# Phase Sensitivity Improvement in Correlation-Enhanced Nonlinear Interferometers

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## Abstract

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## 1. Introduction

## 2. SU(1,1) Interferometer vs. SU(2) Interferometer

#### 2.1. Input–Output Relations

#### 2.2. Phase Measurement

## 3. Phase Sensitivity of SU(1,1) Interferometers

#### 3.1. Phase Sensitivity

#### 3.2. Quantum Cramér–Rao Bound

#### 3.3. Signal Enhanced and Noise Reduced by Destructive Quantum Interference

## 4. Various Types of SU(1,1) Interferometers

#### 4.1. Atomic SU(1,1) Interferometer

#### 4.2. Atom–Light Hybrid SU(1,1) Interferometer

#### 4.3. Deformation of Nonlinear SU(1,1) Interferometers

#### 4.3.1. Truncated SU(1,1) Interferometer

#### 4.3.2. PA + BS Type of SU(1,1) Interferometer

#### 4.3.3. Phase-Sensing Amplified SU(2)-SU(1,1) Hybrid Interferometer

#### 4.3.4. SU(1,1) Interferometer with Feedback and SU(1,1) Interferometer with Multi-Stage

#### 4.3.5. SU(1,1) Interferometer with Nonlinear Phase Shifter

## 5. Phase Sensitivity Improvement via Gain Unbalance in Lossy Interferometers

#### 5.1. Normal All-Optical SU(1,1) Interferometer

#### 5.2. Atom–Light Hybrid SU(1,1) Interferometer

#### 5.3. SU(2) Nested SU(1,1) Interferometer

## 6. Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MZI | Mach–Zehnder interferometer |

SNL | Shot noise limit |

SQL | Standard quantum limit |

HL | Heisenberg limit |

SUI | SU(1,1) interferometer |

OPA | Optical parametric amplifier |

FWM | Four-wave mixing |

PDC | Parametric down-conversion |

BS | Beam splitter |

SMD | Spin-mixing dynamics |

BEC | Bose–Einstein condensate |

QND | Non-demolition measurement |

QCRB | Quantum Cramér–Rao bound |

QFI | Quantum Fisher information |

IDS | Intensity-difference squeezing |

ID | Intensity detection |

HD | Homdyne detection |

PLO | Photon level operation |

SNR | Signal-to-noise ration |

## Appendix A

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**Figure 1.**(Color online) Input–output noise diagrams of (

**a**) BS and (

**b**) PA, respectively. ${a}_{in}$ and ${b}_{in}$ are two input states. ${a}_{out}$ and ${b}_{out}$ are two output states. The colored area represents quadrature distribution.

**Figure 2.**(Color online) (

**a**) A conventional SU(2) interferometer. (

**b**) A nonlinear SU(1,1) interferometer. (

**c**) Interference patterns of SU(1,1) and SU(2) interferometers with the same phase-sensing photon number [41]. The blue line represents SU(1,1), while the red represents SU(2). In Figure (

**b**), it can be seen that the noise between the two arms of the SUI is correlated. After a second PA process, the noise is reduced to vacuum level while the signal is amplified.

**Figure 3.**(Color online) Schematic of atomic SUI [49].

**Figure 4.**(Color online) Experimental sketch for the atom–light hybrid SUI and the physical model is shown as inset picture [56]. A Stokes field ${S}_{0}$ and a pump field ${W}_{1}$ are sent into the $R{b}^{87}$ atomic cell to produce a Stokes field ${S}_{1}$ and the atomic spin wave ${S}_{a}$ via the Raman process, which acts as atom–light wave-splitting process. Then, ${S}_{a}$ stays at the cell and ${S}_{1}$ goes out of the cell. After coupling, the phase shift by PZT, the Stokes field ${S}_{1}^{{}^{\prime}}$ and a write field ${W}_{2}$ go back to the atomic cell and interact with ${S}_{a}$ via the second Raman process, which acts as atom–light wave recombination. Finally, the interference output Stokes field ${S}_{2}$ is detected by the detector D.

**Figure 5.**(Color online) Schematic diagrams of deformation of SUI. (

**a**) A truncated SUI, (

**b**) PA + BS types of SUI, (

**c**) SU(2) nested SUI, (

**d**) SUI with feedback, (

**e**) multi-stage SUI, (

**f**) SUI with a nonlinear phase shifter.

**Figure 6.**(Color online) (

**a**) Optimal phase sensitivity $\mathsf{\Delta}{\varphi}_{\mathrm{min}}$ normalized to $\mathsf{\Delta}{\varphi}_{\mathrm{SNL}}$ and (

**b**) the supersensitive phase range $\mathsf{\Delta}$ of unseeded degenerated SUI with direct ID as functions of the gain ${r}_{2}$ of the second amplifier. (

**c**,

**e**) Optimal phase sensitivity $\mathsf{\Delta}{\varphi}_{\mathrm{min}}$ normalized to $\mathsf{\Delta}{\varphi}_{\mathrm{SNL}}$ and (

**d**,

**f**) the super-sensitive phase range $\mathsf{\Delta}$ of seeded degenerated SUI with ( HD, direct ID) as functions of the gain ${r}_{2}$. The values of the detection efficiency $\eta $: blue line $\eta =1$, black dashed line $\eta =0.3$, and red dotted line $\eta =0.1$. The gain of the first amplifier is ${r}_{1}=1.15$ and the internal transmission is $\mu =0.9$ [105].

**Figure 7.**(Color online) Effect of the internal loss ${R}_{d}$ on the inverse Fano factor ${N}_{d,min}/\mathsf{\Delta}{N}_{d,min}^{2}$ in (

**a**) and on the phase sensitivity without external loss $\mathsf{\Delta}{\phi}_{d,min}^{2}$ in (

**b**) [106]. ${V}_{A}$ means the gain of PA1 and ${V}_{B}$ means the gain of PA2. Fano factor ${N}_{d,min}/\mathsf{\Delta}{N}_{d,min}^{2}$ comes from the equation of the phase sensitivity including extra loss $\mathsf{\Delta}{\phi}_{l}^{2}=\mathsf{\Delta}{\phi}^{2}\left(1+\frac{1-\eta}{\eta}\frac{{N}_{d}}{\mathsf{\Delta}{N}_{d}^{2}}\right)$, where ${N}_{d}$ is the output total mean photon number of an interferometer with variance $\mathsf{\Delta}{N}_{d}^{2}$. It shows that the phase sensitivity in presence of external loss $\mathsf{\Delta}{\phi}_{l}^{2}$ strictly depends on the inverse Fano factor $\frac{{N}_{d}}{\mathsf{\Delta}{N}_{d}^{2}}$. The lower the Fano factor, the more external loss resistance.

**Figure 8.**(Color online) (

**a**–

**c**) The visibilities of the light–atom hybrid interferometer (left-hand vertical axis) before optimization ${G}_{2}$ (blue dash-dotted curve) and after optimization ${G}_{2}$ (red dashed curve). (

**d**–

**f**) The SNR of the light–atom hybrid interferometer (left-hand vertical axis) before and after optimized ${G}_{2}$. The black lines represent the case of MZI. The orange dotted curve is the value of ${G}_{2}/{G}_{1}$ after optimizing ${G}_{2}$ for the largest ${V}_{SU}$ in (

**a**–

**c**) (right-hand vertical axis). The pink circles mark the value of ${G}_{2}/{G}_{1}$ after optimizing ${G}_{2}$ for the best $SN{R}_{SU}$ (right-hand vertical axis) [108].

**Figure 9.**(Color online) The visibility value as a function of the loss rate l (

**a**) before optimization ${G}_{2}$ and (

**b**) after optimization ${G}_{2}$. The blue squares are the interference fringes of the atom–light hybrid interferometer with fixed ${G}_{1}=3,\eta =0.4$, and ${G}_{2}=5$. The black triangles are the interference fringes of the MZI under the same operating conditions. The red dots are the values of optimal visibility with optimization ${G}_{2}$ [108].

**Figure 10.**(Color online) (

**a**) The phase sensitivity as function of external losses and (

**b**) the corresponding relation between ${({G}_{2}/{G}_{1})}_{opt}$ and external losses. (

**c**) The phase sensitivity as function of internal losses and (

**d**) the corresponding relation between ${({G}_{2}/{G}_{1})}_{opt}$ and internal losses [98]. The red dotted lines (SQL1) denote the SQL with optimized ${G}_{2}$ and the black solid lines (SQL0) denote the SQL when ${G}_{2}={G}_{1}$. Upper right corner and within the SQL lines indicate that the phase sensitivity can beat the SQL.

**Table 1.**The optimal phase sensitivity $\mathsf{\Delta}\phi $ of the SUI under gain-balanced condition and the QCRB of the SUI with different input states. The SUI is operated at the optimal phase ${\phi}_{\mathrm{opt}}$. The mean photon number of injected coherent state $\left|\alpha \right.\u232a$ is ${\left|\alpha \right|}^{2}\equiv {N}_{\alpha}$. The mean photon number of injected squeezed vacuum state $|0,\zeta \rangle $ is ${N}_{s}$. The case of coherent ⊗ squeezed vacuum state can be reduced to the case of coherent ⊗ vacuum state when the squeezed degree is $r=0$. For convenience, the sign $\mathcal{K}=$${N}_{\mathrm{OPA}}\left({N}_{\mathrm{OPA}}+2\right)$ where ${N}_{\mathrm{OPA}}=2{sinh}^{2}\xi $ indicates the amount of light PA1 would emit with vacuum inputs [113,114].

Input States | Product | Intensity | Parity | Homodyne | QCRB |
---|---|---|---|---|---|

$|0\rangle \otimes |0\rangle $ | $1/{\mathcal{K}}^{1/2}$ | $1/{\mathcal{K}}^{1/2}$ | $1/{\mathcal{K}}^{1/2}$ | Not available | $1/{\mathcal{K}}^{1/2}$ |

$|\alpha \rangle \otimes |0\rangle $ | $1/{\left[\mathcal{K}({N}_{\alpha}+1)\right]}^{1/2}$ | $\mathsf{\Delta}{\phi}_{\mathrm{I},\mathrm{coh}}$^{a} | $1/{[\mathcal{K}{N}_{\alpha}+1]}^{1/2}$ | $1/{\left[\mathcal{K}{N}_{\alpha}\right]}^{1/2}$ | $\begin{array}{c}1/[\mathcal{K}(2{N}_{\alpha}+1)\hfill \\ +2{N}_{\alpha}({N}_{\mathrm{OPA}}+2){]}^{1/2}\hfill \end{array}$ |

$|\frac{i\alpha}{\sqrt{2}}\rangle \otimes |\frac{\alpha}{\sqrt{2}}\rangle $ | $\begin{array}{c}\hfill 1/\{{N}_{\alpha}[({N}_{\mathrm{OPA}}+2)\\ \hfill (\sqrt{\mathcal{K}}+1)+\mathcal{K}{]+\mathcal{K}\}}^{1/2}\end{array}$ | $1/{\left[\mathcal{K}{N}_{\alpha}\right]}^{1/2}$ | $\mathsf{\Delta}{\phi}_{\mathrm{coh}}$^{b} | $\approx 1/{\left[2\mathcal{K}{N}_{\alpha}\right]}^{1/2}$ | $\begin{array}{c}1/\{2{N}_{\alpha}[({N}_{\mathrm{OPA}}+1)\sqrt{\mathcal{K}}\hfill \\ {+\mathcal{K}+1]+\mathcal{K}\}}^{1/2}\hfill \end{array}$ |

$|\alpha \rangle \otimes |0,\zeta \rangle $ | $1/{\left[\mathcal{K}({N}_{\alpha}{e}^{2r}+coshr{e}^{r})\right]}^{1/2}$ | $\mathsf{\Delta}{\phi}_{I,\mathrm{coh}\&\mathrm{Sqz}}$^{c} | $1/{\left[\mathcal{K}({N}_{\alpha}{e}^{2r}+{cosh}^{2}r)\right]}^{1/2}$ | $1/{\left[\mathcal{K}{N}_{\alpha}{e}^{2r}\right]}^{1/2}$ | $\begin{array}{c}\hfill 1/[2{N}_{\alpha}({N}_{\mathrm{OPA}}+2)\\ \hfill +{N}_{\mathrm{OPA}}^{2}{sinh}^{2}\left(2r\right)/2\\ +\mathcal{K}(2{N}_{\alpha}coshr{e}^{r}\hfill \\ +{cosh}^{2}r{\left)\right]}^{1/2}\hfill \end{array}$ |

$|0,\zeta \rangle \otimes |0,\zeta \rangle $ | $1/{\left[\mathcal{K}(2{N}_{s}+1)\right]}^{1/2}$ | $\mathsf{\Delta}{\phi}_{\mathrm{Sqz}\&\mathrm{Sqz}}$^{d} | $1/{\left[\mathcal{K}(2{N}_{s}+1)\right]}^{1/2}$ | Not available | $\begin{array}{c}1/[{(1+{N}_{\mathrm{OPA}})}^{2}\hfill \\ {cosh4r-1]}^{1/2}\hfill \end{array}$ |

^{a,b,c,d}See Appendix A.

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**MDPI and ACS Style**

Liang, X.; Yu, Z.; Yuan, C.-H.; Zhang, W.; Chen, L.
Phase Sensitivity Improvement in Correlation-Enhanced Nonlinear Interferometers. *Symmetry* **2022**, *14*, 2684.
https://doi.org/10.3390/sym14122684

**AMA Style**

Liang X, Yu Z, Yuan C-H, Zhang W, Chen L.
Phase Sensitivity Improvement in Correlation-Enhanced Nonlinear Interferometers. *Symmetry*. 2022; 14(12):2684.
https://doi.org/10.3390/sym14122684

**Chicago/Turabian Style**

Liang, Xinyun, Zhifei Yu, Chun-Hua Yuan, Weiping Zhang, and Liqing Chen.
2022. "Phase Sensitivity Improvement in Correlation-Enhanced Nonlinear Interferometers" *Symmetry* 14, no. 12: 2684.
https://doi.org/10.3390/sym14122684