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Laser Physics Letters

Photon subtraction as a way to increase sensitivity of the Mach–Zehnder interferometer

Alina S Fassakhova, Alexander A Koksharov, Dmitry Kuts, Mikhail S. Podoshvedov

Published Mar 19, 2025• 1 citation
Publisher

Abstract

Abstract Choice of the quantum probe state is of key importance to enhance the sensitivity of the Mach–Zehnder (MZ) interferometer beyond the classical one and even approach the quantum Cramér–Rao bound. According to the work of Caves (Caves 1981 Phys. Rev. D 23 1693), the quantum MZ interferometer is a phase-dependent one operating with single-mode squeezed vacuum (SMSV) light coupled with a coherent state. We use the SMSV state with squeezing <10 dB and report a gain in sensitivity of the phase-dependent MZ interferometer of more than 10 dB compared to the original result (Pezze and Smerzi 2008 Phys. Rev. Lett. 100 073601) via photon subtraction. The gain in sensitivity is also observed when measuring the difference in output intensities of the SMSV state with squeezing of < 3 dB from which two, four, and six photons are subtracted and a large coherent state. Overall, subtracting photons from the initially weakly squeezed light can prove to be a more efficient strategy in quantum MZ interferometry compared to highly squeezed SMSV state generation.

DOI
10.1088/1612-202x/adbd21

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Abstract Choice of the quantum probe state is of key importance to enhance the sensitivity of the Mach–Zehnder (MZ) interferometer beyond the classical one and even approach the quantum Cramér–Rao bound. According to the work of Caves (Caves 1981 Phys. Rev. D 23 1693), the quantum MZ interferometer is a phase-dependent one operating with single-mode squeezed vacuum (SMSV) light coupled with a coherent state. We use the SMSV state with squeezing <10 dB and report a gain in sensitivity of the phase-dependent MZ interferometer of more than 10 dB compared to the original result (Pezze and Smerzi 2008 Phys. Rev. Lett. 100 073601) via photon subtraction. The gain in sensitivity is also observed when measuring the difference in output intensities of the SMSV state with squeezing of < 3 dB from which two, four, and six photons are subtracted and a large coherent state. Overall, subtracting photons from the initially weakly squeezed light can prove to be a more efficient strategy in quantum MZ interferometry compared to highly squeezed SMSV state generation.
Quantum Cramer-Rao (QCR) bound is attached to a particular nonclassical state, therefore appropriate choice of the probe state is of the key importance to enhance sensitivity beyond classical one. Since the work of C.M. Caves (Phys. Rev. D 23 1693 (1981)) Mach-Zehnder (MZ) interferometry operates with single-mode squeezed vacuum (SMSV) light coupled with a coherent state. We report the gain sensitivity of the phase-dependent MZ interferometer by more than 10 dB compared to the original result (Phys. Rev. Lett. 100, 073601 (2008)) by using SMSV state with squeezing <10 dB from which a certain number of photons was initially subtracted. The gain sensitivity is also observed when measuring the difference of output intensities of the SMSV state with squeezing <3 dB from which 2,4,6 photons are subtracted and large coherent state. Overall, subtracting photons from the initially weakly squeezed light can prove to be a more efficient strategy in the quantum MZ interferometry compared to highly squeezed SMSV state generation.
Abstract The phase sensitivity of a Mach–Zehnder (MZ) interferometer with a two-mode squeezed vacuum (TMSV) probe state is studied. At the initial stage, the TMSV state is deterministically converted into two single-mode squeezed vacuum (SMSV) states, from each of which photons are subtracted via photon-number resolving measurement in auxiliary measurement modes. The new probe state can already demonstrate gain sensitivity of more than 20 dB with input squeezing of 5 dB and follow Heisenberg scaling. The phase sensitivity of the MZ interferometer, estimated by measuring the intensity difference of two measurement-induced continuous variable states, can surpass the ultimate one with SMSV probe states, at least, with squeezing less than 5 dB. In general, the strategy with preliminary subtraction of photons significantly increases the estimate potential of weakly squeezed states as the probe in MZ phase-dependent interferometry; in particular, it is more effective compared to generating highly squeezed TMSV states or SMSV states.
We study sensitivity of phase estimation of Mach-Zehnder (MZ) interferometer with original two-mode squeezed vacuum (TMSV) state. At the initial stage, the TMSV state is converted into two single-mode squeezed vacuum (SMSV) states, from each of which photons are subtracted by measurement by photon-number resolving (PNR) detector in auxiliary modes. New measurement-induced continuous variable (CV) states of a certain parity can already demonstrate gain sensitivity more than 20 dB in relation to the initial SMSV states at the output from the MZ interferometer and follow to Heisenberg scaling in the case of subtracting a large number of photons in the measuring channels for practical values of the SMSV squeezing 5 dB>. Using only one measurement-induced CV state of a certain parity together with the SMSV state shows an increase in sensitivity of no more than 11 dB. We show that the sensitivity of the phase estimation obtained by measuring the intensity difference of two measurement-induced CV states in two arms of the MZ interferometer can surpass quantum Cramer-Rao (QCR) boundary of the original two SMSV states just in the practical range of input squeezing 5 dB>. In general, the strategy with preliminary subtraction of photons from two SMSV enables greatly enhance the sensitivity of the MZ interferometer in the practical case of small values of squeezing.
We investigate the sensitivity of phase estimation in a Mach–Zehnder interferometer with photon-subtracted two-mode squeezed vacuum states. Our results show that, for given initial squeezing parameter, both symmetric and asymmetric photon subtractions can further improve the quantum Cramér–Rao bound (i.e., the ultimate phase sensitivity), especially for single-mode photon subtraction. On the other hand, the quantum Cramér–Rao bound can be reached by parity detection for symmetric photon-subtracted two-mode squeezed vacuum states at particular values of the phase shift, but it is not valid for asymmetric photon-subtracted two-mode squeezed vacuum states. In addition, compared with the two-mode squeezed vacuum state, the phase sensitivity via parity detection with asymmetric photon-subtracted two-mode squeezed vacuum states will be getting worse. Thus, parity detection may not always be the optimal detection scheme for nonclassical states of light when they are considered as the interferometer states.
Abstract We propose a photon subtraction scheme within the Mach–Zehnder interferometer using a coherent state and a squeezed vacuum state as inputs. By systematically comparing intensity detection and homodyne detection methods, we demonstrate that strategically applied photon subtraction ( a -mode subtraction) significantly enhances the phase sensitivity of all detection methods. Under this configuration, homodyne detection achieves superior phase sensitivity and loss robustness compared to intensity detection. Remarkably, under 30% photon loss, the system surpasses the Heisenberg limit, enabled by the synergy among the squeezed vacuum state, non-Gaussian operations, and detection optimization. Furthermore, we quantify performance through the quantum Fisher information and the quantum Cramér-Rao bound analyses under both ideal and lossy conditions.
Based on the conventional Mach-Zehnder interferometer, we propose a metrological scheme to improve phase sensitivity. In this scheme, we use a coherent state and a squeezed vacuum state as input states, employ multi-photon-subtraction operations and make intensity-detection or homodyne-detection. We study phase sensitivity, quantum Fisher information and quantum Cramér-Rao bound under both ideal and lossy conditions. The results indicate that choosing an appropriate detection method and photon subtraction scheme can significantly enhance the phase sensitivity and robustness against photon losses. Even under lossy conditions, the multi-photon subtraction schemes can surpass the standard quantum limit. Notably, the homodyne detection method can even break through the Heisenberg limit. Moreover, increasing the number of photon-subtracted can enhance both phase sensitivity and quantum Fisher information. This research highlights the significant value of this scheme in quantum precision measurement.