Phase estimation in an SU(1,1) interferometer with displaced squeezed states

Sushovit Adhikari; Narayan Bhusal; Chenglong You; Hwang Lee; Jonathan P. Dowling

doi:10.1364/OSAC.1.000438

OSA Continuum
Vol. 1,
Issue 2,
pp. 438-450
(2018)
•https://doi.org/10.1364/OSAC.1.000438

Phase estimation in an SU(1,1) interferometer with displaced squeezed states

Sushovit Adhikari, Narayan Bhusal, Chenglong You, Hwang Lee, and Jonathan P. Dowling

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Sushovit Adhikari, Narayan Bhusal, Chenglong You, Hwang Lee, and Jonathan P. Dowling, "Phase estimation in an SU(1,1) interferometer with displaced squeezed states," OSA Continuum 1, 438-450 (2018)

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- Quantum Optics and Quantum Information
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: July 18, 2018
- Revised Manuscript: August 20, 2018
- Manuscript Accepted: August 22, 2018
- Published: September 18, 2018

Abstract

In this paper, we study the phase sensitivity of an SU(1,1) interferometer with coherent and displaced-squeezed-vacuum (DSV) states as inputs, and parity and on-off as detection strategies. Our scheme with parity is sub-shotnoise limited and approaches the Heisenberg limit with increasing squeezing strength of the optical parametric amplifier (OPA). Also, for the on-off detection scheme, we show that sub-shotnoise sensitivity is possible by increasing the squeezing strength of the OPA.

1. Introduction

Optical interferometry is widely used in studying both theoretical limits and practical applications, ranging from small scale to big experiments such as in gravitational observatories like LIGO [1]. A conventional strategy in a Mach-Zehnder Interferometer (MZI), with coherent input in one arm and vacuum in the other is shotnoise-limited (SNL), with a phase sensitivity given by $1 / \sqrt{\bar{n}}$ , where n̄ is the mean number of photons participating in the phase measurement. This limit can be surpassed by taking advantage of the quantum nature of light. Several schemes with different input states and detection strategies have been proposed, not only to beat the SNL, but also to approach the Heisenberg Limit (HL), which is the fundamental limit given by 1/n̄ [2–8].

A different interferometer used in phase estimation is a SU(1,1) interferometer, proposed by Yurke et al., in 1986 [9]. This interferometer consists of active nonlinear elements (such as an OPA or a four-wave mixer) instead of passive beamsplitters. This interferometic scheme has gained considerable theoretical and experimental attention in the last few years. Plick et al. used a modified scheme using coherent states in both input arms of the interferometer and showed a sub-shotnoise scaling in the high intensity regime, and this scheme was experimentally demonstrated by Ou [10, 11]. Li et al. used a combination of a coherent state and a squeezed vacuum state, with homodyne measurement, to reach HL sensitivity [12]. Later, Li et al. modified the detection strategy to parity measurement and showed Heisenberg-like scaling in the optimal case [13]. In Ref. [14], phase estimation with a coherent state and a displaced-squeezed-vacuum state (DSV) with homodyne detection was studied and Heisenberg-like scaling in sensitivity was shown. It was also shown that these DSV states perform better than Li et al.’s scheme with coherent and squeezed vacuum with Homodyne detection. Other phase estimation strategies with displaced states were also studied in [15, 16]. In Ref. [17–19], the effect of loss on the sensitivity of these interferometers was studied. In Ref. [20], the authors used a thermal state and a squeezed-vacuum state with parity detection and showed that it can beat the SNL. More recently, there has been increased interest in the study of variants of this interferometer. In Ref. [21], a modified setup of SU(1,1), where all the input particles participate in the phase measurement, was proposed with a suggested implementation in spinor Bose-Einstein condensates and hybrid atom-light systems. In Ref. [22], authors introduced a truncated SU(1,1) interferometer, where a single nonlinear element was used.

In this paper, motivated by the recent theoretical and experimental work in SU(1,1) with different input states, we study phase estimation in a SU(1,1) interferometer using coherent and displaced-squeezed-vacuum states as inputs, and parity and on-off detection as measurement. The article is organized as follows: In Sec. 2, we introduce the model and discuss the propagation of the input light fields using characteristic functions and detection strategies. In Sec. 3, we discuss the sensitivity of the SU(1,1) interferometer using displaced-squeezed-vacuum with parity and on-off detection schemes. Lastly, we conclude with a summary in Sec. 4.

2. SU(1,1) Interferometer

A SU(1,1) interferometer is shown in Fig. 1. After the first optical parametric amplifier (OPA), one of the arms undergoes a ϕ phase shift and the other arm is used as a reference. The modes are recombined in the second OPA, and the output depends on the phase shift ϕ. Parity measurement is performed on mode b and for on-off detection, signals are measured in both output modes to estimate the phase difference between the two modes.

Fig. 1 A schematic of a SU(1,1) interferometer. Two OPAs with the same squeezing parameter g is used. The pump field between the two OPAs has a π phase difference. Parity measurement is performed in mode b, and the on-off detection is done in both modes a and b.

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In this paper, we will use the characteristic-function formalism to describe the propagation of the input states through the interferometer. Also known as the symplectic formalism, this approach makes calculation much simpler and intuitive for Gaussian states [23].

Let â (â^†), b̂ (b̂^†) be the annihilation (creation) operators of the upper and lower modes respectively. We can define the quadrature operators of the modes as [23]:

{\hat{x}}_{a_{k}} = {\hat{a}}_{k} + {\hat{a}}_{k}^{†}, {\hat{p}}_{a_{k}} = - i ({\hat{a}}_{k} - {\hat{a}}_{k}^{†}),

{\hat{x}}_{b_{k}} = {\hat{b}}_{k} + {\hat{b}}_{k}^{†}, {\hat{p}}_{a_{k}} = - i ({\hat{b}}_{k} - {\hat{b}}_{k}^{†}) .

A column vector of quadrature operators can be written as:

X_{k} = {({\hat{X}}_{k, 1}, {\hat{X}}_{k, 2}, {\hat{X}}_{k, 3}, {\hat{X}}_{k, 4})}^{T} = {({\hat{x}}_{a_{k}}, {\hat{p}}_{a_{k}}, {\hat{x}}_{b_{k}}, {\hat{p}}_{b_{k}})}^{T} .

The mean and the covariance of quadrature operators are given by:

{\bar{X}}_{k} = {(〈 {\bar{X}}_{k, 1} 〉, 〈 {\bar{X}}_{k, 2} 〉, 〈 {\bar{X}}_{k, 3} 〉, 〈 {\bar{X}}_{k, 4} 〉)}^{T},

Γ_{k}^{m n} = Tr [(Δ {\hat{X}}_{k, m} Δ {\hat{X}}_{k, n} + Δ {\hat{X}}_{k, n} Δ {\hat{X}}_{k, m}) ρ],

where, ΔX̂_k,m = X̂_k,m − 〈X̂_k,m〉, ΔX̂_k,n = X̂_k,n − 〈X̂_k,n〉 and ρ is a density matrix of the input state.

Using the mean and covariance matrix, the input Wigner function can be written as:

W (X_{0}) = \frac{exp [- {(X_{0} - {\bar{X}}_{0})}^{T} \cdot {(Γ_{0})}^{- 1} \cdot (X_{0} - {\bar{X}}_{0})]}{\sqrt{| Γ_{0} |}} .

Here, W(X₀) = W_|α〉 × W_|DSV〉, where W_|α〉 and W_|DSV〉 are the Wigner functions of the coherent state and displaced-squeezed-states respectively.

The symplectic representation of the components of the SU(1,1) interferometer, namely, the first OPA, phase shifter, and the second OPA are described in phase space by:

S_{OPA 1} = (\begin{matrix} cosh (g) & 0 & sinh (g) & 0 \\ 0 & cosh (g) & 0 & - sinh (g) \\ sinh (g) & 0 & cosh (g) & 0 \\ 0 & - sinh (g) & 0 & cosh (g) \end{matrix}),

S_{ϕ} = (\begin{matrix} cos (ϕ) & - sin (ϕ) & 0 & 0 \\ sin (ϕ) & cos (ϕ) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}),

S_{OPA 2} = (\begin{matrix} cosh (g) & 0 & - sinh (g) & 0 \\ 0 & cosh (g) & 0 & sinh (g) \\ - sinh (g) & 0 & cosh (g) & 0 \\ 0 & sinh (g) & 0 & cosh (g) \end{matrix}),

where, we have assumed a squeezing strength of g₁ = g₂ = g and a π phase difference between the first and the second OPA. The propagation of the input mean X̄₀, and input covariance Γ₀, through the SU(1,1) interferometer is given by S = S_OPA2S_ϕS_OPA1. Hence, the output mean (X̄₂) and covariance (Γ₂) of the quadrature operators in the SU(1,1) interferometer can be obtained by:

{\bar{X}}_{2} = S {\bar{X}}_{0},

Γ_{2} = S Γ_{0} S^{T} .

3. Measurement strategies

Our first measurement scheme is parity detection. Parity detection was first proposed by Bollinger et al. to study spectroscopy [24]. It was later adopted by Gerry for optical interferometry [25]. Parity detection is a single-mode measurement, and the parity operator on output mode b is given by:

{\hat{Π}}_{b} = {(- 1)}^{{\hat{b}}_{2}^{†} {\hat{b}}_{2}} .

The parity measurement satisfies 〈Π̂_b〉 = πW(0) [26]. That is, the expectation of the parity measurement is given by the value of the Wigner function at the origin of phase space. This property makes it simple to calculate the parity signal.

Using parity measurement, the phase sensitivity is given by:

Δ ϕ = \frac{〈 Δ {\hat{Π}}_{b} 〉}{| \frac{\partial 〈 {\hat{Π}}_{b} 〉}{\partial ϕ} |}

where

〈 Δ {\hat{Π}}_{b} 〉 = \sqrt{〈 {\hat{Π}}_{b}^{2} 〉 - 〈 {\hat{Π}}_{b} 〉} = \sqrt{1 - {〈 {\hat{Π}}_{b} 〉}^{2}}

with

〈 {\hat{Π}}_{b}^{2} 〉 = 1

Similarly, our other measurement scheme is a on-off detector which only discriminates between zero and non-zero photon number. This detector can be mathematically represented by a set of measurement operators:

{\hat{Π}}_{off} = | 0 〉 〈 0 |, {\hat{Π}}_{on} = \hat{I} - | 0 〉 〈 0 |

where Î is an identity operator. For a single mode Gaussian state, the probability of obtaining non-zero photons is given by: [27]

P_{on} = 1 - \frac{2}{\sqrt{det (Γ + I)}}

where Γ is the covariance matrix at the output and I is an indentity matrix with the same dimension as Γ. The phase sensitivity for the on-off scheme can be calculated using Fisher information. The sensitivity is lower bounded by the classical Crámer-Rao bound given by the inequality [28,29]:

Δ ϕ \geq 1 / \sqrt{F},

And the Fisher information is given by [28–30]:

F = \sum \frac{1}{P_{on}} {(\frac{d P_{on}}{d ϕ})}^{2} .

We want to compare the phase sensitivity of our scheme with a standard metric, namely the HL and the SNL. These limits are given by [14]:

Δ ϕ_{HL} = \frac{1}{{\bar{n}}_{total}} = \frac{1}{{\bar{n}}_{1} + {\bar{n}}_{2} + {\bar{n}}_{ξ} + {\bar{n}}_{opa} (1 + {\bar{n}}_{1} + {\bar{n}}_{2} + {\bar{n}}_{ξ}) + 2 \sqrt{{\bar{n}}_{1} {\bar{n}}_{2} {\bar{n}}_{opa} ({\bar{n}}_{opa} + 2)}}

Δ ϕ_{SNL} = \frac{1}{\sqrt{{\bar{n}}_{total}}} = \frac{1}{\sqrt{{\bar{n}}_{1} + {\bar{n}}_{2} + {\bar{n}}_{ξ} + {\bar{n}}_{opa} (1 + {\bar{n}}_{1} + {\bar{n}}_{2} + {\bar{n}}_{ξ}) + 2 \sqrt{{\bar{n}}_{1} {\bar{n}}_{2} {\bar{n}}_{opa} ({\bar{n}}_{opa} + 2)}}}

Here, n̄₁ is the average photon number in coherent state in the first arm. Similarly, n̄₂ and n̄_ξ = sinh² (r) are the average photon number in the displaced and squeezed part of the DSV, with r being the squeezing parameter. And lastly, n̄_opa = 2 sinh² (g) is the average photon number of the two mode squeezer, or equivalently, the OPA, and g is the squeezing strength.

4. Sensitivity with DSV state

We first report the results with parity measurement. Since, parity is a single mode measurement, we only need to calculate the Wigner function at output mode b to get the parity signal. The Wigner function at mode b can be calculated using:

〈 {\hat{Π}}_{b} 〉 = \frac{exp (- {\bar{X}}_{22}^{T} \cdot Γ_{22}^{- 1} \cdot {\bar{X}}_{22})}{\sqrt{| Γ_{22} |}},

where

{\bar{X}}_{22}^{T} = (〈 X_{2, 3} 〉, 〈 X_{2, 4} 〉)

and

Γ_{22} = (\begin{matrix} Γ_{2}^{33} & Γ_{2}^{34} \\ Γ_{2}^{43} & Γ_{2}^{44} \end{matrix})

. The phase sensitivity can be calculated by using the above expression and Eq. (13). The expression for sensitivity is long and not illuminating to report here. Upon examining the sensitivity as a function of ϕ, we find that minimum is not at zero as previously reported [13,20]. Hence, we resort to numerical minimization to find the optimal sensitivity. We set the phases of the coherent state (θ₁) and DSV state (θ₂ and Γ) to zero because these phases only shift the position of the optimal point.

In Fig. 2, we show the effect of the increase in the squeezing parameter r of the DSV state on the sensitivity along with the Heisenberg Limit. We see that with increase in r, the sensitivity of the scheme increases as expected. Similar behavior is observed when increasing the average photon number of the coherent state on the first arm.

Fig. 2 The effect on the phase sensitivity with the increase in the squeezing parameter r. The HL (blue) is given by Eq. 18. Plotted with n̄₁ = 16, n̄₂ = 4, g = 2.

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Next, we present the sensitivity of our scheme and compare it with SNL and HL [Eq. (18), Eq. (19)] introduced in the last section. In Fig. 3, we see that our scheme is sub-shotnoise limited even for small values of the squeezing strength g of the OPA. With higher g, the sensitivity of our scheme keeps increasing and approaches the HL for g ≥ 2. However, we would like to point out that the sensitivity of our scheme never goes below the HL.

Fig. 3 Phase sensitivity as a function of the squeezing strength g of the OPA. The sensitivity with coherent and DSV (pink) is obtained by numerically optimizing ϕ. The HL (blue) and SNL (red) are given by Eqs. (18) and (19). Plotted with n̄₁ = 16, n̄₂ = 4 and r = 2.

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Parity measurement requires a photon-number-resolving detector (PNRD). These detectors are very costly and difficult to implement in an experimental setup. For Gaussian states, there has been a proposal of obtaining the parity signal without the use of photon number resolving detectors but it requires post-processing [31]. Here, we implement a simple detection scheme which gives sub-shotnoise sensitivity using on-off detectors, which only discriminates between zero and non-zero photons, as discussed in previous section. Although Homodyne measurement is an efficient technique for sub-shotnoise sensitivity but on-off detection scheme are easier to implement in some laboratory. Figure 4 shows the sensitivity of our scheme using an on-off detector. We can see that sub-shotnoise sensitivity can be achieved for g ≤ 2. Thus, if only sub-shotnoise sensitivity is desired for a particular application, our simple measurement setup with on-off detector will suffice preventing the use of complicated PNRD to obtain the parity signal.

Fig. 4 Phase sensitivity with coherent and DSV state with on-off detector. The sensitivity (green) is obtained by numerically optimizing ϕ from the classical CRB. The SNL (red), parity (pink) and HL (blue) are also shown for comparison. Plotted with n̄₁ = 16, n̄₂ = 4 and r = 2.

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5. Conclusion

In this paper, we studied the phase sensitivity in a SU(1,1) interferometer with a coherent and a displaced-squeezed-vacuum state with two different detection scheme. We showed that the sensitivity of our scheme with parity detection is sub-shotnoise limited and approaches Heisenberg sensitivity with increasing squeezing strength g of the OPA. We also studied the sensitivity using on-off detection strategy. We showed that sub-shotnoise limited sensitivity can be achieved even for g ≤ 2 using a simple on-off detector.

Appendix

The mean and covariance matrix of the coherent state is given by:

X_{coh} = (\begin{matrix} \sqrt{2} α_{1} cos (θ_{1}) \\ \sqrt{2} α_{1} sin (θ_{1}) \end{matrix}),

Γ_{coh} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) .

Similarly, the mean and covariance of the displaced squeezed vacuum (DSV) is given by:

X_{DSV} = (\begin{matrix} \sqrt{2} α_{2} cos (θ_{2}) \\ \sqrt{2} α_{2} sin (θ_{2}) \end{matrix}),

Γ_{DSV} = (\begin{matrix} {(cosh (r) + sinh (r))}^{2} & 0 \\ 0 & {(cosh (r) - sinh (r))}^{2} \end{matrix}) .

The combined input mean and covariance is given by:

X_{0} = X_{coh} \oplus X_{DSV} = (\begin{matrix} \sqrt{2} α_{1} cos (θ_{1}) \\ \sqrt{2} α_{1} sin (θ_{1}) \\ \sqrt{2} α_{2} cos (θ_{2}) \\ \sqrt{2} α_{2} sin (θ_{2}) \end{matrix}),

Γ_{0} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {(cosh (r) + sinh (r))}^{2} & 0 \\ 0 & 0 & 0 & {(cosh (r) - sinh (r))}^{2} \end{matrix}) .

The input Wigner function can easily be constructed using these input mean and covariance matrices by plugging them into Eq. (6). We do not need to calculate the input Wigner function and it just suffices to propagate these mean and covariance matrices through the SU(1,1) interferometer. After propagation through the SU(1,1) interferometer given by Eqs. (10) and (11), the input mean and covariance matrix evolves to output mean and output covariance matrix given by:

X_{2} = (\begin{matrix} \sqrt{2} α_{1} ({cos}^{2} (g) cos (θ_{1} + ϕ) - {sin}^{2} (g) cos (θ_{1})) - \sqrt{2} α_{2} sinh (2 g) sin (\frac{ϕ}{2}) sin (\frac{ϕ}{2} - θ_{2}) \\ \sqrt{2} α_{1} ({cosh}^{2} (g) sin (θ_{1} + ϕ) - {sin}^{2} (g) sin (θ_{1})) + \sqrt{2} α_{2} sinh (2 g) sin (\frac{ϕ}{2}) cos (\frac{ϕ}{2} - θ_{2}) \\ \sqrt{2} α_{1} sinh (2 g) sin (\frac{ϕ}{2}) sin (θ_{1} + \frac{ϕ}{2}) + \sqrt{2} α_{2} ({cosh}^{2} (g) cos (θ_{2}) - {sinh}^{2} (g) cos (ϕ - θ_{2})) \\ \sqrt{2} α_{1} sinh (2 g) sin (\frac{ϕ}{2}) cos (θ_{1} + \frac{ϕ}{2}) + \sqrt{2} α_{2} ({sinh}^{2} (g) sin (ϕ - θ_{2}) + {cosh}^{2} (g) sin (θ_{2})) \end{matrix}) .

And the covariance is given by:

Γ_{2} = (\begin{matrix} Γ_{2}^{11} & Γ_{2}^{12} & Γ_{2}^{13} & Γ_{2}^{14} \\ Γ_{2}^{21} & Γ_{2}^{22} & Γ_{2}^{23} & Γ_{2}^{24} \\ Γ_{2}^{31} & Γ_{2}^{32} & Γ_{2}^{33} & Γ_{2}^{34} \\ Γ_{2}^{41} & Γ_{2}^{42} & Γ_{2}^{43} & Γ_{2}^{44} \end{matrix}) .

With each element of the covariance matrix is given by:

Γ_{2}^{11} = e^{2 r} {sin}^{2} (2 g) {sin}^{4} (\frac{ϕ}{2}) + {sinh}^{4} (g) + {cosh}^{4} (g) + {sinh}^{2} (g) {cosh}^{2} (g) (e^{- 2 r} {sin}^{2} (ϕ) - 2 cos (ϕ)),

Γ_{2}^{12} = 4 {sinh}^{2} (g) {cosh}^{2} (g) sinh (r) cosh (r) sin (ϕ) (cos (ϕ) - 1),

\begin{array}{l} Γ_{2}^{12} & = - \frac{1}{4} e^{- 2 r} (e^{2 r} - 1) (e^{2 r} + 1) sinh g cosh (g) (2 {sinh}^{2} (g) cos (2 ϕ + 1)) \\ - \frac{1}{4} e^{- 2 r} (e^{2 r} + 1) sinh (g) cosh (g) cosh (2 g) (- 4 e^{2 r} cos (ϕ) + 3 e^{2 r} + 1), \end{array}

\begin{array}{l} Γ_{2}^{14} & = sinh g cosh (g) sin (ϕ) ({sinh}^{2} (r) + {cosh}^{2} (r) + 1) \\ - sinh (g) cosh (g) sinh (2 r) sin (ϕ) (cosh (2 g) - 2 {sinh}^{2} (g) cos (ϕ)), \end{array}

Γ_{2}^{12} = Γ_{2}^{12} = 4 {sinh}^{2} (g) {cosh}^{2} (g) sinh (r) cosh (r) sin (ϕ) (cos (ϕ) - 1),

\begin{array}{l} Γ_{2}^{22} & = e^{- 2 r} {sinh}^{2} (2 g) {sin}^{4} (\frac{ϕ}{2}) - {sinh}^{2} (g) {cosh}^{2} (g) (2 cos (ϕ) - e^{2 r} {sin}^{2} (ϕ)) \\ + {sinh}^{4} (g) + {cosh}^{4} (g), \end{array}

\begin{array}{l} Γ_{2}^{23} & = sinh (g) cosh (g) sin (ϕ) sinh (2 r) (cosh (2 g) - 2 {sinh}^{2} (g) cos (ϕ)) \\ + sinh (g) cosh (g) sin (ϕ) ({sinh}^{2} (r) + {cosh}^{2} (r) + 1), \end{array}

\begin{array}{l} Γ_{2}^{24} & = \frac{1}{2} e^{- 2 r} (e^{2 r} + 1) sinh (g) {sinh}^{2} (g) cosh (g) (2 e^{2 r} {sin}^{2} (ϕ) + cos (2 ϕ)) \\ + \frac{1}{4} e^{- 2 r} (e^{2 r} + 1) sinh (g) cosh (g) cosh (2 g) (3 - 4 cos (ϕ)) + 1), \end{array}

\begin{array}{l} Γ_{2}^{31} = Γ_{2}^{13} & = - \frac{1}{4} e^{- 2 r} (e^{2 r} - 1) (e^{2 r} + 1) sinh (g) cosh (g) (2 {sinh}^{2} (g) cos (2 ϕ + 1)) \\ - \frac{1}{4} e^{- 2 r} (e^{2 r} + 1) sinh (g) cosh (g) cosh (2 g) (- 4 e^{2 r} cos (ϕ) + 3 e^{2 r} + 1), \end{array}

\begin{array}{l} Γ_{2}^{32} = Γ_{2}^{23} & = sinh (g) cosh (g) sin (ϕ) sinh (2 r) (cosh (2 g) - 2 {sinh}^{2} (g) cos (ϕ)) \\ + sinh (g) cosh (g) sin (ϕ) ({sinh}^{2} (r) + {cosh}^{2} (r) + 1), \end{array}

Γ_{2}^{33} = e^{- 2 r} {sinh}^{4} (g) {sin}^{2} (ϕ) + e^{2 r} {({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ))}^{2} + {sinh}^{2} (2 g) {sin}^{2} (\frac{ϕ}{2}),

Γ_{2}^{34} = 4 {sinh}^{2} (g) sinh (r) cosh (r) sin (ϕ) ({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ)),

\begin{array}{l} Γ_{2}^{41} & = sinh (g) cosh (g) sin (ϕ) ({sinh}^{2} (r) + {cosh}^{2} (r) + 1) \\ - sinh (g) cosh (g) sinh (2 r) sin (ϕ) (cosh (2 g) - 2 {sinh}^{2} (g) cos (ϕ)), \end{array}

\begin{array}{l} Γ_{2}^{42} = Γ_{2}^{24} & = \frac{1}{2} e^{- 2 r} (e^{2 r} + 1) sinh (g) {sinh}^{2} (g) cosh (g) (2 e^{2 r} {sin}^{2} (ϕ) + cos (2 ϕ)) \\ + \frac{1}{4} e^{- 2 r} (e^{2 r} + 1) sinh (g) cosh (g) cosh (2 g) (3 - 4 cos (ϕ)) + 1), \end{array}

Γ_{2}^{43} = Γ_{2}^{34} = 4 {sinh}^{2} (g) sinh (r) cosh (r) sin (ϕ) ({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ)),

Γ_{2}^{44} = e^{2 r} {sinh}^{4} (g) {sin}^{2} (ϕ) + e^{- 2 r} {({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ))}^{2} + {sinh}^{2} (2 g) {sin}^{2} (\frac{ϕ}{2}) .

Using this, the parity signal is calculated to be:

〈 \hat{Π} 〉 = \frac{8}{v 6} exp [\frac{128 (v 1 \times v 2 - v 3 \times v 4)}{v 5}]

with,

v 1 = α_{1} sinh (2 g) sin (\frac{ϕ}{2}) cos (θ_{1} + \frac{ϕ}{2}) + α_{2} ({sinh}^{2} (g) sin (ϕ - θ_{2}) + {cosh}^{2} (g) sin (θ_{2}))

\begin{array}{l} v 2 & = α_{1} sinh (2 g) sin (\frac{ϕ}{2}) sin (θ_{1} + \frac{ϕ}{2}) + α_{2} ({cosh}^{2} (g) cos (θ_{2}) - {sinh}^{2} (g) cos (ϕ - θ_{2})) \\ \times 4 {sinh}^{2} (g) sinh (r) cosh (r) sin (ϕ) ({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ)) \\ - (α_{1} sinh (2 g) sin (\frac{ϕ}{2}) cos (θ_{1} + \frac{ϕ}{2}) + α_{2} ({sinh}^{2} (g) sin (ϕ - θ_{2}) + {cosh}^{2} (g) sin (θ_{2}))) \\ \times (e^{- 2 r} {sinh}^{4} (g) {sin}^{2} (ϕ) + e^{2 r} {({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ))}^{2} + {sinh}^{2} (2 g) {sin}^{2} (\frac{ϕ}{2})) \end{array}

v 3 = α_{1} sinh (2 g) sin (\frac{ϕ}{2}) sin (θ_{1} + \frac{ϕ}{2}) + α_{2} ({cosh}^{2} (g) cos (θ_{2}) - {sinh}^{2} (g) cos (ϕ - θ_{2}))

\begin{array}{l} v 4 & = (e^{2 r} {sin}^{4} (g) {sin}^{2} (ϕ) + e^{- 2 r} {({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ))}^{2} + {sinh}^{2} (2 g) {sin}^{2} (\frac{ϕ}{2})) \\ \times (α_{1} sinh (2 g) sin (\frac{ϕ}{2}) sin (θ_{1} + \frac{ϕ}{2}) + α_{2} ({cosh}^{2} (g) cos (θ_{2}) - {sinh}^{2} (g) cos (ϕ - θ_{2}))) \\ - (4 {sinh}^{2} (g) sinh (r) cosh (r) sin (ϕ)) ({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ)) \\ \times (α_{1} sinh (2 g) sin (\frac{ϕ}{2}) cos (θ_{1} + \frac{ϕ}{2}) + α_{2} ({sinh}^{2} (g) sin (ϕ - θ_{2}) + {cosh}^{2} (g) sin (θ_{2}))) \end{array}

\begin{array}{l} v 5 & = 32 {cosh}^{2} (r) ({sinh}^{4} (2 g) cos (2 ϕ) - {sinh}^{2} (4 g) cos (ϕ)) + 4 cosh (4 g - 2 r) + 3 cosh (8 g - 2 r) \\ + 8 cosh (4 g) + 6 cosh (8 g) + 50 + 4 cosh (2 (2 g + r)) + 3 cosh (2 (4 g + r)) - 14 cosh (2 r) \end{array}

\begin{array}{l} v 6 & = \sqrt{8 {sinh}^{4} (2 g) cos (2 ϕ) - 8 {sinh}^{2} (4 g) cos (ϕ) + 4 cosh (4 g) + 3 cosh (8 g)} \\ \times \sqrt{4 cosh (r) - 14 cosh (2 r) + 50} \end{array}

Similarly, we can calculate the Fisher information using the probabilities at the output modes a and b.

The probability at output mode a is given by:

P_{on}^{a} = 1 - \frac{16}{\sqrt{q 1 + q 2}}

With,

\begin{array}{l} q 1 & = - 8 cosh (4 g) (- 18 {cosh}^{2} (r) + 8 cos (ϕ) + cos (2 ϕ)) - 78 cosh (2 r) \\ + 72 cos (ϕ) + 6 cos (2 ϕ) + 242 + 2 cosh (8 g) (6 {cosh}^{2} (r) - 4 cos (ϕ) + cos (2 ϕ)), \end{array}

And,

q 2 = 16 {sinh}^{2} (2 g) (cosh (2 r) ({sinh}^{2} (2 g) cos (2 ϕ) - 2 (cosh (4 g) + 5) cos (ϕ)))

- 16 {sinh}^{2} (2 g) (16 sinh (2 r) {sin}^{2} (\frac{ϕ}{2}) sin (Γ - ϕ)) .

Similarly, the probability at output mode b is given by:

P_{on}^{b} = 1 - \frac{16}{\sqrt{t 1 + t 2 + t 3 + t 4 + t 5}} .

With,

\begin{array}{l} t 1 & = - 8 cosh (4 g) (- 18 {cosh}^{2} (r) + 8 cos (ϕ) + cos (2 ϕ)) - 32 cos (ϕ) cosh (4 g - 2 r) \\ - 4 cos (ϕ) cosh (8 g - 2 r) - 4 cos (2 ϕ) cosh (4 g - 2 r) + cos (2 ϕ) cosh (8 g - 2 r) \\ + 72 cos (ϕ) + 6 cos (2 ϕ) - 14, \end{array}

\begin{array}{l} t 2 & = 2 cosh (8 g) (6 {cosh}^{2} (r) - 4 cos (ϕ) + cos (2 ϕ)) - 32 cos (ϕ) cosh (4 g + 2 r) \\ - 4 cos (2 ϕ) cosh (4 g + 2 r) - 4 cos (ϕ) cosh (8 g + 2 r) + cos (2 ϕ) cosh (8 g + 2 r) \\ + 72 cosh (2 r) cos (ϕ) + 6 cosh (2 r) cos (2 ϕ) + 178 cosh (2 r), \end{array}

\begin{array}{l} t 3 & = 64 sin (Γ) sinh (2 g - 2 r) + 64 cos (Γ) sin (2 ϕ) sinh (2 g - 2 r) + 16 sin (Γ) sinh (4 g - 2 r) \\ - 32 sin (Γ) cos (ϕ) sinh (4 g - 2 r) - 64 sin (Γ) cos (2 ϕ) sinh (2 g - 2 r), \end{array}

\begin{array}{l} t 4 & = 16 sin (Γ) cos (2 ϕ) sinh (4 g - 2 r) + 32 cos (Γ) sin (ϕ) sinh (4 g - 2 r) - 96 sin (Γ) sinh (2 r) \\ + 64 cos (Γ) sinh (2 r) - 64 sin (Γ) sinh (2 r) cos (2 ϕ) - 96 sin (Γ) sinh (2 r) cos (2 ϕ) \\ - 16 cos (Γ) sin (2 ϕ) sinh (4 g - 2 r), \end{array}

\begin{array}{l} t 5 & = 64 {sin}^{2} (\frac{ϕ}{2}) sin (Γ - ϕ) sinh (4 g + 2 r) - 128 sin (ϕ) cos (Γ - ϕ) sinh (2 (g + r)) \\ + 96 cos (Γ) sinh (2 r) sin (2 ϕ), \end{array}

Funding

Army Research Office; Air Force Office of Scientific Research; Defense Advanced Research Projects Agency; National Science Foundation; Economic Development Assistantship, Louisiana State University; Department of Physics and Astronomy, Louisiana State University.

Acknowledgement

S.A. would like to acknowledge support from ARO and AFOSR. N.B. acknowledges support from Department of Physics and Astronomy at Louisiana State University. C.Y. would like to acknowledge support from Economic Development Assistantship from the Louisiana State University System Board of Regents. H.L. and J.P.D would like to acknowledge support from ARO, AFOSR, DARPA and NSF.

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Fig. 2 The effect on the phase sensitivity with the increase in the squeezing parameter r. The HL (blue) is given by Eq. 18. Plotted with n̄₁ = 16, n̄₂ = 4, g = 2.

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Equations (61)

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{\hat{x}}_{a_{k}} = {\hat{a}}_{k} + {\hat{a}}_{k}^{†}, {\hat{p}}_{a_{k}} = - i ({\hat{a}}_{k} - {\hat{a}}_{k}^{†}),

{\hat{x}}_{b_{k}} = {\hat{b}}_{k} + {\hat{b}}_{k}^{†}, {\hat{p}}_{a_{k}} = - i ({\hat{b}}_{k} - {\hat{b}}_{k}^{†}) .

X_{k} = {({\hat{X}}_{k, 1}, {\hat{X}}_{k, 2}, {\hat{X}}_{k, 3}, {\hat{X}}_{k, 4})}^{T} = {({\hat{x}}_{a_{k}}, {\hat{p}}_{a_{k}}, {\hat{x}}_{b_{k}}, {\hat{p}}_{b_{k}})}^{T} .

{\bar{X}}_{k} = {(〈 {\bar{X}}_{k, 1} 〉, 〈 {\bar{X}}_{k, 2} 〉, 〈 {\bar{X}}_{k, 3} 〉, 〈 {\bar{X}}_{k, 4} 〉)}^{T},

Γ_{k}^{m n} = Tr [(Δ {\hat{X}}_{k, m} Δ {\hat{X}}_{k, n} + Δ {\hat{X}}_{k, n} Δ {\hat{X}}_{k, m}) ρ],

W (X_{0}) = \frac{exp [- {(X_{0} - {\bar{X}}_{0})}^{T} \cdot {(Γ_{0})}^{- 1} \cdot (X_{0} - {\bar{X}}_{0})]}{\sqrt{| Γ_{0} |}} .

S_{OPA 1} = (\begin{matrix} cosh (g) & 0 & sinh (g) & 0 \\ 0 & cosh (g) & 0 & - sinh (g) \\ sinh (g) & 0 & cosh (g) & 0 \\ 0 & - sinh (g) & 0 & cosh (g) \end{matrix}),

S_{ϕ} = (\begin{matrix} cos (ϕ) & - sin (ϕ) & 0 & 0 \\ sin (ϕ) & cos (ϕ) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}),

S_{OPA 2} = (\begin{matrix} cosh (g) & 0 & - sinh (g) & 0 \\ 0 & cosh (g) & 0 & sinh (g) \\ - sinh (g) & 0 & cosh (g) & 0 \\ 0 & sinh (g) & 0 & cosh (g) \end{matrix}),

{\bar{X}}_{2} = S {\bar{X}}_{0},

Γ_{2} = S Γ_{0} S^{T} .

{\hat{Π}}_{b} = {(- 1)}^{{\hat{b}}_{2}^{†} {\hat{b}}_{2}} .

Δ ϕ = \frac{〈 Δ {\hat{Π}}_{b} 〉}{| \frac{\partial 〈 {\hat{Π}}_{b} 〉}{\partial ϕ} |}

{\hat{Π}}_{off} = | 0 〉 〈 0 |, {\hat{Π}}_{on} = \hat{I} - | 0 〉 〈 0 |

P_{on} = 1 - \frac{2}{\sqrt{det (Γ + I)}}

Δ ϕ \geq 1 / \sqrt{F},

F = \sum \frac{1}{P_{on}} {(\frac{d P_{on}}{d ϕ})}^{2} .

Δ ϕ_{HL} = \frac{1}{{\bar{n}}_{total}} = \frac{1}{{\bar{n}}_{1} + {\bar{n}}_{2} + {\bar{n}}_{ξ} + {\bar{n}}_{opa} (1 + {\bar{n}}_{1} + {\bar{n}}_{2} + {\bar{n}}_{ξ}) + 2 \sqrt{{\bar{n}}_{1} {\bar{n}}_{2} {\bar{n}}_{opa} ({\bar{n}}_{opa} + 2)}}

Δ ϕ_{SNL} = \frac{1}{\sqrt{{\bar{n}}_{total}}} = \frac{1}{\sqrt{{\bar{n}}_{1} + {\bar{n}}_{2} + {\bar{n}}_{ξ} + {\bar{n}}_{opa} (1 + {\bar{n}}_{1} + {\bar{n}}_{2} + {\bar{n}}_{ξ}) + 2 \sqrt{{\bar{n}}_{1} {\bar{n}}_{2} {\bar{n}}_{opa} ({\bar{n}}_{opa} + 2)}}}

〈 {\hat{Π}}_{b} 〉 = \frac{exp (- {\bar{X}}_{22}^{T} \cdot Γ_{22}^{- 1} \cdot {\bar{X}}_{22})}{\sqrt{| Γ_{22} |}},

X_{coh} = (\begin{matrix} \sqrt{2} α_{1} cos (θ_{1}) \\ \sqrt{2} α_{1} sin (θ_{1}) \end{matrix}),

Γ_{coh} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) .

X_{DSV} = (\begin{matrix} \sqrt{2} α_{2} cos (θ_{2}) \\ \sqrt{2} α_{2} sin (θ_{2}) \end{matrix}),

Γ_{DSV} = (\begin{matrix} {(cosh (r) + sinh (r))}^{2} & 0 \\ 0 & {(cosh (r) - sinh (r))}^{2} \end{matrix}) .

X_{0} = X_{coh} \oplus X_{DSV} = (\begin{matrix} \sqrt{2} α_{1} cos (θ_{1}) \\ \sqrt{2} α_{1} sin (θ_{1}) \\ \sqrt{2} α_{2} cos (θ_{2}) \\ \sqrt{2} α_{2} sin (θ_{2}) \end{matrix}),

Γ_{0} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {(cosh (r) + sinh (r))}^{2} & 0 \\ 0 & 0 & 0 & {(cosh (r) - sinh (r))}^{2} \end{matrix}) .

X_{2} = (\begin{matrix} \sqrt{2} α_{1} ({cos}^{2} (g) cos (θ_{1} + ϕ) - {sin}^{2} (g) cos (θ_{1})) - \sqrt{2} α_{2} sinh (2 g) sin (\frac{ϕ}{2}) sin (\frac{ϕ}{2} - θ_{2}) \\ \sqrt{2} α_{1} ({cosh}^{2} (g) sin (θ_{1} + ϕ) - {sin}^{2} (g) sin (θ_{1})) + \sqrt{2} α_{2} sinh (2 g) sin (\frac{ϕ}{2}) cos (\frac{ϕ}{2} - θ_{2}) \\ \sqrt{2} α_{1} sinh (2 g) sin (\frac{ϕ}{2}) sin (θ_{1} + \frac{ϕ}{2}) + \sqrt{2} α_{2} ({cosh}^{2} (g) cos (θ_{2}) - {sinh}^{2} (g) cos (ϕ - θ_{2})) \\ \sqrt{2} α_{1} sinh (2 g) sin (\frac{ϕ}{2}) cos (θ_{1} + \frac{ϕ}{2}) + \sqrt{2} α_{2} ({sinh}^{2} (g) sin (ϕ - θ_{2}) + {cosh}^{2} (g) sin (θ_{2})) \end{matrix}) .

Γ_{2} = (\begin{matrix} Γ_{2}^{11} & Γ_{2}^{12} & Γ_{2}^{13} & Γ_{2}^{14} \\ Γ_{2}^{21} & Γ_{2}^{22} & Γ_{2}^{23} & Γ_{2}^{24} \\ Γ_{2}^{31} & Γ_{2}^{32} & Γ_{2}^{33} & Γ_{2}^{34} \\ Γ_{2}^{41} & Γ_{2}^{42} & Γ_{2}^{43} & Γ_{2}^{44} \end{matrix}) .

Γ_{2}^{11} = e^{2 r} {sin}^{2} (2 g) {sin}^{4} (\frac{ϕ}{2}) + {sinh}^{4} (g) + {cosh}^{4} (g) + {sinh}^{2} (g) {cosh}^{2} (g) (e^{- 2 r} {sin}^{2} (ϕ) - 2 cos (ϕ)),

Γ_{2}^{12} = 4 {sinh}^{2} (g) {cosh}^{2} (g) sinh (r) cosh (r) sin (ϕ) (cos (ϕ) - 1),

\begin{array}{l} Γ_{2}^{12} & = - \frac{1}{4} e^{- 2 r} (e^{2 r} - 1) (e^{2 r} + 1) sinh g cosh (g) (2 {sinh}^{2} (g) cos (2 ϕ + 1)) \\ - \frac{1}{4} e^{- 2 r} (e^{2 r} + 1) sinh (g) cosh (g) cosh (2 g) (- 4 e^{2 r} cos (ϕ) + 3 e^{2 r} + 1), \end{array}

\begin{array}{l} Γ_{2}^{14} & = sinh g cosh (g) sin (ϕ) ({sinh}^{2} (r) + {cosh}^{2} (r) + 1) \\ - sinh (g) cosh (g) sinh (2 r) sin (ϕ) (cosh (2 g) - 2 {sinh}^{2} (g) cos (ϕ)), \end{array}

Γ_{2}^{12} = Γ_{2}^{12} = 4 {sinh}^{2} (g) {cosh}^{2} (g) sinh (r) cosh (r) sin (ϕ) (cos (ϕ) - 1),

\begin{array}{l} Γ_{2}^{22} & = e^{- 2 r} {sinh}^{2} (2 g) {sin}^{4} (\frac{ϕ}{2}) - {sinh}^{2} (g) {cosh}^{2} (g) (2 cos (ϕ) - e^{2 r} {sin}^{2} (ϕ)) \\ + {sinh}^{4} (g) + {cosh}^{4} (g), \end{array}

\begin{array}{l} Γ_{2}^{23} & = sinh (g) cosh (g) sin (ϕ) sinh (2 r) (cosh (2 g) - 2 {sinh}^{2} (g) cos (ϕ)) \\ + sinh (g) cosh (g) sin (ϕ) ({sinh}^{2} (r) + {cosh}^{2} (r) + 1), \end{array}

\begin{array}{l} Γ_{2}^{24} & = \frac{1}{2} e^{- 2 r} (e^{2 r} + 1) sinh (g) {sinh}^{2} (g) cosh (g) (2 e^{2 r} {sin}^{2} (ϕ) + cos (2 ϕ)) \\ + \frac{1}{4} e^{- 2 r} (e^{2 r} + 1) sinh (g) cosh (g) cosh (2 g) (3 - 4 cos (ϕ)) + 1), \end{array}

\begin{array}{l} Γ_{2}^{31} = Γ_{2}^{13} & = - \frac{1}{4} e^{- 2 r} (e^{2 r} - 1) (e^{2 r} + 1) sinh (g) cosh (g) (2 {sinh}^{2} (g) cos (2 ϕ + 1)) \\ - \frac{1}{4} e^{- 2 r} (e^{2 r} + 1) sinh (g) cosh (g) cosh (2 g) (- 4 e^{2 r} cos (ϕ) + 3 e^{2 r} + 1), \end{array}

\begin{array}{l} Γ_{2}^{32} = Γ_{2}^{23} & = sinh (g) cosh (g) sin (ϕ) sinh (2 r) (cosh (2 g) - 2 {sinh}^{2} (g) cos (ϕ)) \\ + sinh (g) cosh (g) sin (ϕ) ({sinh}^{2} (r) + {cosh}^{2} (r) + 1), \end{array}

Γ_{2}^{33} = e^{- 2 r} {sinh}^{4} (g) {sin}^{2} (ϕ) + e^{2 r} {({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ))}^{2} + {sinh}^{2} (2 g) {sin}^{2} (\frac{ϕ}{2}),

Γ_{2}^{34} = 4 {sinh}^{2} (g) sinh (r) cosh (r) sin (ϕ) ({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ)),

\begin{array}{l} Γ_{2}^{41} & = sinh (g) cosh (g) sin (ϕ) ({sinh}^{2} (r) + {cosh}^{2} (r) + 1) \\ - sinh (g) cosh (g) sinh (2 r) sin (ϕ) (cosh (2 g) - 2 {sinh}^{2} (g) cos (ϕ)), \end{array}

\begin{array}{l} Γ_{2}^{42} = Γ_{2}^{24} & = \frac{1}{2} e^{- 2 r} (e^{2 r} + 1) sinh (g) {sinh}^{2} (g) cosh (g) (2 e^{2 r} {sin}^{2} (ϕ) + cos (2 ϕ)) \\ + \frac{1}{4} e^{- 2 r} (e^{2 r} + 1) sinh (g) cosh (g) cosh (2 g) (3 - 4 cos (ϕ)) + 1), \end{array}

Γ_{2}^{43} = Γ_{2}^{34} = 4 {sinh}^{2} (g) sinh (r) cosh (r) sin (ϕ) ({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ)),

Γ_{2}^{44} = e^{2 r} {sinh}^{4} (g) {sin}^{2} (ϕ) + e^{- 2 r} {({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ))}^{2} + {sinh}^{2} (2 g) {sin}^{2} (\frac{ϕ}{2}) .

〈 \hat{Π} 〉 = \frac{8}{v 6} exp [\frac{128 (v 1 \times v 2 - v 3 \times v 4)}{v 5}]

v 1 = α_{1} sinh (2 g) sin (\frac{ϕ}{2}) cos (θ_{1} + \frac{ϕ}{2}) + α_{2} ({sinh}^{2} (g) sin (ϕ - θ_{2}) + {cosh}^{2} (g) sin (θ_{2}))

\begin{array}{l} v 2 & = α_{1} sinh (2 g) sin (\frac{ϕ}{2}) sin (θ_{1} + \frac{ϕ}{2}) + α_{2} ({cosh}^{2} (g) cos (θ_{2}) - {sinh}^{2} (g) cos (ϕ - θ_{2})) \\ \times 4 {sinh}^{2} (g) sinh (r) cosh (r) sin (ϕ) ({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ)) \\ - (α_{1} sinh (2 g) sin (\frac{ϕ}{2}) cos (θ_{1} + \frac{ϕ}{2}) + α_{2} ({sinh}^{2} (g) sin (ϕ - θ_{2}) + {cosh}^{2} (g) sin (θ_{2}))) \\ \times (e^{- 2 r} {sinh}^{4} (g) {sin}^{2} (ϕ) + e^{2 r} {({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ))}^{2} + {sinh}^{2} (2 g) {sin}^{2} (\frac{ϕ}{2})) \end{array}

v 3 = α_{1} sinh (2 g) sin (\frac{ϕ}{2}) sin (θ_{1} + \frac{ϕ}{2}) + α_{2} ({cosh}^{2} (g) cos (θ_{2}) - {sinh}^{2} (g) cos (ϕ - θ_{2}))

\begin{array}{l} v 4 & = (e^{2 r} {sin}^{4} (g) {sin}^{2} (ϕ) + e^{- 2 r} {({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ))}^{2} + {sinh}^{2} (2 g) {sin}^{2} (\frac{ϕ}{2})) \\ \times (α_{1} sinh (2 g) sin (\frac{ϕ}{2}) sin (θ_{1} + \frac{ϕ}{2}) + α_{2} ({cosh}^{2} (g) cos (θ_{2}) - {sinh}^{2} (g) cos (ϕ - θ_{2}))) \\ - (4 {sinh}^{2} (g) sinh (r) cosh (r) sin (ϕ)) ({cosh}^{2} (g) - {sinh}^{2} (g) cos (ϕ)) \\ \times (α_{1} sinh (2 g) sin (\frac{ϕ}{2}) cos (θ_{1} + \frac{ϕ}{2}) + α_{2} ({sinh}^{2} (g) sin (ϕ - θ_{2}) + {cosh}^{2} (g) sin (θ_{2}))) \end{array}

\begin{array}{l} v 5 & = 32 {cosh}^{2} (r) ({sinh}^{4} (2 g) cos (2 ϕ) - {sinh}^{2} (4 g) cos (ϕ)) + 4 cosh (4 g - 2 r) + 3 cosh (8 g - 2 r) \\ + 8 cosh (4 g) + 6 cosh (8 g) + 50 + 4 cosh (2 (2 g + r)) + 3 cosh (2 (4 g + r)) - 14 cosh (2 r) \end{array}

\begin{array}{l} v 6 & = \sqrt{8 {sinh}^{4} (2 g) cos (2 ϕ) - 8 {sinh}^{2} (4 g) cos (ϕ) + 4 cosh (4 g) + 3 cosh (8 g)} \\ \times \sqrt{4 cosh (r) - 14 cosh (2 r) + 50} \end{array}

P_{on}^{a} = 1 - \frac{16}{\sqrt{q 1 + q 2}}

\begin{array}{l} q 1 & = - 8 cosh (4 g) (- 18 {cosh}^{2} (r) + 8 cos (ϕ) + cos (2 ϕ)) - 78 cosh (2 r) \\ + 72 cos (ϕ) + 6 cos (2 ϕ) + 242 + 2 cosh (8 g) (6 {cosh}^{2} (r) - 4 cos (ϕ) + cos (2 ϕ)), \end{array}

q 2 = 16 {sinh}^{2} (2 g) (cosh (2 r) ({sinh}^{2} (2 g) cos (2 ϕ) - 2 (cosh (4 g) + 5) cos (ϕ)))

- 16 {sinh}^{2} (2 g) (16 sinh (2 r) {sin}^{2} (\frac{ϕ}{2}) sin (Γ - ϕ)) .

P_{on}^{b} = 1 - \frac{16}{\sqrt{t 1 + t 2 + t 3 + t 4 + t 5}} .

\begin{array}{l} t 1 & = - 8 cosh (4 g) (- 18 {cosh}^{2} (r) + 8 cos (ϕ) + cos (2 ϕ)) - 32 cos (ϕ) cosh (4 g - 2 r) \\ - 4 cos (ϕ) cosh (8 g - 2 r) - 4 cos (2 ϕ) cosh (4 g - 2 r) + cos (2 ϕ) cosh (8 g - 2 r) \\ + 72 cos (ϕ) + 6 cos (2 ϕ) - 14, \end{array}

\begin{array}{l} t 2 & = 2 cosh (8 g) (6 {cosh}^{2} (r) - 4 cos (ϕ) + cos (2 ϕ)) - 32 cos (ϕ) cosh (4 g + 2 r) \\ - 4 cos (2 ϕ) cosh (4 g + 2 r) - 4 cos (ϕ) cosh (8 g + 2 r) + cos (2 ϕ) cosh (8 g + 2 r) \\ + 72 cosh (2 r) cos (ϕ) + 6 cosh (2 r) cos (2 ϕ) + 178 cosh (2 r), \end{array}

\begin{array}{l} t 3 & = 64 sin (Γ) sinh (2 g - 2 r) + 64 cos (Γ) sin (2 ϕ) sinh (2 g - 2 r) + 16 sin (Γ) sinh (4 g - 2 r) \\ - 32 sin (Γ) cos (ϕ) sinh (4 g - 2 r) - 64 sin (Γ) cos (2 ϕ) sinh (2 g - 2 r), \end{array}

\begin{array}{l} t 4 & = 16 sin (Γ) cos (2 ϕ) sinh (4 g - 2 r) + 32 cos (Γ) sin (ϕ) sinh (4 g - 2 r) - 96 sin (Γ) sinh (2 r) \\ + 64 cos (Γ) sinh (2 r) - 64 sin (Γ) sinh (2 r) cos (2 ϕ) - 96 sin (Γ) sinh (2 r) cos (2 ϕ) \\ - 16 cos (Γ) sin (2 ϕ) sinh (4 g - 2 r), \end{array}

\begin{array}{l} t 5 & = 64 {sin}^{2} (\frac{ϕ}{2}) sin (Γ - ϕ) sinh (4 g + 2 r) - 128 sin (ϕ) cos (Γ - ϕ) sinh (2 (g + r)) \\ + 96 cos (Γ) sinh (2 r) sin (2 ϕ), \end{array}

Abstract

1. Introduction

2. SU(1,1) Interferometer

3. Measurement strategies

4. Sensitivity with DSV state

5. Conclusion

Appendix

Funding

Acknowledgement

References

Cited By

Figures (4)

Equations (61)

OSA Continuum