The impact of parametric amplifiers on the phase sensitivity for an unbalanced interferometer

Improving interferometric phase sensitivity is crucial for high-precision measurements in rapidly developing quantum technologies. The Mach–Zehnder interferometer (MZI) is a versatile tool for analyzing this phenomenon. By splitting and recombining a light beam using beam splitters, the MZI allows for precise phase sensitivity analysis using tools like the quantum Cramér–Rao bound (QCRB) and the quantum Fisher information (QFI). This paper analyzes the phase sensitivity of an MZI in various scenarios using different detection schemes and input states. We compare the single- and two-parameter quantum estimation and their associated QCRB for three phase-shift situations: in both arms; only in the upper arm (asymmetric); and in both arms symmetrically. We then investigate the phase sensitivity under three detection schemes: intensity difference; single-mode intensity; and balanced homodyne. Additionally, we explore the use of Perelomov and Barut–Girardello coherent states, two types of SU(1,1) coherent states, in all scenarios. Notably, we demonstrate that, under optimal conditions, all detection schemes can achieve the QCRB by utilizing SU(1,1) coherent states as input states.

Phase estimation has a wide range of applications. Over the years, several strategies have been studied to improve precision in phase estimation. These strategies include using exotic quantum states to quantum detection schemes. This dissertation summarizes my effort in improving the precision of phase estimation with a linear and nonlinear interferometer. Chapter 1 introduces quantum optics and quantum metrology. I introduce all relevant quantum states of light used. We also look into tools and terminologies of quantum metrology such as Fisher information, shot-noise limit, Heisenberg limit, etc., along with examples of phase estimation with a Mach-Zehnder interferometer. In Chapter 2, I discuss multiple phase estimation using a multimode interferometer. Building upon previous work, our scheme consists of a multimode interferometer with single-photon inputs. By using a quantum Fisher information analysis, we show that our scheme gives a constant improvement over other schemes. We also show that our scheme with photon-number-resolving detection approaches the quantum Cram\'er-Rao bound. Moreover, we also consider the probabilistic nature of photon emission at the input, and we study its effect on phase sensitivity. I discuss phase estimation with SU(1,1) interferometer in Chapter 3. We look at phase sensitivity in this interferometer with different input states. Namely, we consider two different phase estimation scheme, one using thermal and squeezed states, and others using coherent and displaced squeezed states with parity and on-off as a detection scheme. We also look into the effect of photon loss inside the interferometer. In Chapter 4, we revisit phase estimation in SU(1,1) interferometer from the perspective of quantum Fisher information. I discuss in detail a longstanding confusion regarding the use of quantum Fisher information in SU(1,1) interferometer. We show that phase averaging or quantum Fisher information matrix method is needed in general for calculating the phase sensitivity which resolves inconsistencies reported in previously published articles.

In this work we address the problem of phase sensitivity optimization for an unbalanced Mach-Zehnder interferometer. While the quantum Fisher information can be employed in order to obtain the optimum transmission coefficient for the first beam splitter, this is no longer true for the second one, whose optimization is detection-scheme dependent. We thus consider three commonly used detection schemes and provide the optimal solution for each case. We also provide applications of the optimization method showing that for some input states a non-balanced Mach-Zehnder interferometer can outperform its balanced counterpart in terms of phase sensitivity.

Quantum metrology is to estimate accurately the value of an unknown parameter with the assistance of the quantum effects, in order to break through the standard quantum limit, even reach the Heisenberg limit. In this work, we study the performance of a general photon-added two-mode squeezed vacuum state that is taken as a detection state of a Mach-Zehnder interferometer. Based on quantum Fisher information, within the constraint on the total mean photon number, symmetric and asymmetric photon addition cannot improve the ultimate phase sensitivity. However, for a given initial squeezing parameter, on this occasion, the symmetric and asymmetric photon addition can improve the ultimate phase sensitivity. Compared with the asymmetric photon-added two-mode squeezed vacuum state, the symmetric one can well improve the ultimate phase sensitivity. This may be because it is always better to implement the symmetric photon addition rather than the asymmetric one in order to increase the mean photon number of the resulting state. On the other hand, via parity detection, the symmetric and asymmetric photon-added two-mode squeezed vacuum state can indeed improve the phase sensitivity of a Mach-Zehnder interferometer for a given initial squeezing parameter. Based on the parity detection, within a constraint on the mean photon number, although the two-mode squeezed vacuum state can give the better phase sensitivity at the optimal phase shift (<i>φ</i> = 0), the phase sensitivity offered by the symmetric and asymmetric photon-added two-mode squeezed vacuum states are both more stable around <i>φ</i> = 0 than by the two-mode squeezed vacuum state. In addition, we show that for the symmetric photon-added two-mode squeezed vacuum state, parity detection is an optimal detection only when the optimal phase shift approaches to zero. When the phase shift slightly deviates from zero, the parity detection is not an optimal detection scheme. Finally, for all values of the phase shift, our results also clearly show that the parity detection is not an optimal detection scheme for the asymmetric photon-added two-mode squeezed vacuum state serving as an interferometer state.

In this paper we address the problem of optimizing an unbalanced Mach-Zehnder interferometer, for a given pure input state and considering a specific detection scheme. While the optimum transmission coefficient of the first beam splitter can be uniquely determined via the quantum Fisher information only [Ataman, Phys. Rev. A 105, 012604 (2022)], the second beam-splitter transmission coefficient is detection-scheme dependent, too. We systematically give analytic solutions for the optimum transmission coefficient of the second beam splitter for three types of widely used detection schemes. We provide detailed examples including both Gaussian and non-Gaussian input states, showing when an unbalanced Mach-Zehnder interferometer can outperform its balanced counterpart in terms of phase sensitivity.

We study the phase super–sensitivity of a Mach–Zehnder interferometer (MZI) with the squeezed Kerr state (SKS) and coherent state as the inputs. We discuss the lower bound in phase sensitivity by considering the quantum Fisher information and the corresponding quantum Cramér–Rao bound. With the help of single intensity detection, intensity difference detection, and homodyne detection schemes, we find that our scheme gives a better sensitivity under both lossless and lossy conditions as compared to the well–known results of the combinations of inputs, such as coherent plus vacuum, coherent plus squeezed vacuum, and double coherent states. Because of the possibility of the generation of SKS with the present available quantum optical techniques, we expect that SKS may be an alternative nonclassical resource for the improvement in the phase super–sensitivity of the MZI in realistic scenarios.

Quantum phase estimation and realistic detection schemes in Mach-Zehnder interferometer using SU(2) coherent states

The Cram\'er-Rao bound and the quantum Fisher information have been tools used extensively for interferometric phase sensitivity. Most scenarios considering a Mach-Zehnder interferometer with two input sources focused on the phase-matched case, when the Fisher information is maximal. Under this constraint, the best sensitivity is achieved for a balanced (50/50) input beam splitter. In this paper, we take a different approach: we allow the beam splitter transmission coefficient as well as the input phase mismatch to be variable parameters. We then search for a pair of these parameters that maximizes the Fisher information. We find that for the double coherent input the maximum Fisher information can always be reached in the unbalanced case for a carefully chosen input phase mismatch. For the coherent plus squeezed vacuum case we find that under certain circumstances a threshold phase mismatch exists, beyond which the optimum Fisher information is found for the degenerate case. For the squeezed-coherent plus squeezed vacuum case we find that the optimum actually occurs when the squeezing angles of the two inputs are in anti-phase.

Quantum Fisher information (QFI) may exhibit the irregular behavior at the critical point of phase transitions of a physical system and be very sensitive to slight variations of some controlling parameters. This parameter sensitivity may be used for quantum parameter estimation or quantum sensing. In this study, taking the quantum Rabi model as an example, we investigate the critical properties of the QFI for the parameter estimation at the critical point of the SU(1,1) dynamic systems. We show that the QFI goes divergently in the sixth power law $({T}^{6})$ of the parameter coding time around the critical point. After taking into the consumption of energy during the dynamic evolution, we find that the variation of the QFI around the critical point is scaled by the Heisenberg scaling ${T}^{2}$. It is noticed that for nonclassical initial probe states the scaling of QFI can beat the standard quantum limit $({n}_{0})$ as a function of the initial mean phonon number ${n}_{0}$. The homodyne and phonon-number measurement schemes are compared. We find that the quantum Cram\'er-Rao bound can be reached by use of the phonon-number detection scheme. However, it is more sensitive to the noise than the homodyne detection scheme. We extend the investigation to a two-mode non-Hermitian system and show that the QFI exhibits the same irregular properties at the exceptional point, revealing that for the SU(1,1) dynamic systems the QFI universally diverges as ${T}^{6}$ at the critical point.

Optimal phase measurements in a lossy Mach–Zehnder interferometer with coherent input light

We review various schemes of quantum-enhanced optical interferometers, both linear (SU(2)) and non-linear (SU(1,1)) ones, as well as hybrid SU(2)/SU(1,1) options, using the unified modular approach based on the Quantum Cramèr–Rao bound (QCRB), and taking into account the practical limitations pertinent to all real-world highly-sensitive interferometers. We focus on three important cases defined by the interferometer symmetry: (i) the asymmetric single-arm interferometer; (ii) the symmetric two-arm interferometer with the antisymmetric phase shifts in the arms; and (iii) the symmetric two-arm interferometer with the symmetric phase shifts in the arms. We show that while the optimal regimes for these cases differ significantly, their QCRBs asymptotically correspond to the same squeezing-enhanced shot noise limit (2), which first appeared in the pioneering work by C. Caves in 1981.We show also that in all considered cases the QCRB can be asymptotically saturated by the standard (direct or homodyne) detection schemes.

The effect of temporal synchronization between the chirped signal pulse and the pumping pulse in an optical parametric chirped pulse amplification laser system is researched theoretically and experimentally. The results show that the gain of optical parametric amplification is sensitive to the temporal synchronization. Therefore, accurate temporal synchronization between the chirped signal pulse and the pumping pulse is essential to obtain high optical parametric amplification gain and stable output from an optical parametric chirped pulse amplification laser. Based on our 16.7-TW/120-fs optical parametric chirped pulse amplification laser system with similar to1-ns pumping pulse duration and <10-ps time jitter between the signal and pumping pulse, the effect of the temporal synchronization on optical parametric chirped pulse amplification is demonstrated. The experimental results agree with the calculation. (C) 2004 Society of Photo-Optical Instrumentation Engineers.

q-deformed photon states arise from generalized oscillator algebras in which the standard bosonic commutation relations are modified by a deformation parameter q. They effectively model realistic nonideal photon statistics, including deviations such as sub- and super-Poissonian distributions. We investigate quantum phase estimation in a Mach–Zehnder interferometer using q-deformed photon states, including q-coherent and q-cat states, which model realistic deviations from ideal light sources. By deriving closed-form photon count likelihoods via the Jordan–Schwinger mapping, we compute the quantum and classical Fisher information and perform Bayesian inference on simulated detector data. Our results show that photon counting remains an optimal measurement strategy even for deformed states, with classical and quantum Fisher information in exact agreement. Furthermore, the phase sensitivity improves with increasing q-deformation, indicating enhanced metrological performance driven by nonclassical photon statistics. These findings highlight the utility of q-deformed states in quantum sensing.

Photon addition operations applied to squeezed states have been shown to significantly enhance phase sensitivity. In this work, we extend this approach by applying photon addition not only to coherent states but also within a Mach–Zehnder interferometer, using coherent and squeezed vacuum states as input. Both intensity‐difference and homodyne detection are used to evaluate photon addition schemes, and their phase sensitivities are compared under ideal and lossy conditions. We also analyze the quantum Fisher information of these two schemes. Results show that both schemes improve phase sensitivity, quantum Fisher information, and robustness against losses. In particular, photon addition within the interferometer performs better. Homodyne detection outperforms intensity difference detection under photon losses. Notably, the two schemes exhibit distinct parameter dependencies, rendering them suitable for different operational scenarios. When the squeezing parameter is small, photon addition employed at the coherent input with intensity difference detection can approach the Heisenberg limit in ideal conditions and can exceed the standard quantum limit in high‐loss conditions. The proposed scheme represents a valuable method for quantum precision measurements. PACS : 03.67.‐a, 05.30.‐d, 42.50,Dv, 03.65.Wj

Summary form only given. The coherent synthesis of custom-tailored, intense, sub-cycle optical waveforms is promising for attosecond science and strong-field physics, e.g., for precision control of strong-field interactions in atoms, molecules and solids, for the generation of intense isolated attosecond pulses, and for attosecond pump-probe spectroscopy.Recently, coherent pulse synthesis based on supercontinuum generation in a hollow-core fiber compressor allowed the generation of sub-cycle -300-μJ optical pulses. However, in this case ionization losses in the gas medium prevent further scaling to the mJ level. In contrast, parametric synthesizers do not face an energy scaling limit, and allow for spectral extension into the particularly appealing MIR region. In previous works we demonstrated coherent pulse synthesis of two optical parametric chirped-pulse amplifiers and of two optical parametric amplifiers (OPAs) on the few-μJ level [3]. Here we present the ongoing development of a novel 3-channel parametric synthesizer for generating a 2-octave-wide spectrum, which can easily be upscaled to the mJ-level. We start from a cryogenically cooled Ti:sapphire chirped-pulse amplifier (150 fs, 22 mJ, 0.8 μm, 1 kHz) and generate a CEP-stable continuum (0.5-2.3 μm) [5], by white-light generation in a YAG crystal pumped by the second harmonic (1.06 μm) of the CEP-stable idler of a NIR OPA. The continuum is split with custom-designed dichroic beam splitters (which will also be used for the final beam recombination) and seeds three OPAs, a VIS noncollinear OPA (NOPA), a NIR and an IR degenerate OPA (DOPA), each composed of 2 (later 3) amplification stages, and pumped by the pulses at 0.8 μm (IR DOPA), and by its second harmonic at 0.4 μm (VIS NOPA, NIR DOPA). To synthesize a coherent ultrashort pulse from these three OPAs, the relative timing of the pulses will be tightly locked using feedback loops with balanced optical cross-correlators, that can achieve sub-cycle synchronization with s0-as RMS timing jitter [2,3]. Figure 1(a) shows the measured output spectra and the energies from the second amplification stages operating in parallel. The transform-limited (TL) pulse duration from the synthesis of these spectra is 1.9 fs FWHM, corresponding to 0.7 optical cycles at 785 nm center wavelength. Our current work aims to scale up the energies (by a third amplification stage) to -0.5 mJ for the VIS NOPA/NIR DOPA and -2 mJ for the IR DOPA using the 18.5 mJ of pump remaining after the first two amplification stages. State-of-the-art double-chirped mirror pairs (at present in fabrication) will allow for the final pulse compression with ultralow residual ripple in the resulting total group-delay dispersion over the full bandwidth (0.52-2.3μm). Temporal characterization of the synthesized two-octave-spanning optical waveforms will be performed by two-dimensional spectral shearing interferometry (2DSI). We foresee that our 3-channel synthesizer, when further scaled in energy to the mJ level, will become a versatile tool for controlling strong-field interactions in atoms, molecules and solids and for attosecond pumpprobe spectroscopy employing ultrashort pulses in the VIS/IR and XUV/soft-X-ray regions.