[Paper Review] Double balanced homodyne detection

K. Nakamura, Masa‐Katsu Fujimoto|arXiv (Cornell University)|Nov 10, 2017

Pulsars and Gravitational Waves Research1 references3 citations

TL;DR

This paper proposes a double balanced homodyne detection scheme that enables direct measurement of the expectation value of the linear combination of photon annihilation and creation operators, $\cos\theta\hat{b}_1 + \sin\theta\hat{b}_2$, by properly controlling the complex amplitude of the local oscillator. The method overcomes a fundamental limitation in conventional homodyne detection and is realized via an eight-port interferometer configuration, offering a path to beat the standard quantum limit in future gravitational-wave detectors.

ABSTRACT

In the context of the readout scheme for gravitational-wave detectors, the "double balanced homodyne detection" proposed in [K.~Nakamura and M.-K.~Fujimoto, arXiv:1709.01697.] is discussed in detail. This double balanced homodyne detection enables us to measure the expectation values of the photon creation and annihilation operators. Although it has been said that the operator $\hat{b}_θ:=\cosθ\hat{b}_{1}+\sinθ\hat{b}_{2}$ can be measured through the homodyne detection in literature, we first show that the expectation value of the operator $\hat{b}_θ$ cannot be measured as the linear combination of the upper- and lower-sidebands from the output of the balanced homodyne detection. Here, the operators $\hat{b}_{1}$ and $\hat{b}_{2}$ are the amplitude and phase quadrature in the two-photon formulation, respectively. On the other hand, it is shown that the above double balanced homodyne detection enables us to measure the expectation value of the operator $\hat{b}_θ$ if we can appropriately prepare the complex amplitude of the coherent state from the local oscillator. It is also shown that the interferometer set up of the eight-port homodyne detection realizes our idea of the double balanced homodyne detection. We also evaluate the noise-spectral density of the gravitational-wave detectors when our double balanced homodyne detection is applied as their readout scheme. Some requirements for the coherent state from the local oscillator to realize the double balanced homodyne detection are also discussed.

Motivation & Objective

• To resolve the long-standing issue that conventional homodyne detection cannot directly measure $\langle \cos\theta\hat{b}_1 + \sin\theta\hat{b}_2 \rangle$ as a linear combination of upper- and lower-sideband outputs.
• To establish a theoretical and experimental framework for measuring arbitrary quadrature expectation values in gravitational-wave detectors using a modified homodyne scheme.
• To demonstrate that the eight-port homodyne interferometer setup realizes the proposed double balanced homodyne detection.
• To evaluate the noise spectral density of gravitational-wave detectors when this scheme is applied as a readout method.
• To identify the precise requirements on the local oscillator's complex amplitude for successful implementation.

Proposed method

• The method employs a double balanced homodyne detection setup using two coherent states with controlled complex amplitudes $\gamma_+$ and $\gamma_-$ from the local oscillator.
• It derives the signal outputs $\frac{1}{\sqrt{2}}\left(\frac{\langle\hat{s}_+\rangle}{|\gamma_+|} + \frac{\langle\hat{s}_-\rangle}{|\gamma_-|}\right)$ and $\frac{1}{\sqrt{2}i}\left(\frac{\langle\hat{s}_+\rangle}{|\gamma_+|} - \frac{\langle\hat{s}_-\rangle}{|\gamma_-|}\right)$ as key operators for quadrature measurement.
• By setting the phases of $\gamma_\pm$ such that $\theta_+ = \theta_-$ and $|\gamma_\pm| \neq 0$, the scheme achieves the desired expectation value $\langle |\cos\theta|(\hat{b}_1 + \hat{b}_1^\dagger) + |\sin\theta|(\hat{b}_2 + \hat{b}_2^\dagger) \rangle$.
• The interferometer configuration in Fig. 3 implements the double balanced homodyne detection by combining signals from two balanced homodyne detectors with appropriate phase and amplitude control.
• The method uses the linear combination of outputs to extract $\langle \cos\theta\hat{b}_1 + \sin\theta\hat{b}_2 \rangle$, correcting for phase ambiguities in standard homodyne detection.
• Theoretical analysis confirms that the scheme allows measurement of $\langle \cos\theta\hat{b}_1 + \sin\theta\hat{b}_2 \rangle$ when $\theta_+ = \theta_-$ and $|\gamma_\pm| \neq 0$, enabling full quadrature control.

Experimental results

Research questions

• RQ1 Can the expectation value $\langle \cos\theta\hat{b}_1 + \sin\theta\hat{b}_2 \rangle$ be measured using standard homodyne detection via upper- and lower-sideband outputs?
• RQ2 What conditions on the local oscillator's complex amplitude are necessary to realize the measurement of $\langle \cos\theta\hat{b}_1 + \sin\theta\hat{b}_2 \rangle$?
• RQ3 How does the double balanced homodyne detection scheme improve noise performance in gravitational-wave detectors compared to conventional readout schemes?
• RQ4 Can the eight-port homodyne interferometer configuration realize the proposed double balanced detection scheme?
• RQ5 Is it possible to simultaneously measure both the amplitude and phase quadrature expectation values using this method?

Key findings

• The expectation value $\langle \cos\theta\hat{b}_1 + \sin\theta\hat{b}_2 \rangle$ cannot be measured as a linear combination of upper- and lower-sideband outputs in standard homodyne detection, contrary to prior assumptions.
• The proposed double balanced homodyne detection enables direct measurement of $\langle \cos\theta\hat{b}_1 + \sin\theta\hat{b}_2 \rangle$ when the local oscillator's complex amplitude satisfies $|\gamma_+| = |\gamma_-|$ and $\theta_+ = \theta_-$.
• The eight-port homodyne interferometer configuration is shown to realize the double balanced homodyne detection by properly combining signals from two balanced homodyne detectors.
• The method allows the measurement of both $\langle \cos\theta(\hat{b}_1 + \hat{b}_1^\dagger) + \sin\theta(\hat{b}_2 + \hat{b}_2^\dagger) \rangle$ and $\langle \cos\theta(\hat{b}_1 - \hat{b}_1^\dagger) + \sin\theta(\hat{b}_2 - \hat{b}_2^\dagger) \rangle$ simultaneously through appropriate signal combination.
• The noise spectral density of gravitational-wave detectors using this scheme is evaluated, showing potential for improved sensitivity beyond the standard quantum limit.
• The key requirement for the local oscillator is that $|\gamma_\pm| \neq 0$ and $\theta_+ = \theta_-$, ensuring coherent control of the quadrature measurement.