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Input spectrum for heterodyne detection in advanced gravitational wave detectors without non-stationary shot-noise

Future interferometric gravitational wave detectors will make use of the coupling between shot noise and radiation pressure noise that produces a squeezed output for the quantum noise at the dark-port of the interferometer allowing these interferometers to operate with a sensitivity that exceeds the standard quantum limit at certain frequencies. The ability of the detector to take advantage of this squeezed output state depends in part on the readout system used. With the phase modulated laser input planned for these interferometers the quantum-noise limited sensitivity is slightly better when homodyne (rather than heterodyne) detection is used. We show that by modifying the laser input spectrum for these interferometers, readout with heterodyne detection can be made to have the same quantum-noise limited sensitivity as readout with homodyne detection.


I. INTRODUCTION
The prospect of operating interferometric gravitational wave detectors for quantum non-demolition measurements [1] has lead to investigation into the details of the interferometer read-out scheme [3,4]. Two readout schemes relevant for gravitational wave detection are heterodyne readout (often called RF readout) and homodyne readout (often called DC readout). The readout scheme converts the field at the output port of the interferometer into a signal. In heterodyne readout a local oscillator field or fields that have a frequency difference from the field being measured interfere with the output field to produce an intensity that has a component which beats at the frequency difference of the fields. The phase of the beat note is a measure of the phase of the output field. In homodyne detection a local oscillator field at the same frequency as the field being measured interferes with the output field to produce constructive or destructive interference depending on the relative phase of the output field.
A principle issue for interferometric gravitational wave detectors is the difference in the quantum-noise-limited sensitivity of an interferometer with homodyne readout versus heterodyne readout. Buonanno, Chen and Mavalvala [3] show that an interferometer with homodyne readout has better quantum-noise limited sensitivity than with heterodyne readout. Although other technical tradeoffs exist between homodyne and heterodyne detection [5], we show that with a change to the laser input field assumed by Buonanno, Chen and Mavalvala it is possible for an interferometer with heterodyne detection to reach the same quantum-noise-limited sensitivity that can be achieved with homodyne detection.

II. INPUT SPECTRUM
The conventional laser input spectrum for interferometric gravitational wave detectors is shown in the left-hand panel of figure 1. It consists of a high-power carrier at frequency ω 0 which is resonant in the arms and interferes destructively at the dark-port of the interferometer, along with a pair of sidebands at ω 0 ± ω m that do not resonate in the arms and are made to transmit efficiently to the dark-port of the interferometer via an asymmetry in the arm lengths. Depending on the implementation there may be additional pairs of sidebands at other frequencies that are used for alignment sensing and control, which are not relevant for the discussion of the readout noise and which we shall ignore in our analysis. In heterodyne detection one or both of the sidebands at ω 0 ± ω m are transmitted to the dark port of the interferometer where they interfere with the gravitational wave signal sidebands of the carrier field at frequency ω 0 ± Ω gw , where Ω gw ≪ ω m is the gravitational wave frequency which is small compared to the modulation frequency. The sidebands serve as the local oscillator for heterodyne detection of signals in a frequency band centered on the carrier frequency. With this input laser spectrum the signal sidebands at frequencies near ω 0 are detected, along with the vacuum fluctuations (quantum noise) that have frequencies in a band centered on ω 0 , as well as the The input spectrum commonly used in gravitational wave interferometers. It has a high power component at frequency ω0 which resonates in the arms of the interferometer and low power modulation sidebands at ω0 ± ωm which transmit to the dark port and are used as a local oscillator for heterodyne detection. We refer to this as the "conventional input spectrum". Right: A "suppressed carrier input spectrum" with a low power component at frequency ω0 which transmit to the dark port to be used as a local oscillator for heterodyne detection, and high power modulation sidebands at ω0 ± ωm which resonate in the arms of the interferometer fluctuations that have a frequency in either of the frequency bands centered on ω 0 ± 2ω m . Since the spectral region around ω 0 ± 2ω m contributes noise when it mixes with the local oscillator fields at ω 0 ± ω m and is demodulated, but does not contribute any signal, it degrades the quantum-noise-limited sensitivity of the interferometer. By contrast with homodyne detection a field at the carrier frequency ω 0 is used as the local oscillator so that only the signal and quantum noise in the spectral region around ω 0 contribute. Thus with homodyne detection there is less quantum-noise added to the signal.
The carrier-suppressed input spectrum shown in the right-hand panel of Figure 1 leads to a difference in the way vacuum fluctuations affect the sensitivity. It consists of a low powered local oscillator at ω 0 and two high-powered sidebands at ω 0 ± ω m . The sidebands are assumed to be resonant in the arm cavities while the local oscillator at frequency ω 0 may or may not be resonant in the arm cavities depending on the implementation. Like with the conventional input spectrum, the Michelson interferometer's interference condition is set so that the high power components that resonate in the arm cavities interfere destructively at the dark-port while the low-power component that is used as a local oscillator is transmitted to the dark port of the interferometer. For comparison to the case of heterodyne readout with the conventional input spectrum or homodyne readout, we shall assume the same total power in the input spectrum is the same, so that the sidebands of our input spectrum each have half the power of the carrier in the conventional input spectrum (and the low power local oscillator components are assumed to have negligible power). With the suppressed-carrier input spectrum the local oscillator detects the gravitational wave signal sidebands in the frequency bands centered on ω 0 ± ω m as well as the vacuum fluctuations at these same frequencies. Unlike heterodyne detection with the conventional input spectrum, there is no additional quantum noise from frequency regions that do not contribute to the readout of the gravitational wave signal. To analyze the quantum-noise limited sensitivity of a configuration with this supressed-carrier input spectrum we must consider the input-output relations for quantum noise in frequency bands centered on both the upper and lower laser sideband frequencies. We extend the analysis of the input-output relations for a single frequency range done by Buononno and Chen [2] in which the quantum noise is represented by a 2-element vector and the interferometer is represented by a 2x2 matrix by describing the quantum noise with a 4-element vector and representing the interferometer by a 4x4 matrix.

III. INPUT-OUTPUT RELATIONS
We consider vacuum fluctuations at the dark port of the interferometer as the source for the quantum noise that is read out. Specifically we look at the components of the vacuum fluctuations at frequencies ω 0 ± ω m which are in-phase and quadrature-phase with the high-power sideband at each of those frequencies. We call the amplitude of the vacuum fluctuations that have a frequency in a band centered on the lower laser sideband frequency ω 0 − ω m and are are in-phase and quadrature phase with this sideband a 1 and a 2 respectively. We let a 3 and a 4 represent the components of the vacuum fluctuations at frequencies in a band centered on the upper sideband frequency of ω 0 + ω m that are in-phase and quadrature phase respectively relative to this sideband. With this notation, we consider the fields that contribute to the signal readout by heterodyne detection as a 4-element vector. The interferoemter can then be described as a 4 × 4 matrix that transforms the (unsqueezed) input quantum noise into a squeezed state at the output. By considering the action of a gravitational wave induced motion of the arm cavity mirrors on the output fields we relate the quantum-noise after demodulation to an equivalent signal magnitude to determine the quantum-noise limited sensitivity.
[2] for a signal-recycled interferometer with a conventional input laser spectrum. To investigate the quantum-noise limted sensitivity of an interferometer with this readout scheme we must first consider how these output fields are converted to a signal through heterodyne detection and demodulation.

IV. DEMODULATION
The output field of the interferometer is composed of two parts, a local oscillator and a signal (which includes the quantum noise) and can be written as Where where h.c. stands for the hermitian conjugate, and L describes the amplitude and phase of the local oscillator. We follow the analysis of reference [3] and express the signal as where Λ is the demodulation bandwidth. For simplicity we omit terms that are at frequencies that will not contribute to the final expression. The detected photocurrent is proportional to the square of the field giving of which only the cross term 2L(t)S(t) has significant amplitude at the modulation frequency which is detected by demodulation and low pass filtering. This modulated component of the photocurrent is The demodulated output signal comes from mixing (i.e. multiplying) the modulated photocurrent with cos (ω m t + φ d ) and low pass filtering with a cut-off frequency of Λ giving where T = 1/Λ is the time constant associated with the low pass filter and we approximate the low pass filter as being a perfect step-function filter passing only frequencies within its bandwidth. Using the relations we can rewrite the double-sided frequency integral in equation 31 as the single-sided integral which in the frequency domain is Setting ζ ± = ζ 0 ∓ φ d − π/2 where ζ 0 ≡ arg (L) we can express this as and with the definitions which represent the signal due to the output field in a band centered at ω 0 ± ω m in the ζ ± quadratures respectively, the demodulated signal is The values of b i in expression 38 are given by the input-output relations of equations 10 and 11. Since the output fields b ζ± each can contain vacuum fluctuations and signals from gravitational waves, the quantum-noise limited output is unlike that of equation 13 of reference [ [3]] for heterodyne readout of an interferometer with a conventional input spectrum for which the frequency regions around ω 0 ± 2ω m contribute noise (called non-stationary shot-noise) but not signal. Thus we have shown that heterodyne detection with a suppressed carrier input spectrum is not subject to non-stationary shot-noise.

V. NOISE SPECTRAL DENSITY AND SENSITIVITY
To compute the noise spectral density we express the magnitude of the noise by the equivalent signal level h n . From equation 10 we have We find the noise spectral density by Equation.
(22) of Ref. [1] and assuming the input field a is in the vacuum state such that the noise spectral density is The A i terms in the numerator represent the noise contributions from each of the 4 input 'a' fields and are defined as We then define the sensitivity as the square root of the noise spectral density. For reference we reference this to the sensitivity associated with the standard quantum limit, h sql

A. Balanced laser sidebands
We now consider the special case of balanced laser sidebands such that κ − = κ + . With this assumption C 11 = C 22 = C 33 = C 44 , C 12 = C 34 , C 13 = C 24 = C 31 = C 42 , C 14 = C 32 , C 21 = C 43 , C 23 = C 41 D 1 = D 3 and D 2 = D 4 . With these simplifications, the noise spectral density is an even function of the demodulation phase φ d and is minimized for φ d → 0. The noise spectral density is plotted in figure 3 and compared to the noise spectral density with homodyne readout and with heterodyne readout using a conventional input spectrum. The noise spectral density for this heterodyne readout with a carrier-suppressed input spectrum is identical to that with a homodyne readout scheme and is quieter than that for heterodyne readout with a conventional input spectrum. The optimal value for

B. Unbalanced laser sidebands
In the general case of unbalanced laser sidebands, the lack of symmetry causes the optimal demodulation phase to differ from zero in a frequency dependent manner, however, unlike the phase of the local oscillator which is fixed, several different demodulation phases can be used to readout the signal in parallel channels, allowing for a noise spectral density that approximates that which would be achieved by having the optimal demodulation phase at each frequency. The noise spectral density of the readout with unbalanced sidebands and frequency optimized demodulation angle is always between the best case of balanced sidebands and the worst case of totally unbalanced sidebands. The quantum noise ellipse [8] is shown in figure 6 for the totally unbalanced case and the balanced case. The total quantum noise ellipse (outer shaded ellipse) and its components from frequencies near the lower sideband (dark shaded inner ellipse) and the upper sideband (light shaded inner ellipse). The ellipse representing a signal of magnitude h sql is shown for comparison. On the left is the noise ellipse for unbalanced (I− = I sql , I+ = 0) sidebands on the right is the noise ellipse for balanced sidebands (I− = I+ = I sql /2). For both cases ρ = 0.9, τ = 1 − ρ 2 , Ω = γ/2, φ = π/2 − 0.47, and φ d = 0. If homodyne detection were used the noise ellipse would be equivalent to the contribution from the lower frequency sideband in the unbalanced case (inner dark ellipse on the left) and the ellipse would be identical the total noise ellipse for balanced detection (outer ellipse on the right).
It may seem that the ability to generate an output spectrum with a frequency optimized demodulation angle is a benefit of having unbalanced sidebands, however the optimal demodulation angle with balanced sidebands is frequency independent, so there is no disadvantage to balanced sidebands VI. CONCLUSION Future gravitational wave interferometric detectors will be limited by quantum-noise over much of the detection band. The details of how the readout scheme affects the quantum-noise limited sensitivity are an important consideration for the detector design . With the conventional input spectrum the quantum-noise with heterodyne readout is worse than with homodyne readout due to the presence of non-stationary shot noise -vacuum fluctuations at twice the modulation frequency which are down converted into the output signal by mixing with the modulation sidebands.
With the suppressed-carrier input spectrum presented here the quantum-noise limited sensitivity with heterodyne readout is equivalent to that with homodyne readout because the non-stationary shot noise is not present. Although both heterodyne readout schemes and homodyne readout schemes have other strengths and weaknesses, we show that the use of heterodyne detection does not necessarily lead to an increase in the quantum-limited noise floor.