Robust phase sensitivity in Mach-Zehnder interferometer using photon added and subtracted squeezed coherent state

Abstract: For the precision-based measurements, Mach-Zehnder interferometry is a widely used technique. There are various ways to enhance the precision of Mach-Zehnder interferometer (MZI), e.g., having a non-classical input state is one of the ways to enhance the precision of the phase estimation performed by MZI. The phase estimation performed by MZI is investigated here by considering that the input states of MZI are different combinations of photon added and subtracted squeezed coherent states (PASCS and PSSCS). Using quantum Fisher information, it is shown that the use of PASCS in both the input modes of MZI, provides the most precise estimate of the unknown phase. This system is also analyzed in two different measurement scenarios -- single intensity detection (SID) and intensity difference detection (IDD). Systematic analysis has established that the intensity measurement might not be an optimal measurement scheme for phase estimation in MZI as phase and intensity correspond to non-commuting observables. The impact of the photon loss on the MZI-based phase estimation setup is also studied and it is found that PASCS is robust against photon loss, when the loss in MZI is low.

• The paper shows that using photon added squeezed coherent states in both ports maximizes quantum Fisher information, pushing sensitivity toward the Heisenberg limit.
• It details how photon addition/subtraction balance and detection schemes influence phase estimation in a Mach-Zehnder interferometer.
• The study demonstrates robust phase sensitivity under photon loss, underscoring the promise of non-Gaussian states for quantum metrology.

Robust Phase Sensitivity Enhancement in Mach-Zehnder Interferometry via Photon Added and Subtracted Squeezed Coherent States

Introduction and Motivation

The precision of phase estimation in optical interferometry is fundamentally constrained by quantum mechanical limits, namely the standard quantum limit (SQL) and the Heisenberg limit (HL). Classical inputs yield sensitivity scaling as $\Delta \phi_{SQL} \sim 1/\sqrt{\langle N \rangle}$, while non-classical states can breach SQL, approaching HL with scaling $\Delta\phi_{HL} \sim 1/\langle N \rangle$, contingent on the input state's quantum character and detection protocol. Prior work has established the superiority of non-Gaussian states for optimal parameter estimation, though their practical generation is often impeded by susceptibility to environmental decoherence and photon loss.

This study addresses the enhancement of phase sensitivity in the Mach-Zehnder interferometer (MZI) via engineered non-Gaussian states—specifically photon added squeezed coherent states (PASCS) and photon subtracted squeezed coherent states (PSSCS). The work systematically evaluates the impact of these states and their combinations on phase estimation, scrutinizes the role of detection schemes (single and difference intensity detection), and quantifies robustness against photon loss, with explicit dependence on quantum Fisher information (QFI) and the quantum Cramer-Rao bound (QCRB).

Formalism: Mach-Zehnder Interferometer with Engineered Input States

The analysis models the MZI with standard $50:50$ beam splitters and phase shifters. Input states are constructed as combinations of PASCS and PSSCS in both arms, defined as:

• PASCS: $$|\psi_{+}\rangle = N_{+}(\alpha,z,m) \hat{a}_1^{\dagger m} |\alpha, z\rangle$$
• PSSCS: $$|\psi_{-}\rangle = N_{-}(\beta,z',m') \hat{a}_2^{m'} |\beta, z'\rangle$$

Parameters include displacement operators ($\alpha$, $\beta$), squeezing magnitude ($r$, $r'$), squeezing phase ($\phi$, $\Delta\phi_{HL} \sim 1/\langle N \rangle$0), and photon addition/subtraction numbers ($\Delta\phi_{HL} \sim 1/\langle N \rangle$1, $\Delta\phi_{HL} \sim 1/\langle N \rangle$2). The output state evolution is governed by the unitary transformations induced by the BS and phase shifters.

Quantum Fisher Information and Ultimate Phase Sensitivity

The lower bound of phase sensitivity is determined via QFI, which is computed for single parameter estimation (relative phase $\Delta\phi_{HL} \sim 1/\langle N \rangle$3) as:

$\Delta\phi_{HL} \sim 1/\langle N \rangle$4

where $\Delta\phi_{HL} \sim 1/\langle N \rangle$5 captures the covariance and correlations associated with the photon statistic moments of the engineered input states.

Key findings include:

• Maximum QFI is achieved when PASCS populates both input ports.
• Minimum QFI occurs with PSSCS at both ports.
• Mixed input (PASCS-PSSCS) yields intermediate, oscillatory sensitivity profiles, contingent on photon addition/subtraction balance.
• Numerical and analytic results demonstrate that increasing the photon addition number in PASCS directly drives phase sensitivity toward HL.

Detection Schemes: Single and Difference Intensity Measurement

The practical phase sensitivity depends not only on the input states but also on the selected detection protocol:

• IDD (Intensity Difference Detection): Optimal when relative phase $\Delta\phi_{HL} \sim 1/\langle N \rangle$6 is integer multiples of $\Delta\phi_{HL} \sim 1/\langle N \rangle$7, with a divergence at odd multiples of $\Delta\phi_{HL} \sim 1/\langle N \rangle$8. Achieves QCRB under certain parameter regimes.
• SID (Single Intensity Detection): Minimum sensitivity occurs within $\Delta\phi_{HL} \sim 1/\langle N \rangle$9 windows that vary based on detector configuration, but overall less informative than IDD due to partial loss of mode-specific photon statistics.

Intensity-based detection schemes are shown to be suboptimal in regimes of large squeezing, due to the uncertainty relation $50:50$0. Precise intensity measurement ultimately increases phase estimation uncertainty, especially as squeezing amplitude $50:50$1 grows.

Dependence on State Parameters

Systematic exploration of state parameters reveals the following:

• Displacement parameters ($50:50$2, $50:50$3): Sensitivity improves monotonically with increased coherent amplitude.
• Squeezing phase ($50:50$4, $50:50$5): Minimum sensitivity at integral multiples of $50:50$6.
• Squeezing amplitude ($50:50$7, $50:50$8): For QCRB, increased $50:50$9 reduces phase sensitivity. For intensity-based detection (IDD/SID), the opposite holds, due to the anti-commuting nature of phase and intensity.

Robustness Against Photon Loss

Photon loss is introduced through a fictitious beam splitter with transmission $$|\psi_{+}\rangle = N_{+}(\alpha,z,m) \hat{a}_1^{\dagger m} |\alpha, z\rangle$$0. The modified QFI is:

$$|\psi_{+}\rangle = N_{+}(\alpha,z,m) \hat{a}_1^{\dagger m} |\alpha, z\rangle$$1

Sensitivity remains robust (QFI near ideal) for $$|\psi_{+}\rangle = N_{+}(\alpha,z,m) \hat{a}_1^{\dagger m} |\alpha, z\rangle$$2 (small losses). In PASCS inputs, higher photon number due to the addition process ensures resilience to photon loss, preserving enhanced phase sensitivity in realistic noisy environments.

Practical and Theoretical Implications

The study demonstrates that MZI phase sensitivity with PASCS inputs reaches the theoretical maximum set by QCRB under specific parameter regimes, surpassing classical and Gaussian non-classical inputs. The robustness to photon loss positions PASCS as a promising candidate for quantum-enhanced metrological applications, including quantum radar and sensing in lossy environments. Intensity measurements, while convenient, are fundamentally limited—highlighting the necessity for optimal quantum detection strategies. Future research should focus on adaptive schemes, feedback-based interferometry, and the exploration of further non-Gaussian resources to unlock deeper quantum advantage in metrology.

Conclusion

Photon added squeezed coherent states as inputs in Mach-Zehnder interferometry enable robust, near-Heisenberg-limited phase sensitivity, contingent on state parameters and detection protocol. The quantum Fisher information analysis establishes PASCS as the optimal resource, with high resilience to photon loss. This work sets the stage for practical deployment of non-Gaussian states in quantum metrology and prompts further investigation into optimal measurement and adaptive techniques for enhanced parameter estimation in noisy systems (2607.02984).