Squeezed States of Light
Definition
In the 1980s, quantum optics theory revealed the possibility of squeezed states of light, previously unconsidered and unobserved. Beyond novelty, these states offered the potential to surpass the standard quantum limit in measurement precision, a long-held fundamental boundary.
Squeezed states are defined by generalizing the quasi-classical state definition. A quasi-classical state | \boldsymbol \alpha_\lambda\rangle is an eigenstate of the annihilation operator \mathbf{a}_\lambda:
\mathbf a_\lambda | \boldsymbol \alpha_\lambda\rangle = \alpha_{\lambda R} | \boldsymbol \alpha_\lambda\rangle, \quad \alpha_\lambda \in \mathbb C
Analogously, a single-mode squeezed state is defined as an eigenstate of a generalized annihilation operator \mathbf{A}_{R_\lambda}, a linear combination of the annihilation \mathbf{a}_\lambda and creation \mathbf{a}^\dag_\lambda operators of the mode, with coefficients \cosh \left(R_\lambda\right) and \sinh \left(R_\lambda\right), where R_\lambda is a real number:
\begin{aligned} & \mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = \alpha_{\lambda R} | \boldsymbol \alpha_\lambda, R_\lambda \rangle, \quad \alpha_\lambda \in \mathbb C, \; R_\lambda \in \mathbf R \\ & \mathbf{A}_{R_\lambda} =\mathbf a_\lambda \cosh \left(R_\lambda\right) + \mathbf a_\lambda^\dag\sinh \left(R_\lambda\right) \\ & \mathbf{A}_{R_\lambda}^\dag =\mathbf a_\lambda^\dag \cosh \left(R_\lambda\right) + \mathbf a_\lambda\sinh \left(R_\lambda\right) \end{aligned}
Similar to \mathbf{a}_\lambda, \mathbf{A}_{R_\lambda} is non-Hermitian and possesses complex eigenvalues. The commutator is:
\begin{aligned} \left[\mathbf{A}_{R_\lambda}, \mathbf{A}_{R_\lambda}^\dag\right] = & \left[\mathbf a_\lambda \cosh \left(R_\lambda\right) + \mathbf a_\lambda^\dag\sinh \left(R_\lambda\right), \mathbf a_\lambda^\dag \cosh \left(R_\lambda\right) + \mathbf a_\lambda\sinh \left(R_\lambda\right)\right] \\ = & \left[\mathbf a_\lambda \cosh \left(R_\lambda\right), \mathbf a_\lambda^\dag \cosh \left(R_\lambda\right) + \mathbf a_\lambda\sinh \left(R_\lambda\right)\right] \\ & + \left[\mathbf a_\lambda^\dag\sinh \left(R_\lambda\right), \mathbf a_\lambda^\dag \cosh \left(R_\lambda\right) + \mathbf a_\lambda\sinh \left(R_\lambda\right)\right] \\ = & \left[\mathbf a_\lambda \cosh \left(R_\lambda\right), \mathbf a_\lambda^\dag \cosh \left(R_\lambda\right)\right] + \left[\mathbf a_\lambda \cosh \left(R_\lambda\right), \mathbf a_\lambda\sinh \left(R_\lambda\right)\right] \\ & + \left[\mathbf a_\lambda^\dag\sinh \left(R_\lambda\right), \mathbf a_\lambda^\dag \cosh \left(R_\lambda\right)\right] + \left[\mathbf a_\lambda^\dag\sinh \left(R_\lambda\right), \mathbf a_\lambda\sinh \left(R_\lambda\right)\right] \\ = & \cosh^2 \left(R_\lambda\right) \left[\mathbf a_\lambda, \mathbf a_\lambda^\dag\right] + \cosh \left(R_\lambda\right) \sinh \left(R_\lambda\right) \left[\mathbf a_\lambda, \mathbf a_\lambda\right] \\ & + \sinh \left(R_\lambda\right) \cosh \left(R_\lambda\right) \left[\mathbf a_\lambda^\dag, \mathbf a_\lambda^\dag\right] + \sinh^2 \left(R_\lambda\right) \left[\mathbf a_\lambda^\dag, \mathbf a_\lambda\right] \\ = & \cosh^2 \left(R_\lambda\right) \left[\mathbf a_\lambda, \mathbf a_\lambda^\dag\right] + \sinh^2 \left(R_\lambda\right) \left[\mathbf a_\lambda^\dag, \mathbf a_\lambda\right] \\ = & \cosh^2 \left(R_\lambda\right) \left[\mathbf a_\lambda, \mathbf a_\lambda^\dag\right] - \sinh^2 \left(R_\lambda\right) \left[\mathbf a_\lambda, \mathbf a_\lambda^\dag\right] \\ = & \left(\cosh^2 \left(R_\lambda\right) - \sinh^2 \left(R_\lambda\right)\right) \left[\mathbf a_\lambda, \mathbf a_\lambda^\dag\right] \\ = & 1 \end{aligned}
justifying the designation of \mathbf{A}_{R_\lambda} and \mathbf{A}_{R_\lambda}^\dag as generalized annihilation and creation operators.
The expressions in terms of \mathbf{a} and \mathbf{a}^\dagger are invertible:
\begin{aligned} \mathbf a_\lambda & = \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \\ \mathbf a_\lambda^\dag & = \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \end{aligned}
Average
We will now proceed to calculate the average electric field in a squeezed state | \boldsymbol \alpha_\lambda, R_\lambda \rangle.
The electric field can be written as:
\begin{aligned} \mathbf{E}_{\lambda}(\mathbf r,t) = & i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left( \mathbf{a}_{\lambda} e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} - \mathbf{a}_{\lambda}^{\dagger} e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right) \\ = & i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left[ \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right. \\ & - \left. \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right) e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right] \\ = & i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left[ \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} - \text{h.c.} \right] \end{aligned}
The average can be computing since \mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = \alpha_{\lambda R} | \boldsymbol \alpha_\lambda, R_\lambda \rangle:
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{E}_{\lambda}(\mathbf r,t) | \boldsymbol \alpha_\lambda, R_\lambda \rangle & = i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \langle \boldsymbol \alpha_\lambda, R_\lambda \left[ \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right. \\ & - \left. \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right) e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right]| \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ = & i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left[ \langle \boldsymbol \alpha_\lambda, R_\lambda | \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right. \\ & - \left. \langle \boldsymbol \alpha_\lambda, R_\lambda | \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right) e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right] \\ = & i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left[ e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \langle \boldsymbol \alpha_\lambda, R_\lambda | \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right. \\ & - \left. e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \langle \boldsymbol \alpha_\lambda, R_\lambda | \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right) | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right] \\ = & i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left[ e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \left( \cosh \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle - \sinh \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right) \right. \\ & - \left. e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \left( \cosh \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag | \boldsymbol \alpha_\lambda, R_\lambda \rangle - \sinh \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right) \right] \\ = & i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left[ e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \left( \cosh \left(R_\lambda\right) \alpha_\lambda - \sinh \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right) \right. \\ & - \left. e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \left( \cosh \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag | \boldsymbol \alpha_\lambda, R_\lambda \rangle - \sinh \left(R_\lambda\right) \alpha_{\lambda R} \right) \right] \\ = & i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left[ e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \left( \cosh \left(R_\lambda\right) \alpha_{\lambda R} - \sinh \left(R_\lambda\right) \bar\alpha_{\lambda R} \right) \right. \\ & \left. - e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \left( \cosh \left(R_\lambda\right) \bar \alpha_{\lambda R} - \sinh \left(R_\lambda\right) \alpha_{\lambda R} \right) \right] \\ = & i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left[ e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \left( \cosh \left(R_\lambda\right) \alpha_{\lambda R} - \sinh \left(R_\lambda\right) \bar\alpha_{\lambda R} \right) - \text{c.c.} \right] \end{aligned}
This computation shows that a squeezed state is the same as a semi-classical state |\boldsymbol \alpha_\lambda^\prime\rangle with:
\alpha_\lambda^\prime = \cosh \left(R_\lambda\right) \alpha_{\lambda R} - \sinh \left(R_\lambda\right) \bar\alpha_{\lambda R}
and the average of the electric field is then:
\langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{E}_{\lambda}(\mathbf r,t) | \boldsymbol \alpha_\lambda, R_\lambda \rangle = \langle \boldsymbol \alpha_\lambda^\prime | \mathbf{E}_{\lambda}(\mathbf r,t) | \boldsymbol \alpha_\lambda^\prime \rangle
In the case the complex number \alpha_{\lambda R} is real (\alpha_\lambda \in \mathbb R) then also \alpha_\lambda^\prime is real and simplifies to:
\begin{aligned} \alpha_\lambda^\prime & = \alpha_{\lambda R}\left[\cosh \left(R_\lambda\right) - \sinh \left(R_\lambda\right) \right] \\ & = \alpha_{\lambda R}\left[\frac{e^{R_\lambda} + e^{-R_\lambda}}{2} - \frac{e^{R_\lambda} - e^{-R_\lambda}}{2} \right] = \alpha_{\lambda R} e^{-R_\lambda} \end{aligned}
Variance
Let’s now compute the variance of the field, which, as often in quantum mechanics, can be computed as the difference between the the square minus the square of the average of the observable:
\left(\Delta E_{\lambda}(\mathbf r,t)\right)^2 = \langle \left(\mathbf{E}_{\lambda}(\mathbf r,t) \right)^2 \rangle - \left(\langle \mathbf{E}_{\lambda}(\mathbf r,t) \rangle \right)^2
We can compute the average of the square of the electric field \langle \left(\mathbf{E}_{\lambda}(\mathbf r,t) \right)^2 \rangle:
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{E}_{\lambda}(\mathbf r,t)^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle = & - \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \langle \boldsymbol \alpha_\lambda, R_\lambda | \left[ \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right. \\ & \left. - \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right) e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right]^2| \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ = & - \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \langle \boldsymbol \alpha_\lambda, R_\lambda | \left\{ \left[ \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right]^2 \right. \\ & - \left[ \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right] \left[ \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right) e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right] \\ & - \left[ \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right) e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right] \left[ \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right] \\ & \left. + \left[ \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right) e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right]^2 \right\} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ = & - \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \left\{ e^{2i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \langle \boldsymbol \alpha_\lambda, R_\lambda | \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right)^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right. \\ & - \langle \boldsymbol \alpha_\lambda, R_\lambda | \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right) | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ & - \langle \boldsymbol \alpha_\lambda, R_\lambda | \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right) \left( \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right) | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ & \left. + e^{-2i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \langle \boldsymbol \alpha_\lambda, R_\lambda | \left( \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right)^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right\} \\ = & - \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \left\{ e^{2i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \langle \mathbf{X}^2 \rangle - \langle \mathbf{X} \mathbf{Y} \rangle - \langle \mathbf{Y} \mathbf{X} \rangle + e^{-2i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \langle \mathbf{Y}^2 \rangle \right\} \end{aligned}
where \mathbf{X} = \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) and \mathbf{Y} = \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right).
We can write:
\langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{E}_{\lambda}(\mathbf r,t)^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle = \left(\langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{E}_{\lambda}(\mathbf r,t) | \boldsymbol \alpha_\lambda, R_\lambda \rangle\right)^2 + \Delta E^2
where:
\Delta E^2 = - \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \left\{ e^{2i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} (\langle \mathbf{X}^2 \rangle - \langle \mathbf{X} \rangle^2) - (\langle \mathbf{X} \mathbf{Y} \rangle + \langle \mathbf{Y} \mathbf{X} \rangle - 2 \langle \mathbf{X} \rangle \langle \mathbf{Y} \rangle) + e^{-2i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} (\langle \mathbf{Y}^2 \rangle - \langle \mathbf{Y} \rangle^2) \right\}
We can compute:
\begin{aligned} & \langle \mathbf{X}^2 \rangle - \langle \mathbf{X} \rangle^2 = - \frac{1}{2} \sinh(2R_\lambda) \\ & \langle \mathbf{Y}^2 \rangle - \langle \mathbf{Y} \rangle^2 = - \frac{1}{2} \sinh(2R_\lambda) \\ & \langle \mathbf{X} \mathbf{Y} \rangle + \langle \mathbf{Y} \mathbf{X} \rangle - 2 \langle \mathbf{X} \rangle \langle \mathbf{Y} \rangle = \cosh(2R_\lambda) \end{aligned}
Therefore:
\begin{aligned} \Delta E^2 = & - \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \left\{ e^{2i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \left( - \frac{1}{2} \sinh(2R_\lambda) \right) - \cosh(2R_\lambda) + e^{-2i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \left( - \frac{1}{2} \sinh(2R_\lambda) \right) \right\} \\ = & \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \left\{ \cosh(2R_\lambda) + \frac{1}{2} \sinh(2R_\lambda) \left( e^{2i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} + e^{-2i(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)} \right) \right\}\\ = & \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \left\{ \cosh(2R_\lambda) + \sinh(2R_\lambda) \cos(2(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)) \right\}\\ = & \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \left\{ \cosh(2R_\lambda) + \sinh(2R_\lambda) (\cos^2(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t) - \sin^2(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t)) \right\}\\ = & \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \left\{ (\cosh(R_\lambda) + \sinh(R_\lambda))^2 \cos^2(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t) \right. \\ & + \left. (\cosh(R_\lambda) - \sinh(R_\lambda))^2 \sin^2(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t) \right\} \\ = & \left[\mathscr{E}_{\lambda}^{(1)}\right]^2 \mathbf e_\lambda \mathbf e_\lambda \left\{ e^{2R_\lambda} \cos^2(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t) + e^{-2R_\lambda} \sin^2(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t) \right\} \end{aligned}
Since \mathbf e_\lambda \cdot \mathbf e_\lambda = 1
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{E}_{\lambda}(\mathbf r,t)^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle = & \left(\langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{E}_{\lambda}(\mathbf r,t) | \boldsymbol \alpha_\lambda, R_\lambda \rangle\right)^2 \\ & + \left[\mathscr{E}_{\lambda}^{(1)} \right]^2 \left( e^{2 R_\lambda} \cos^2\left(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t \right) + e^{-2 R_\lambda} \sin^2\left(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t \right) \right) \end{aligned}
Then the variance result:
\begin{aligned} \left(\Delta E_{\lambda}(\mathbf r,t)\right)^2 & = \langle \left(\mathbf{E}_{\lambda}(\mathbf r,t) \right)^2 \rangle - \left(\langle \mathbf{E}_{\lambda}(\mathbf r,t) \rangle \right)^2 \\ & = \left[\mathscr{E}_{\lambda}^{(1)} \right]^2 \left( e^{2 R_\lambda} \cos^2\left(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t \right) + e^{-2 R_\lambda} \sin^2\left(\mathbf{k}_{\lambda} \cdot \mathbf{r_\lambda} - \omega_{\lambda} t \right) \right) \end{aligned}
Which, compared to the quasi-classical state result:
\left[\mathscr{E}_{\lambda}^{(1)} \right]^2
is no longer constant and it varies both as function of the position and as function of time.
Complex plane representation
In order to discuss the behavior, let’s fix a position in the space and, without loss of generality we take \mathbf r = \mathbf 0 (a translation of axes can be used if \mathbf r \ne \mathbf 0):
\left(\Delta E_{\lambda}(\mathbf r,t)\right)^2 = \left[\mathscr{E}_{\lambda}^{(1)} \right]^2 \left( e^{2 R_\lambda} \cos^2\left( - \omega_{\lambda} t \right) + e^{-2 R_\lambda} \sin^2\left( - \omega_{\lambda} t \right) \right)
It means that there are state where the variance is less than \mathscr{E}_{\lambda}^{(1)} square, which is the variance of a classical state. This happen when:
\left( e^{2 R_\lambda} \cos^2\left( - \omega_{\lambda} t \right) + e^{-2 R_\lambda} \sin^2\left( - \omega_{\lambda} t \right) \right) < 1
It is necessary to distinguish the case where R_\lambda is positive from the case where is negative.
Case negative R_\lambda
Let’s consider the case first R_\lambda < 0. In such cases, when \sin^2\left( - \omega_{\lambda} t \right)=0, then variance is less than \mathscr{E}_{\lambda}^{(1)} square. To have an expression easy to manipulate, let’s also consider the case \alpha_{\lambda R} \in \mathbb R so that:
\alpha_\lambda^\prime = \alpha_{\lambda R} e^{-R_\lambda} \; \in \mathbb R
Then the average of the electric field evolves as:
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{E}_{\lambda}(\mathbf r,t) | \boldsymbol \alpha_\lambda, R_\lambda \rangle & = \langle \boldsymbol \alpha_\lambda^\prime | \mathbf{E}_{\lambda}(\mathbf r,t) | \boldsymbol \alpha_\lambda^\prime \rangle \\ & = i \mathscr{E}_{\lambda}^{(1)}\alpha'\left[e^{-i\omega_\lambda t} - e^{i\omega_\lambda t} \right] \\ & = -2i\mathscr{E}_{\lambda}^{(1)} \alpha'\sin(-i\omega_\lambda t) \end{aligned}
We can then plot this curve, along with the standard deviation of the squeezed state in orange and the standard deviation of the correspondent semi-classical state in red dashed lines.
It is clear that the dispersion for a squeezed state is bigger at \pi/2, 3\pi/2, ... and smaller at 0, 2\pi, ... Zooming on a section around 3\pi, it is clear that the orange line is closer to the curve than the red line.
The complex plane representation is useful for understanding the electric field average and dispersion evolution in squeezed states. Similar to quasi-classical states, the average electric field is obtained by projecting -2\alpha'\mathscr{E}_{\lambda}^{(1)} e^{-i\omega_\lambda t} onto the imaginary axis, where \alpha' is a complex number related to the squeezed state amplitude.
However, the dispersion in squeezed states differs significantly from quasi-classical states. Instead of a dispersion disk of diameter 2\mathscr{E}_{\lambda}^{(1)} (as in quasi-classical states, represented by the dotted black circle), squeezed states exhibit an elliptical dispersion. This ellipse, with a major axis larger and a minor axis smaller than 2\mathscr{E}_{\lambda}^{(1)}, rotates with the complex amplitude.
Projecting this rotating ellipse onto the imaginary axis results in a modulated dispersion, varying between maximum and minimum values. The dispersion is minimized at angles 0, 2\pi, ... and maximized at \pi/2, 3\pi/2, ....
I created a video illustrating the time evolution of a squeezed state with a negative squeezing factor. On the left, the electric field rotates in the complex plane, while on the right, its progression over time is displayed.
The maximum dispersion value is \Delta E_{max} = \mathscr{E}_{\lambda}^{(1)} e^R_\lambda and the minimum is \Delta E_{min} = \mathscr{E}_{\lambda}^{(1)} e^{-R_\lambda}. Their product is:
\Delta E_{max}\cdot \Delta E_{min} = \left[\mathscr{E}_{\lambda}^{(1)} e^{R_\lambda}\right] \cdot \left[\mathscr{E}_{\lambda}^{(1)} e^{-R_\lambda}\right] = \left[\mathscr{E}_{\lambda}^{(1)}\right]^2
representing the minimum dispersion allowed by the Heisenberg uncertainty principle, considering the non-zero commutator of field observables at time intervals of a quarter period of oscillation.
Case positive R_\lambda
Let’s consider now the case R_\lambda > 0. For positive R_\lambda and real positive \alpha_{\lambda R}, the electric field evolution and dispersion are depicted. The standard deviation is maximized at \omega_\lambda t = 0, 2\pi, ... and minimized at \omega_\lambda t = \pi/2, 3\pi/2, .... Dashed red lines represent the constant standard deviation of a quasi-classical state with the same average field evolution.
A magnified view around \omega_\lambda t = \pi/2 \pmod{\pi} highlights the reduced fluctuations compared to the quasi-classical state.
This evolution is represented by a rotating ellipse in the complex plane, centered at a complex number of modulus \alpha_\lambda^\prime. The ellipse’s long axis is tangent to the rotation path, while its short axis is radial. Projecting this ellipse onto the imaginary axis yields the field average and dispersion.
The dotted black circle illustrates the constant standard deviation for a quasi-classical state with amplitude \alpha_\lambda^\prime.
I also created for a positive squeezing factor a video, showing on the left the electric field rotates in the complex plane, while on the right, its progression over time.
The product of the maximum and minimum dispersions in the squeezed state reaches the Heisenberg limit, indicating that squeezed states, like quasi-classical states, are minimum dispersion states.
Measurements below the SQL
Before the theoretical discovery of squeezed states, the shot noise inherent in stable beams was considered the fundamental limit to optical measurement precision, termed the standard quantum limit (SQL).
SQL was associated with fluctuations in quasi-classical states, representing perfectly stable beams with constant photon detection probability. However, in the early 1980s, it was theorized that squeezed states of light could surpass this SQL, particularly in the context of gravitational wave detection.
The term “squeezed” refers to the reduction of fluctuations in specific observables. While the previous section demonstrated that the electric field dispersion in squeezed states can be temporarily lower than the quasi-classical limit, direct observation of the oscillating electric field at optical frequencies is technologically infeasible at the present time.
However, quadrature components, time-independent observables measurable via balanced homodyne detection, offer a solution. Squeezed states exhibit reduced dispersion in certain quadratures, enabling amplitude or phase measurements of optical waves with accuracy exceeding the standard quantum limit.
In squeezed states with positive R, the electric field dispersion is minimized when the average field amplitude is maximized, specifically at \omega_\lambda t = \pi/2, 3\pi/2, .... Measurements taken at these times allow for determining the field amplitude with an accuracy surpassing the standard quantum limit. Balanced homodyne detection of the \mathbf Q_{\lambda} or -\mathbf Q_{\lambda} quadrature enables field sampling precisely at these optimal times. The subsequent step involves calculating the average value and dispersion of the \mathbf Q_{\lambda} quadrature for a squeezed state with positive R_\lambda:
| \boldsymbol \alpha_\lambda, R_\lambda \rangle, \quad R_\lambda >0
and it is an eigenvalue of the generalized destruction operator \mathbf{A}_{R_\lambda}:
\mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = \alpha_{\lambda R} | \boldsymbol \alpha_\lambda, R_\lambda \rangle
We also consider \alpha_{\lambda R} \in \mathbb R so that:
\alpha_\lambda^\prime = \alpha_{\lambda R} e^{-R_\lambda} \quad \alpha_\lambda \in \mathbb R
We can express the quadrature \mathbf Q_{\lambda} as:
\begin{aligned} \mathbf Q_{\lambda} = & \sqrt{\frac{\hbar}{2}}\left( \mathbf a_\lambda + \mathbf a_\lambda ^\dag \right) = \sqrt{\frac{\hbar}{2}}e^{-R_\lambda}\left( \mathbf{A}_{R_\lambda} + \mathbf{A}_{R_\lambda} ^\dag \right) \end{aligned}
The average of \mathbf Q_{\lambda} is given by:
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = & \langle \boldsymbol \alpha_\lambda, R_\lambda | \sqrt{\frac{\hbar}{2}}e^{-R_\lambda}\left( \mathbf{A}_{R_\lambda} + \mathbf{A}_{R_\lambda} ^\dag \right)| \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ = & \sqrt{\frac{\hbar}{2}}e^{-R_\lambda}\left(\alpha_{\lambda R} + \bar \alpha_{\lambda R} \right) \\ = & \sqrt{\frac{\hbar}{2}}e^{-R_\lambda} 2\alpha_{\lambda R} \end{aligned}
The dispersion of \mathbf Q_\lambda is given as usual as the square of the average minus the average of the square:
\begin{aligned} \Delta Q = & \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle - \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle \end{aligned}
The square of the average is:
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle = & \langle \boldsymbol \alpha_\lambda, R_\lambda | \left[ \sqrt{\frac{\hbar}{2}}e^{-R_\lambda}\left( \mathbf{A}_{R_\lambda} + \mathbf{A}_{R_\lambda} ^\dag \right) \right]^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ = & \frac{\hbar}{2}e^{-2R_\lambda} \langle \boldsymbol \alpha_\lambda, R_\lambda | \left( \mathbf{A}_{R_\lambda} + \mathbf{A}_{R_\lambda} ^\dag \right)^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ = & \frac{\hbar}{2}e^{-2R_\lambda} \langle \boldsymbol \alpha_\lambda, R_\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) | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ = & \frac{\hbar}{2}e^{-2R_\lambda} \left( \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle + \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^{\dag 2} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right. \\ & \left. + \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda}^\dag | \boldsymbol \alpha_\lambda, R_\lambda \rangle + \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right) \\ = & \frac{\hbar}{2}e^{-2R_\lambda} \left( \alpha_{\lambda R}^2 + \bar \alpha_{\lambda R}^2 + \langle \boldsymbol \alpha_\lambda, R_\lambda | (\mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} + \mathbf {I}) | \boldsymbol \alpha_\lambda, R_\lambda \rangle + \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right) \\ = & \frac{\hbar}{2}e^{-2R_\lambda} \left(\alpha_{\lambda R}^2 + \bar \alpha_{\lambda R}^2 + \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle + 1 + \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right) \\ = & \frac{\hbar}{2}e^{-2R_\lambda} \left( \alpha_{\lambda R}^2 + \bar \alpha_{\lambda R}^2 + 2 \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle + 1 \right) \\ = & \frac{\hbar}{2}e^{-2R_\lambda} \left( \alpha_{\lambda R}^2 + \bar \alpha_{\lambda R}^2 + 2 |\alpha_{\lambda R}|^2 + 1 \right) \\ \end{aligned}
Since \alpha_{\lambda R}\in \mathbb R, we have \bar \alpha_{\lambda R} = \alpha_{\lambda R}\alpha_\lambda = |\alpha_{\lambda R}|:
\langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle = \frac{\hbar}{2}e^{-2R_\lambda} \left( \alpha_{\lambda R}^2 + \alpha_{\lambda R}^2 + 2 \alpha_{\lambda R}^2 + 1 \right) = \frac{\hbar}{2}e^{-2R_\lambda} \left( 4 \alpha_{\lambda R}^2 + 1 \right)
We know that \langle \mathbf{Q}_{\lambda} \rangle = \sqrt{\frac{\hbar}{2}}e^{-R_\lambda} 2\alpha_{\lambda R}.
\left( \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right)^2 = \left( \sqrt{\frac{\hbar}{2}}e^{-R_\lambda} 2\alpha_{\lambda R}\right)^2 = \frac{\hbar}{2}e^{-2R_\lambda} 4\alpha_{\lambda R}^2 = 2\hbar e^{-2R_\lambda} \alpha_{\lambda R}^2
Therefore the dispersion is:
\begin{aligned} \Delta Q_\lambda^2 = & \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle - \left( \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right)^2 \\ = & \frac{\hbar}{2}e^{-2R_\lambda} \left( 4 \alpha_{\lambda R}^2 + 1 \right) - 2\hbar e^{-2R_\lambda} \alpha_{\lambda R}^2 \\ = & \frac{\hbar}{2}e^{-2R_\lambda} 4\alpha_{\lambda R}^2 + \frac{\hbar}{2}e^{-2R_\lambda} - 2\hbar e^{-2R_\lambda} \alpha_{\lambda R}^2 \\ = & \frac{\hbar}{2}e^{-2R_\lambda} \end{aligned}
And the standard deviation:
\Delta Q_\lambda = \sqrt{\frac{\hbar}{2}}e^{-R_\lambda}
which is equal to the standard quantum limit reduced by a factor exponential of -R_\lambda.
We can also express the quadrature \mathbf P_{\lambda} as:
\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}
The average of \mathbf P_{\lambda} is given by:
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{P}_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = & \langle \boldsymbol \alpha_\lambda, R_\lambda | -i\sqrt{\frac{\hbar}{2}}e^{R_\lambda}\left( \mathbf{A}_{R_\lambda} - \mathbf{A}_{R_\lambda} ^\dag \right)| \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ = & -i\sqrt{\frac{\hbar}{2}}e^{R_\lambda}\left(\alpha_{\lambda R}- \bar \alpha_{\lambda R} \right) \\ = & -i\sqrt{\frac{\hbar}{2}}e^{R_\lambda} \left(\alpha_{\lambda R}- \alpha_{\lambda R} \right) \\ = & 0 \end{aligned}
The zero average for \mathbf P_{\lambda} arises from the assumption that \alpha_\lambda \in \mathbb{R_\lambda}. If \alpha_\lambda were complex, the average of \mathbf P_{\lambda} would generally be non-zero.
The square of the operator is:
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf P_{\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle = & \langle \boldsymbol \alpha_\lambda, R_\lambda | \left[ -i\sqrt{\frac{\hbar}{2}}e^{R_\lambda}\left( \mathbf{A}_{R_\lambda} - \mathbf{A}_{R_\lambda} ^\dag \right) \right]^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle\\ = & \langle \boldsymbol \alpha_\lambda, R_\lambda |-\frac{\hbar}{2}e^{2R_\lambda} \left( \mathbf{A}_{R_\lambda} - \mathbf{A}_{R_\lambda} ^\dag \right)^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle\\ = & \langle \boldsymbol \alpha_\lambda, 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) | \boldsymbol \alpha_\lambda, R_\lambda \rangle\\ = & \langle \boldsymbol \alpha_\lambda, 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) | \boldsymbol \alpha_\lambda, R_\lambda \rangle\\ = & \langle \boldsymbol \alpha_\lambda, R_\lambda |\frac{\hbar}{2}e^{2R_\lambda} \left( - \mathbf{A}_{R_\lambda}^2 - \mathbf{A}_{R_\lambda}^{\dag 2} + (\mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} + \mathbf{I}) + \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} \right) | \boldsymbol \alpha_\lambda, R_\lambda \rangle\\ = & \langle \boldsymbol \alpha_\lambda, R_\lambda |\frac{\hbar}{2}e^{2R_\lambda} \left( - \mathbf{A}_{R_\lambda}^2 - \mathbf{A}_{R_\lambda}^{\dag 2} + 2 \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} + \mathbf{I} \right)| \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ = & \frac{\hbar}{2}e^{2R_\lambda} \left( - \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle - \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^{\dag 2} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right. \\ & \left. + 2 \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle + \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{I} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right) \\ = & \frac{\hbar}{2}e^{2R_\lambda} \left( - \alpha_{\lambda R}^2 - \bar \alpha_{\lambda R}^2 + 2 |\alpha_{\lambda R}|^2 + 1 \right) \end{aligned}
Since \alpha_\lambda \in \mathbb R, we have \bar \alpha_{\lambda R} =\alpha_{\lambda R} = |\alpha_{\lambda R}|:
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{P}_{\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle = & \frac{\hbar}{2}e^{2R_\lambda} \left( - \alpha_{\lambda R}^2 - \alpha_{\lambda R}^2 + 2 \alpha_{\lambda R}^2 + 1 \right) \\ = & \frac{\hbar}{2}e^{2R_\lambda} \left( -2 \alpha_{\lambda R}^2 + 2 \alpha_{\lambda R}^2 + 1 \right) \\ = & \frac{\hbar}{2}e^{2R_\lambda} \end{aligned}
The dispersion of \mathbf P_\lambda is given as usual as the average of the square minus the square of the average:
\begin{aligned} \Delta P_\lambda^2 = & \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{P}_{\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle - \left( \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{P}_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \right)^2 \\ = & \frac{\hbar}{2}e^{2R_\lambda} - 0^2 \\ = & \frac{\hbar}{2}e^{2R_\lambda} \end{aligned}
And the standard deviation:
\Delta P_\lambda = \sqrt{\frac{\hbar}{2}}e^{R_\lambda}
which is equal to the standard quantum limit increased by a factor exponential of R.
The product of the two quadratures is:
\Delta Q_\lambda \Delta P_\lambda = \frac{\hbar}{2}
as expected from the Heisenberg uncertainty principle. It also shows that the squeezed state is a minimum dispersion state.
It is not possible to squeeze the two quadratures \mathbf Q_\lambda and \mathbf P_\lambda at the same time: for example, reducing the dispersion \Delta \mathbf Q_\lambda by a factor e^{−R_\lambda} < 1 (with R positive) induces an increase of \Delta \mathbf P_\lambda by a factor e^{R_\lambda}>1.
It is possible to represent the quadratures in the complex plane as done for quasi-classical states.
We can associate to the squeezed state an ellipse such that the projection on the horizontal axis yields the \mathbf Q_\lambda quadrature with its dispersion. The projection on the vertical axis yields the \mathbf P_\lambda quadrature with its dispersion. A projection onto an axis at an angle \theta yields the average and the dispersion of \mathbf Q_\lambda(\theta). If \alpha_\lambda^\prime \in \mathbf R (which means \varphi_\lambda=0) the graph takes the shape below.
While resembling the field evolution diagram, the quadrature representation diagram depicts the ensemble of measurement outcomes for quadratures \mathbf Q_\lambda, \mathbf P_\lambda, and \mathbf Q_\lambda(\theta), which are time-independent and measurable.
Amplitude-squeezed light is proposed for enhancing measurements of minute absorption in delicate samples, like biological specimens, that are susceptible to damage from high light power. Conventional methods using quasi-classical states require increasing beam intensity to improve measurement accuracy within a fixed experiment duration. However, sample damage and laser power limitations restrict this approach. Squeezed light offers an alternative: it enhances amplitude measurement accuracy, and thus absorption measurement precision, without raising the average beam power.
For amplitude measurements employing balanced homodyne detection of the \mathbf Q_\lambda quadrature in a squeezed state with positive R and \alpha_\lambda, the relative measurement uncertainty is reduced by a factor of e^{-R_\lambda} compared to quasi-classical state measurements:
\frac{\Delta Q_\lambda}{\langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle} = \frac{e^{-R_\lambda}}{2\alpha_\lambda^\prime}
While for a quasi-classical state:
\frac{\Delta Q_\lambda}{\langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle} = \frac{1}{2\alpha_\lambda^\prime}
This demonstrates the enhanced relative accuracy of \mathbf Q_\lambda quadrature measurements using squeezed states.
The comparison with a quasi-classical state of amplitude \alpha_\lambda^\prime is justified by the fact that both states have approximately the same beam power.
To compare the power of different light states, we compare their average photon number within the same quantization volume.
For a quasi-classical state |\boldsymbol \alpha'_\lambda\rangle, the average photon number is |\alpha_\lambda^\prime|^2.
To calculate the average photon number for a squeezed state |\boldsymbol \alpha_\lambda, R\rangle, we express \mathbf a_\lambda and \mathbf a_\lambda^\dagger in terms of the generalized annihilation and creation operators \mathbf{A}_{R_\lambda} and \mathbf{A}_{R_\lambda} ^\dag:
\begin{aligned} \mathbf N_\lambda = & \mathbf a_\lambda^\dagger \mathbf a_\lambda \\ = & \left[ \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \right] \left[ \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \right] \\ = & \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \\ & - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) + \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \\ = & \cosh^2 \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^{\dag 2} \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^2 + \sinh^2 \left(R_\lambda\right) \mathbf{A}_{R_\lambda} \mathbf{A}_{R_\lambda}^\dag \\ = & \cosh^2 \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^{\dag 2} \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^2 + \sinh^2 \left(R_\lambda\right) (1 + \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda}) \\ = & \cosh^2 \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^{\dag 2} \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^2 + \sinh^2 \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \mathbf{A}_{R_\lambda}^\dag \mathbf{A}_{R_\lambda} \\ = & \mathbf{A}_{R_\lambda}^\dag\mathbf{A}_{R_\lambda} \left[ \cosh^2 \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \right] - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right)\left[ \mathbf{A}_{R_\lambda}^2 + \mathbf{A}_{R_\lambda}^{\dag 2} \right] + \sinh^2 \left(R_\lambda\right) \end{aligned}
It is now possible to complete the calculation, since |\boldsymbol \alpha_\lambda, R_\lambda \rangle is an eigenstate of \mathbf{A}_{R_\lambda}:
\mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = \alpha_{\lambda R}| \boldsymbol \alpha_\lambda, R_\lambda \rangle
Therefore the average number of photon is:
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf N_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = & \langle \boldsymbol \alpha_\lambda, R_\lambda | \left[ \mathbf{A}_{R_\lambda}^\dag\mathbf{A}_{R_\lambda} \left[ \cosh^2 \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \right] \right. \\ & \left. - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right)\left[ \mathbf{A}_{R_\lambda}^2 + \mathbf{A}_{R_\lambda}^{\dag 2} \right] + \sinh^2 \left(R_\lambda\right) \right] | \boldsymbol \alpha_\lambda, R_\lambda \rangle\\ = & \left[ \cosh^2 \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \right] \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag\mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^2 | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^{\dag 2} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ & + \sinh^2 \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ = & \left[ \cosh^2 \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \right] \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda}^\dag \alpha_{\lambda R}| \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{A}_{R_\lambda} \alpha_{\lambda R} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \langle \boldsymbol \alpha_\lambda, R_\lambda | \bar \alpha_{\lambda R}\mathbf{A}_{R_\lambda}^{\dag} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ & + \sinh^2 \left(R_\lambda\right) \\ = & \left| \alpha_{\lambda R}\right|^2 \left[ \cosh^2 \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \right] \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \alpha_{\lambda R} \langle \boldsymbol \alpha_\lambda, R_\lambda | \alpha_{\lambda R} | \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \bar \alpha_{\lambda R} \langle \boldsymbol \alpha_\lambda, R_\lambda | \bar \alpha_{\lambda R}| \boldsymbol \alpha_\lambda, R_\lambda \rangle \\ & + \sinh^2 \left(R_\lambda\right) \\ = & \left| \alpha_{\lambda R}\right|^2 \left[ \cosh^2 \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \right] \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \left(\alpha_{\lambda R}\right)^2 \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) \left( \bar \alpha_{\lambda R}\right)^2 \\ & + \sinh^2 \left(R_\lambda\right) \\ = & \left| \alpha_{\lambda R}\right|^2 \left[ \cosh^2 \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \right] \\ & - \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right)\left[ \left(\bar\alpha_{\lambda R}\right)^2 + \left( \alpha_{\lambda R}\right)^2 \right] + \sinh^2 \left(R_\lambda\right) \end{aligned}
If \alpha_{\lambda R} \in \mathbb R then it simplifies to:
\begin{aligned} \langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf N_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = & \alpha_{\lambda R}^2 \left[ \cosh^2 \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \right] \\ & - 2\alpha_{\lambda R}^2 \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \\ = & \alpha_{\lambda R}^2 \left[ \cosh^2 \left(R_\lambda\right) - 2 \cosh \left(R_\lambda\right)\sinh \left(R_\lambda\right) + \sinh^2 \left(R_\lambda\right) \right] + \sinh^2 \left(R_\lambda\right) \\ = & \alpha_{\lambda R}^2 \left[ \cosh \left(R_\lambda\right) - \sinh \left(R_\lambda\right) \right]^2 + \sinh^2 \left(R_\lambda\right) \\ = & \alpha_{\lambda R}^2 \left[ \frac{e^{R_\lambda} + e^{-R_\lambda}}{2} - \frac{e^{R_\lambda} - e^{-R_\lambda}}{2} \right]^2 + \sinh^2 \left(R_\lambda\right) \\ = & \alpha_{\lambda R}^2 \left[ \frac{2e^{-R_\lambda}}{2} \right]^2 + \sinh^2 \left(R_\lambda\right) \\ = & \alpha_{\lambda R}^2 \left[ e^{-R_\lambda} \right]^2 + \sinh^2 \left(R_\lambda\right) \\ = & \alpha_{\lambda R}^2 e^{-2R_\lambda} + \sinh^2 \left(R_\lambda\right) \\ = & \left(\alpha_\lambda^\prime\right)^2 + \sinh^2 \left(R_\lambda\right) \end{aligned}
The average photon number in the squeezed state | \boldsymbol \alpha_\lambda, R_\lambda \rangle is \left(\alpha_\lambda^\prime\right)^2 + \sinh^2 \left(R_\lambda\right). If \alpha_\lambda^\prime is large compared to one, the term \sinh^2 \left(R_\lambda\right) becomes negligible, and the average power of the squeezed state |\alpha, R\rangle is nearly identical to that of the quasi-classical state | \boldsymbol \alpha_\lambda^\prime \rangle.
Therefore, the two state to compare when assessing the accuracy improvement for the same beam power are the squeezed state | \boldsymbol \alpha_\lambda, R_\lambda \rangle and a quasi-classical state | \boldsymbol \alpha_\lambda^\prime \rangle. The improved relative accuracy of the \mathbf Q_\lambda quadrature measurement using squeezed states, compared to a quasi-classical state of the same power, demonstrates measurement beyond the standard quantum limit.
For negative R_\lambda with real \alpha_{\lambda R}, the dispersion around zero field values is reduced. Determining the times \omega_\lambda t when the field is zero corresponds to measuring the field’s phase. These squeezed states, therefore, offer improved phase measurement accuracy compared to quasi-classical states of the same amplitude, and are termed phase-squeezed states.
Phase squeezing is visualized in the complex plane: for negative R_\lambda, the squeezed quadrature is \mathbf P_\lambda, with its dispersion reduced by a factor e^{R_\lambda} < 1. In terms of wave parameter measurements, this corresponds to phase squeezing.
Consider measuring the phase of a complex number within the green ellipse, the phase uncertainty is:
\Delta \varphi = \frac{\Delta P_\lambda}{\sqrt{2\hbar}} \frac{1}{\alpha_\lambda^\prime} = \frac{e^{R_\lambda}}{2\alpha_{\lambda R}}
Reducing \Delta P_\lambda means reducing phase dispersion. Phase-squeezed states achieve phase determination with uncertainty below the standard quantum limit. Consequently, they are used for enhancing interferometric measurements of small phase variations.
Fragility
Squeezing is a state which is fragile and requires significant technical expertise, limiting its use to very specific scenarios, as detection of gravitational waves.
The fragility of squeezing arise from its susceptibility to losses; these losses, encompassing absorption, stray reflections, and imperfect detector quantum efficiency, diminish the squeezing and consequently the potential improvement over the standard quantum limit.
Losses can be effectively modeled as a beam splitter that reflects a portion of the beam corresponding to the loss magnitude, followed by ideal detection of the squeezed quadrature. The reflected beam in output channel (3), representing undetected losses, is ignored. Conversely, the empty input channel (2) will be relevant.
To understand the impact of losses on a squeezed state, consider a squeezed state | \boldsymbol \alpha_\lambda, R_\lambda \rangle_1 entering channel (1) of a beam splitter, with channel (2) in a vacuum state | \mathbf 0 \rangle_2.
The goal is to describe the output state in channel four as a function of the input states. Therefore, we express the annihilation operator \mathbf a_{\lambda_4} in terms of \mathbf a_{\lambda_1} and \mathbf a_{\lambda_2}:
\mathbf a_{\lambda_4} = t\mathbf a_{\lambda_1} - r \mathbf a_{\lambda_2}
The amplitude reflection and transmission coefficients, r and t, are real and positive.
Given a squeezed state | \boldsymbol \alpha_\lambda, R_\lambda \rangle_1 in channel one, it is useful to express \mathbf a_{\lambda_1} in terms of the operator \mathbf{A}_{R_{\lambda_1}} for which \alpha_{{\lambda R}_1} is an eigenstate:
\mathbf a_{\lambda_1} = \mathbf{A}_{R_{\lambda_1}} \cosh \left(R_{\lambda_1}\right) - \mathbf{A}_{R_{\lambda_1}}^\dag\sinh \left(R_{\lambda_1}\right)
We consider real and positive \alpha_{\lambda R} and R_\lambda, implying that the squeezed quadrature is \mathbf Q_\lambda:
\mathbf Q_{\lambda 4} = \sqrt{\frac{\hbar}{2}}\left(\mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag \right)
We want to compute its average and its standard deviation. First it is necessary to express as function of the inputs:
\begin{aligned} \mathbf a_{\lambda_4} & = t\mathbf a_{\lambda_1} - r \mathbf a_{\lambda_2} \\ & = t\left[\mathbf{A}_{R_{\lambda_1}} \cosh \left(R_{\lambda_1}\right) - \mathbf{A}_{R_{\lambda_1}}^\dag\sinh \left(R_{\lambda_1}\right)\right] - r \mathbf a_{\lambda_2} \end{aligned}
Then:
\begin{aligned} \mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag = & t\left[\mathbf{A}_{R_{\lambda_1}} \cosh \left(R_{\lambda_1}\right) - \mathbf{A}_{R_{\lambda_1}}^\dag\sinh \left(R_{\lambda_1}\right)\right] - r \mathbf a_{\lambda_2} \\ & + t\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] - r \mathbf a_{\lambda_2}^\dag \\ =& t\left[\mathbf{A}_{R_{\lambda_1}} \cosh \left(R_{\lambda_1}\right) - \mathbf{A}_{R_{\lambda_1}} \sinh \left(R_{\lambda_1}\right) + \mathbf{A}_{R_{\lambda_1}}^\dag \cosh \left(R_{\lambda_1}\right) - \mathbf{A}_{R_{\lambda_1}}^\dag\sinh \left(R_{\lambda_1}\right) \right] \\ & - r \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) \\ =& t\left[\left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) \cosh \left(R_{\lambda_1}\right) - \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right)\sinh \left(R_{\lambda_1}\right) \right] \\& - r \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) \\ =& t\left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) \left[ \cosh \left(R_{\lambda_1}\right) - \sinh \left(R_{\lambda_1}\right) \right] - r \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) \\ =& t e^{-R_{\lambda_1}} \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) - r \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) \end{aligned}
Considering the the input state with a squeezed state with eigenvalue \alpha_{{\lambda R}_1} \in \mathbb R and the vacuum:
| \boldsymbol \Psi_{\text{in}} \rangle = | \boldsymbol \alpha_\lambda, R_\lambda \rangle_1 \otimes | \mathbf 0 \rangle_2
The average to compute:
\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} | \mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag | \boldsymbol \Psi_{\text{in}} \rangle = & \langle \boldsymbol \Psi_{\text{in}} | \left[ t e^{-R_{\lambda_1}} \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) - r \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) \right] | \boldsymbol \Psi_{\text{in}} \rangle \\ = & t e^{-R_{\lambda_1}} \langle \boldsymbol \Psi_{\text{in}} | \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) | \boldsymbol \Psi_{\text{in}} \rangle - r \langle \boldsymbol \Psi_{\text{in}} | \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) | \boldsymbol \Psi_{\text{in}} \rangle \end{aligned}
For the first term:
\begin{aligned} t e^{-R_{\lambda_1}} \langle \boldsymbol \Psi_{\text{in}} | \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) | \boldsymbol \Psi_{\text{in}} \rangle &= t e^{-R_{\lambda_1}} \langle \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} |_1 \langle \mathbf 0 |_2 \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) | \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} \rangle_1 \otimes | \mathbf 0 \rangle_2 \\ &= t e^{-R_{\lambda_1}} \langle \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} | \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) | \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} \rangle \langle \mathbf 0 |_2 | \mathbf 0 \rangle_2 \\ &= t e^{-R_{\lambda_1}} \langle \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} | \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) | \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} \rangle\\ &= t e^{-R_{\lambda_1}} \left( \alpha_{{\lambda R}_1} + \alpha_{{\lambda R}_1} \right) \\ &= 2 t e^{-R_{\lambda_1}} \alpha_{{\lambda R}_1} \end{aligned}
The second term is null because it acts on the vacuum:
\begin{aligned} - r \langle \boldsymbol \Psi_{\text{in}} | \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) | \boldsymbol \Psi_{\text{in}} \rangle &= - r \langle \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} |_1 \langle \mathbf 0 |_2 \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) | \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} \rangle_1 \otimes | \mathbf 0 \rangle_2 \\ &= - r \langle \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} |_1 | \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} \rangle_1 \langle \mathbf 0 |_2 \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) | \mathbf 0 \rangle_2 \\ &= - r \langle \mathbf 0 | \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) | \mathbf 0 \rangle_2 \\ &= - r \left( \langle \mathbf 0 | \mathbf a_{\lambda_2} | \mathbf 0 \rangle_2 + \langle \mathbf 0 | \mathbf a_{\lambda_2}^\dag | \mathbf 0 \rangle_2 \right) \\ &= - r (0 + 0) = 0 \end{aligned}
The average then takes the simple form:
\langle \boldsymbol \Psi_{\text{in}} | \mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag | \boldsymbol \Psi_{\text{in}} \rangle = 2 t e^{-R_{\lambda_1}} \alpha_{{\lambda R}_1}
So the average of the quadrature \mathbf Q_{\lambda 4} applying the factor:
\langle \mathbf Q_{\lambda 4} \rangle = \sqrt{\frac{\hbar}{2}}\langle\mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag \rangle = \sqrt{2\hbar} t e^{-R_{\lambda_1}} \alpha_{{\lambda R}_1}
is reduced by a factor t compared to its value without losses.
We can now compute the dispersion. First we compute the square of the quantity previously derived:
\begin{aligned} \left(\mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag\right)^2 = & \left[t e^{-R_{\lambda_1}} \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) - r \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right)\right]^2 \\ = & \left[t e^{-R_{\lambda_1}} \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right)\right]^2 + \left[ - r \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right)\right]^2 \\ & + 2 \left[t e^{-R_{\lambda_1}} \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right)\right] \left[ - r \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right)\right] \\ = & t^2 e^{-2R_{\lambda_1}} \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right)^2 + r^2 \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right)^2 \\ & - 2 r t e^{-R_{\lambda_1}} \left(\mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^\dag\right) \left( \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^\dag \right) \\ = & t^2 e^{-2R_{\lambda_1}} \left(\mathbf{A}_{R_{\lambda_1}}^2 + \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} + \mathbf{A}_{R_{\lambda_1}}^{\dag 2}\right) + r^2 \left( \mathbf a_{\lambda_2}^2 + \mathbf a_{\lambda_2} \mathbf a_{\lambda_2}^\dag + \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} + \mathbf a_{\lambda_2}^{\dag 2} \right) \\ & - 2 r t e^{-R_{\lambda_1}} \left( \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2} + \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2}^\dag + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2} + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2}^\dag \right) \\ = & t^2 e^{-2R_{\lambda_1}} \left(\mathbf{A}_{R_{\lambda_1}}^2 + \mathbf{A}_{R_{\lambda_1}}^{\dag 2} + \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \right) + r^2 \left( \mathbf a_{\lambda_2}^2 + \mathbf a_{\lambda_2}^{\dag 2} + \mathbf a_{\lambda_2} \mathbf a_{\lambda_2}^\dag + \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right) \\ & - 2 r t e^{-R_{\lambda_1}} \left( \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2} + \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2}^\dag + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2} + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2}^\dag \right) \end{aligned}
Then we take the average again with the same input states:
\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} | \left( \mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag\right)^2 | \boldsymbol \Psi_{\text{in}} \rangle = & \langle \boldsymbol \Psi_{\text{in}} | \left[t^2 e^{-2R_{\lambda_1}} \left(\mathbf{A}_{R_{\lambda_1}}^2 + \mathbf{A}_{R_{\lambda_1}}^{\dag 2} + \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \right) \right. \\ & \left.+ r^2 \left( \mathbf a_{\lambda_2}^2 + \mathbf a_{\lambda_2}^{\dag 2} + \mathbf a_{\lambda_2} \mathbf a_{\lambda_2}^\dag + \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right) \right. \\ & \left. - 2 r t e^{-R_{\lambda_1}} \left( \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2} + \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2}^\dag + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2} + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2}^\dag \right) \right] | \boldsymbol \Psi_{\text{in}} \rangle \\ = & t^2 e^{-2R_{\lambda_1}} \langle \boldsymbol \Psi_{\text{in}} | \left(\mathbf{A}_{R_{\lambda_1}}^2 + \mathbf{A}_{R_{\lambda_1}}^{\dag 2} + \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \right) | \boldsymbol \Psi_{\text{in}} \rangle \\ & + r^2 \langle \boldsymbol \Psi_{\text{in}} | \left( \mathbf a_{\lambda_2}^2 + \mathbf a_{\lambda_2}^{\dag 2} + \mathbf a_{\lambda_2} \mathbf a_{\lambda_2}^\dag + \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right) | \boldsymbol \Psi_{\text{in}} \rangle \\ & - 2 r t e^{-R_{\lambda_1}} \langle \boldsymbol \Psi_{\text{in}} | \left( \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2} + \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2}^\dag + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2} + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2}^\dag \right) | \boldsymbol \Psi_{\text{in}} \rangle \end{aligned}
We compute each term separately. The first set of terms is related to input (2):
\begin{aligned} & \langle \boldsymbol \Psi_{\text{in}} | \mathbf{A}_{R_{\lambda_1}}^2 | \boldsymbol \Psi_{\text{in}} \rangle = \langle \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^2 | \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} \rangle = \alpha_{{\lambda R}_1}^2 \\ & \langle \boldsymbol \Psi_{\text{in}} | \mathbf{A}_{R_{\lambda_1}}^{\dag 2} | \boldsymbol \Psi_{\text{in}} \rangle = \langle \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^{\dag 2} | \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} \rangle = (\bar \alpha_{{\lambda R}_1})^2 = \alpha_{{\lambda R}_1}^2 \\ & \langle \boldsymbol \Psi_{\text{in}} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \boldsymbol \Psi_{\text{in}} \rangle = \langle \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag | \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} \rangle = \alpha_{{\lambda R}_1} + 1 \\ &\langle \boldsymbol \Psi_{\text{in}} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} | \boldsymbol \Psi_{\text{in}} \rangle = \langle \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} | \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} | \boldsymbol \alpha_{\lambda_1}, R_{\lambda_1} \rangle =\alpha_{{\lambda R}_1}^2 \end{aligned}
Therefore this set of terms give:
\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} | \left(\mathbf{A}_{R_{\lambda_1}}^2 + \mathbf{A}_{R_{\lambda_1}}^{\dag 2} + \mathbf{A}_{R_{\lambda_1}} \mathbf{A}_{R_{\lambda_1}}^\dag + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf{A}_{R_{\lambda_1}} \right) | \boldsymbol \Psi_{\text{in}} \rangle &= \alpha_{{\lambda R}_1}^2 + \alpha_{{\lambda R}_1}^2 + (\alpha_{{\lambda R}_1} + 1) + \alpha_{{\lambda R}_1}^2 \\ &= 4\alpha_{{\lambda R}_1}^2 + 1 \end{aligned}
The second set of terms involving the vacuum in channel (1) give a single contribution from the term which is not in normal order:
\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} | \left( \mathbf a_{\lambda_2}^2 + \mathbf a_{\lambda_2}^{\dag 2} + \mathbf a_{\lambda_2} \mathbf a_{\lambda_2}^\dag + \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right) | \boldsymbol \Psi_{\text{in}} \rangle &= \langle \mathbf 0 |_2 \left( \mathbf a_{\lambda_2}^2 + \mathbf a_{\lambda_2}^{\dag 2} + \mathbf a_{\lambda_2} \mathbf a_{\lambda_2}^\dag + \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} \right) | \mathbf 0 \rangle_2 \\ &= \langle \mathbf 0 | \mathbf a_{\lambda_2} \mathbf a_{\lambda_2}^\dag + \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} | \mathbf 0 \rangle = \langle \mathbf 0 | \mathbf a_{\lambda_2} \mathbf a_{\lambda_2}^\dag | \mathbf 0 \rangle \\ & = \langle \mathbf 0 | \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} + 1 | \mathbf 0 \rangle = 1 \end{aligned}
The cross terms either have the vacuum state either acting on the left or on the right and do not give any contribution:
\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} | \left( \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2} + \mathbf{A}_{R_{\lambda_1}} \mathbf a_{\lambda_2}^\dag + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2} + \mathbf{A}_{R_{\lambda_1}}^\dag \mathbf a_{\lambda_2}^\dag \right) | \boldsymbol \Psi_{\text{in}} \rangle &= 0 \end{aligned}
Combining all terms:
\begin{aligned} \langle \boldsymbol \Psi_{\text{in}} | \left( \mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag\right)^2 | \boldsymbol \Psi_{\text{in}} \rangle = & t^2 e^{-2R_{\lambda_1}} (4 \alpha_{{\lambda R}_1}^2 + 1) + r^2 (1) - 2 r t e^{-R_{\lambda_1}} (0) \\ = & t^2 e^{-2R_{\lambda_1}} (4 \alpha_{{\lambda R}_1}^2 + 1) + r^2 \end{aligned}
It is now possible to compute the variance:
\begin{aligned} \Delta Q_{\lambda 4} = & \langle \boldsymbol \Psi_{\text{in}} | \left( \mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag\right)^2 | \boldsymbol \Psi_{\text{in}} \rangle - \left(\langle \boldsymbol \Psi_{\text{in}} | \mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag | \boldsymbol \Psi_{\text{in}} \rangle\right)^2 \\ & = t^2 e^{-2R_{\lambda_1}} (4 \alpha_{{\lambda R}_1}^2 + 1) + r^2 - \left(2 t e^{-R_{\lambda_1}} \alpha_{{\lambda R}_1}\right)^2 \\ & = 4 t^2 e^{-2R_{\lambda_1}} \alpha_{{\lambda R}_1}^2 + t^2 e^{-2R_{\lambda_1}} - 4 t^2 e^{-2R_{\lambda_1}} \alpha_{{\lambda R}_1}^2 + r^2 \\ & = t^2 e^{-2R_{\lambda_1}} + r^2 \end{aligned}
This r^2 term signifies a degradation of squeezing compared to an ideal scenario with amplitude t \alpha_{{\lambda R}_1}. It arises from vacuum fluctuations entering through channel two.
While technically derived from the commutator of \mathbf a_{\lambda_2} and \mathbf a_{\lambda_2}^\dag, this commutator embodies vacuum fluctuations. Therefore, vacuum fluctuations entering via channel two introduce additional noise in \mathbf Q_{\lambda 4}, reducing the squeezing benefit.
Since the beam splitter models any loss mechanism, it follows that any loss between the source and detector effectively adds vacuum fluctuations, thereby diminishing the advantages of squeezing. Minimizing all losses, including detection inefficiency, is a significant challenge, achieved only in limited instances.