NOON States Research Articles

Experimental quantum-enhanced estimation of a lossy phase shift

When standard light sources are employed, the precision of the phase determination is limited by the shot noise. Quantum entanglement provides means to exceed this limit with the celebrated example of N00N states that saturate the ultimate Heisenberg limit on precision, but at the same time are extremely fragile to losses. In contrast, we provide experimental evidence that appropriately engineered quantum states outperform both standard and N00N states in the precision of phase estimation when losses are present.

Multi-pass classical vs. quantum strategies in lossy phase estimation

The use of classical multiple-pass approach for phase estimation which mimics the behavior of the N00N states, is compared with quantum techniques. It is shown that in the presence of losses its performance is significantly worse than the one of the optimal quantum strategy.

NOON states from cavity-enhanced down-conversion: high quality and super-resolution

Indistinguishable photons play a key role in quantum optical information technologies. We characterize the output of an ultra-bright photon-pair source using multi-particle tomography [R. B. A. Adamson et al., Phys. Rev. Lett. 98, 043601 (2007)] and separately identify coherent errors, decoherence, and distinguishability. We demonstrate generation of high-quality indistinguishable pairs and polarization NOON states with 99% fidelity to an ideal NOON state. Using a NOON state we perform a super-resolving angular measurement with 90% visibility.

Efficient creation of maximally entangled states by modulation of tunneling rates

For systems described by the two-site Bose-Hubbard Hamiltonian, we show that a sinusoidal modulation of the tunneling matrix element assists higher-order co-tunneling processes. Using this mechanism, we propose an efficient new scheme for creating a coherent superposition of states in which all particles are either on one site or all on the other site, the so-called NOON state. This scheme yields an almost perfect NOON state periodically. For larger numbers of particles, further reduction of the time to create the state is possible if more than one modulation frequency is employed. With this scheme, NOON states with a larger number of particles could be realized with state-of-the-art techniques for cold Bose gases in a double-well potential.

Heralded Generation of an Atomic NOON State

We report the heralded generation of an atomic NOON state by observation of phase super resolution in a motion-sensitive spin-wave (SW) interferometer. The SW interferometer is implemented by generating a superposition of two SWs and observing the interference between them, where the interference fringe is sensitive to the atomic collective motion. By heralded generation of a second order NOON state in the SW interferometer, we observe the interference pattern which provides strong evidence of phase super resolution. The demonstrated SW interferometer can in principle be scaled up to a highly entangled state, and thus is of fundamental importance, and might be used as an inertial sensor.

Optimizing the multiphoton absorption properties of maximally path-entangled number states

In this paper we examine the N-photon absorption properties of maximally path-entangled number states (N00N states). We consider two cases. The first involves the N-photon absorption properties of the ideal N00N state, one that does not include spectral information. We study how the N-photon absorption probability of this state scales with N, confirming results presented by others in a previous paper by a different method. We compare this to the absorption probability of various other states. The second case is that of two-photon absorption for an N=2 N00N state generated from a type-II spontaneous down-conversion event. In this situation we find that the absorption probability is both better than analogous coherent light (due to frequency entanglement) and highly dependent on the optical setup. We show that the poor production rates of quantum states of light may be partially mitigated by adjusting the spectral parameters to improve their two-photon absorption rates. This work has application to quantum imaging, particularly quantum lithography, where the N-photon absorbing process in the lithographic resist must be optimized for practical applications.

An engineering two-mode field NOON state in cavity QED

We generate highly non-classical entangled two-mode field states of the type by utilizing an atomic analogue of the Mach–Zehnder interferometer, where quantized fields in the high-Q cavities act as beam splitters and mirrors. We discuss that the probability for the production of the desired states may approach a value close to unity under presently available experimental conditions.

Optimization of quantum interferometric metrological sensors in the presence of photon loss

We optimize two-mode, entangled, number states of light in the presence of loss in order to maximize the extraction of the available phase information in an interferometer. Our approach optimizes over the entire available input Hilbert space with no constraints, other than fixed total initial photon number. We optimize to maximize the Fisher information, which is equivalent to minimizing the phase uncertainty. We find that in the limit of zero loss the optimal state is the so-called N00N state, for small loss, the optimal state gradually deviates from the N00N state, and in the limit of large loss the optimal state converges to a generalized two-mode coherent state, with a finite total number of photons. The results provide a general protocol for optimizing the performance of a quantum optical interferometer in the presence of photon loss, with applications to quantum imaging, metrology, sensing, and information processing.

How to perform the most accurate possible phase measurements

We present the theory of how to achieve phase measurements with the minimum possible variance in ways that are readily implementable with current experimental techniques. Measurements whose statistics have high-frequency fringes, such as those obtained from maximally path-entangled $(|N,0⟩+|0,N⟩)/\sqrt{2}$ (``NOON'') states, have commensurately high information yield (as quantified by the Fisher information). However, this information is also highly ambiguous because it does not distinguish between phases at the same point on different fringes. We provide schemes to eliminate this phase ambiguity in a highly efficient way, providing phase estimates with uncertainty that is within a small constant factor of the Heisenberg limit, the minimum allowed by the laws of quantum mechanics. These techniques apply to NOON state and multipass interferometry, as well as phase measurements in quantum computing. We have reported the experimental implementation of some of these schemes with multipass interferometry elsewhere. Here, we present the theoretical foundation and also present some additional experimental results. There are three key innovations to the theory in this paper. First, we examine the intrinsic phase properties of the sequence of states (in multiple time modes) via the equivalent two-mode state. Second, we identify the key feature of the equivalent state that enables the optimal scaling of the intrinsic phase uncertainty to be obtained. This enables us to identify appropriate combinations of states to use. The remaining difficulty is that the ideal phase measurements to achieve this intrinsic phase uncertainty are often not physically realizable. The third innovation is to solve this problem by using realizable measurements that closely approximate the optimal measurements, enabling the optimal scaling to be preserved. We consider both adaptive and nonadaptive measurement schemes.

Quantum-circuit guide to optical and atomic interferometry

Atomic (qubit) and optical or microwave (modal) phase-estimation protocols are placed on the same footing in terms of quantum-circuit diagrams. Circuit equivalences are used to demonstrate the equivalence of protocols that achieve the Heisenberg limit by employing entangled superpositions of Fock states, such as N00N states. The key equivalences are those that disentangle a circuit so that phase information is written exclusively on a mode or modes or on a qubit. The Fock-state-superposition phase-estimation circuits are converted to use entangled coherent-state superpositions; the resulting protocols are more amenable to realization in the lab, particularly in a qubit/cavity setting at microwave frequencies.

Three-photon N00N states generated by photon subtraction from double photon pairs

We describe an experimental demonstration of a novel three-photon N00N state generation scheme using a single source of photons based on spontaneous parametric down-conversion (SPDC). The three-photon entangled state is generated when a photon is subtracted from a double pair of photons and detected by a heralding counter. Interference fringes measured with an emulated three-photon detector reveal the three-photon de Broglie wavelength and exhibit visibility > 70% without background subtraction.

Effective Two-State Model and NOON States for Double-Well Bose-Einstein Condensates in Strong-Interaction Regime

The model of double-well Bose–Einstein condensates in the strong-interaction regime is shown to reduce adiabatically to an effective two-state model describing the Rabi oscillations between the two atomic Fock states |N, 0〉 and |0, N〉, and the NOON states of arbitrary ultracold atoms can therefore be generated periodically from the initial state of either one of the Fock states.

Demonstrating Heisenberg-limited unambiguous phase estimation without adaptive measurements

We derive, and experimentally demonstrate, an interferometric scheme for unambiguous phase estimation with precision scaling at the Heisenberg limit that does not require adaptive measurements. That is, with no prior knowledge of the phase, we can obtain an estimate of the phase with a standard deviation that is only a small constant factor larger than the minimum physically allowed value. Our scheme resolves the phase ambiguity that exists when multiple passes through a phase shift, or NOON states, are used to obtain improved phase resolution. Like a recently introduced adaptive technique (Higgins et al 2007 Nature 450 393), our experiment uses multiple applications of the phase shift on single photons. By not requiring adaptive measurements, but rather using a predetermined measurement sequence, the present scheme is both conceptually simpler and significantly easier to implement. Additionally, we demonstrate a simplified adaptive scheme that also surpasses the standard quantum limit for single passes.

Quantum target detection using entangled photons

We investigate performances of pure continuous variable states in discriminating thermal and identity channels by comparing their $M$-copy error-probability bounds. This offers us a simplified mathematical analysis for quantum target detection with slightly modified features: the object---if it is present---perfectly reflects the signal beam irradiating it, while thermal noise photons are returned to the receiver in its absence. This model facilitates us to obtain analytic results on error-probability bounds, i.e., the quantum Chernoff bound and the lower bound constructed from the Bhattacharya bound on $M$-copy discrimination error probabilities of some important quantum states, like photon number states, $\text{N}$-photon maximally entangled (N00N) states, coherent states and the entangled photons obtained from spontaneous parametric down conversion (SPDC). Comparing the $M$-copy error-bounds, we identify that path-entangled states indeed offer enhanced sensitivity than the photon number state system, when average signal photon number is small compared to the thermal noise level. However, in the high signal-to-noise scenario, N00N states fail to be advantageous than the photon number states. Entangled SPDC photon pairs too outperform conventional coherent state system in the low signal-to-noise case. On the other hand, conventional coherent state system surpasses the performance sensitivity offered by entangled photon pair, when the signal intensity is much above that of thermal noise. We find an analogous performance regime in the lossy target detection (where the target is modeled as a weakly reflecting object) in a high signal-to-noise scenario.

Amplification of polarization NOON states

We analyze the quantum states obtained by optical parametric amplification of polarization NOON states. First we study, theoretically and experimentally, the amplification of a two-photon state by a collinear quantum injected optical parametric amplifier (QIOPA). We compare the stimulated emission regime with the spontaneous one, studied by Sciarrino et al. [Phys. Rev. A77, 012324 (2008)]. As a second step, we show that the collinear amplifier cannot be successfully used for amplifying N-photon states with N>2, and we propose to adopt a different scheme, based on a noncollinear QIOPA. We show that the state obtained by the latter amplification process preserves the λ/N feature and exhibits a high resilience to losses. Furthermore, measurement of part of the output state can be adopted to increase the pattern visibility.

Quantum phase measurements and a general method for the simulation of random processes

Quantum interferences offer the potential for improving the effective resolution wavelength of many measurements by a factor of N . Coincidence detection methods to date have been limited to N = 4 due to an increase in complexity of the apparatus with N . We offer an alternative method of extracting this higher-order phase information from a standard (first-order) interferometer. This “phase function fitting” algorithm also eliminates the need to null the interferometer. We compare the interferometer’s statistics to those of the quantum phase measurement, which we also utilize to derive Heisenberg limits for the quantum noon states. States which surpass this noon state limit are then demonstrated; and we discuss how state optimization for various signal environments might proceed. The phase representation is also shown to yield computationally efficient and conceptually revealing forms for calculating the statistics of the interferometer itself. We define a static limit in which the signal varies slowly enough that we can make the integration time long enough to approach perfect measurement of the interferometer statistics. We discuss the sense in which the phase function fitting algorithm can approach zero error in this limit, while still at finite N —satisfying the power constraint (but approaching infinite energy as we collect over longer times). To incorporate the signal dynamics we present a general method in which we can approximately prescribe the autocorrelation, as well as the probability distribution, of a random process; and illustrate how probabilistic effects can mitigate the spectral distortions of a nonlinear mapping. Mathematical challenges in applying these techniques to the simulation of the quantum phase measurement are circumvented—enabling the simulation of a variety of quantum algorithms for the estimation and tracking of various signal models in this exciting application.

Nonexistence of entanglement sudden death in dephasing of high NOON states

We study the dynamics of entanglement in continuous variable quantum systems. Specifically, we study the phenomena of entanglement sudden death (ESD) in general two-mode-N-photon states undergoing pure dephasing. We show that for these circumstances, ESD never occurs. These states are generalizations of the so-called high NOON states (i.e., a superposition of N photons in the first mode, O in the second, with O photons in the first, N in the second), shown to decrease the Rayleigh limit of lambda to lambda/N, which promises great improvement in resolution of interference patterns if states with large N are physically realized [Phys. Rev. Lett.85, 2733 (2000)]. However, we show that in dephasing NOON states, the time to reach some critical visibility Vcrit, scales inversely with N2. On the practical level, this shows that as N increases, the visibility degrades much faster, which is likely to be a considerable drawback for any practical application of these states.

TOWARD REAL NOON-STATE SOURCES

Path-entangled N-photon systems described by NOON states are the main ingredient of many quantum information and quantum imaging protocols. Our analysis aims to lead the way toward the implementation of both NOON-state sources and their applications. To this end, we study the functionality of "real" NOON-state sources by quantifying the effect real experimental apparatuses have on the actual generation of the desired NOON state. In particular, since the conditional generation of NOON states strongly relies on photon counters, we evaluate the dependence of both the reliability and the signal-to-noise ratio of "real" NOON-state sources on detection losses. We find a surprising result: NOON-state sources relying on nondetection are much more reliable than NOON-state sources relying on single-photon detection. Also the comparison of the resources required to implement these two protocols comes out to be in favor of NOON-state sources based on nondetection. A scheme to improve the performances of "real" NOON-state sources based on single-photon detection is also proposed and analyzed.

The Role of Entanglement in Quantum Lithography

We investigate the origin of the resolution improvement beyond the classical diffraction limit in quantum lithography. We show that the degree of the resolution improvement persists even when the pure entangled state of light initially prepared (the NOON state) is transformed to a mixed entangled state caused by phase and amplitude damping, regardless of the degree of entanglement remaining in the state. We also show that the resolution improvement beyond the classical diffraction limit can be achieved with a particular class of separable states of light. The resolution improvement in this case can be traced to entanglement (of the NOON type) that may be considered to be embedded in the separable state. We conclude that it is the existence of entanglement (of the NOON type) in the state of light used, not details of entanglement such as the degree of entanglement and relative weight of entanglement in the state, that determines the degree of the resolution improvement in quantum lithography. The visibility of the pattern and the total deposition rate, however, depend on the details of entanglement.

Aspects of practical remote quantum sensing

We consider the viability of photonic N00N states for practical remote quantum sensing in realistic atmospheres. This is done by analyzing the phase estimation performance of N-photon N00N states propagating in an attenuating medium. It is shown that the 1/N Heisenberg limit is never achieved. Moreover, an attenuated N00N state will actually produce a worse phase estimate than an equally attenuated N-photon separable state unless the transmittance of the medium is sufficiently high. The transmittance threshold for which the N00N states perform better than separable states increases as the number of signal photons increases. The amount of photon attenuation that arises in realistic, turbulent atmospheres is generally far below this threshold even for low N N00N states. Photonic N00N states are thus not viable for use in practical remote quantum sensing.