1. Introduction

Many high precision experiments, such as gravitational wave detectors (GWDs), employ intricate optical readout schemes. These commonly rely on lasers with high power stability, as laser power noise can couple to the readout channel and limit the experiment’s sensitivity. Usually, active noise suppression has to be applied and in order to ensure that the stability demand is met, power noise monitoring is necessary. Both require high precision power noise sensing, as the noise of the detector used for stabilization (in-loop sensor) dictates the achievable power stability and the detectors used for monitoring (out-of-loop sensors) must be at least as sensitive as the in-loop sensor to verify the control loop performance. The conventional approach of measuring power noise via a photodetector is fundamentally limited by the shot noise of the detected beam. Thus, power stabilization using conventional photodetection can only achieve a stability equal to the relative shot noise (RSN) of the in-loop beam, which scales proportionally to $P^{-1/2}$ (detected power $P$). For applications such as the aLIGO GWDs [1], which need a power stability of $2\times 10^{-9}\, \mathrm {Hz}^{-1/2}$ at $10\, \mathrm {Hz}$, this means a detected power of more than $100\, \mathrm {mW}$, which is beyond the power tolerance of suitable photodetectors. For the current generation of GWDs this is solved by splitting the light onto an array of photodetectors [2], but as the requirements for future generations are likely to be roughly an order of magnitude more stringent, this approach would result in massive technical effort. Therefore, alternative approaches of power noise sensing have been investigated, including carrier attenuation by measuring the power noise in reflection of an optical resonator [3], injecting squeezed light into the detection port [4] and detecting power noise via radiation pressure [5]. While these methods are promising for in-loop detection, they still require considerable effort, and as the out-of-loop detectors have to match the in-loop detector’s sensitivity, such an implementation for all detectors would overall result in an enormous technical challenge. In this article we propose an alternative for out-of-loop detection at a reduced technical effort.

The idea elaborated here, called quantum correlation measurement (QCM), is an approach to significantly reduce the complexity of out-of-loop sensors as less power needs to be detected compared to the in-loop detector. Also, the QCM method is superior in applications where only a limited amount of laser power can be spared for monitoring. It can as well be used in the characterization of bright squeezed light, where standard homodyne detection is limited by local oscillator power noise that beats with the strong carrier of the bright squeezed light field. In the QCM technique, a single photodetector is substituted by a pair of photodetectors, which allows to deduce the power noise from correlating the two detectors’ signals. By cancellation of quantum noise contributions in a cross spectral density measurement this enables measurements below the RSN of the detected power, which is impossible with the conventional method. Additionally, the QCM scheme is less sensitive to the electronic dark noise of the detectors, as the individual dark noise contributions are uncorrelated to one another. In this article, a comprehensive, quantum mechanical description of the QCM technique is given and its advantages over the conventional concept are discussed. Finally, this theory is confirmed by experimental results from a test experiment, in which we demonstrate a power noise measurement with a sensitivity of one order of magnitude below the RSN of the total detected power.

2. Theoretical description

Without loss of generality, the laser light will be described as a plane, quasi-monochromatic, linearly-polarized electromagnetic wave in vacuum and represented by its electric field Heisenberg operator $\hat {E}(t)$ [6], which can be described in the quadrature picture. There it is decomposed into two orthogonal components: the amplitude quadrature (c subscript; ’cosine quadrature’) and the phase quadrature (s subscript; ’sine quadrature’):

The laser power can be defined as the mean energy per time interval carried by the electric field, which can be derived from an integral of the time-averaged Poynting vector [7] over the effective beam cross-section:

 

Fig. 1. Schematic of a conventional RPN measurement comprising a laser pick-off port via a low-transmission beamsplitter and a single photodetector (PD).

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$\hat {{\boldsymbol {f}}}$ represents the input laser field and thus can contain both technical and quantum noise contributions. $\hat {{\boldsymbol {s}}}$ only carries quantum noise originating from the vacuum fluctuations, which are independent from the carrier field. They appear as a constant contribution to the PSD of the detection field’s amplitude quadrature. In this formalism, one finds for a field $\hat {{\boldsymbol {a}}}$ in a vacuum state that $S^{\hat {{\boldsymbol {a}}}}_{\mathrm {cc}}(\Omega ) = S^{\hat {{\boldsymbol {a}}}}_{\mathrm {ss}}(\Omega ) = 1$ [6]. Thus, one can define the relative shot noise $\mathrm {RSN}$ for a power of $P_0$ in analogy to Eq. (6) as:

The QCM technique can circumvent this limitation. Similar approaches have been used before, for example for residual noise characterization of the LIGO Livingston detector [9], measurements of liquid surface fluctuations [10], or spectroscopy of atomic vapors [11]. Here, this method shall be used in a power noise monitoring scheme, which surpasses the shot-noise limit of the conventional scheme for the same detected power. Consider Fig. 2. The first part of this setup is identical to the conventional approach. But instead of sending field $\hat {{\boldsymbol {d}}}$ to one photodetector, the light is split by a beamsplitter (amplitude reflectivity $\rho$ and transmissivity $\tau$) into two beams, that each are incident on a photodetector ($\mathrm {PD}_1$ and $\mathrm {PD}_2$ respectively). In the open port, vacuum fluctuations enter in form of the field $\hat {{\boldsymbol {v}}}$. Together this results in the output fields:

 

Fig. 2. Schematic of the QCM configuration where a beamsplitter and two photodetectors are placed in the detection path.

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This creates the opportunity to measure the CSD of the two photodetectors’ signals, which are proportional to the amplitude quadratures of the two fields $\hat {{\boldsymbol {p}}}$ and $\hat {{\boldsymbol {q}}}$ respectively. Equations (5) and (10) yield

In summary, this means that the QCM technique is ideally only sensitive to the technical power noise of the input laser field. As such it is advantageous to the conventional method which is limited by the detected beam’s RSN. Furthermore, this alternative can be implemented with minimal technical effort by adding a beamsplitter and a second photodetector. Another benefit can be found considering incoherent noise in the two photodetector signals, like the individual electronic dark noise. As these noise components only add uncorrelated crossterms to Eq. (11), which average out to zero, they theoretically have no influence on the CSD. In a regular PSD measurement, however, dark noise poses a lower limit of the detection sensitivity.

3. Experimental results

So far in this article the QCM technique was analyzed theoretically. To demonstrate it in practice, a test experiment was set up to compare the RPN of a laser beam measured by the conventional scheme (Fig. 1) and by the QCM technique (Fig. 2). A Nd:YAG solid-state laser in a non-planar ring oscillator configuration with an operating wavelength of $1064\, \mathrm {nm}$ and a full beam power (representing field $\hat {f}$) of $770\, \mathrm {mW}$ fabricated by the company Coherent was used. In both configurations this power was attenuated at the first beamsplitter to $4\, \mathrm {mW}$ in the detection field $\hat {{\boldsymbol {d}}}$ in order to demonstrate a shot noise limited detection for the conventional scheme. For the QCM method $\hat {{\boldsymbol {d}}}$ further was incident on a 50:50 beamsplitter. This resulted in an effective QCM power of $2\, \mathrm {mW}$, which was sufficient for this setup. In general a higher $P_\mathrm {QCM}$ increases the signal-to-noise ratio, when considering noise sources that couple coherently in both signals (like electronic interference). Therefore the best results should be found for a beamsplitter with equal transmitted and reflected power, as it is used here.

With both methods, the free running power noise of the laser was measured in a frequency band from $200\, \mathrm {kHz}$ to $100\, \mathrm {MHz}$. The PSD for the conventional measurement was recorded with a spectrum analyzer (Agilent/Keysight 4395A). For the QCM method, the analog signals of both photodetectors were digitized and recorded simultaneously with a digital oscilloscope (Agilent/Keysight MSOX4054A) with a sampling rate of $2.5 \times 10^{9}\, \mathrm {samples/s}$ and then processed via the MATLAB function ’cpsd’. This function is based on Welch’s method for power spectral density estimation [12] and calculates the CSD for two signals by first segmenting the respective time series and determining the fast Fourier transforms (FFTs) for both signals in each segment. The CSD estimate is then obtained by multiplying the FFT of one signal with the complex conjugated FFT of the other signal and averaging this product over all segments. Given sufficient measurement time, this estimate approaches the theoretical CSD well. As such, this method is analogous to the way spectrum analyzers like the one used here perform PSD measurements.

Figure 3 illustrates the results of these measurements calibrated to RPN. In red the conventional RPN measurement is depicted. The blue curve shows the square root of the MCSD obtained from the QCM technique, which according to Eq. (13) should coincide with the technical power noise of the laser. In the black dashed curve, the RSN of the detection beam at $4\, \mathrm {mW}$ is given. This measurement band was chosen since the technical power noise follows a distinct and well-known behavior dictated by the relaxation oscillation of the laser source, which results in a sharp resonant peak (in this case at roughly $540\, \mathrm {kHz}$) followed by an approximate $1 / f^{2}$ decline for larger frequencies [13,14]. Both the red and blue curve portray this well up to $4\, \mathrm {MHz}$ where they closely agree. The difference at the resonance peak stems from different resolution bandwidths ($1.25\, \mathrm {kHz}$ for the conventional and $5\, \mathrm {kHz}$ for the QCM technique) and is irrelevant here.

 

Fig. 3. Comparison of the RPN of the free running laser noise obtained from a conventional power noise measurement and from the QCM scheme.

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After the steep decline, the red curve levels on a value of $10^{-8}\, \mathrm {Hz}^{-1/2}$ from about $10\, \mathrm {MHz}$ to $30\, \mathrm {MHz}$. This directly corresponds to the RSN of the detection beam and indicates that the conventional method is shot noise limited in this frequency range. For higher frequencies, the electronic noise of the photodetector becomes larger than the shot noise, which results in the rising noise towards the end of the measurement band. This illustrates the typical limitation of the conventional method both due to shot noise and electronic dark noise as discussed above.

For the QCM technique, this is different. Like the red curve, the QCM result in blue shows the expected technical laser noise for lower frequencies, but it continues the $1/f^{2}$ behavior even below the shot noise level of the detection beam, which it meets at around $7\, \mathrm {MHz}$. At roughly $50\, \mathrm {MHz}$ it finally reaches the lowest point at a sensitivity of about $10^{-9}\, \mathrm {Hz}^{-1/2}$, which is one order of magnitude below the RSN. For frequencies above $50\, \mathrm {MHz}$, the QCM curve is limited by the projected dark noise equivalent for this measurement, which is illustrated in green. This dark noise measurement was obtained by performing a measurement without laser power incident on the detectors, while all other circumstances (such as position of the detectors and the measurement electronics, background lighting of the laboratory, etc.) were identical. The term ’equivalent’ is used in order to emphasize that this differs from a conventional dark noise measurement (i.e. a PSD for a single photodetector), as this curve is also obtained from an MCSD.

4. Summary and conclusion

Overall, the QCM technique provides a power noise measurement significantly below the detection beam’s RSN. The RPN obtained this way matches the behavior expected from the laser source, which supports the above derived formalism. Furthermore, it indicates the feasibility of the QCM technique as a sub shot noise power noise monitor under realistic circumstances. The limitation of the measurement shown here is the QCM dark noise equivalent. In an infinitely long measurement, this dark noise contribution should tend to zero due to the uncorrelated nature of the sources. However, for realistic measurement times, the contribution to the CSD is finite and expected to scale inversely with the measurement time (due to the derivation of the CSD from Welch’s method). With longer measurements this could be reduced to an arbitrarily low level, as long as the corresponding noise in both photodetectors is truly uncorrelated. The setup used here was limited by the processing speed of the available recording and evaluation hardware. A future implementation of an FFT algorithm via a field programmable gate array could result in quasi real-time measurements.

In this article the quantum correlation measurement technique was investigated as an alternative to conventional power noise monitoring, which is both powerful and easily implemented. Its theoretical basis was laid and compared to the conventional concept. Subsequently, a test experiment was described and its results were presented, which support the established theory. In conclusion, the QCM scheme was demonstrated to allow for a sub shot noise measurement of power fluctuations, which could be beneficial for applications with extremely high power stability demands or with limited light power available for monitoring purposes. Another promising application of this technique could be as a bright squeezed light detector, which becomes apparent when considering Eq. (12) for an input field $\hat {{\boldsymbol {f}}}$ with non-classical noise properties like (anti-) squeezed light.