Mach-Zehnder interferometer with squeezed vacuum (average)
The Mach-Zehnder interferometer is a fundamental tool in quantum optics. Let’s consider a setup where input channel (1) receives a squeezed vacuum state | \mathbf 0, R_\lambda \rangle_1, and input channel (2) is fed with a quasi-classical coherent state | \boldsymbol \alpha_\lambda \rangle_2 with real amplitude \alpha_\lambda. The input state to the interferometer is a tensor product:
| \boldsymbol \Psi_{\text{in}} \rangle = | \mathbf 0, R_\lambda \rangle_1 \otimes | \boldsymbol \alpha_{\lambda} \rangle_2
We are interested in the difference between the photon number operators at the output ports, \mathbf N_6 and \mathbf N_5. For a phase shift \delta = k(L_3 - L_4) = \frac{\pi}{2} + \varepsilon, where \varepsilon represents a small deviation from \pi/2, the difference in output signals is given by:
\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)
To evaluate the expectation value of this difference for our input state, I need to express the operators \mathbf a_{\lambda_1} and \mathbf a_{\lambda_1}^\dag in terms of the annihilation and creation operators for the squeezed vacuum, \mathbf{A}_{R_{\lambda_1}} and \mathbf{A}_{R_{\lambda_1}}^\dag. The transformations are:
\begin{aligned} \mathbf a_{\lambda_1} &= \mathbf{A}_{R_{\lambda_1}} \cosh(R_{\lambda_1}) - \mathbf{A}_{R_{\lambda_1}}^\dag \sinh(R_{\lambda_1}) \\ \mathbf a_{\lambda_1}^\dag &= \mathbf{A}_{R_{\lambda_1}}^\dag \cosh(R_{\lambda_1}) - \mathbf{A}_{R_{\lambda_1}} \sinh(R_{\lambda_1}) \end{aligned}
Substituting these expressions into the output signal difference and taking the expectation value with respect to the input state | \boldsymbol \Psi_{\text{in}} \rangle, we leverage the properties of squeezed vacuum and coherent states. Specifically, for the squeezed vacuum state | \mathbf 0, R_{\lambda_1} \rangle_1, we have \mathbf{A}_{R_{\lambda_1}} | \mathbf 0, R_{\lambda_1} \rangle_1 = 0. For the coherent state | \boldsymbol \alpha_\lambda \rangle_2, we use \langle \boldsymbol \alpha_{\lambda} |_2 \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} | \boldsymbol \alpha_{\lambda} \rangle_2 = \alpha_\lambda^2.
After performing the expectation value calculation, we find that most terms vanish due to the properties of the vacuum and coherent states, except for two contributions:
- the term related to the squeezed vacuum input, {}_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, which arises from the non-normal ordering and the commutator relation,
- the term related to the coherent state input, -{}_2\langle \boldsymbol \alpha_{\lambda} | \mathbf a_{\lambda_2}^\dag \mathbf a_{\lambda_2} | \boldsymbol \alpha_{\lambda} \rangle_2 = - \alpha_\lambda^2.
Combining these results, the expectation value of the output signal difference simplifies to:
\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]
For small deviations \varepsilon, we can approximate \sin(\varepsilon) \approx \varepsilon. If the squeezed vacuum contribution is small compared to the coherent state amplitude (\alpha_\lambda^2 \gg \sinh^2(R_{\lambda_1})), the result further approximates to:
\langle \boldsymbol \Psi_{\text{in}} |\mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} | \boldsymbol \Psi_{\text{in}} \rangle \approx \varepsilon \alpha_{\lambda}^2 = \varepsilon \langle \mathbf N_{\lambda_2} \rangle
This result indicates that even with a squeezed vacuum input in one channel, the output signal difference is primarily determined by the coherent state amplitude and the phase deviation, similar to the case with classical inputs, although the squeezed vacuum introduces a quantum correction term -\sinh^2(R_{\lambda_1}).
For more insights into this topic, you can find the details here.