techniques work together. In this letter we report our results obtained 2 for interferometers operating in the broadband mode. We arrive at a complete expression for the variance of the photon number fluctuations which is found to have a time-dependent component. The presence of sidebands significantly alters the noise. We calculate the minimum detectable gravitational wave amplitude as a function of its frequency as well as squeezing and recycling parameters and conclude that that the broad-band operation of dual recycling is compatible with the squeezed light technique and can therefore be used to enhance the sensitivity. We first evaluate the minimum detectable phase difference with both squeezing and dual recycling without considering gravitational waves. Referring to Fig.1, monochromatic light beams (of angular frequency ω0) in coherent and squeezed vacuum states enter ports 1 and 2 respectively. The annihilation operators a and b represent light in the coherent and squeezed modes respectively. One may now write down equations for the intra-cavity electric field operators, Ea′ and Eb′ . We assume that the distance between the recycling mirrors and the 50:50 beam-splitter has been adjusted such that Ea′ and Eb′ add in phase with the ingoing modes, Ea and Eb respectively. We also assume that the beam-splitter introduces no phase shift upon reflection for a wave incident on the side of port 2 and a phase shift π for a wave reflected on the side of port 1. The quantities t1 (t2) and r1 (r2) represent the transmission and reflection coeff! icients of the power (signal) rec One can obtain expressions for the annihilation operators a and b in terms of the input modes a and b. Then the annihilation operator describing the mode at the output of port 2 can be given as Out2 := t2b ′ − r2b = 1 M [iat1t2 sin θ + b{t2(cos θ − r1)− r2M}], (1) where M = 1 + r1r2 − (r1 + r2) = (1− r1)(1− r2) (2) and θ is the phase difference of light between the two arms of the interferometer (at dark fringe, θ = 0). Then the mean and the rms value of the photon number at the output port 3 2 are found to be N = t1t 2 2 sin 2 θ M2 n̄ (3) and ∆N = √ n̄ t1t2 sin θ M2 [ t1t 2 2 sin 2 θ + (t2 cos θ − r1t2 − r2M)e 1/2 (4) respectively, where n̄ is the mean number of photons in the coherent beam and r is the squeeze factor. In these expressions, we have neglected terms with coefficients sinh r since n̄ ≫ sinh r. At the dark fringe most of the laser light escapes towards port 1. However, for a very small phase shift δθ, we obtain a very small change in the mean number of photons, δN(θ) at the output port 2. Equating the change to the rms value, we, therefore obtain the minimum detectable phase δθ at a dark fringe to be δθ = e √ n̄ [ t2(1− r1)− r2M 2t1t2 ] . (5) As can be easily seen, for values of r1 and r2 close to unity and a large squeeze factor, δθ is considerably reduced. This shows that squeezing and recycling are compatible with each other and can be used together to increase the sensitivity of the interferometer. We now examine the case when a gravity wave of dimensionless amplitude, h(t) = h0 sinωgt, propagating along the z-axis impinges on an interferometer whose arms are oriented along the x and y axes. If the gravity wave interacts with the laser beam of frequency ω0, propagating along the y-axis for a time τ then the phase picked up by light as a function of time is δφ(t) = ω0h0 2ωg ∫ t t−τ sin(ωgt) dt = ǫg sinωg(t− τ 2 ), (6) where ω0 is the laser light frequency and ǫg = ω0h0 2ωg sin ωgτ 2 . (7) Due to the quadrupolar nature of the gravity wave the phase acquired by the laser beam travelling along the x-axis is (−δφ). The gravity wave thus modulates the phase of light in the two arms which gives rise to a time-dependent intensity [5]. 4 The positive frequency part of the electric field operator propagating along the y-axis can be written as