Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Tutorials Issues in Progress Current Issue All Issues Early Posting Feature Issues

Pictorial interpretation of quantum-enhanced measurements with wave functions

Yuki Sano and Yasuyuki Ozeki

Open Access Open Access

Abstract

Quantum-enhanced measurements using squeezed states of light provide sensitivities beyond the shot noise limit. Although the mechanism of quantum enhancement is well described by quantum mechanics, it is difficult to intuitively understand how squeezed states are exploited. In this paper, we present a pictorial interpretation of quantum-enhanced measurements using wave functions, which provides intuitive pictures of squeezed states in various experimental systems.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

The sensitivity of optical measurements with classical light is restricted by the shot noise resulting from the quantum nature of light. To break the shot noise limit, ultrasensitive measurement methods [1] with squeezed states of light [2] have been extensively developed in various fields such as gravitational-wave detection [3], spectroscopy [4], optical displacement measurements [5,6], and biological measurement and imaging [7–11].

Although the characteristics of squeezed states of light are well described by quantum mechanics [11–15], it is difficult to intuitively understand how squeezed states are utilized. This is partly because squeezed states are often analyzed in the Heisenberg picture, which greatly simplifies the calculation but obscures the evolution of quantum states. On the other hand, in the Schrödinger picture, squeezed states of light have literally squeezed wave functions, providing us with an intuitive picture of squeezed states. Indeed, two-mode wave functions were used to explain two-mode squeezing [16,17], the Hong–Ou–Mandel state [18], and quantum-enhanced balanced detection [11].

This paper introduces pictorial interpretations of various quantum-enhanced measurements based on two-mode wave functions. Specifically, we present wave functions in the detection of gravitational waves, the generation of an amplitude-squeezed state with an asymmetric beam splitter (BS), and balanced homodyne detection of a squeezed state. In contrast to previous studies where quantum enhancement was investigated by following the evolution of the Wigner function with a BS [19–22], this study focuses on the two-mode wave function, which enables us to see the entanglement between different modes intuitively. In Appendix A, we describe the wave functions in the detection of gravitational waves under the effect of radiation pressure, and summarize the properties of wave functions of coherent states and squeezed states as well as their evolution by various operations such as phase shift and beam splitting.

2. WAVE FUNCTIONS IN VARIOUS QUANTUM-ENHANCED MEASUREMENTS

In this section, we describe the evolutions of wave functions in quantum-enhanced measurements, including gravitational-wave detection, amplitude squeezing, and balanced homodyne detection.

A. Gravitational-Wave Detection

The detection of gravitational waves with a Michelson interferometer is a representative example of quantum-enhanced measurements utilizing squeezed states of light. Figure 1 shows a simplified schematic of a gravitational-wave detector [3], where squeezed vacuum is injected from the output port of the Michelson interferometer via a circulator, which consists of a polarizing BS (PBS) and a Faraday rotator. When gravitational waves arrive, one arm of the Michelson interferometer contracts and the other arm extends, causing an opposite phase shift to each beam. By measuring this phase shift, gravitational waves can be detected.

 figure: Fig. 1.

Fig. 1. Schematic of a gravitational-wave detector. A squeezed vacuum is injected from the output port of a Michelson interferometer. PD, photodiode; PBS, polarizing beam splitter.

Download Full Size | PDF

 figure: Fig. 2.

Fig. 2. Evolution of the wave function in gravitational-wave detection. The left figure shows the beams under consideration, and the middle figure and right figure show the wave function in the ${x_a}$-${x_b}$ plane and ${p_a}$-${p_b}$ plane, respectively. (a) Wave function at the input of the 50:50 BS, where a squeezed vacuum and a coherent state are combined. (b) Wave function at the output of the 50:50 BS, which is rotated by ${+}{{45}}^\circ$ from (a). (c) Wave function with a small opposite phase shift given in each arm due to gravitational waves. (d) Wave function of the final state, rotated by ${-}{{45}}^\circ$ from (c). The detected signal at PD is $x_b^2 + p_b^2 - 1/2 \sim p_b^2$, whose fluctuation is squeezed as seen from the right figure in (d). Saturation and hue represent the absolute value and phase of the two-mode wave function, respectively, as shown in the scale bar in the upper right corner. The phase stripes on the $p$ plane should actually be more frequent, but they are drawn less for ease of viewing. PD, photodiode; PBS, polarizing beam splitter.

Download Full Size | PDF

By introducing the squeezed vacuum that is anti-squeezed in $x$ into the output port, the shot noise can be reduced [1]. Figure 2 represents the evolution of the wave function in the Michelson interferometer, where the left figure shows the beams under consideration, and the middle figure and right figure show the two-mode wave functions in the ${x_a}$-${x_b}$ plane and ${p_a}$-${p_b}$ plane, respectively. Saturation and hue represent the absolute value and phase of the wave function, respectively, as shown in the scale bar in the upper right corner. For simplicity, we neglect the constant phase shift caused by light propagation in the two arms of the interferometer.

First, we assume that a coherent state with an amplitude ${x_0}$ and anti-squeezed vacuum with a squeezing parameter of ${-}r$ are combined by a 50:50 BS [Fig. 2(a)]. We will see later that the anti-squeezing leads to quantum enhancement. According to Eqs. (A20), (A24), and (A25) in Appendix A, the wave function ${\psi _1}({x_a},{x_b})$ at the input of BS can be derived as

$$\begin{split} {{\psi _1}({x_a},{x_b})}&= (2/\pi {)^{1/4}}\exp \left[{- {{({x_a} - {x_0})}^2}} \right]{{(2/\pi)}^{1/4}}\\&\quad\times\exp \left[{- \frac{r}{2} - {e^{- 2r}}x_b^2} \right]\\&= \sqrt {\frac{2}{\pi}} \exp \left[{- \frac{r}{2} - {{({x_a} - {x_0})}^2} - {e^{- 2r}}x_b^2} \right].\end{split}$$
This represents an ellipse centered at $({x_a},{x_b}) = ({x_0},0)$ and elongated in the ${x_b}$ direction. Its Fourier transform ${\tilde \psi _1}({p_a},{p_b})$ [see Eq. (A16) in Appendix A, Section A.2] is given by
$${\tilde \psi _1}({p_a},{p_b}) = \sqrt {\frac{2}{\pi}} \exp \left[{\frac{r}{2} - 2i{x_0}{p_a} - p_a^2 - {e^{2r}}p_b^2} \right].$$
This is also an ellipse centered at the origin and shrunk in the ${p_b}$ direction, and has a linear phase shift in the ${-}{p_a}$ direction that corresponds to the displacement in ${x_a}$. At the BS output, these wave functions are rotated by 45° (see Appendix A, Section A.6) as shown in Fig. 2(b). Therefore, the wave function ${\psi _2}({x_a},{x_b})$ and its Fourier transform ${\tilde \psi _2}({p_a},{p_b})$ at the BS output are given by
$$\begin{split}{\psi _2}({x_a},{x_b}) &= \sqrt {\frac{2}{\pi}} \exp \left[- \frac{r}{2} - \frac{1}{2}{{\left({{x_a} + {x_b} - \sqrt 2 {x_0}} \right)}^2}\right. \\&\quad- \left.\frac{{{e^{- 2r}}}}{2}{{(- {x_a} + {x_b})}^2} \right],\end{split}$$
$$\begin{split}{\tilde \psi _2}({p_a},{p_b})& = \sqrt {\frac{2}{\pi}} \exp \left[\vphantom{\frac{1}{2}\frac{{{e^{2r}}}}{2}}\frac{r}{2} - \sqrt 2 i{x_0}({p_a} + {p_b}) \right.\\&\quad-\left. \frac{1}{2}{{({p_a} + {p_b})}^2} - \frac{{{e^{2r}}}}{2}{{(- {p_a} + {p_b})}^2} \right].\end{split}$$

When gravitational waves arrive at the interferometer, a tiny phase shift $\Delta \phi$ is given in each arm oppositely [Fig. 2(c)]. According to Eq. (A22), the phase shift changes the wave function as

$$\begin{split}{\psi _3}({x_a},{x_b})& = \sqrt {\frac{2}{\pi}} \exp \left[\!\vphantom{\frac{1}{2}\frac{{{e^{2r}}}}{2}}- \frac{r}{2} - 2i{x_0}\Delta \phi (- {x_a} + {x_b}) \right.\\&\quad- \left.\frac{1}{2}{{\left({{x_a} + {x_b} - \sqrt 2 {x_0}} \right)}^2} - \frac{{{e^{- 2r}}}}{2}{{(- {x_a} + {x_b})}^2} \right],\end{split}$$
$$\begin{split}{\tilde \psi _3}({p_a},{p_b}) &= \sqrt {\frac{2}{\pi}} \exp \left[\vphantom{\frac{1}{2}\frac{{{e^{2r}}}}{2}}\frac{r}{2} - \sqrt 2 i{x_0}({p_a} + {p_b}) \right.\\&\quad-\left. \frac{1}{2}{{({p_a} + {p_b}\!)}^2} - \frac{{{e^{2r}}}}{2}{{(- {p_a} + {p_b} + 2{x_0}\Delta \phi)}^2} \!\right]\!.\end{split}$$
Subsequently, beams are combined again at the BS, and the wave function is rotated by −45° [Fig. 2(d)], resulting in the wave functions given by
$$\begin{split}{\psi _4}({x_a},{x_b}) &= \sqrt {\frac{2}{\pi}} \exp \left[- \frac{r}{2} - 2\sqrt 2 i{x_0}\Delta \phi {x_b} \right.\\&\quad-\left. {{({{x_a} - {x_0}} )}^2} - {e^{- 2r}}x_b^2\vphantom{\frac{r}{2}} \right],\end{split}$$
$$\begin{split}{\tilde \psi _4}({p_a},{p_b})& = \sqrt {\frac{2}{\pi}} \exp \left[\vphantom{\left({{p_b} + \sqrt 2 {x_0}\Delta \phi} \right)^2}\frac{r}{2} - 2i{x_0}{p_a} - p_a^2\right. \\&\quad-\left. {e^{2r}}{{\left({{p_b} + \sqrt 2 {x_0}\Delta \phi} \right)}^2} \right].\end{split}$$
Considering $x_b^2 \sim 0$ as seen in the middle figure in Fig. 2(d), the number of photons detected by the photodiode is $x_b^2 + p_b^2 - 1/2 \sim p_b^2$. The right figure in Fig. 2(d) clearly shows that the fluctuation in ${p_b}$ is squeezed, which enables sensitive measurement by quantum enhancement.

It is worth mentioning that the fluctuations of ${x_a}$ and ${x_b}$ are anti-correlated as shown in the middle figure of Fig. 2(b), which increases radiation pressure noise and degrades sensitivity in the low-frequency regime, necessitating frequency-dependent squeezing for more sensitive detection. In Appendix A, Section A.1, we describe the evolution of the wave function when radiation pressure is significant.

B. Generation of Amplitude-Squeezed States with an Asymmetric Beam Splitter

Amplitude-squeezed states, which have lower intensity noise than the shot-noise limit, have been widely used for sensitive measurements [4,8–10]. Amplitude-squeezed states can be generated in two ways: deamplification of a coherent state by optical parametric amplification (OPA) [10] or combining a squeezed vacuum state and a coherent state with an asymmetric BS [4,9]. The former method can be understood straightforwardly because OPA directly provides the squeezing operation, which is described in Appendix A, Section A.4. On the other hand, the latter method does not provide a pure amplitude-squeezed state because the asymmetric BS with a coherent state can act as a displacement operator, while squeezing level is decreased because some of the quantum fluctuation of the coherent state is mixed to the squeezed state by the BS. Here, we look into this point by using wave functions.

Figure 3 shows the evolution of the wave function in the generation of an amplitude-squeezed state with a 99:1 BS. When a coherent state $a$ with an amplitude ${x_0}$ and squeezed vacuum $b$ with a squeezing parameter $r$ are combined by a 99:1 BS [Fig. 3(a)], the wave function ${\psi _5}({x_a},{x_b})$ of the input beams is given by

$${\psi _5}({x_a},{x_b}) = \sqrt {\frac{2}{\pi}} \exp \left[{\frac{r}{2} - {{({x_a} - {x_0})}^2} - {e^{2r}}x_b^2} \right].$$
At the output of BS, this is rotated by ${\arcsin}\sqrt {0.01}$ as shown in Fig. 3(b). The resultant wave function ${\psi _6}({x_a},{x_b})$ and its Fourier transform ${\tilde \psi _6}({p_a},{p_b})$ are given by
$$\begin{split} {\psi _6}({x_a},{x_b}) &= \sqrt {\frac{2}{\pi}} \exp \left[\frac{r}{2} - {(\sqrt {0.99} {x_a} + \sqrt {0.01} {x_b} - {x_0})^2}\right.\\[-2pt]& \quad-\left. {e^{2r}}{(- \sqrt {0.01} {x_a} + \sqrt {0.99} {x_b})^2}\right],\end{split}$$
$$\begin{split}{{\tilde \psi}_6}({p_a},{p_b})& = \sqrt {\frac{2}{\pi}} \exp \left[\!\vphantom{{\left({- \sqrt {0.01} {p_a} + \sqrt {0.99} {p_b}} \right)^2}}- \frac{r}{2} - 2i{x_0}\left({\sqrt {0.99} {p_a} + \sqrt {0.01} {p_b}} \right) \right.\\[-2pt]&\quad - {{\left({\sqrt {0.99} {p_a} + \sqrt {0.01} {p_b}} \right)}^2} \\&\quad-\left.{e^{- 2r}}{{\left({- \sqrt {0.01} {p_a} + \sqrt {0.99} {p_b}} \right)}^2} \right],\end{split}$$
respectively. From Fig. 3(b), the product of the probability densities of ${x_b}$ and ${p_b}$ can be depicted as Fig. 3(c), which can be considered as the Wigner function of the output beam on side $b$, where squeezed vacuum is shifted in ${x_b}$, which is the same form as the amplitude-squeezed state. Note that the resulting amplitude-squeezed state has an anti-squeezing level and a squeezing level that are smaller than the original squeezed vacuum. This can be understood as a result of the tilted elliptical distribution as shown in Fig. 3(b), which is the consequence of the mixing of quantum fluctuation by the BS.
 figure: Fig. 3.

Fig. 3. Evolution of the wave function in the generation of an amplitude-squeezed state with a 99:1 BS. The left figure shows the beams under consideration, and the middle figure and right figure show the wave functions in the ${x_a}$-${x_b}$ plane and ${p_a}$-${p_b}$ plane, respectively. (a) Input beams of the 99:1 BS, where $a$ is a coherent state and $b$ is a squeezed vacuum. (b) Output beams of the 99:1 BS, where the wave function is rotated slightly by ${\arcsin}\sqrt {0.01}$ from (a). (c) Product of the probability densities of ${x_{{b^\prime}}}$ and ${p_{{b^\prime}}}$, which can be considered as the Wigner function of the output beam of ${b^\prime}$. The distribution is shifted from the origin in the ${x_{{b^\prime}}}$ direction. Note that the resulting amplitude-squeezed state has a smaller anti-squeezing level and squeezing level than that of the original squeezed vacuum since the elliptical distribution is slightly tilted in (b). Saturation and hue represent the absolute value and phase of the two-mode wave function, respectively, as shown in the scale bar in the lower right corner. The phase stripes on the $p$ plane should actually be more frequent, but they are drawn less for ease of viewing.

Download Full Size | PDF

 figure: Fig. 4.

Fig. 4. Evolution of the wave function in the balanced homodyne detection of a squeezed state. The left figure shows the beams under consideration, and the middle figure and right figure show the wave functions in the ${x_a}$-${x_b}$ plane and in the ${p_a}$-${p_b}$ plane, respectively. (a) Input beam of the 50:50 BS, where $a$ is a coherent state and $b$ is a squeezed vacuum. (b) Output beam of the 50:50 BS, where the wave function is rotated by 45° from (a). We can see that the fluctuations in ${x_{{a^\prime}}}$ and ${x_{{b^\prime}}}$ are correlated, resulting in a smaller fluctuation in the output of the balanced homodyne detector. Saturation and hue represent the absolute value and phase of the two-mode wave function, respectively, as shown in the scale bar in the upper right corner. The phase stripes on the $p$ plane should have higher frequency in practice, but they are drawn less for clarity. PD, photodiode.

Download Full Size | PDF

C. Balanced Homodyne Detection

Squeezed states of light are often measured by the balanced homodyne detection technique. Here, we explore the evolution of the quantum state in balanced homodyne detection. The same wave function is used in some types of quantum-enhanced measurements such as [6] and [11].

When a coherent state and a squeezed vacuum are merged at a 50:50 BS [Fig. 4(a)], the wave function is the same as Eq. (9). This wave function is rotated by 45° at the output of the BS as shown in Fig. 4(b). The resultant wave function is given by

$$\begin{split}&{\psi _7}({x_a},{x_b}) = \sqrt {\frac{2}{\pi}} \exp \\&\left[{\frac{r}{2} - \frac{1}{2}{{\left({({x_a} + {x_b}) - \sqrt 2 {x_0}} \right)}^2} - \frac{{{e^{2r}}}}{2}{{(- {x_a} + {x_b})}^2}} \right].\end{split}$$
As seen in Fig. 4(b), the fluctuation in ${-}{x_a} + {x_b}$ is squeezed, and therefore the fluctuations in ${x_a}$ and ${x_b}$ are correlated. Consequently, the photocurrents of the two photodiodes are correlated, resulting in small fluctuations in the output of the balanced homodyne detector, i.e., the difference between photocurrents. Thus, the balanced homodyne detection of squeezed states can be regarded as the result of the correlation (entanglement) in the quantum fluctuation between the two output ports of the BS.

3. CONCLUSION

In this paper, we have shown the pictorial interpretations of various quantum-enhanced measurements. We presented the evolution of wave functions in gravitational-wave detection, the generation of amplitude-squeezed states by a BS, and balanced homodyne detection. We anticipate that these pictures provide intuitive interpretations of quantum states in various quantum-enhanced measurements and facilitate the applications of quantum optics.

APPENDIX A

A.1. Gravitational-Wave Detection when Radiation Pressure Noise Is Significant

The discussion in Section 2.A is true only when the shot noise is dominant. When considering the effect of radiation pressure, the optimal squeezing angle changes [23–25]. Here, we introduce two-mode wave functions and their analytical expressions assuming that the effect of radiation pressure noise is significant.

At the intermediate frequencies where the shot noise and radiation pressure noise are balanced, a squeezing angle of 45° is optimal, where $(x + p)/\sqrt 2$ is squeezed and $(x - p)/\sqrt 2$ is anti-squeezed. Figure 5 shows the evolution of the two-mode wave function in the gravitational-wave detection when 45° squeezing is applied as a mode $\hat b$.

 figure: Fig. 5.

Fig. 5. Evolution of the two-mode wave function in gravitational-wave detection when 45° squeezing is applied as mode $b$. (a) Wave function at the input of the 50:50 BS, where a 45° squeezed vacuum and a coherent state are combined. The fluctuations are large in both ${x_b}$ and ${p_b}$ directions, and ${\psi _1}({x_a},{x_b})$ has a phase proportional to $x_b^2$. (b) Wave function at the output of the 50:50 BS, which is rotated by 45° from (a). (c) Wave function after radiation pressure is applied. An optical phase shift proportional to the number of photons introduces the phase shift of $\Delta {\phi _{{\rm{rad}}}}$ into the wave function, which cancels out the original phase proportional to ${({x_a} - {x_b})^2}$, resulting in suppression of the fluctuation in $({p_a} - {p_b})/\sqrt 2$ direction. (d) Wave function with a small opposite phase shift given in each arm due to gravitational waves. (e) Wave function of the final state, rotated by ${-}{{45}}^\circ$ from (c). The detected signal at PD is $x_b^2 + p_b^2 - 1/2 \sim p_b^2$. Saturation and hue represent the absolute value and phase of the two-mode wave function, respectively, as shown in the scale bar in the upper right corner. The phase stripes should actually be more frequent, but they are drawn less for ease of viewing. PD, photodiode; PBS, polarizing beam splitter.

Download Full Size | PDF

The wave function of the input of BS is represented as

$$\begin{split}{\psi _1}({x_a},{x_b})&= \sqrt {\frac{{2\sqrt {{\rm{sech}}2r}}}{\pi}} \\[-2pt]&\quad\times{\exp}\big(\!-\! {{({x_a} - {x_0})}^2} - \big({{\rm{sech}}2r + i \tanh 2r} \big)x_b^2 \big),\end{split}$$
$$\begin{split}{\tilde \psi _1}({p_a},{p_b}) &= \sqrt {\frac{{2\sqrt {{\rm{sech}}2r}}}{\pi}} \\[-2pt]&\quad\times\exp \big({- p_a^2 - \big({{\rm{sech}}2r - i \tanh 2r} \big)p_b^2} \big),\end{split}$$
where the fluctuations are larger than that of a coherent state in both ${x_b}$ and ${p_b}$ directions, and ${\psi _1}({x_a},{x_b})$ has a phase proportional to $x_b^2$ as shown in Eq. (A1) and Fig. 5(a). The wave function is rotated by ${+}{{45}}^\circ$ at 50:50 BS as shown in Fig. 5(b), and becomes
$$\begin{split}{\psi _2}({x_a},{x_b})& = \sqrt {\frac{{2\sqrt {{\rm{sech}}2r}}}{\pi}} \exp\left({- \frac{1}{2}{{({x_a} + {x_b} - \sqrt 2 {x_0})}^2}} \right.\\[-2pt]&\quad\left. {- \frac{1}{2}\!\left({{\rm{sech}}2r + i\tanh 2r} \right){{(- {x_a} + {x_b})}^2}} \right),\end{split}$$
$$\begin{split}{{\tilde \psi}_2}({p_a},{p_b})& = \sqrt {\frac{{2\sqrt {{\rm{sech}}2r}}}{\pi}} \exp\left({- \frac{1}{2}{{({p_a} + {p_b})}^2}} \right.\\&\quad- \left. {\frac{1}{2}\big({{\rm{sech}}2r - i\tanh 2r} \big){{(- {p_a} + {p_b})}^2}} \right).\end{split}$$
The radiation pressure leads to an optical phase shift that is proportional to the number of photons. Assuming that $|{p_a}| \ll |{x_a}|$ and $|{p_b}| \ll |{x_b}|$, this leads to the phase shift of the two-mode wave function in proportion to ${\hat a^\dagger}\hat a{\hat a^\dagger}\hat a + {\hat b^\dagger}\hat b{\hat b^\dagger}\hat b \simeq \hat x_a^4 + \hat x_b^4$. By taking the Taylor expansion around $({x_0}/\sqrt 2 ,{x_0}/\sqrt 2)$,
$$\begin{split}{\phi _{{\rm{rad}}}}& \propto \frac{{x_0^4}}{2} + \sqrt 2 x_0^3\left({({x_a} - {x_0}/\sqrt 2) + ({x_b} - {x_0}/\sqrt 2)} \right)\\&\quad + 3x_0^2\left({{{({x_a} - {x_0}/\sqrt 2)}^2} + {{({x_b} - {x_0}/\sqrt 2)}^2}} \right).\end{split}$$
Since the average displacement in ${p_a}$ and ${p_b}$ due to the radiation pressure is cancelled by the control system of the interferometer, we ignore the second term of Eq. (A5). We also ignore the first term Eq. (A5) because it is a constant phase. Therefore, we obtain
$${\phi _{{\rm{rad}}}} \propto 3x_0^2\left({{{({x_a} - {x_0}/\sqrt 2)}^2} + {{({x_b} - {x_0}/\sqrt 2)}^2}} \right).$$
When the radiation pressure noise is equal to the shot noise,
$${\phi _{{\rm{rad}}}} = ({x_a} - {x_0}/\sqrt 2 {)^2} + {({x_b} - {x_0}/\sqrt 2)^2}.$$
By applying this phase shift to ${\psi _2}({x_a},{x_b})$,
$$\begin{split}{{\psi _3}({x_a},{x_b})}&= {\psi _2}({x_a},{x_b})\exp ({i{\phi _{{\rm{rad}}}}} )\\&= \sqrt {\frac{{2\sqrt {{\rm{sech}}2r}}}{\pi}} \exp \left(\frac{{- 1 + i}}{2}{{({x_a} + {x_b} - \sqrt 2 {x_0})}^2}\right. \\&\quad-\left. \frac{1}{2}\big({{\rm{sech}}2r + i(1 - \tanh 2r)} \big){{(- {x_a} + {x_b})}^2}\! \right).\end{split}$$
Since $\tanh 2r \sim 1$ when $r \gt 1$, we can approximate ${\psi _3}$ as
$$\begin{split}{\psi _3}({x_a},{x_b}) &\sim \sqrt {\frac{{2\sqrt {{\rm{sech}}2r}}}{\pi}} \exp \left(\frac{{- 1 + i}}{2}{{({x_a} + {x_b} - \sqrt 2 {x_0})}^2}\right.\\&\quad -\left. \frac{{{\rm{sech}}2r}}{2}{{(- {x_a} + {x_b})}^2}\! \right).\end{split}$$
As a result, the phase ${\phi _{{\rm{rad}}}}$ of the radiation pressure cancels out the original phase proportional to ${(- {x_a} + {x_b})^2}$. ${\tilde \psi _3}({p_a},{p_b})$ is obtained by Fourier transform of ${\psi _3}({x_a},{x_b})$:
$$\begin{split}{\tilde \psi _3}({p_a},{p_b})& = \sqrt {\frac{{1 + i}}{{\pi \sqrt {{\rm{sech}}2r}}}} \exp \left(- \frac{{1 + i}}{4}{{({p_a} + {p_b})}^2}\right.\\&\quad -\left. i{x_0}({p_a} + {p_b}) - \frac{{\cosh 2r}}{2}{{(- {p_a} + {p_b})}^2} \right).\end{split}$$
As can be seen in Eq. (A10) and Fig. 5(c), the fluctuation in the $(- {p_a} + {p_b})/\sqrt 2$ direction becomes smaller because the phase proportional to ${(- {x_a} + {x_b})^2}$ is negligible.

The rest of the calculations are the same as in Section 2.A. When gravitational waves arrive at the interferometer, a phase shift $\Delta \phi$ is given in each arm oppositely [Fig. 5(d)], and the wave function is

$$\begin{split}&{\psi _4}({x_a},{x_b}) = \sqrt {\frac{{2\sqrt {{\rm{sech}}2r}}}{\pi}}\exp \left(\vphantom{\frac{{- 1 + i}}{2}}- 2i{x_0}\Delta \phi (- {x_a} + {x_b}) \right.\\&\quad+\left. \frac{{- 1 + i}}{2}{{({x_a} + {x_b} - \sqrt 2 {x_0})}^2} - \frac{{{\rm{sech}}2r}}{2}{{(- {x_a} + {x_b})}^2} \right),\end{split}$$
$$\begin{split}&{{\tilde \psi}_4}({p_a},{p_b}) = \sqrt {\frac{{1 + i}}{{\pi \sqrt {{\rm{sech}}2r}}}} \exp\left(- \frac{{1 + i}}{4}{{({p_a} + {p_b})}^2}\right. \\[-2pt]&\quad-\left. i{x_0}({p_a} + {p_b}) - \frac{{\cosh 2r}}{2}{{(- {p_a} + {p_b} + 2{x_0}\Delta \phi)}^2} \right).\end{split}$$
Then the wave function is rotated by ${-}{{45}}^\circ$ by 50:50 BS [Fig. 5(e)] and represented by
$$\begin{split}{\psi _5}({x_a},{x_b}) &= \sqrt {\frac{{2\sqrt {{\rm{sech}}2r}}}{\pi}} \exp \left(- 2\sqrt 2 i{x_0}\Delta \phi {x_b}\right.\\[-2pt]&\quad +\left. ({- 1 + i} ){{({x_a} - {x_0})}^2} - {\rm{sech}}2rx_b^2\vphantom{\left(- 2\sqrt 2 i{x_0}\Delta \phi {x_b}\right)} \right),\end{split}$$
$$\begin{split}\!\!\!{{\tilde \psi}_5}({p_a},{p_b}) &= \sqrt {\frac{{1 + i}}{{\pi \sqrt {{\rm{sech}}2r}}}} \exp \left({- \frac{{1 + i}}{2}p_a^2 - i2{x_0}{p_a}} \right.\!\!\!\\[-2pt]&\quad\left. {- \cosh 2r{{({p_b} + \sqrt 2 {x_0}\Delta \phi)}^2}} \vphantom{\frac{{1 + i}}{2}}\right).\end{split}$$
As the wave function is rotated by ${-}{{45}}^\circ$, the fluctuation in the $(- {p_a} + {p_b})/\sqrt 2$ direction in Eq. (A10) eventually becomes the fluctuation in the ${p_b}$ direction in Eq. (A14), which is detected by the photodiode. Importantly, the phase ${\phi _{{\rm{rad}}}}$ due to the radiation pressure cancels the phase proportional to ${(- {x_a} + {x_b})^2}$ due to the 45° squeezing to achieve the Fourier transform limit and suppress the fluctuation in ${p_b}$. In this way, we can see how the phase curvature of wave functions are corrected to reduce the fluctuation in the output signal ${p_b}$ by the 45° squeezing when the radiation pressure noise and shot noise are balanced.

At low frequencies where the radiation noise is dominant, ${\phi _{{\rm{rad}}}}$ becomes larger. In this case, the optimal squeezing angle is 0°, i.e., $x$ is squeezed and $p$ is anti-squeezed to minimize the effect of ${p_b}$ fluctuations being increased by ${\phi _{{\rm{rad}}}}$.

A.2. Quantum Harmonic Oscillators and Wave Function

Here, we summarize the treatment of a quantum state of light with a wave function [11,16,26,27]. We utilize a quantum harmonic oscillator as an analog of a lightwave in a single spatial mode and a single time–frequency mode. The quantum harmonic oscillator can be described by the position operator $\hat x$ and momentum operator $\hat p$, which satisfy the commutation relation $[\hat x,\hat p] = i/2$ so that $\hat p = - (i/2)(d/dx)$ in the $x$ representation. Using the annihilation operator $\hat a = \hat x + i\hat p$ and creation operator ${\hat a^\dagger} = \hat x - i\hat p$ that satisfy $[\hat a,{\hat a^\dagger}] = 1$, the Hamiltonian of the quantum harmonic oscillator is described as $\hat H = \hbar \omega ({\hat x^2} + {\hat p^2}) = \hbar \omega ({{{\hat a}^\dagger}\hat a + 1/2}),$ where $\hbar$ is Planck’s constant divided by $2\pi$, and $\omega$ is the angular frequency of light under consideration. Assuming a linearly polarized lightwave propagating along the $z$ axis with a homogeneous electric field and a mode volume of $V$, the electric field operator is given by $\hat E(z,t) = i\sqrt {\hbar \omega /2{\varepsilon _0}V} (\hat a{e^{i(kz - \omega t)}} - {\hat a^\dagger}{e^{- i(kz - \omega t)}})$. More complex lightwaves can be described as a linear combination of various modes. The quantum state of such a linear combination can be discussed by the same concept as the beam combining with a BS.

The $n$-photon state is given by $|n\rangle = (n{!)^{- 1/2}}{({a^\dagger})^n}|0\rangle$, and any pure quantum state $|\psi \rangle$ can be expressed as a linear combination of $|n\rangle$, i.e.,

$$|\psi \rangle = \sum\limits_n {c_n}|n\rangle ,$$
where ${c_n} = \langle n|\psi \rangle$. The wave function of the quantum state is its $x$ representation, i.e., $\psi (x) = \langle x|\psi \rangle$. Its Fourier transform gives the $p$ representation as follows:
$$\begin{split}\tilde \psi (p) &= \langle p|\psi \rangle = \int_{- \infty}^\infty \langle p|x\rangle \langle x|\psi \rangle {\rm{d}}x \\[-2pt]&= {\pi ^{- 1/2}}\int_{- \infty}^\infty \psi (x){e^{- 2ipx}}{\rm{d}}x,\end{split}$$
where we used $\langle x|p\rangle = {e^{2ipx}}/\sqrt \pi$. The ground state $\langle x|0\rangle$ of the harmonic oscillator is given by $\langle x|0\rangle = (2/\pi {)^{1/4}}{e^{- {x^2}}}$ because it satisfies $\langle x|\hat a|0\rangle = \{x + i(1/2i)(d/dx)\} {e^{- {x^2}}} = 0$ and $\parallel |0\rangle {\parallel ^2} = \int_{- \infty}^\infty |\langle x|0\rangle {|^2}{\rm{d}}x = 1$.

The time evolution of a wave function $\psi (x,t)$ is given by

$$\psi (x,t) = \sum\limits_{n = 0}^\infty {e^{- i\omega (n + 1/2)t}}\langle x|n\rangle \langle n|\psi \rangle ,$$
because the eigenenergy of $|n\rangle$ is ${E_n} = \hbar \omega (n + 1/2)$. Furthermore, we can show that the time evolution over a quarter period is given by the Fourier transform with a phase factor. This is because $\psi (x,t + \pi /2\omega)$ is the same form as $\tilde \psi (p,t)$:
$$\begin{split}{\psi (x,t + \pi /2\omega)}&= {e^{- i\pi /4}}\sum\limits_{n = 0}^\infty {{(- i)}^n}\langle x|n\rangle \langle n|\psi \rangle\\[-2pt]&= {e^{- i\pi /4}}\sum\limits_{n = 0}^\infty \frac{{{{(- i)}^n}}}{{\sqrt {n!}}}\langle x|({{\hat a}^\dagger}{)^n}|0\rangle \langle n|\psi \rangle\\[-2pt]&= {e^{- i\pi /4}}\sum\limits_{n = 0}^\infty \frac{{{{(- i)}^n}}}{{\sqrt {n!}}}\langle x|(\hat x - i\hat p{)^n}|0\rangle \langle n|\psi \rangle\\[-2pt]&= {e^{- i\pi /4}}\sum\limits_{n = 0}^\infty \frac{{{{(- i)}^n}}}{{\sqrt {n!}}}{{\left({x - \frac{1}{2}\frac{\partial}{{\partial x}}} \right)}^n}\langle x|0\rangle \langle n|\psi \rangle ,\end{split}$$
whereas
$$\begin{split}{\tilde \psi (p,t)}&= \sum\limits_{n = 0}^\infty \langle p|n\rangle \langle n|\psi \rangle\\[-2pt]&= \sum\limits_{n = 0}^\infty \frac{1}{{\sqrt {n!}}}\langle p|({{\hat a}^\dagger}{)^n}|0\rangle \langle n|\psi \rangle\\[-2pt]{}&= \sum\limits_{n = 0}^\infty \frac{1}{{\sqrt {n!}}}\langle p|(\hat x - i\hat p{)^n}|0\rangle \langle n|\psi \rangle\\[-2pt]&= \sum\limits_{n = 0}^\infty \frac{1}{{\sqrt {n!}}}{{\left({\frac{i}{2}\frac{\partial}{{\partial p}} - i\hat p} \right)}^n}\langle p|0\rangle \langle n|\psi \rangle\\[-2pt]&= \sum\limits_{n = 0}^\infty \frac{{{{(- i)}^n}}}{{\sqrt {n!}}}{{\left({p - \frac{1}{2}\frac{\partial}{{\partial p}}} \right)}^n}\langle p|0\rangle \langle n|\psi \rangle .\end{split}$$

Thus, $\psi (x)$, $\tilde \psi (x)$, $\psi (- x)$, and $\tilde \psi (- x)$ are four snapshots of $\psi (x,t)$ in a one period, where $\tilde \psi (x)$ is defined by $\tilde \psi (p{)|_{p = x}}$.

A.3. Coherent State and Its Phase Shift

The wave function of a coherent state with a complex amplitude of $\alpha = {x_0} + i{p_0}$ is given by

$$\begin{split}{{\psi _\alpha}(x)}&= \langle x|\alpha \rangle = \langle x|\hat D({x_0} + i{p_0})|0\rangle = \langle x|\hat D({x_0})\hat D(i{p_0}){e^{i{p_0}{x_0}}}|0\rangle\\&= (2/\pi {)^{1/4}}\exp \left[{2i{p_0}(x - {x_0}) + i{p_0}{x_0} - {{(x - {x_0})}^2}} \right],\end{split}$$
where $\hat D(\alpha) = {e^{\alpha {{\hat a}^\dagger} - {\alpha ^*}\hat a}}$ is the displacement operator. In the calculation, we used its $x$ representation, i.e., $\hat D({x_0}) = {e^{- {x_0}\frac{d}{{dx}}}}$, which translates the wave function by ${x_0}$, and $\hat D(i{p_0}) = {e^{2i{p_0}x}}$, which applies an $x$-dependent phase shift. In the Schrödinger picture, the wave function of the coherent state evolves over time according to
$$\begin{split}{{\psi _\alpha}(x,t)}&= \sum\limits_{n = 0}^\infty {e^{- i(n + 1/2)\omega t}}\langle x|n\rangle \langle n|\alpha \rangle\\&= \sum\limits_{n = 0}^\infty {e^{- i\omega t/2}}\langle x|n\rangle \langle n|\alpha {e^{- i\omega t}}\rangle ,\end{split}$$
which is known as a Gaussian wave packet oscillating at an angular frequency of $\omega$ with an oscillation amplitude of $|\alpha |$.

By substituting $\alpha = {x_0}$ and $t = - \Delta \theta /\omega$ in Eq. (A21), we can calculate a coherent state when the phase shift of $\Delta \theta$ is applied. Assuming $|\Delta \theta | \ll 1$, the wave function can be approximated as

$$\begin{split}{\psi _\alpha}(x, - \Delta \theta /\omega)&\simeq {{(2/\pi)}^{1/4}}\exp \big[2i{x_0}\Delta \theta (x - {x_0})\\&\quad + i\Delta \theta (x_0^2 + 1/2) - {{(x - {x_0})}^2} \big].\end{split}$$

A.4. Squeezing

The squeezing operator with a squeezing angle of 0° is given by $\hat S(r) = {e^{r(\hat a\hat a - {{\hat a}^\dagger}{{\hat a}^\dagger})/2}}$, where $r$ denotes the squeezing parameter. It shrinks the wave function by ${e^{- r}}$ times. This point is understood by considering its $x$ representation:

$$\hat S(r) = {e^{ir(\hat x\hat p + \hat p\hat x)}} = {e^{ir(2\hat x\hat p - i/2)}} = {e^{rx\frac{d}{{dx}}}}{e^{r/2}}.$$
Here, ${e^{rx\frac{d}{{dx}}}}$ shrinks the wave function because ${e^{\frac{d}{{dx}}}} = {e^{2i\hat p}}$ shifts the function in the ${-}x$ direction, and therefore ${e^{x\frac{d}{{dx}}}}$ gives an $x$-dependent shift. The norm of the wave function is preserved by ${e^{r/2}}$. Consequently, the wave function of the squeezed coherent state is given by
$$\begin{split}{\psi _s}(x)& = (2/\pi {)^{1/4}}\exp \\&\quad\left[\frac{r}{2} + 2i{p_0}({e^r}x - {x_0}) + i{p_0}{x_0} - {{({e^r}x - {x_0})}^2} \right].\end{split}$$
The Wigner functions before and after the squeezing of a coherent state are illustrated in Fig. 6. Through squeezing, the Wigner function is shrunk in $x$ and expanded in $p$ from Figs. 6(a) to 6(b), and consequently the intensity fluctuation is squeezed.
 figure: Fig. 6.

Fig. 6. Wigner functions of a (a) coherent state and (b) squeezed coherent state.

Download Full Size | PDF

A.5. Two-Mode Wave Function

We consider two modes of light and denote them as $a$ and $b$. The corresponding harmonic oscillators are characterized by positions ${x_a}$ and ${x_b}$ and momenta ${p_a}$ and ${p_b}$. The quantum state $|\psi \rangle$ can be represented by the two-mode wave function $\psi ({x_a},{x_b}) = \langle {x_a},{x_b}|\psi \rangle$, where $|{x_a},{x_b}\rangle = |{x_a}{\rangle _a}|{x_b}{\rangle _b}$. If the two modes are not entangled, the two-mode wave function is the product of wave functions of $a$ and $b$:

$$\psi ({x_a},{x_b}) = {\psi _a}({x_a}){\psi _b}({x_b}).$$
By taking the Fourier transform, we can express the quantum state using ${p_a}$ or ${p_b}$ such as $\langle {p_a},{x_b}|\psi \rangle$, $\langle {x_a},{p_b}|\psi \rangle$, and $\langle {p_a},{p_b}|\psi \rangle$. When the number of modes $m$ is larger than two, we can consider a wave function in $m$ dimensions to describe the quantum state.

A.6. Beam Splitting

As described in [13], the operation of BS is characterized by the SU(2) group. Assuming mode $a$ and mode $b$ as $x$ polarization and $y$ polarization, respectively, we can introduce the Stokes parameter operators given by [16]:

$$\begin{split}{{{\hat S}_1}}&= {{\hat a}^\dagger}\hat a - {{\hat b}^\dagger}\hat b,\\{{{\hat S}_2}}&= {{\hat a}^\dagger}\hat b + \hat a{{\hat b}^\dagger},\\{{{\hat S}_3}}&= i(\hat a{{\hat b}^\dagger} - {{\hat a}^\dagger}\hat b).\end{split}$$
Any type of mode coupling can be described by the combination of unitary transformation using ${\hat S_1}$, ${\hat S_2}$, and ${\hat S_3}$ as an interaction Hamiltonian. Among them, ${\hat S_1}$ merely applies different phase shifts for mode $a$ and mode $b$, while ${\hat S_2}$ and ${\hat S_3}$ can couple mode $a$ and mode $b$. Throughout this paper, we assume that the BS operation is described by ${e^{- i\Theta {S_3}}}$, which leads to a transmittance of $\mathop {\cos}\nolimits^2 \Theta$ and a reflectance of $\mathop {\sin}\nolimits^2 \Theta$. As shown in Fig. 7(a), ${e^{- i\Theta {S_3}}}$ rotates the wave function $\psi ({x_a},{x_b})$ in the ${x_a}$-${x_b}$ plane [Fig. 7(b)] by $\Theta$ in counterclockwise direction as shown in Fig. 7(c).
 figure: Fig. 7.

Fig. 7. (a) Beam splitter with a power transmittance of $\mathop {\cos}\nolimits^2 \Theta$ and power reflectance $\mathop {\sin}\nolimits^2 \Theta$, respectively, where mode $a$ and mode $b$ are coupled. (b) Wave function of the input beams in the ${x_a}$-${x_b}$ plane. (c) Wave function of the output beams in the ${x_{{a^\prime}}}$-${x_{{b^\prime}}}$ plane, which is rotated by $\Theta$ from (b).

Download Full Size | PDF

The rotation of the wave function can be understood by calculating the counterclockwise rotating wave function in the ${x_a}$-${x_b}$ plane with respect to $\Theta$, and taking a derivative as follows:

$$\begin{split}&\left.\frac{d}{{d\Theta}}\psi ({x_a}\cos \Theta + {x_b}\sin \Theta , - {x_a}\sin \Theta + {x_b}\cos \Theta )\right|_{\Theta = 0}\\&= {x_b}\frac{\partial}{{\partial {x_a}}}\psi - {x_a}\frac{\partial}{{\partial {x_b}}}\psi\\&= - 2i({{\hat x}_a}{{\hat p}_b} - {{\hat p}_a}{{\hat x}_b})\psi\\&= (\hat a{{\hat b}^\dagger} - {{\hat a}^\dagger}\hat b)\psi,\end{split}$$
which leads to $\psi ({x_a}\cos \Theta + {x_b}\sin \Theta , - {x_a}\sin \Theta + {x_b}\cos \Theta) = {e^{\Theta (\hat a{{\hat b}^\dagger} - {{\hat a}^\dagger}\hat b)}}\psi ({x_a},{x_b})$. In a similar manner, we can show that ${e^{- i\Theta {S_2}}}$ gives the clockwise rotation of the wave function $\tilde \psi ({x_a},{p_b})$ Fourier-transformed only in mode $b$ by $\Theta$ in the ${x_a}$-${p_b}$ plane.

Funding

Core Research for Evolutional Science and Technology (JPMJCR1872).

Acknowledgment

The authors thank Dr. Kenichi Oguchi and Yoshitaka Taguchi for fruitful discussions, and anonymous reviewers for insightful comments that improved the quality of the paper.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES

1. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981). [CrossRef]  

2. R. Slusher, L. Hollberg, B. Yurke, J. Mertz, and J. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985). [CrossRef]  

3. J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, R. Adhikari, and C. Affeldt, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013). [CrossRef]  

4. E. Polzik, J. Carri, and H. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992). [CrossRef]  

5. N. Treps, N. Grosse, W. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003). [CrossRef]  

6. H. Sun, Z. Liu, K. Liu, R. Yang, J. Zhang, and J. Gao, “Experimental demonstration of a displacement measurement of an optical beam beyond the quantum noise limit,” Chin. Phys. Lett. 31, 084202 (2014). [CrossRef]  

7. N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Matre, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002). [CrossRef]  

8. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Subdiffraction-limited quantum imaging within a living cell,” Phys. Rev. X 4, 011017 (2014). [CrossRef]  

9. R. B. de Andrade, H. Kerdoncuff, K. Berg-Sørensen, T. Gehring, M. Lassen, and U. L. Andersen, “Quantum-enhanced continuous-wave stimulated Raman scattering spectroscopy,” Optica 7, 470–475 (2020). [CrossRef]  

10. C. A. Casacio, L. S. Madsen, A. Terrasson, M. Waleed, K. Barnscheidt, B. Hage, M. A. Taylor, and W. P. Bowen, “Quantum-enhanced nonlinear microscopy,” Nature 594, 201–206 (2021). [CrossRef]  

11. Y. Ozeki, Y. Miyawaki, and Y. Taguchi, “Quantum-enhanced balanced detection for ultrasensitive transmission measurement,” J. Opt. Soc. Am. B 37, 3288–3295 (2020). [CrossRef]  

12. R. S. Bondurant and J. H. Shapiro, “Squeezed states in phase-sensing interferometers,” Phys. Rev. D 30, 2548–2554 (1984). [CrossRef]  

13. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986). [CrossRef]  

14. M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016). [CrossRef]  

15. B. J. Lawrie, P. D. Lett, A. M. Marino, and R. C. Pooser, “Quantum sensing with squeezed light,” ACS Photon. 6, 1307–1318 (2019). [CrossRef]  

16. U. Leonhardt, Essential Quantum Optics: From Quantum Measurements to Black Holes (Cambridge University, 2010).

17. A. I. Lvovsky, “Squeezed light,” in Photonics: Scientific Foundations, Technology and Applications (2015), Vol. 1, pp. 121–163.

18. K. Makino, Y. Hashimoto, J.-I. Yoshikawa, H. Ohdan, T. Toyama, P. van Loock, and A. Furusawa, “Synchronization of optical photons for quantum information processing,” Sci. Adv. 2, e1501772 (2016). [CrossRef]  

19. W. Winkler, G. Wagner, and G. Leuchs, “Interferometric detection of gravitational radiation and nonclassical light,” in Fundamentals of Quantum Optics II (Springer Verlag, 1987), pp. 92–108.

20. G. Leuchs, T. Ralph, C. Silberhorn, and N. Korolkova, “Scheme for the generation of entangled solitons for quantum communication,” J. Mod. Opt. 46, 1927–1939 (1999). [CrossRef]  

21. G. Leuchs, “Precision in length,” in Laser Physics at the Limits (Springer Verlag, 2002), pp. 209–221.

22. B. Stiller, U. Seyfarth, and G. Leuchs, “Temporal and spectral properties of quantum light,” arXiv 1411.3765 (2014).

23. W. Unruh, “Quantum noise in the interferometer detector,” in Quantum Optics, Experimental Gravitation, and Measurement Theory (Plenum, 1983), pp. 647–660.

24. M. Jaeckel and S. Reynaud, “Quantum limits in interferometric measurements,” Europhys. Lett. 13, 301–306 (1990). [CrossRef]  

25. A. Luis and L. Sánchez-Soto, “Breaking the standard quantum limit for interferometric measurements,” Opt. Commun. 89, 140–144 (1992). [CrossRef]  

26. R. Loudon and P. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987). [CrossRef]  

27. U. Leonhardt, “Quantum statistics of a lossless beam splitter: SU(2) symmetry in phase space,” Phys. Rev. A 48, 3265–3277 (1993). [CrossRef]  

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)
Fig. 1.
Fig. 1. Schematic of a gravitational-wave detector. A squeezed vacuum is injected from the output port of a Michelson interferometer. PD, photodiode; PBS, polarizing beam splitter.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Evolution of the wave function in gravitational-wave detection. The left figure shows the beams under consideration, and the middle figure and right figure show the wave function in the ${x_a}$-${x_b}$ plane and ${p_a}$-${p_b}$ plane, respectively. (a) Wave function at the input of the 50:50 BS, where a squeezed vacuum and a coherent state are combined. (b) Wave function at the output of the 50:50 BS, which is rotated by ${+}{{45}}^\circ$ from (a). (c) Wave function with a small opposite phase shift given in each arm due to gravitational waves. (d) Wave function of the final state, rotated by ${-}{{45}}^\circ$ from (c). The detected signal at PD is $x_b^2 + p_b^2 - 1/2 \sim p_b^2$, whose fluctuation is squeezed as seen from the right figure in (d). Saturation and hue represent the absolute value and phase of the two-mode wave function, respectively, as shown in the scale bar in the upper right corner. The phase stripes on the $p$ plane should actually be more frequent, but they are drawn less for ease of viewing. PD, photodiode; PBS, polarizing beam splitter.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Evolution of the wave function in the generation of an amplitude-squeezed state with a 99:1 BS. The left figure shows the beams under consideration, and the middle figure and right figure show the wave functions in the ${x_a}$-${x_b}$ plane and ${p_a}$-${p_b}$ plane, respectively. (a) Input beams of the 99:1 BS, where $a$ is a coherent state and $b$ is a squeezed vacuum. (b) Output beams of the 99:1 BS, where the wave function is rotated slightly by ${\arcsin}\sqrt {0.01}$ from (a). (c) Product of the probability densities of ${x_{{b^\prime}}}$ and ${p_{{b^\prime}}}$, which can be considered as the Wigner function of the output beam of ${b^\prime}$. The distribution is shifted from the origin in the ${x_{{b^\prime}}}$ direction. Note that the resulting amplitude-squeezed state has a smaller anti-squeezing level and squeezing level than that of the original squeezed vacuum since the elliptical distribution is slightly tilted in (b). Saturation and hue represent the absolute value and phase of the two-mode wave function, respectively, as shown in the scale bar in the lower right corner. The phase stripes on the $p$ plane should actually be more frequent, but they are drawn less for ease of viewing.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Evolution of the wave function in the balanced homodyne detection of a squeezed state. The left figure shows the beams under consideration, and the middle figure and right figure show the wave functions in the ${x_a}$-${x_b}$ plane and in the ${p_a}$-${p_b}$ plane, respectively. (a) Input beam of the 50:50 BS, where $a$ is a coherent state and $b$ is a squeezed vacuum. (b) Output beam of the 50:50 BS, where the wave function is rotated by 45° from (a). We can see that the fluctuations in ${x_{{a^\prime}}}$ and ${x_{{b^\prime}}}$ are correlated, resulting in a smaller fluctuation in the output of the balanced homodyne detector. Saturation and hue represent the absolute value and phase of the two-mode wave function, respectively, as shown in the scale bar in the upper right corner. The phase stripes on the $p$ plane should have higher frequency in practice, but they are drawn less for clarity. PD, photodiode.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. Evolution of the two-mode wave function in gravitational-wave detection when 45° squeezing is applied as mode $b$. (a) Wave function at the input of the 50:50 BS, where a 45° squeezed vacuum and a coherent state are combined. The fluctuations are large in both ${x_b}$ and ${p_b}$ directions, and ${\psi _1}({x_a},{x_b})$ has a phase proportional to $x_b^2$. (b) Wave function at the output of the 50:50 BS, which is rotated by 45° from (a). (c) Wave function after radiation pressure is applied. An optical phase shift proportional to the number of photons introduces the phase shift of $\Delta {\phi _{{\rm{rad}}}}$ into the wave function, which cancels out the original phase proportional to ${({x_a} - {x_b})^2}$, resulting in suppression of the fluctuation in $({p_a} - {p_b})/\sqrt 2$ direction. (d) Wave function with a small opposite phase shift given in each arm due to gravitational waves. (e) Wave function of the final state, rotated by ${-}{{45}}^\circ$ from (c). The detected signal at PD is $x_b^2 + p_b^2 - 1/2 \sim p_b^2$. Saturation and hue represent the absolute value and phase of the two-mode wave function, respectively, as shown in the scale bar in the upper right corner. The phase stripes should actually be more frequent, but they are drawn less for ease of viewing. PD, photodiode; PBS, polarizing beam splitter.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. Wigner functions of a (a) coherent state and (b) squeezed coherent state.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. (a) Beam splitter with a power transmittance of $\mathop {\cos}\nolimits^2 \Theta$ and power reflectance $\mathop {\sin}\nolimits^2 \Theta$, respectively, where mode $a$ and mode $b$ are coupled. (b) Wave function of the input beams in the ${x_a}$-${x_b}$ plane. (c) Wave function of the output beams in the ${x_{{a^\prime}}}$-${x_{{b^\prime}}}$ plane, which is rotated by $\Theta$ from (b).

View in Article | Download Full Size | PDF

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

ψ 1 ( x a , x b ) = ( 2 / π ) 1 / 4 exp ⁡ [ − ( x a − x 0 ) 2 ] ( 2 / π ) 1 / 4 × exp ⁡ [ − r 2 − e − 2 r x b 2 ] = 2 π exp ⁡ [ − r 2 − ( x a − x 0 ) 2 − e − 2 r x b 2 ] .
ψ ~ 1 ( p a , p b ) = 2 π exp ⁡ [ r 2 − 2 i x 0 p a − p a 2 − e 2 r p b 2 ] .
ψ 2 ( x a , x b ) = 2 π exp ⁡ [ − r 2 − 1 2 ( x a + x b − 2 x 0 ) 2 − e − 2 r 2 ( − x a + x b ) 2 ] ,
ψ ~ 2 ( p a , p b ) = 2 π exp ⁡ [ 1 2 e 2 r 2 r 2 − 2 i x 0 ( p a + p b ) − 1 2 ( p a + p b ) 2 − e 2 r 2 ( − p a + p b ) 2 ] .
ψ 3 ( x a , x b ) = 2 π exp ⁡ [ 1 2 e 2 r 2 − r 2 − 2 i x 0 Δ ϕ ( − x a + x b ) − 1 2 ( x a + x b − 2 x 0 ) 2 − e − 2 r 2 ( − x a + x b ) 2 ] ,
ψ ~ 3 ( p a , p b ) = 2 π exp ⁡ [ 1 2 e 2 r 2 r 2 − 2 i x 0 ( p a + p b ) − 1 2 ( p a + p b ) 2 − e 2 r 2 ( − p a + p b + 2 x 0 Δ ϕ ) 2 ] .
ψ 4 ( x a , x b ) = 2 π exp ⁡ [ − r 2 − 2 2 i x 0 Δ ϕ x b − ( x a − x 0 ) 2 − e − 2 r x b 2 r 2 ] ,
ψ ~ 4 ( p a , p b ) = 2 π exp ⁡ [ ( p b + 2 x 0 Δ ϕ ) 2 r 2 − 2 i x 0 p a − p a 2 − e 2 r ( p b + 2 x 0 Δ ϕ ) 2 ] .
ψ 5 ( x a , x b ) = 2 π exp ⁡ [ r 2 − ( x a − x 0 ) 2 − e 2 r x b 2 ] .
ψ 6 ( x a , x b ) = 2 π exp ⁡ [ r 2 − ( 0.99 x a + 0.01 x b − x 0 ) 2 − e 2 r ( − 0.01 x a + 0.99 x b ) 2 ] ,
ψ ~ 6 ( p a , p b ) = 2 π exp ⁡ [ ( − 0.01 p a + 0.99 p b ) 2 − r 2 − 2 i x 0 ( 0.99 p a + 0.01 p b ) − ( 0.99 p a + 0.01 p b ) 2 − e − 2 r ( − 0.01 p a + 0.99 p b ) 2 ] ,
ψ 7 ( x a , x b ) = 2 π exp [ r 2 − 1 2 ( ( x a + x b ) − 2 x 0 ) 2 − e 2 r 2 ( − x a + x b ) 2 ] .
ψ 1 ( x a , x b ) = 2 s e c h 2 r π × exp ( − ( x a − x 0 ) 2 − ( s e c h 2 r + i tanh ⁡ 2 r ) x b 2 ) ,
ψ ~ 1 ( p a , p b ) = 2 s e c h 2 r π × exp ⁡ ( − p a 2 − ( s e c h 2 r − i tanh ⁡ 2 r ) p b 2 ) ,
ψ 2 ( x a , x b ) = 2 s e c h 2 r π exp ⁡ ( − 1 2 ( x a + x b − 2 x 0 ) 2 − 1 2 ( s e c h 2 r + i tanh ⁡ 2 r ) ( − x a + x b ) 2 ) ,
ψ ~ 2 ( p a , p b ) = 2 s e c h 2 r π exp ⁡ ( − 1 2 ( p a + p b ) 2 − 1 2 ( s e c h 2 r − i tanh ⁡ 2 r ) ( − p a + p b ) 2 ) .
ϕ r a d ∝ x 0 4 2 + 2 x 0 3 ( ( x a − x 0 / 2 ) + ( x b − x 0 / 2 ) ) + 3 x 0 2 ( ( x a − x 0 / 2 ) 2 + ( x b − x 0 / 2 ) 2 ) .
ϕ r a d ∝ 3 x 0 2 ( ( x a − x 0 / 2 ) 2 + ( x b − x 0 / 2 ) 2 ) .
ϕ r a d = ( x a − x 0 / 2 ) 2 + ( x b − x 0 / 2 ) 2 .
ψ 3 ( x a , x b ) = ψ 2 ( x a , x b ) exp ⁡ ( i ϕ r a d ) = 2 s e c h 2 r π exp ⁡ ( − 1 + i 2 ( x a + x b − 2 x 0 ) 2 − 1 2 ( s e c h 2 r + i ( 1 − tanh ⁡ 2 r ) ) ( − x a + x b ) 2 ) .
ψ 3 ( x a , x b ) ∼ 2 s e c h 2 r π exp ⁡ ( − 1 + i 2 ( x a + x b − 2 x 0 ) 2 − s e c h 2 r 2 ( − x a + x b ) 2 ) .
ψ ~ 3 ( p a , p b ) = 1 + i π s e c h 2 r exp ⁡ ( − 1 + i 4 ( p a + p b ) 2 − i x 0 ( p a + p b ) − cosh ⁡ 2 r 2 ( − p a + p b ) 2 ) .
ψ 4 ( x a , x b ) = 2 s e c h 2 r π exp ⁡ ( − 1 + i 2 − 2 i x 0 Δ ϕ ( − x a + x b ) + − 1 + i 2 ( x a + x b − 2 x 0 ) 2 − s e c h 2 r 2 ( − x a + x b ) 2 ) ,
ψ ~ 4 ( p a , p b ) = 1 + i π s e c h 2 r exp ⁡ ( − 1 + i 4 ( p a + p b ) 2 − i x 0 ( p a + p b ) − cosh ⁡ 2 r 2 ( − p a + p b + 2 x 0 Δ ϕ ) 2 ) .
ψ 5 ( x a , x b ) = 2 s e c h 2 r π exp ⁡ ( − 2 2 i x 0 Δ ϕ x b + ( − 1 + i ) ( x a − x 0 ) 2 − s e c h 2 r x b 2 ( − 2 2 i x 0 Δ ϕ x b ) ) ,
ψ ~ 5 ( p a , p b ) = 1 + i π s e c h 2 r exp ⁡ ( − 1 + i 2 p a 2 − i 2 x 0 p a − cosh ⁡ 2 r ( p b + 2 x 0 Δ ϕ ) 2 1 + i 2 ) .
| ψ ⟩ = ∑ n c n | n ⟩ ,
ψ ~ ( p ) = ⟨ p | ψ ⟩ = ∫ − ∞ ∞ ⟨ p | x ⟩ ⟨ x | ψ ⟩ d x = π − 1 / 2 ∫ − ∞ ∞ ψ ( x ) e − 2 i p x d x ,
ψ ( x , t ) = ∑ n = 0 ∞ e − i ω ( n + 1 / 2 ) t ⟨ x | n ⟩ ⟨ n | ψ ⟩ ,
ψ ( x , t + π / 2 ω ) = e − i π / 4 ∑ n = 0 ∞ ( − i ) n ⟨ x | n ⟩ ⟨ n | ψ ⟩ = e − i π / 4 ∑ n = 0 ∞ ( − i ) n n ! ⟨ x | ( a ^ † ) n | 0 ⟩ ⟨ n | ψ ⟩ = e − i π / 4 ∑ n = 0 ∞ ( − i ) n n ! ⟨ x | ( x ^ − i p ^ ) n | 0 ⟩ ⟨ n | ψ ⟩ = e − i π / 4 ∑ n = 0 ∞ ( − i ) n n ! ( x − 1 2 ∂ ∂ x ) n ⟨ x | 0 ⟩ ⟨ n | ψ ⟩ ,
ψ ~ ( p , t ) = ∑ n = 0 ∞ ⟨ p | n ⟩ ⟨ n | ψ ⟩ = ∑ n = 0 ∞ 1 n ! ⟨ p | ( a ^ † ) n | 0 ⟩ ⟨ n | ψ ⟩ = ∑ n = 0 ∞ 1 n ! ⟨ p | ( x ^ − i p ^ ) n | 0 ⟩ ⟨ n | ψ ⟩ = ∑ n = 0 ∞ 1 n ! ( i 2 ∂ ∂ p − i p ^ ) n ⟨ p | 0 ⟩ ⟨ n | ψ ⟩ = ∑ n = 0 ∞ ( − i ) n n ! ( p − 1 2 ∂ ∂ p ) n ⟨ p | 0 ⟩ ⟨ n | ψ ⟩ .
ψ α ( x ) = ⟨ x | α ⟩ = ⟨ x | D ^ ( x 0 + i p 0 ) | 0 ⟩ = ⟨ x | D ^ ( x 0 ) D ^ ( i p 0 ) e i p 0 x 0 | 0 ⟩ = ( 2 / π ) 1 / 4 exp ⁡ [ 2 i p 0 ( x − x 0 ) + i p 0 x 0 − ( x − x 0 ) 2 ] ,
ψ α ( x , t ) = ∑ n = 0 ∞ e − i ( n + 1 / 2 ) ω t ⟨ x | n ⟩ ⟨ n | α ⟩ = ∑ n = 0 ∞ e − i ω t / 2 ⟨ x | n ⟩ ⟨ n | α e − i ω t ⟩ ,
ψ α ( x , − Δ θ / ω ) ≃ ( 2 / π ) 1 / 4 exp ⁡ [ 2 i x 0 Δ θ ( x − x 0 ) + i Δ θ ( x 0 2 + 1 / 2 ) − ( x − x 0 ) 2 ] .
S ^ ( r ) = e i r ( x ^ p ^ + p ^ x ^ ) = e i r ( 2 x ^ p ^ − i / 2 ) = e r x d d x e r / 2 .
ψ s ( x ) = ( 2 / π ) 1 / 4 exp [ r 2 + 2 i p 0 ( e r x − x 0 ) + i p 0 x 0 − ( e r x − x 0 ) 2 ] .
ψ ( x a , x b ) = ψ a ( x a ) ψ b ( x b ) .
S ^ 1 = a ^ † a ^ − b ^ † b ^ , S ^ 2 = a ^ † b ^ + a ^ b ^ † , S ^ 3 = i ( a ^ b ^ † − a ^ † b ^ ) .
d d Θ ψ ( x a cos ⁡ Θ + x b sin ⁡ Θ , − x a sin ⁡ Θ + x b cos ⁡ Θ ) | Θ = 0 = x b ∂ ∂ x a ψ − x a ∂ ∂ x b ψ = − 2 i ( x ^ a p ^ b − p ^ a x ^ b ) ψ = ( a ^ b ^ † − a ^ † b ^ ) ψ ,
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use