The simultaneous two-parameter estimation problem in single squeezed-light Mach-Zehnder interferometer with double-port homodyne detection is investigated in this work. The analytical form of the two-parameter quantum Cramer-Bao bound defined by the quantum Fisher information matrix is presented, which shows the ultimate limit of the phase sensitivity will be further approved by the squeezed vacuum state. It can not only surpass the shot-noise limit, but also can even surpass the Heisenberg limit when half of the input intensity of the interferometer is provided by the coherent state and half by the squeezed light. For the double-port homodyne detection, the classical Fisher information matrix is also obtained. Our results show that although the classical Cramer-Rao bound does not saturate the quantum one, it can still asymptotically approach the quantum Cramer -Bao bound when the intensity of the coherent state is large enough. Our results also indicate that the squeezed vacuum state indeed can further improve the phase sensitivity. In addition, when half of the input intensity of the interferometer is provided by the coherent state and half by the squeezed light, the phase sensitivity obtained by the double-port homodyne detection can surpass the Heisenberg limit for a small range of the estimated phase.
We suggest and demonstrate a scheme to reconstruct the symmetric two-mode squeezed thermal states of spectral sideband modes from an optical parametric oscillator. The method is based on a single homodyne detector and active stabilization of the cavity. The measurement scheme have been successfully tested on different two-mode squeezed thermal states, ranging from uncorrelated coherent states to entangled states.
Conditional homodyne detection of quadrature squeezing is compared with standard nonconditional detection. Whereas the latter identifies nonclassicality in a quantitative way, as a reduction of the noise power below the shot noise level, conditional detection makes a qualitative distinction between vacuum state squeezing and squeezed classical noise. Implications of this comparison for the realistic interpretation of vacuum fluctuations (stochastic electrodynamics) are discussed.
A new scheme is described for pulsed squeezed light generation using femtosecond pulses parametrically deamplified through a single pass in a thin (0.1mm) potassium niobate KNbO3 crystal, with a significant deamplification of about -3dB. The quantum noise of each individual pulse is registered in the time domain using a single-shot homodyne detection operated with femtosecond pulses and the best squeezed quadrature variance was measured to be 1.87 dB below the shot noise level. Such a scheme provides the basic ressource for time-resolved quantum communication protocols.
A simple scheme for conditional generation of nonclassical light with sub-Poissonian photon-number statistics is proposed. The method utilizes entanglement of signal and idler modes in two-mode squeezed vacuum state generated in optical parametric amplifier. A quadrature component of the idler mode is measured in balanced homodyne detector and only those experimental runs where the absolute value of the measured quadrature is higher than certain threshold are accepted. If the threshold is large enough then the conditional output state of signal mode exhibits reduction of photon-number fluctuat…
We present a scheme for quantum randomness certification based on quantum steering. The protocol is one-sided device independent, providing high security, but requires only states and measurements that are simple to realise on quantum optics platforms - entangled squeezed vacuum states and homodyne detection. This ease of implementation is demonstrated by certifying randomness in existing experimental data and implies that giga-hertz random bit rates should be attainable with current technology. Furthermore, the steering-based setting represents the closest to full device independence that can…
We theoretically study the phase sensitivity of the SU(1,1) interferometer with a coherent light together with a squeezed vacuum input case using the method of homodyne. We find that the homodyne detection has better sensitivity than the intensity detection under this input case. For a certain intensity of coherent light (squeezed light) input, the relative phase sensitivity is not always better with increasing the squeezed strength (coherent light strength). The phase sensitivity can reach the Heisenberg limit only under a certain moderate parameter interval, which can be realized with curren…
We study the effect of homodyne detector visibility on the measurement of quadrature squeezing for a spatially multi-mode source of two-mode squeezed light. Sources like optical parametric oscillators (OPO) typically produce squeezing in a single spatial mode because the nonlinear medium is within a mode-selective optical cavity. For such a source, imperfect interference visibility in the homodyne detector couples in additional vacuum noise, which can be accounted for by introducing an equivalent loss term. In a free-space multi-spatial-mode system imperfect homodyne detector visibility can co…
The Cramér-Rao bound plays a central role in both classical and quantum parameter estimation, but finding the observable and the resulting inversion estimator that saturates this bound remains an open issue for general multi-outcome measurements. Here we consider multi-outcome homodyne detection in a coherent-light Mach-Zehnder interferometer and construct a family of inversion estimators that almost saturate the Cramér-Rao bound over the whole range of phase interval. This provides a clue on constructing optimal inversion estimators for phase estimation and other parameter estimation in any m…
We calculate the Holevo Cramér-Rao bound for estimation of the displacement experienced by one mode of an two-mode squeezed vacuum state with squeezing r and find that it is equal to 4exp(-2r). This equals the sum of the mean squared error obtained from a dual homodyne measurement, indicating that the bound is tight and that the dual homodyne measurement is optimal.
We study the sensitivity of phase estimation in a lossy Mach-Zehnder interferometer (MZI) using two general, and practical, resources generated by a laser and a nonlinear optical medium with passive optimal elements, which are readily available in the laboratory: One is a two-mode separable coherent and squeezed vacuum state at a beam splitter and the other is a two-mode squeezed vacuum state. In view of the ultimate precision given by quantum Fisher information, we show that the two-mode squeezed vacuum state can achieve a lower bound of estimation error than the coherent and squeezed vacuum …
We propose a feasible optical setup allowing for a loophole-free Bell test with efficient homodyne detection. A non-gaussian entangled state is generated from a two-mode squeezed vacuum by subtracting a single photon from each mode, using beamsplitters and standard low-efficiency single-photon detectors. A Bell violation exceeding 1% is achievable with 6-dB squeezed light and an homodyne efficiency around 95%. A detailed feasibility analysis, based upon the recent generation of single-mode non-gaussian states, confirms that this method opens a promising avenue towards a complete experimental B…
We propose a procedure for tomographic characterization of continuous variable quantum operations which employs homodyne detection and single-mode squeezed probe states with a fixed degree of squeezing and anti-squeezing and a variable displacement and orientation of squeezing ellipse. Density matrix elements of a quantum process matrix in Fock basis can be estimated by averaging well behaved pattern functions over the homodyne data. We show that this approach can be straightforwardly extended to characterization of quantum measurement devices. The probe states can be mixed, which makes the pr…
We present a multimode theory of non-Gaussian operation induced by an imperfect on/off-type photon detector on a splitted beam from a wideband squeezed light. The events are defined for finite time duration $T$ in the time domain. The non-Gaussian output state is measured by the homodyne detector with finite bandwidh $B$. Under this time- and band-limitation to the quantm states, we develop a formalism to evaluate the frequency mode matching between the on/off trigger channel and the conditional signal beam in the homodyne channel. Our formalism is applied to the CW and pulsed schemes. We expl…
Homodyne detection is often used for interferometers based on nonlinear optical gain media. For the configuration of a seeded, 'truncated SU(1,1)' interferometer Anderson et al. (Phys. Rev. A 95, 063843 (2017)) showed how to optimize the homodyne detection scheme and demonstrated theoretically that it can saturate the quantum Cramer-Rao bound for phase estimation. In this work we extend those results by taking into account loss in the truncated SU(1,1) interferometer and determining the optimized homodyne detection scheme for phase measurement. Further, we build a truncated SU(1,1) interferome…
We experimentally verify the quantum non-Gaussian character of a conditionally generated noisy squeezed single-photon state with positive Wigner function. Employing an optimized witness based on probabilities of squeezed vacuum and squeezed single-photon states we prove that the state cannot be expressed as a mixture of Gaussian states. In our experiment, the non-Gaussian state is generated by conditional subtraction of a single photon from squeezed vacuum state. The state is probed with a homodyne detector and the witness is determined by averaging a suitable pattern function over the measure…
We derive the filtering equation for Markovian systems undergoing homodyne measurement in the situation where the output processes being monitored are squeezed. The filtering theory applies to case where the system is driven by Fock noise (that, quantum input processes in a coherent state) and where the output is mixed with a squeezed signal. It also applies to the case of a system driven by squeezed noise, but here there is a physical restriction to emission/absorption coupling only. For the special case of a cavity mode where the dynamics is linear, we are able to derive explicitly the filte…
We provide a detailed analysis of the recently proposed setup for a loophole-free test of Bell inequality using conditionally generated non-Gaussian states of light and balanced homodyning. In the proposed scheme, a two-mode squeezed vacuum state is de-gaussified by subtracting a single photon from each mode with the use of an unbalanced beam splitter and a standard low-efficiency single-photon detector. We thoroughly discuss the dependence of the achievable Bell violation on the various relevant experimental parameters such as the detector efficiencies, the electronic noise and the mixedness …
Compared to the quantum noise in the measurement of the translational motion of a suspended mirror using laser light, the quantum noise in the measurement of the angular motion of a suspended mirror has not been investigated intensively despite its potential importance. In this article, an expression for the quantum noise in the angular motion measurement is explicitly derived. The expression indicates that one quadrature of the vacuum field of the first-order Hermite-Gaussian mode of light causes quantum sensing noise and the other causes quantum backaction noise, or in other words the first-…
In a recent table-top experiment we demonstrated the compatibility of three advanced interferometer techniques for gravitational wave detection, namely power-recycling, detuned signal-recycling and squeezed field injection. The interferometer's signal to noise ratio was improved by up to 2.8 dB beyond the coherent state's shot-noise. This value was mainly limited by optical losses on the squeezed field. We present a detailed analysis of the optical losses of in our experiment and provide an estimation of the possible nonclassical performance of a future squeezed field enhanced GEO600 detector.
We derive sampling functions for estimation of quantum state fidelity with Schrödinger cat-like states, which are defined as superpositions of two coherent states with opposite amplitudes. We also provide sampling functions for fidelity with squeezed Fock states that can approximate the cat-like states and can be generated from Gaussian squeezed states by conditional photon subtraction. The fidelities can be determined by averaging the sampling functions over quadrature statistics measured by homodyne detection. The sampling functions are designed such that they can compensate for losses and i…
A high-sensitive interferometric scheme is presented. It is based on homodyne detection and squeezed vacuum phase properties. The resulting phase sensitivity scales as $δϕ\simeq {1/4} \bar{n}^{-1}$ with respect to input photons number.
A combined ferrite parametric amplifier and millimeter-wave homodyne interferometer are proposed as an ambient temperature Bell Test. It is shown that the non-linear magnetic susceptibility of the yttrium iron garnet (YIG) ferrite, on account of its narrow line-width Larmor precessional resonance, make it an ideal material for the creation of entangled photons. These can be measured using a homodyne interferometer, as the much larger number of thermally generated photons associated with ambient temperature emission can be screened out. The proposed architecture may enable YIG quantum technolog…
We report an experimental demonstration of optimal measurements of small displacement and tilt of a Gaussian beam - two conjugate variables - involving a homodyne detection with a TEM10 local oscillator. We verify that the standard split detection is only 64% efficient. We also show a displacement measurement beyond the quantum noise limit, using a squeezed vacuum TEM10 mode within the input beam.
We assess the security of a quantum key distribution protocol relying on the transmission of Gaussian-modulated coherent states and homodyne detection. This protocol is shown to be equivalent to a squeezed state protocol based on a CSS code construction, and is thus provably secure against any eavesdropping strategy. We also briefly show how this protocol can be generalized in order to improve the net key rate.
We introduce a formalism of nonlinear canonical transformations for general systems of multiphoton quantum optics. For single-mode systems the transformations depend on a tunable free parameter, the homodyne local oscillator angle; for n-mode systems they depend on n heterodyne mixing angles. The canonical formalism realizes nontrivial mixings of pairs of conjugate quadratures of the electromagnetic field in terms of homodyne variables for single-mode systems; and in terms of heterodyne variables for multimode systems. In the first instance the transformations yield nonquadratic model Hamilton…
We present a scheme for the amplification of Schrodinger cats that collapses two smaller states onto their constructive interference via a homodyne projection. We analyze the performance of the amplification in terms of fidelity and success rate when the input consists of either exact coherent state superpositions or of photon-subtracted squeezed vacua. The impact of imprecise homodyne detection and of impure squeezing is quantified. We also assess the scalability of iterated amplifications.
According to quantum theory the interactions between physical systems are quantized. As a direct consequence, measurement sensitivities are fundamentally limited by quantization noise, or just `quantum noise' in short. Furthermore, Heisenberg's Uncertainty Principle demands measurement back-action for some observables of a system if they are measured repeatedly. In both respects, squeezed states are of high interest since they show a `squeezed' uncertainty, which can be used to improve the sensitivity of measurement devices beyond the usual quantum noise limits including those impacted by quan…
We address the experimental estimation of Gaussian quantum discord for two-mode squeezed thermal state, and demonstrate a measurement scheme based on a pair of homodyne detectors assisted by Bayesian analysis which provides nearly optimal estimation for small value of discord. Besides, though homodyne detection is not optimal for Gaussian discord, the noise ratio to the ultimate quantum limit, as dictacted by the quantum Cramer-Rao bound, is limited to about 10 dB.
We suggest an interferometric scheme assisted by squeezing and linear feedback to realize the whole class of field-quadrature quantum nondemolition measurements, from Von Neumann projective measurement to fully non-destructive non-informative one. In our setup, the signal under investigation is mixed with a squeezed probe in an interferometer and, at the output, one of the two modes is revealed through homodyne detection. The second beam is then amplitude-modulated according to the outcome of the measurement, and finally squeezed according to the transmittivity of the interferometer. Using str…
We report on the experimental combination of three advanced interferometer techniques for gravitational wave detection, namely power-recycling, detuned signal-recycling and squeezed field injection. For the first time we experimentally prove the compatibility of especially the latter two. To achieve a broadband non-classical sensitivity improvement we applied a filter cavity for compensation of quadrature rotation. Signal to noise ratio was improved by up to 2.8 dB beyond the coherent state's shot noise. The complete set-up was stably locked for arbitrary times and characterized by injected si…
Two mode squeezed states can be used to achieve Heisenberg limit scaling in interferometry: a phase shift of $δϕ\approx 2.76 / < N >$ can be resolved. The proposed scheme relies on balanced homodyne detection and can be implemented with current technology. The most important experimental imperfections are studied and their impact quantified.
A Michelson-type interferometer with two-mode squeezed coherent state input is considered. Such an interferometer has a better phase sensitivity over the shot-noise limit by a factor of $e^{2r}$, where $r$ is the squeezing parameter [Phys. Rev. A 102,022614 (2020)]. We show that when photon loss and noise in the two arms is asymmetric an optimal choice of the squeezing angle can allow improvement in phase sensitivity without any increase in input or pump power. In particular, when loss occurs only in one arm of the interferometer, we can have improvement in phase sensitivity for photon loss up…
We present the observation of optical fields carrying squeezed vacuum states at sideband frequencies from 10Hz to above 35MHz. The field was generated with type-I optical parametric oscillation below threshold at 1064nm. A coherent, unbalanced classical modulation field at 40MHz enabled the generation of error signals for stable phase control of the squeezed vacuum field with respect to a strong local oscillator. Broadband squeezing of approximately -4dB was measured with balanced homodyne detection. The spectrum of the squeezed field allows a quantum noise reduction of ground-based gravitatio…
A method is introduced which allows to measure normal-ordered moments of the displaced photon-number operator up to high orders. It is based on unbalanced homodyne correlation measurements, the local oscillator being replaced by a displaced dephased laser. The measured moments yield a simple approximation of the full quantum state, which is obtained without the need of photon-number resolution. Quantum properties of light are efficiently certified through the direct detection of normal-ordered observables, which is illustrated for a weakly squeezed vacuum and a single-photon-added thermal stat…
The eight-port homodyne detection apparatus is analyzed in the framework of the operational theory of quantum measurement. For an arbitrary quantum noise leaking through the unused port of the beam splitter, the positive operator valued measure and the corresponding operational homodyne observables are derived. It is shown that such an eight-port homodyne device can be used to construct the operational quantum trigonometry of an optical field. The quantum trigonometry and the corresponding phase space Wigner functions are derived for a signal field probed by a classical local oscillator and a …
Control over the spectral properties of the bright squeezed vacuum (BSV), a highly multimode non-classical macroscopic state of light that can be generated through high-gain parametric down conversion, is crucial for many applications. In particular, in several recent experiments BSV is generated in a strongly pumped SU(1,1) interferometer to achieve phase supersensitivity, perform broadband homodyne detection, or tailor the frequency spectrum of squeezed light. In this work, we present an analytical approach to the theoretical description of BSV in the frequency domain based on the Bloch-Mess…
We introduce a new family of quantum circuits in Continuous Variables and we show that, relying on the widely accepted conjecture that the polynomial hierarchy of complexity classes does not collapse, their output probability distribution cannot be efficiently simulated by a classical computer. These circuits are composed of input photon-subtracted (or photon-added) squeezed states, passive linear optics evolution, and eight-port homodyne detection. We address the proof of hardness for the exact probability distribution of these quantum circuits by exploiting mappings onto different architectu…
We report on the successful implementation of a new approach to locking the frequencies of an OPO-based squeezed-vacuum source and its driving laser. The technique allows the simultaneous measurement of the phase-shifts induced by a cavity, which may be used for the purposes of frequency-locking, as well as the simultaneous measurement of the sub-quantum-noise-limited (sub-QNL) phase quadrature output of the OPO. The homodyne locking technique is cheap, easy to implement and has the distinct advantage that subsequent homodyne measurements are automatically phase-locked. The homodyne locking te…
We report on the generation of continuous-wave squeezed vacuum states of light at the telecommunication wavelength of 1550 nm. The squeezed vacuum states were produced by type I optical parametric amplification (OPA) in a standing-wave cavity built around a periodically poled potassium titanyl phosphate (PPKTP) crystal. A non-classical noise reduction of 5.3 dB below the shot noise was observed by means of balanced homodyne detection.
We derive various sampling functions for multimode homodyne tomography with a single local oscillator. These functions allow us to sample multimode s-parametrized quasidistributions, density matrix elements in Fock basis, and s-ordered moments of arbitrary order directly from the measured quadrature statistics. The inevitable experimental losses can be compensated by proper modification of the sampling functions. Results of Monte Carlo simulations for squeezed three-mode state are reported and the feasibility of reconstruction of the three-mode Q-function and s-ordered moments from 10^7 sample…
We address phase-shift estimation by means of squeezed vacuum probe and homodyne detection. We analyze Bayesian estimator, which is known to asymptotically saturate the classical Cramer-Rao bound to the variance, and discuss convergence looking at the a posteriori distribution as the number of measurements increases. We also suggest two feasible adaptive methods, acting on the squeezing parameter and/or the homodyne local oscillator phase, which allow to optimize homodyne detection and approach the ultimate bound to precision imposed by the quantum Cramer-Rao theorem. The performances of our t…
We design and demonstrate on-chip homodyne detection operating in the quantum regime, i.e. able to detect genuine nonclassical features. Our setup exploits a glass-integrated homodyne analyzer (IHA) entirely fabricated by femtosecond laser micromachining. The IHA incorporates on the same chip a balanced waveguide beam splitter and a thermo-optic phase shifter, allowing us to record homodyne traces at different phases and to perform reliable quantum state tomography. In particular, we show that the IHA allows for the detection of nonclassical features of continuous-variable quantum states, such…
Over the years, quite a few proposals have been put forward by various groups to exploit the Stern-Gerlach effect to create stable macroscopic spatial superpositions between micron-sized neutral test masses over appreciably long time scales. One such proposal put forward by Bose et al. and co-workers in 2017 uses this idea to show that two masses cannot be gravitationally entangled if not for the presence of a quantum coherent mediator. A key aspect of this approach involves the measure of the visibility of the SG-interferometer, a quantity that provides an estimate of the degree of spin coher…
A probe light in a squeezed vacuum state was injected into cold 87 $Rb atoms with an intense control light in a coherent state. A sub-MHz window was created due to electromagnetically induced transparency, and the incident squeezed vacuum could pass through the cold atoms without optical loss, as was successfully monitored using a time-domain homodyne method.
Direct photon detection is experimentally implemented to measure the squeezing and purity of a single-mode squeezed vacuum state without an interferometric homodyne detection. Following a recent theoretical proposal [arXiv quant-ph/0311119], the setup only requires a tunable beamsplitter and a single-photon detector to fully characterize the generated Gaussian states. The experimental implementation of this procedure is discussed and compared with other reference methods.
We present a Gaussian state description of squeezed light generated in an optical parametric oscillator. Using the Gaussian state description we describe the dynamics of the system conditioned on homodyne detection on the output field. Our theory shows that the output field is squeezed only if observed for long enough times or by a detector with finite bandwidth. As an application of the present approach we consider the use of finite bandwidth squeezed light together with a sample of spin-polarized atoms to estimate a magnetic field.
We have succeeded in observing ultraslow propagation of squeezed vacuum pulses with electromagnetically induced transparency. Squeezed vacuum pulses (probe lights) were incident on a laser cooled 87Rb gas together with an intense coherent light (control light). A homodyne method sensitive to the vacuum state was employed for detecting the probe pulse passing through the gas. A delay of 3.1us was observed for the probe pulse having a temporal width of 10 us.
We propose a method of an improving quality of a ring cavity which is imperfect due to non-unit mirror reflectivity. The method is based on using squeezed states of light pulses illuminating the mirror and gradual homodyne detection of a radiation escaping from the cavity followed by single displacement and single squeezing operation performed on the released state. We discuss contribution of this method in process of storing unknown coherent and known squeezed state and generation of squeezing in the optical ring cavities.
We present the first experimental observation of quantum fluctuation spectra in two coupled optical cavities with an injected squeezed vacuum light. The quadrature components of the reflected squeezed vacuum spectra are measured by phase sensitive homodyne detector. The experimental results demonstrate coupled-resonator-induced transparency in the quantum regime, in which electromagnetically-induced-transparency-like characteristic of the absorption and dispersion properties of the coupled optical cavities determines the line-shape of the reflected quantum noise spectra.
The two modes of the Einstein-Podolsky-Rosen quadrature entangled state generated by parametric down-conversion interfere on a beam splitter of variable splitting ratio. Detection of a photon in one of the beam splitter output channels heralds preparation of a signal state in the other, which is characterized using homodyne tomography. By controlling the beam splitting ratio, the signal state can be chosen anywhere between the single-photon and squeezed state.
Photon counting measurement has been regarded as the optimal measurement scheme for phase estimation in the squeezed-state interferometry, since the classical Fisher information equals to the quantum Fisher information and scales as $\bar{n}^2$ for given input number of photons $\bar{n}$. However, it requires photon-number-resolving detectors with a large enough resolution threshold. Here we show that a collection of $N$-photon detection events for $N$ up to the resolution threshold $\sim \bar{n}$ can result in the ultimate estimation precision beyond the shot-noise limit. An analytical formul…
We show a protocol achieving the ultimate Heisenberg-scaling sensitivity in the estimation of a parameter encoded in a generic linear network, without employing any auxiliary networks, and without the need of any prior information on the parameter nor on the network structure. As a result, this protocol does not require a prior coarse estimation of the parameter, nor an adaptation of the network. The scheme we analyse consists of a single-mode squeezed state and homodyne detectors in each of the $M$ output channels of the network encoding the parameter, making it feasible for experimental appl…
The LIGO-II gravitational-wave interferometers (ca. 2006--2008) are designed to have sensitivities at about the standard quantum limit (SQL) near 100 Hz. This paper describes and analyzes possible designs for subsequent, LIGO-III interferometers that can beat the SQL. These designs are identical to a conventional broad-band interferometer (without signal recycling), except for new input and/or output optics. Three designs are analyzed: (i) a "squeezed-input interferometer" (conceived by Unruh based on earlier work of Caves) in which squeezed vacuum with frequency-dependent (FD) squeeze angle i…
Iterative $C_Z$-gate-based protocol for squeezed Schrödinger cat state engineering
ITMO University（ITMO大学）；SMARTS-Quanttelecom LLC（SMARTS-Quanttelecom有限责任公司）；Saint-Petersburg State Institute of Technology（圣彼得堡国立技术学院）；Research and Education Center of Photonics and Optical IT, ITMO University（ITMO大学光子学与光学信息技术研究与教育中心）
A method for measuring the transmittivity of optical samples by using squeezed--vacuum radiation is illustrated. A squeezed vacuum field generated by a below--threshold optical parametric oscillator is propagated through a nondispersive medium and detected by a homodyne apparatus. The variance of the detected quadrature is used for measuring the transmittivity. With this method it is drastically reduced the number of photons passing through the sample during the measurement interval. The results of some tests are reported.
