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### Abstract

Very low intensity and phase fluctuations are present in a bright light field such as a laser beam. These subtle quantum fluctuations may be used to encode quantum information. Although intensity is easily measured with common photodetectors, accessing the phase information requires interference experiments. We introduce one such technique, the rotation of the noise ellipse of light, which employs an optical cavity to achieve the conversion of phase to intensity fluctuations. We describe the quantum noise of light and how it can be manipulated by employing an optical resonance technique and compare it to similar techniques, such as Pound–Drever–Hall laser stabilization and homodyne detection.

The author gratefully thank Paulo Nussenzveig and Marcelo Martinelli for their advice throughout my student years, and for posing to me the questions discussed in this article: Thinking about their intuitive meaning has provided me a lot of the joy of physics. I dedicate this work to them. I also thank Katiuscia N. Cassemiro for valuable suggestions, Wolfgang Schleich for his encouragement, and Gerd Leuchs for his interest in this manuscript. This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and the European Commission through the SCALA network.

I. INTRODUCTION

II. QUANTUM NOISE AND FIELD QUADRATURES

III. OPTICAL CAVITY

IV. CAVITY EFFECT ON THE FIELD QUADRATURES

V. ROTATION OF THE NOISE ELLIPSE

VI. CLASSROOM EXPERIMENT

VII. CONCLUSION

### Key Topics

- Optical resonators
- 27.0
- Photons
- 15.0
- Quantum noise
- 14.0
- Photodetectors
- 13.0
- Homodyne
- 11.0

## Figures

(a) In the Mach-Zehnder interferometer the relative phase between the two possible paths determines the intensity difference between the two output ports. (b) Homodyne detection is performed by interfering in a balanced beam splitter (BS 50/50) the field to be measured (dotted arrow) and a strong local oscillator field (LO, solid arrow). The subtraction of the photocurrents gives information about the phase fluctuations of the target beam when the mean relative phase is .

(a) In the Mach-Zehnder interferometer the relative phase between the two possible paths determines the intensity difference between the two output ports. (b) Homodyne detection is performed by interfering in a balanced beam splitter (BS 50/50) the field to be measured (dotted arrow) and a strong local oscillator field (LO, solid arrow). The subtraction of the photocurrents gives information about the phase fluctuations of the target beam when the mean relative phase is .

Representation of the light field in the complex plane and its fluctuations. The vector is the complex field mean amplitude . The decomposition of the fluctuation in the direction of the mean value and in the orthogonal one defines the amplitude and phase quadratures, respectively. In the time domain a field presenting the amplitude fluctuations would resemble the inset (a), and the phase fluctuations would correspond to (b).

Representation of the light field in the complex plane and its fluctuations. The vector is the complex field mean amplitude . The decomposition of the fluctuation in the direction of the mean value and in the orthogonal one defines the amplitude and phase quadratures, respectively. In the time domain a field presenting the amplitude fluctuations would resemble the inset (a), and the phase fluctuations would correspond to (b).

Linear optical cavity with coupling mirror showing the reflectivity and the output mirror with reflectivity . The reflected field amplitude is the sum of the incident amplitude with the vacuum coupled by the output mirror and losses.

Linear optical cavity with coupling mirror showing the reflectivity and the output mirror with reflectivity . The reflected field amplitude is the sum of the incident amplitude with the vacuum coupled by the output mirror and losses.

Squared modulus (continuous line) and phase (dashed line) of as functions of the carrier-cavity detuning parameter relative to the cavity bandwidth. The numerical values and were used.

Squared modulus (continuous line) and phase (dashed line) of as functions of the carrier-cavity detuning parameter relative to the cavity bandwidth. The numerical values and were used.

Representation of the light field frequency components, carrier and sidebands (upper curve), and cavity resonance transmission profile (lower curve). The cavity is either resonant with (a) one sideband or (b) the carrier.

Representation of the light field frequency components, carrier and sidebands (upper curve), and cavity resonance transmission profile (lower curve). The cavity is either resonant with (a) one sideband or (b) the carrier.

Noise ellipse representation in the complex plane (shaded ellipse). The dotted circle represents the shot noise. The ellipse size compared to the mean value is exaggerated for ease of visualization.

Noise ellipse representation in the complex plane (shaded ellipse). The dotted circle represents the shot noise. The ellipse size compared to the mean value is exaggerated for ease of visualization.

The rotation angle between the field carrier and the noise ellipse introduced by the cavity as a function of for .

The rotation angle between the field carrier and the noise ellipse introduced by the cavity as a function of for .

Phase rotation of the noise ellipse as a function of the carrier-cavity detuning parameter relative to the cavity bandwidth. The central curve is the reflected beam amplitude noise [Eq. (25)], and the frames around it represent, for the corresponding numbered detuning, the reflected field in the complex plane (see Fig. 6). For , , , and .

Phase rotation of the noise ellipse as a function of the carrier-cavity detuning parameter relative to the cavity bandwidth. The central curve is the reflected beam amplitude noise [Eq. (25)], and the frames around it represent, for the corresponding numbered detuning, the reflected field in the complex plane (see Fig. 6). For , , , and .

Detunings for which has zero derivatives as a function of the analysis frequency . Each symbol represents a different zero derivative point. At a single detuning with zero derivative (open circles) gives rise to three such points as increases, because the complete conversion of phase to amplitude noise begins to be possible (triangles). The gray curves help to visualize the asymptotic behavior. Losses are assumed to be zero .

Detunings for which has zero derivatives as a function of the analysis frequency . Each symbol represents a different zero derivative point. At a single detuning with zero derivative (open circles) gives rise to three such points as increases, because the complete conversion of phase to amplitude noise begins to be possible (triangles). The gray curves help to visualize the asymptotic behavior. Losses are assumed to be zero .

Schematic of the experimental setup required to perform a classroom demonstration of the effect. PM: phase modulator.

Schematic of the experimental setup required to perform a classroom demonstration of the effect. PM: phase modulator.

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