Measuring Squeezed States Of Light

Measuring Squeezed States of Light

Mach-Zehnder Interferometer

Vacuum in Channel One

Generic State in Channel One

Squeezed State in Channel One

Squeezed state in channel one

In the complex plane representation of vacuum quadratures, the result suggests a method to reduce quantum noise in balanced interferometers, using squeezed light. Specifically, \mathbf P_{\lambda}-squeezed states with negative R_\lambda can reduce fluctuations. This representation depicts a special case, \mathbf P_{\lambda}-squeezed vacuum.

Introducing \mathbf P_{\lambda}-squeezed vacuum at input channel (1) we could reduce phase measurement fluctuations below the standard quantum noise limit.

We can define the squeezed vacuum as:

\mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = \alpha_{\lambda R} | \boldsymbol \alpha_\lambda, R_\lambda \rangle, \quad \alpha_{\lambda R} = 0

We indicate it as | \mathbf 0, R_\lambda \rangle and as a consequence:

\alpha_\lambda^\prime = \alpha_{\lambda R}\left[\cosh \left(R_\lambda\right) - \sinh \left(R_\lambda\right) \right] = 0

Therefore the expectation of the electric field, as well the expectation of the quadratures, are null:

\begin{aligned} & \langle \mathbf 0, R_\lambda | \mathbf{E}_{\lambda}(\mathbf r,0) | \mathbf 0, R_\lambda \rangle = 0 \\ & \langle \mathbf 0, R_\lambda | \mathbf{Q}_{\lambda} | \mathbf 0, R_\lambda \rangle = 0 \\ & \langle \mathbf 0, R_\lambda | \mathbf{P}_{\lambda} | \mathbf 0, R_\lambda \rangle = 0 \end{aligned}

We express \mathbf P_{\lambda} as function of \mathbf{A}_{R_\lambda} and \mathbf{A}_{R_\lambda}^\dag:

\begin{aligned} \mathbf P_{\lambda} = & -i\sqrt{\frac{\hbar}{2}}\left( \mathbf a_\lambda - \mathbf a_\lambda ^\dag \right) = -i\sqrt{\frac{\hbar}{2}}e^{R_\lambda}\left( \mathbf{A}_{R_\lambda} - \mathbf{A}_{R_\lambda} ^\dag \right) \end{aligned}

We can then calculate the variance which is the expectation of \mathbf P_{\lambda}^2:

\begin{aligned} \langle \mathbf 0, R_\lambda | \mathbf P_{\lambda}^2 | \mathbf 0, R_\lambda \rangle = & \langle \mathbf 0, R_\lambda | \left[ -i\sqrt{\frac{\hbar}{2}}e^{R_\lambda}\left( \mathbf{A}_{R_\lambda} - \mathbf{A}_{R_\lambda} ^\dag \right) \right]^2 | \mathbf 0, R_\lambda \rangle\\ = & \langle \mathbf 0, R_\lambda |-\frac{\hbar}{2}e^{2R_\lambda} \left( \mathbf{A}_{R_\lambda} - \mathbf{A}_{R_\lambda} ^\dag \right)^2 | \mathbf 0, R_\lambda \rangle\\ = & \langle \mathbf 0, R_\lambda |-\frac{\hbar}{2}e^{2R_\lambda} \left( \mathbf{A}_{R_\lambda}^2 + \mathbf{A}_{R_\lambda}^{\dag 2} - \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda}^\dag - \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} \right) | \mathbf 0, R_\lambda \rangle\\ = & -\frac{\hbar}{2}e^{2R_\lambda} \left[ 0 + 0 - 1 - 0 \right] \\ = & \frac{\hbar}{2}e^{2R_\lambda} \end{aligned}

Using \mathbf{A}_{R_\lambda} | \mathbf 0, R_\lambda \rangle = 0 and \langle \mathbf 0, R_\lambda | \mathbf{A}_{R_\lambda}^\dag = 0 and the commutator relation \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda}^\dag = 1 + \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda}.

If R_\lambda < 0, the the fluctuations of \mathbf P_{\lambda} are below the standard quantum limit.

Squeezed vacuum, unlike ordinary vacuum, exhibits a non-zero average photon number despite having a zero average field and a minimal product of quadrature standard deviations:

\Delta Q_\lambda \Delta P_\lambda = \frac{\hbar}{2}

We express \mathbf N_{\lambda} as function of \mathbf{A}_{R_\lambda} and \mathbf{A}_{R_\lambda}^\dag:

\begin{aligned} \mathbf N_{\lambda} = & \mathbf a_\lambda^\dag \mathbf a_\lambda \\ =& \left( \mathbf{A}_{R_\lambda}^\dag \cosh(R_\lambda) - \mathbf{A}_{R_\lambda} \sinh(R_\lambda) \right) \left( \mathbf{A}_{R_\lambda} \cosh(R_\lambda) - \mathbf{A}_{R_\lambda}^\dag \sinh(R_\lambda) \right) \\ =& \cosh^2(R_\lambda) \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} - \cosh(R_\lambda)\sinh(R_\lambda) \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda}^\dag \\ & - \sinh(R_\lambda)\cosh(R_\lambda) \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda} + \sinh^2(R_\lambda) \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda}^\dag \\ =& \cosh^2(R_\lambda) \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} + \sinh^2(R_\lambda) \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda}^\dag \\ & - \sinh(R_\lambda)\cosh(R_\lambda) \left( \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda}^\dag + \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda} \right) \\ =& \cosh^2(R_\lambda) \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} + \sinh^2(R_\lambda) \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda}^\dag - \\ & \sinh(R_\lambda)\cosh(R_\lambda) \left( \mathbf{A}_{R_\lambda}^{\dag 2} + \mathbf{A}_{R_\lambda}^2 \right) \end{aligned}

We can then calculate the expectation:

\begin{aligned} \langle \mathbf 0, R_\lambda | \mathbf N_{\lambda} | \mathbf 0, R_\lambda \rangle = & \langle \mathbf 0, R_\lambda | \left[ \cosh^2(R_\lambda) \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} + \sinh^2(R_\lambda) \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda}^\dag - \right.\\ & \left. \sinh(R_\lambda)\cosh(R_\lambda) \left( \mathbf{A}_{R_\lambda}^{\dag 2} + \mathbf{A}_{R_\lambda}^2 \right) \right] | \mathbf 0, R_\lambda \rangle\\ = & \cosh^2(R_\lambda) \langle \mathbf 0, R_\lambda | \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} | \mathbf 0, R_\lambda \rangle + \sinh^2(R_\lambda) \langle \mathbf 0, R_\lambda | \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda}^\dag | \mathbf 0, R_\lambda \rangle - \\ & \sinh(R_\lambda)\cosh(R_\lambda) \left( \langle \mathbf 0, R_\lambda | \mathbf{A}_{R_\lambda}^{\dag 2} | \mathbf 0, R_\lambda \rangle + \langle \mathbf 0, R_\lambda | \mathbf{A}_{R_\lambda}^2 | \mathbf 0, R_\lambda \rangle \right) \\ = & \cosh^2(R_\lambda) \langle \mathbf 0, R_\lambda | \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} | \mathbf 0, R_\lambda \rangle + \sinh^2(R_\lambda) \langle \mathbf 0, R_\lambda | (1 + \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda}) | \mathbf 0, R_\lambda \rangle - \\ & \sinh(R_\lambda)\cosh(R_\lambda) \left( \langle \mathbf 0, R_\lambda | \mathbf{A}_{R_\lambda}^{\dag 2} | \mathbf 0, R_\lambda \rangle + \langle \mathbf 0, R_\lambda | \mathbf{A}_{R_\lambda}^2 | \mathbf 0, R_\lambda \rangle \right) \\ = & \cosh^2(R_\lambda) \times 0 + \sinh^2(R_\lambda) \times \left (1 + 0 \right) - \sinh(R_\lambda)\cosh(R_\lambda) \left( 0 + 0 \right) \\ = & \sinh^2(R_\lambda) \end{aligned}

The average number of photons is equal to \sinh^2(R_\lambda) small but non-null photon number is detectable with sensitive photoelectric detectors.

We can again consider a Mach-Zehnder interferometer where the state in input channel (1) is a squeezed vacuum | \mathbf 0, R_\lambda \rangle and we have the same quasi-classical state | \boldsymbol \alpha_\lambda \rangle_2 in input channel (2), with \alpha_\lambda \in \mathbb R (^{}\langle \boldsymbol \alpha_{\lambda} | \mathbf a_{\lambda}^\dag = \alpha_{\lambda} \langle \boldsymbol \alpha_{\lambda} | and \mathbf a_\lambda | \boldsymbol \alpha_\lambda \rangle = \alpha_{\lambda} | \boldsymbol \alpha_\lambda \rangle):

| \boldsymbol \Psi_{\text{in}} \rangle = | \mathbf 0, R_\lambda \rangle_1 \otimes | \boldsymbol \alpha_{\lambda} \rangle_2

The difference between the two output signals \mathbf N_6 - \mathbf N_5 for a \delta:

\delta = k\left(L_3 - L_4 \right) = \frac{\pi}{2} + \varepsilon

around the dephasing variation \varepsilon was previously calculated:

\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} = -\sin(\varepsilon) \left( \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} - \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right) + i \cos(\varepsilon) \left( \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_2} - \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_1} \right)

We can express \mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} replacing \mathbf a_{\lambda_1}^\dag and \mathbf a_{\lambda_1} as function of \mathbf{A}_{R_{\lambda_1}}, \mathbf{A}_{R_{\lambda_1}}^\dag since input channel (1) is a squeezed state, while \mathbf a_{\lambda_2}^\dag and \mathbf a_{\lambda_2} are unchanged since input channel (2) is semi-classical state:

\begin{aligned} \mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} = & -\sin(\varepsilon) \left( \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} - \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right) + i \cos(\varepsilon) \left( \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_2} - \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_1} \right) \\ = & -\sin(\varepsilon) \left[ \left[ \mathbf{A}_{R_{\lambda_1}}^\dag \cosh(R_{\lambda_1}) - \mathbf{A}_{R_{\lambda_1}} \sinh(R_{\lambda_1}) \right] \right. \\ & \cdot \left.\left[ \mathbf{A}_{R_{\lambda_1}} \cosh(R_{\lambda_1}) - \mathbf{A}_{R_{\lambda_1}}^\dag \sinh(R_{\lambda_1}) \right] - \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right] \\ & + i \cos(\varepsilon) \left[ \left[ \mathbf{A}_{R_{\lambda_1}}^\dag \cosh(R_{\lambda_1}) - \mathbf{A}_{R_{\lambda_1}} \sinh(R_{\lambda_1}) \right] \mathbf a_{\lambda_2} \right. \\ & \left .- \mathbf a_{\lambda_2}^\dag \left[ \mathbf{A}_{R_{\lambda_1}} \cosh(R_{\lambda_1}) - \mathbf{A}_{R_{\lambda_1}}^\dag \sinh(R_{\lambda_1}) \right] \right] \\ = & -\sin(\varepsilon) \left[ \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. - \sinh(R_{\lambda_1})\cosh(R_{\lambda_1}) \left( \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag + \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} \right) - \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right] \\ & + i \cos(\varepsilon) \left[ \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2} - \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2} \right. \\ & \left. - \cosh(R_{\lambda_1}) \mathbf a_{\lambda_2}^\dag \mathbf{A}_{R_{\lambda_1}} + \sinh(R_{\lambda_1}) \mathbf a_{\lambda_2}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right] \end{aligned}

We can then calculate the expectation:

\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} |\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} | \boldsymbol \Psi_{\text{in}} \rangle = & {}_1\langle \mathbf 0, R_{\lambda_1} | -\sin(\varepsilon) \left[ \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. - \sinh(R_{\lambda_1})\cosh(R_{\lambda_1}) \left( \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag + \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} \right) - \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right] \\ & + i \cos(\varepsilon) \left[ \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2} - \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2} \right. \\ & \left. - \cosh(R_{\lambda_1}) \mathbf a_{\lambda_2}^\dag \mathbf{A}_{R_{\lambda_1}} + \sinh(R_{\lambda_1}) \mathbf a_{\lambda_2}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right] | {}_1\mathbf 0, R_\lambda \rangle_1\\ = & -\sin(\varepsilon) \left[ \cosh^2(R_{\lambda_1}) {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_\lambda \rangle_1 +\right. \\ & \left. \sinh^2(R_{\lambda_1}) {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_\lambda \rangle_1 \right. \\ & \left. - \sinh(R_{\lambda_1})\cosh(R_{\lambda_1}) \left( {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag | \boldsymbol \Psi_{\text{in}} \rangle \right. \right. \\ & + \left. \left. {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_\lambda \rangle_1 \right) \right. \\ & \left.- {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} | \mathbf 0, R_\lambda \rangle_1 \right] \\ & + i \cos(\varepsilon) \left[ \cosh(R_{\lambda_1}) {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2} | \mathbf 0, R_\lambda \rangle_1 \right. \\ & \left.- \sinh(R_{\lambda_1}) {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2} | \mathbf 0, R_\lambda \rangle_1 \right. \\ & \left. - \cosh(R_{\lambda_1}){}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf a_{\lambda_2}^\dag \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_\lambda \rangle_1 \right. \\ & \left.+ \sinh(R_{\lambda_1}) {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf a_{\lambda_2}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_\lambda \rangle_1 \right] \\ = & -\sin(\varepsilon) \left[ \cosh^2(R_{\lambda_1}) \times 0 + \sinh^2(R_{\lambda_1}) \times 1\right. \\ & \left. - \sinh(R_{\lambda_1})\cosh(R_{\lambda_1}) \left( 0 + 0 \right) - \alpha_\lambda^2 \right] \\ & + i \cos(\varepsilon) \left[ \cosh(R_{\lambda_1}) \times 0 - \sinh(R_{\lambda_1}) \times 0\right. \\ & \left. - \cosh(R_{\lambda_1}) \times 0 + \sinh(R_{\lambda_1}) \times 0 \right] \\ = & \sin(\varepsilon) \left[ \alpha_\lambda^2 - \sinh^2(R_{\lambda_1}) \right] \end{aligned}

The only two terms which are contributing are {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_\lambda \rangle_1 and -{}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} | \mathbf 0, R_\lambda \rangle_1, all the other gives 0 (the sign are then reversed by -\sin(\varepsilon)).

For the squeezed Vacuum Term the term \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag is not in normal order with respect to the operators \mathbf{A}_{R_{\lambda_1}} and \mathbf{A}_{R_{\lambda_1}}^\dag. Normal ordering would require all annihilation operators to be on the right of all creation operators. Here, the annihilation operator \mathbf{A}_{R_{\lambda_1}} is on the left of the creation operator \mathbf{A}_{R_{\lambda_1}}^\dag.

Because it’s not in normal order, when we evaluate its expectation value with respect to the squeezed vacuum state | \mathbf 0, R_{\lambda_1} \rangle, we use the commutator relation [\mathbf{A}_{R_{\lambda_1}}, \mathbf{A}_{R_{\lambda_1}}^\dag] = 1 to rewrite \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag = 1 + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}. This leads to:

\begin{aligned} {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 = & {}_1\langle \mathbf 0, R_{\lambda_1} | (1 + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}) | \mathbf 0, R_{\lambda_1} \rangle_1 \\ = & 1 + {}_1\langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_{\lambda_1} \rangle_1 = 1 + 0 = 1 \end{aligned}

For the coherent state term the term \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} is in normal order with respect to the operators \mathbf a_{\lambda_2} and \mathbf a_{\lambda_2}^\dag. The creation operator \mathbf a_{\lambda_2}^\dag is on the left of the annihilation operator \mathbf a_{\lambda_2}.

For a coherent state | \boldsymbol \alpha_{\lambda} \rangle_2, the expectation value of the normally ordered number operator \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} directly gives the average number of photons, which is the square of the amplitude \alpha_\lambda^2:

\langle \boldsymbol \alpha_{\lambda} |_2 \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} | \boldsymbol \alpha_{\lambda} \rangle_2 = |\alpha_\lambda|^2 = \alpha_\lambda^2

This term represents the contribution from the semi-classical coherent state, where the photon number is determined by the amplitude of the classical field.

So the expectation of \mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} is:

\langle \boldsymbol \Psi_{\text{in}} |\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} | \boldsymbol \Psi_{\text{in}} \rangle = \sin(\varepsilon) \left[ \alpha_\lambda^2 - \sinh^2(R_{\lambda_1}) \right] \approx \alpha_{\lambda}^2 \varepsilon = \varepsilon \langle \mathbf N_{\lambda_2} \rangle

as small variations \varepsilon, we can approximate \sin(\varepsilon) \approx \varepsilon and the number of photons of the squeezed vacuum is negligible compared to \alpha_\lambda^2. So the results is again the average number of photons entering input channel (2).

If we wanted to compute the expectation of the squared balanced signal:

\begin{aligned} \left(\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5}\right)^2 = & \left\{-\sin(\varepsilon) \left[ \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right.\right. \\ & \left. \left. - \sinh(R_{\lambda_1})\cosh(R_{\lambda_1}) \left( \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag + \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} \right) - \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right] \right.\\ & \left. + i \cos(\varepsilon) \left[ \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2} - \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2} \right.\right. \\ & \left. \left. - \cosh(R_{\lambda_1}) \mathbf a_{\lambda_2}^\dag \mathbf{A}_{R_{\lambda_1}} + \sinh(R_{\lambda_1}) \mathbf a_{\lambda_2}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right] \right\}^2 \end{aligned}

It would require the computation of a long expression in one step. To split in three parts, we can use We have already computed for a generic state in input channel (1), and we can use that result:

\begin{aligned} {}_2 \langle \boldsymbol \alpha_\lambda | \left(\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} \right)^2 | \boldsymbol \alpha_\lambda \rangle_2 = & \sin^2(\varepsilon) \left[ \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} - 2 \alpha_{\lambda_2}^2 \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ & - \cos^2(\varepsilon) \left[ \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag)^2 - \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} (\alpha_{\lambda_2}^2 + 1) - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}\mathbf a_{\lambda_1}^\dag) + \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1})^2 \right] \\ & - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left[ (\mathbf a_{\lambda_1}^\dag)^2 \mathbf a_{\lambda_1} - \mathbf a_{\lambda_1}^\dag (\mathbf a_{\lambda_1})^2 - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag - \mathbf a_{\lambda_1}) \right] \\ = & \sin^2(\varepsilon) \left[ \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} - 2 \alpha_{\lambda_2}^2 \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ & + \cos^2(\varepsilon) \left[ -\alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag)^2 + \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} (\alpha_{\lambda_2}^2 + 1) + \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}\mathbf a_{\lambda_1}^\dag) - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1})^2 \right] \\ & - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left[ (\mathbf a_{\lambda_1}^\dag)^2 \mathbf a_{\lambda_1} - \mathbf a_{\lambda_1}^\dag (\mathbf a_{\lambda_1})^2 - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag - \mathbf a_{\lambda_1}) \right] \end{aligned}

We can now express \mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} replacing \mathbf a_{\lambda_1}^\dag and \mathbf a_{\lambda_1} as function of \mathbf{A}_{R_{\lambda_1}}, \mathbf{A}_{R_{\lambda_1}}^\dag since input channel (1) is a squeezed state.

We start with the term with \sin^2(\varepsilon):

\sin^2(\varepsilon) \left[ \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} - 2 \alpha_{\lambda_2}^2 \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right]

We compute \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1}:

\begin{aligned} \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} & = \left( \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag - \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \right) \left( \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} - \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \right) \\ = & \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \left( \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} - \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \right) \\ & - \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \left( \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} - \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \right) \\ = & \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \\ & - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \end{aligned}

We compute \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} = (\mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1}) (\mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1}) = (\mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1})^2:

\begin{aligned} \left( \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} \right)^2 = & \left( \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right)^2 \end{aligned}

Putting it all together:

\begin{aligned} & \sin^2(\varepsilon) \left[ \left( \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right.\\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right)^2 \\ & \left. - 2 \alpha_{\lambda_2}^2 \left( \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right.\\ & \left.\left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right) + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \end{aligned}

Now we compute the term with \cos^2(\varepsilon):

\cos^2(\varepsilon) \left[ -\alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag)^2 + \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} (\alpha_{\lambda_2}^2 + 1) + \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}\mathbf a_{\lambda_1}^\dag) - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1})^2 \right]

We compute (\mathbf a_{\lambda_1}^\dag)^2:

\begin{aligned} (\mathbf a_{\lambda_1}^\dag)^2 & = \left( \mathbf{A}_{R_{\lambda_1}}^\dag \cosh \left(R_{\lambda_1}\right) - \mathbf{A}_{R_{\lambda_1}}\sinh \left(R_{\lambda_1}\right) \right)^2 \\ & = \cosh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \end{aligned}

\mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} was previously computed.

We compute \mathbf a_{\lambda_1}\mathbf a_{\lambda_1}^\dag:

\begin{aligned} \mathbf a_{\lambda_1}\mathbf a_{\lambda_1}^\dag = & \left( \mathbf{A}_{R_{\lambda_1}} \cosh(R_{\lambda_1}) - \mathbf{A}_{R_{\lambda_1}}^\dag \sinh(R_{\lambda_1}) \right) \left( \mathbf{A}_{R_{\lambda_1}}^\dag \cosh(R_{\lambda_1}) - \mathbf{A}_{R_{\lambda_1}} \sinh(R_{\lambda_1}) \right)^\dag \\ = & \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \\ & - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \end{aligned}

We compute \mathbf a_{\lambda_1}^2:

\begin{aligned} (\mathbf a_{\lambda_1})^2 & = \left( \mathbf{A}_{R_{\lambda_1}} \cosh(R_{\lambda_1}) - \mathbf{A}_{R_{\lambda_1}}^\dag \sinh(R_{\lambda_1}) \right)^2\\ & = \cosh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag + \sinh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \end{aligned}

Substituting all terms we get:

\begin{aligned} & \cos^2(\varepsilon) \left\{ -\alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \right. \right. \\ &\left. \left.+ \sinh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \right] \right. \\ & + (\alpha_{\lambda_2}^2 + 1) \left[ \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right] \\ & + \alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \right. \\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \right] \\ & \left. - \alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right. \\ &\left. \left. + \sinh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \right] \right\} \end{aligned}

Now we compute the term with \sin(\varepsilon) \cos(\varepsilon):

- 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left[ (\mathbf a_{\lambda_1}^\dag)^2 \mathbf a_{\lambda_1} - \mathbf a_{\lambda_1}^\dag (\mathbf a_{\lambda_1})^2 - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag - \mathbf a_{\lambda_1}) \right]

We compute (\mathbf a_{\lambda_1}^\dag)^2 \mathbf a_{\lambda_1}:

\begin{aligned} (\mathbf a_{\lambda_1}^\dag)^2 \mathbf a_{\lambda_1} = & \left[ \cosh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \right.\\ &\left. + \sinh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \right] \left( \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} - \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \right) \\ = & \cosh^3(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \mathbf{A}_{R_{\lambda_1}} - \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^3 \\ & - 2 \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} \\ & + 2 \cosh(R_{\lambda_1}) \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \\ & + \sinh^2(R_{\lambda_1}) \cosh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \mathbf{A}_{R_{\lambda_1}} - \sinh^3(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \mathbf{A}_{R_{\lambda_1}}^\dag \end{aligned}

We compute \mathbf a_{\lambda_1}^\dag (\mathbf a_{\lambda_1})^2:

\begin{aligned} \mathbf a_{\lambda_1}^\dag (\mathbf a_{\lambda_1})^2 = & \left( \mathbf{A}_{R_{\lambda_1}}^\dag \cosh(R_{\lambda_1}) - \mathbf{A}_{R_{\lambda_1}} \sinh(R_{\lambda_1}) \right) \left[ \cosh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \right.\\ &\left. - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag + \sinh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \right] \\ = & \cosh^3(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag (\mathbf{A}_{R_{\lambda_1}})^2 - 2 \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \\ & + \cosh(R_{\lambda_1}) \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \\ & - \sinh(R_{\lambda_1}) \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} (\mathbf{A}_{R_{\lambda_1}})^2 \\ & + 2 \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \\ & - \sinh^3(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \end{aligned}

We compute \mathbf a_{\lambda_1}^\dag - \mathbf a_{\lambda_1}:

\mathbf a_{\lambda_1}^\dag - \mathbf a_{\lambda_1} = (\cosh(R_{\lambda_1}) + \sinh(R_{\lambda_1})) (\mathbf{A}_{R_{\lambda_1}}^\dag - \mathbf{A}_{R_{\lambda_1}})

Substituting all terms we get:

\begin{aligned} - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} & \left\{ \left[ \cosh^3(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \mathbf{A}_{R_{\lambda_1}} - \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^3 \right. \right. \\ & \left. - 2 \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag + 2 \cosh(R_{\lambda_1}) \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. + \sinh^2(R_{\lambda_1}) \cosh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \mathbf{A}_{R_{\lambda_1}} - \sinh^3(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \mathbf{A}_{R_{\lambda_1}}^\dag \right] \\ & - \left[ \cosh^3(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag (\mathbf{A}_{R_{\lambda_1}})^2 - 2 \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. + \cosh(R_{\lambda_1}) \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 - \sinh(R_{\lambda_1}) \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} (\mathbf{A}_{R_{\lambda_1}})^2 \right. \\ & \left. + 2 \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag - \sinh^3(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \right] \\ & \left. - \alpha_{\lambda_2}^2 (\cosh(R_{\lambda_1}) + \sinh(R_{\lambda_1})) (\mathbf{A}_{R_{\lambda_1}}^\dag - \mathbf{A}_{R_{\lambda_1}}) \right\} \end{aligned}

We can now replace in the original expression:

\begin{aligned} {}_2 \langle \boldsymbol \alpha_\lambda | \left(\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} \right)^2 | \boldsymbol \alpha_\lambda \rangle_2 = & \sin^2(\varepsilon) \left[ \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} - 2 \alpha_{\lambda_2}^2 \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ & + \cos^2(\varepsilon) \left[ -\alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag)^2 + \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} (\alpha_{\lambda_2}^2 + 1) + \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}\mathbf a_{\lambda_1}^\dag) - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1})^2 \right] \\ & - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left[ (\mathbf a_{\lambda_1}^\dag)^2 \mathbf a_{\lambda_1} - \mathbf a_{\lambda_1}^\dag (\mathbf a_{\lambda_1})^2 - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag - \mathbf a_{\lambda_1}) \right] \\ = & \sin^2(\varepsilon) \left[ \left( \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right.\\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right)^2 \\ & \left. - 2 \alpha_{\lambda_2}^2 \left( \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right.\\ & \left.\left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right) + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ & \cos^2(\varepsilon) \left\{ -\alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \right. \right. \\ &\left. \left.+ \sinh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \right] \right. \\ & + (\alpha_{\lambda_2}^2 + 1) \left[ \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right] \\ & + \alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \right. \\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \right] \\ & \left. - \alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right. \\ &\left. \left. + \sinh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \right] \right\} \\ & - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left\{ \left[ \cosh^3(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \mathbf{A}_{R_{\lambda_1}} - \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^3 \right. \right. \\ & \left. - 2 \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag + 2 \cosh(R_{\lambda_1}) \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. + \sinh^2(R_{\lambda_1}) \cosh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \mathbf{A}_{R_{\lambda_1}} - \sinh^3(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \mathbf{A}_{R_{\lambda_1}}^\dag \right] \\ & - \left[ \cosh^3(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag (\mathbf{A}_{R_{\lambda_1}})^2 - 2 \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. + \cosh(R_{\lambda_1}) \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 - \sinh(R_{\lambda_1}) \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} (\mathbf{A}_{R_{\lambda_1}})^2 \right. \\ & \left. + 2 \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag - \sinh^3(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \right] \\ & \left. - \alpha_{\lambda_2}^2 (\cosh(R_{\lambda_1}) + \sinh(R_{\lambda_1})) (\mathbf{A}_{R_{\lambda_1}}^\dag - \mathbf{A}_{R_{\lambda_1}}) \right\} \end{aligned}

We can now compute the expectation of this formula for the state | \mathbf 0, R_{\lambda_1} \rangle.

\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} | \left(\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} \right)^2 | \boldsymbol \Psi_{\text{in}} \rangle = & {}_1 \langle \mathbf 0, R_{\lambda_1} | \sin^2(\varepsilon) \left[ \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} - 2 \alpha_{\lambda_2}^2 \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ & + \cos^2(\varepsilon) \left[ -\alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag)^2 + \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} (\alpha_{\lambda_2}^2 + 1) + \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}\mathbf a_{\lambda_1}^\dag) - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1})^2 \right] \\ & - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left[ (\mathbf a_{\lambda_1}^\dag)^2 \mathbf a_{\lambda_1} - \mathbf a_{\lambda_1}^\dag (\mathbf a_{\lambda_1})^2 - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag - \mathbf a_{\lambda_1}) \right] | \mathbf 0, R_{\lambda_1} \rangle_1 \\ & {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf T_1 + \mathbf T_2 + \mathbf T_3 | \mathbf 0, R_{\lambda_1} \rangle_1 \\ & {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf T_1 | \mathbf 0, R_{\lambda_1} \rangle_1 + {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf T_2 | \mathbf 0, R_{\lambda_1} \rangle_1 + {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf T_3 | \mathbf 0, R_{\lambda_1} \rangle_1 \end{aligned}

We can compute each term separately.

The first term is:

\begin{aligned} {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf T_1 | \mathbf 0, R_{\lambda_1} \rangle_1 = & {}_1 \langle \mathbf 0, R_{\lambda_1} | \sin^2(\varepsilon) \left[ \left( \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right.\\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right)^2 \\ & \left. - 2 \alpha_{\lambda_2}^2 \left( \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right.\\ & \left.\left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right) \right. \\ & \left.+ (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] | \mathbf 0, R_{\lambda_1} \rangle_1 \\ = & {}_1 \langle \mathbf 0, R_{\lambda_1} | \sin^2(\varepsilon) \left[ \left( \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right.\\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right)^2 \\ & \left. - 2 \alpha_{\lambda_2}^2 \left( \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right.\\ & \left.\left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right)\right. \\ & \left. + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] | \mathbf 0, R_{\lambda_1} \rangle_1 \\ = & \sin^2(\varepsilon) \left[ {}_1 \langle \mathbf 0, R_{\lambda_1} | \left( \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right)^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. - 2 \alpha_{\lambda_2}^2 {}_1 \langle \mathbf 0, R_{\lambda_1} | \left( \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right) | \mathbf 0, R_{\lambda_1} \rangle_1 +\right. \\ & \left. (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf 1 | \mathbf 0, R_{\lambda_1} \rangle_1 \right] \\ = & \sin^2(\varepsilon) \left[ \sinh^4(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag)^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left.- 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ = & \sin^2(\varepsilon) \left[ \sinh^4(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left.- 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ = & \sin^2(\varepsilon) \left[ \sinh^4(R_{\lambda_1}) \times 1 - 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) \times 1 + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ = & \sin^2(\varepsilon) \left[ \sinh^4(R_{\lambda_1}) - 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \end{aligned}

The second term is:

\begin{aligned} {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf T_2 | \mathbf 0, R_{\lambda_1} \rangle_1 = & {}_1 \langle \mathbf 0, R_{\lambda_1} | \cos^2(\varepsilon) \left\{ -\alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \right. \right. \\ &\left. \left. - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \right] \right. \\ & + (\alpha_{\lambda_2}^2 + 1) \left[ \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right] \\ & + \alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \right. \\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 + \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \right] \\ & \left. - \alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \right. \\ &\left. \left. + \sinh^2(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \right] \right\} | \mathbf 0, R_{\lambda_1} \rangle_1 \\ = & \cos^2(\varepsilon) \left[ -\alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \right. \\ & \left. \left. - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \right. \\ & \left. + \sinh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}})^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right] \\ & + (\alpha_{\lambda_2}^2 + 1) \left[ \cosh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. + \sinh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 \right] \\ & + \alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. - \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}})^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. - \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left.+ \sinh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_{\lambda_1} \rangle_1 \right] \\ & \left. - \alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}})^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \right. \\ &\left. - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. \left. + \sinh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right] \right] \\ = & \cos^2(\varepsilon) \left[ -\alpha_{\lambda_2}^2 \left[ 0 - 0 + 0 \right] \right. \\ & \left. + (\alpha_{\lambda_2}^2 + 1) \left[ 0 - 0 - 0 + \sinh^2(R_{\lambda_1}) \times 1 \right] \right. \\ & \left. + \alpha_{\lambda_2}^2 \left[ \cosh^2(R_{\lambda_1}) \times 1 - 0 - 0 + 0 \right] \right. \\ & \left. - \alpha_{\lambda_2}^2 \left[ 0 - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \times 1 + 0 \right] \right] \\ = & \cos^2(\varepsilon) \left[ 0 + (\alpha_{\lambda_2}^2 + 1) \sinh^2(R_{\lambda_1}) \right. \\ & \left. + \alpha_{\lambda_2}^2 \cosh^2(R_{\lambda_1})\right. \\ & \left. - \alpha_{\lambda_2}^2 \left[ - 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \right] \right] \\ = & \cos^2(\varepsilon) \left[ (\alpha_{\lambda_2}^2 + 1) \sinh^2(R_{\lambda_1}) \right. \\ & \left.+ \alpha_{\lambda_2}^2 \cosh^2(R_{\lambda_1}) \right. \\ & \left.+ 2 \alpha_{\lambda_2}^2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \right] \\ = & \cos^2(\varepsilon) \left[ \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) \right. \\ & \left.+ \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^2 \cosh^2(R_{\lambda_1})\right. \\ & \left. + 2 \alpha_{\lambda_2}^2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \right] \\ = & \cos^2(\varepsilon) \left[ \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^2 (\sinh^2(R_{\lambda_1}) \right. \\ & \left.+ \cosh^2(R_{\lambda_1}) + 2 \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1})) \right] \\ = & \cos^2(\varepsilon) \left[ \sinh^2(R_{\lambda_1}) + \right. \\ & \left. \alpha_{\lambda_2}^2 (\cosh(2R_{\lambda_1}) + \sinh(2R_{\lambda_1})) \right] \\ = & \cos^2(\varepsilon) \left[ \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^2 e^{2R_{\lambda_1}} \right] \end{aligned}

The third term is:

\begin{aligned} {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf T_3 | \mathbf 0, R_{\lambda_1} \rangle_1 = & {}_1 \langle \mathbf 0, R_{\lambda_1} | - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left\{ \left[ \cosh^3(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \mathbf{A}_{R_{\lambda_1}} \right. \right. \\ & \left. \left. - \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}}^\dag)^3 \right. \right. \\ & \left. - 2 \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag + 2 \cosh(R_{\lambda_1}) \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. + \sinh^2(R_{\lambda_1}) \cosh(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \mathbf{A}_{R_{\lambda_1}} - \sinh^3(R_{\lambda_1}) (\mathbf{A}_{R_{\lambda_1}})^2 \mathbf{A}_{R_{\lambda_1}}^\dag \right] \\ & - \left[ \cosh^3(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag (\mathbf{A}_{R_{\lambda_1}})^2 - 2 \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag \right. \\ & \left. + \cosh(R_{\lambda_1}) \sinh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}}^\dag (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 - \sinh(R_{\lambda_1}) \cosh^2(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} (\mathbf{A}_{R_{\lambda_1}})^2 \right. \\ & \left. + 2 \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag - \sinh^3(R_{\lambda_1}) \mathbf{A}_{R_{\lambda_1}} (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \right] \\ & \left. - \alpha_{\lambda_2}^2 (\cosh(R_{\lambda_1}) + \sinh(R_{\lambda_1})) (\mathbf{A}_{R_{\lambda_1}}^\dag - \mathbf{A}_{R_{\lambda_1}}) \right\} | \mathbf 0, R_{\lambda_1} \rangle_1 \\ = & - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left\{ \left[ \cosh^3(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \right. \\ & \left. \left. - \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}}^\dag)^3 | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \right. \\ & \left. - 2 \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. + 2 \cosh(R_{\lambda_1}) \sinh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. + \sinh^2(R_{\lambda_1}) \cosh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}})^2 \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. - \sinh^3(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}})^2 \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 \right] \\ & - \left[ \cosh^3(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag (\mathbf{A}_{R_{\lambda_1}})^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left.- 2 \cosh^2(R_{\lambda_1}) \sinh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. + \cosh(R_{\lambda_1}) \sinh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left.- \sinh(R_{\lambda_1}) \cosh^2(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} (\mathbf{A}_{R_{\lambda_1}})^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left. + 2 \sinh(R_{\lambda_1}) \cosh(R_{\lambda_1}) \sinh(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \mathbf 0, R_{\lambda_1} \rangle_1 \right. \\ & \left.- \sinh^3(R_{\lambda_1}) {}_1 \langle \mathbf 0, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} (\mathbf{A}_{R_{\lambda_1}}^\dag)^2 | \mathbf 0, R_{\lambda_1} \rangle_1 \right] \\ & \left. - \alpha_{\lambda_2}^2 (\cosh(R_{\lambda_1}) + \sinh(R_{\lambda_1})) {}_1 \langle \mathbf 0, R_{\lambda_1} | (\mathbf{A}_{R_{\lambda_1}}^\dag - \mathbf{A}_{R_{\lambda_1}}) | \mathbf 0, R_{\lambda_1} \rangle_1 \right\} \\ = & - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left\{ \left[ 0 - 0 - 0 + 0 + 0 - 0 \right] \right. \\ & \left.- \left[ 0 - 0 + 0 - 0 + 0 - 0 \right] - \alpha_{\lambda_2}^2 (\cosh(R_{\lambda_1}) + \sinh(R_{\lambda_1})) [0 - 0] \right\} \\ = & - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left\{ 0 - 0 - 0 \right\} \\ = & 0 \end{aligned}

Summing all terms:

\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} | \left(\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} \right)^2 | \boldsymbol \Psi_{\text{in}} \rangle = & {}_1\langle \mathbf 0, R_{\lambda_1} | \sin^2(\varepsilon) \left[ \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} - 2 \alpha_{\lambda_2}^2 \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ & - \cos^2(\varepsilon) \left[ \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag)^2 - \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} (\alpha_{\lambda_2}^2 + 1) - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}\mathbf a_{\lambda_1}^\dag) + \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1})^2 \right] \\ & - 2 i \sin(\varepsilon) \cos(\varepsilon) \alpha_{\lambda_2} \left[ (\mathbf a_{\lambda_1}^\dag)^2 \mathbf a_{\lambda_1} - \mathbf a_{\lambda_1}^\dag (\mathbf a_{\lambda_1})^2 - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag - \mathbf a_{\lambda_1}) \right] | \mathbf 0, R_{\lambda_1} \rangle_1 \\ = & \sin^2(\varepsilon) \left[ \sinh^4(R_{\lambda_1}) - 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ & + \cos^2(\varepsilon) \left[ \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^2 e^{2R_{\lambda_1}} \right] \end{aligned}

We can now compute the variance:

\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} |\left(\Delta_{\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5}}\right)^2 | \boldsymbol \Psi_{\text{in}} \rangle & = \langle \boldsymbol \Psi_{\text{in}} | \left(\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} \right)^2 | \boldsymbol \Psi_{\text{in}} \rangle - \left(\langle \boldsymbol \Psi_{\text{in}} |\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5}| \boldsymbol \Psi_{\text{in}} \rangle\right)^2 \\ = & \left\{\sin^2(\varepsilon) \left[ \sinh^4(R_{\lambda_1}) - 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \right. \\ &\left. + \cos^2(\varepsilon) \left[ \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^2 e^{2R_{\lambda_1}} \right]\right\} \\ & - \left\{\sin(\varepsilon) \left[ \alpha_\lambda^2 - \sinh^2(R_{\lambda_1}) \right]\right\}^2 \\ = & \sin^2(\varepsilon) \left[ \sinh^4(R_{\lambda_1}) - 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ & + \cos^2(\varepsilon) \left[ \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^2 e^{2R_{\lambda_1}} \right] \\ & - \sin^2(\varepsilon) \left[ \alpha_{\lambda_2}^2 - \sinh^2(R_{\lambda_1}) \right]^2 \\ = & \sin^2(\varepsilon) \left[ \sinh^4(R_{\lambda_1}) - 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) + (\alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2) \right] \\ & + \cos^2(\varepsilon) \left[ \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^2 e^{2R_{\lambda_1}} \right] \\ & - \sin^2(\varepsilon) \left[ \alpha_{\lambda_2}^4 - 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) + \sinh^4(R_{\lambda_1}) \right] \\ = & \sin^2(\varepsilon) \left[ \sinh^4(R_{\lambda_1}) - 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^4 + \alpha_{\lambda_2}^2 \right. \\ & \left. - \alpha_{\lambda_2}^4 + 2 \alpha_{\lambda_2}^2 \sinh^2(R_{\lambda_1}) - \sinh^4(R_{\lambda_1}) \right] \\ & + \cos^2(\varepsilon) \left[ \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^2 e^{2R_{\lambda_1}} \right] \\ = & \sin^2(\varepsilon) \left[ \alpha_{\lambda_2}^2 \right] + \cos^2(\varepsilon) \left[ \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^2 e^{2R_{\lambda_1}} \right] \\ = & \alpha_{\lambda_2}^2 \sin^2(\varepsilon) + \cos^2(\varepsilon) \sinh^2(R_{\lambda_1}) + \alpha_{\lambda_2}^2 \cos^2(\varepsilon) e^{2R_{\lambda_1}} \\ = & \alpha_{\lambda_2}^2 \left( \sin^2(\varepsilon) + \cos^2(\varepsilon) e^{2R_{\lambda_1}} \right) + \cos^2(\varepsilon) \sinh^2(R_{\lambda_1}) \end{aligned}

So the variance is:

\alpha_{\lambda_2}^2 \left( \sin^2(\varepsilon) + e^{2R_{\lambda_1}} \cos^2(\varepsilon) \right) + \sinh^2(R_{\lambda_1}) \cos^2(\varepsilon)

We can consider the case \varepsilon = 0, for which case we have:

\alpha_{\lambda_2}^2 e^{2R_{\lambda_1}} = e^{2R_{\lambda_1}} \langle \mathbf N_{\lambda_2} \rangle

which is the expected result, a variance which is reduced by the squeezing factor e^{2R_{\lambda_1}} which less than one for 2R_{\lambda_1} negative.

As double confirmation, we can achieve this results in a similar fashion to what done for the vacuum.

With \varepsilon = 0 the expectation of \mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} is null:

\langle \boldsymbol \Psi_{\text{in}} |\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} | \boldsymbol \Psi_{\text{in}} \rangle = \sin(\varepsilon) \left[ \alpha_\lambda^2 - \sinh^2(R_{\lambda_1}) \right] = 0

we have for a generic |\boldsymbol \Psi_{\text{in}} \rangle:

\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} |\left(\Delta_{\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5}}\right)^2 | \boldsymbol \Psi_{\text{in}} \rangle = & {}_1\langle \mathbf 0, R_{\lambda_1} | -\left[ \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1}^\dag)^2 - \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} (\alpha_{\lambda_2}^2 + 1) \right. \\ & \left. - \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1} \mathbf a_{\lambda_1}^\dag) + \alpha_{\lambda_2}^2 (\mathbf a_{\lambda_1})^2 \right] | \mathbf 0, R_\lambda \rangle_1 - 0 \end{aligned}

If the laser beam in in input channel (2) is intense, then \alpha_{\lambda_2}^2 \gg 1 and therefore (\alpha_{\lambda_2}^2 + 1) \approx \alpha_{\lambda_2}^2:

\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} |\left(\Delta_{\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5}}\right)^2 | \boldsymbol \Psi_{\text{in}} \rangle = & \alpha_{\lambda_2}^2 {}_1\langle \mathbf 0, R_{\lambda_1} | -\left[ (\mathbf a_{\lambda_1}^\dag)^2 - \mathbf a_{\lambda_1}^\dag \mathbf a_{\lambda_1} - \mathbf a_{\lambda_1} \mathbf a_{\lambda_1}^\dag + (\mathbf a_{\lambda_1})^2 \right] | \mathbf 0, R_\lambda \rangle_1\\ = & \alpha_{\lambda_2}^2 {}_1\langle \mathbf 0, R_{\lambda_1} | \left( i (\mathbf a_{\lambda_1}^\dag - \mathbf a_{\lambda_1})\right)^2 | \mathbf 0, R_\lambda \rangle_1 \\ = & \alpha_{\lambda_2}^2 {}_1\langle \mathbf 0, R_{\lambda_1} | \left( \mathbf P_{\lambda_1}\right)^2| \mathbf 0, R_\lambda \rangle_1 \\ \end{aligned}

As we previously computed the variance of the \mathbf P_{\lambda_1} quadrature for a squeezed state here, we have:

\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} |\left(\Delta_{\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5}}\right)^2 | \boldsymbol \Psi_{\text{in}} \rangle = & \alpha_{\lambda_2}^2 {}_1\langle \mathbf 0, R_{\lambda_1} | \left( \mathbf P_{\lambda_1}\right)^2| \mathbf 0, R_\lambda \rangle_1 \\ = & \alpha_{\lambda_2}^2 e^{2R_{\lambda_1}} = e^{2R_{\lambda_1}} \langle \mathbf N_{\lambda_2} \rangle \end{aligned}

which coincide with the analytical result.

The signal to noise ratio:

\text{SNR} = \frac{\langle \mathbf N_{\lambda_5} - \mathbf N_{\lambda_6} \rangle}{\Delta_{\mathbf N_{\lambda_6}-\mathbf N_{\lambda_5}}} = e^{-R_{\lambda_1}} \varepsilon \sqrt{\mathbf N_{\lambda_2}}

is increased by a factor e^{-R_{\lambda_1}} compared to the previous case.

The minimum detectable dephasing for a given signal to noise ratio is reduced by a factor e^{-R_{\lambda_1}} and the sensitivity with a squeezed vacuum is larger than the standard quantum limit.