Post-selected States Research Articles

Enhancement of superluminal weak values under Lorentz boost

The local group velocity defined as the weak value of the velocity operator in (1 + 1)-dimensional Klein-Gordon and Dirac theory is studied. As shown by Berry [J. Phys. A 45, 185308 (2012)], when the pre- and post-selected states for evaluating the weak value are chosen at random from an ensemble of available states, it gives rise to a universal probability distribution for the local group velocity which can have both subluminal and superluminal components. In this work, we explore the possibility of enhancement of the superluminal fraction of this total probability distribution by applying a Lorentz boost and show that it can indeed be enhanced both in the case of Klein–Gordon and Dirac theories.

Quantum Fields and Local Measurements

The process of quantum measurement is considered in the algebraic framework of quantum field theory on curved spacetimes. Measurements are carried out on one quantum field theory, the “system”, using another, the “probe”. The measurement process involves a dynamical coupling of “system” and “probe” within a bounded spacetime region. The resulting “coupled theory” determines a scattering map on the uncoupled combination of the “system” and “probe” by reference to natural “in” and “out” spacetime regions. No specific interaction is assumed and all constructions are local and covariant. Given any initial state of the probe in the “in” region, the scattering map determines a completely positive map from “probe” observables in the “out” region to “induced system observables”, thus providing a measurement scheme for the latter. It is shown that the induced system observables may be localized in the causal hull of the interaction coupling region and are typically less sharp than the probe observable, but more sharp than the actual measurement on the coupled theory. Post-selected states conditioned on measurement outcomes are obtained using Davies–Lewis instruments that depend on the initial probe state. Composite measurements involving causally ordered coupling regions are also considered. Provided that the scattering map obeys a causal factorization property, the causally ordered composition of the individual instruments coincides with the composite instrument; in particular, the instruments may be combined in either order if the coupling regions are causally disjoint. This is the central consistency property of the proposed framework. The general concepts and results are illustrated by an example in which both “system” and “probe” are quantized linear scalar fields, coupled by a quadratic interaction term with compact spacetime support. System observables induced by simple probe observables are calculated exactly, for sufficiently weak coupling, and compared with first order perturbation theory.

Phonon maser stimulated by spin postselection

In a sequence of single spins interacting longitudinally with a mechanical oscillator, and using the micromaser model with random injection, we show that after an appropriate post-selection of each spin, a phonon laser analog with Poisson statistics is created with nearly perfect coherence, evidenced by the second order coherence function that goes asymptotically to one. The non-linear gain of the system depends crucially on the properly post-selected spin state as well as the pump. Our model and results suggest that the mechanism of interaction followed by a post-selected state or partial trace (common in laser/maser theory) of the spins may create the coherent vibrational radiation. However, for situations where the mechanical losses are high and it is imposible to decrease these, then the heralded post-selection can be the only resource to get phonon lasing if compared to the partial trace operation. These ideas and results may be useful for further theoretical and technical developments.

Anomalous amplification in almost-balanced weak measurement for measuring spin Hall effect of light.

In this paper, a method to measure the tiny spin splitting of the spin Hall effect of light (SHEL) using the almost-balanced weak measurement (ABWM) is presented. The ABWM technique uses two orthogonal post-selected states to record all of the information, which is a precise measurement method being different from the standard weak measurement (SWM). The theory model to describe the SHEL measurement based on ABWM is established. As results, the ABWM scheme has a larger amplification factor, reaching ∼105, which is nearly one order of magnitude higher than that of the SWM. When the post-selected angle is less than a certain value, the sensitivity and amplification factor of the ABWM scheme are higher than those of the SWM scheme, while the measurement precision and SNR of the ABWM technique are comparable to those of the SWM scheme. This research may have great potential for the precision metrology or sensing field based on the SHEL measurement.

Generating Postselected Quantum State from Fock State Using Ancillary Squeezed-vacuum State and Continuous-variable Postselection

Injecting a Fock state and an ancillary squeezed vacuum state into a beam-splitter (BS) device and then employing continuous-variable postselection detection in one output port, a postselected quantum state is generated in another output port. We derive the detailed analytical expressions of the density operator for the output state. Then we study the success probability analytically and numerically. In what follows, we study some properties for the generated state, including photon-number representation, Mandel Q parameter and Wigner function. Expressions for every property are given and numerical analyses are employed by changing different interaction parameters. The results show the nonclassical properties will be displayed in proper interaction parameters.

Dual to the anomalous weak-value effect of photon-polarization separation

The quantum Cheshire cat (QCC) thought experiment proposes that a quantum object's property (\textit{e.g} polarisation, spin, etc.) can be separated from its physical body or \textit{disembodied}. This conclusion arose from an argument that interprets a zero weak value (WV) of polarisation as no polarisation. We show that this argument is incomplete in the sense that a zero WV reading could equally be interpreted as linear polarisation. Nevertheless, through a generalisation of the QCC, we complete their argument by excluding the possibility of linear polarisation as a consistent interpretation. We go further, and introduce the dual of the generalised QCC. The dual QCC exhibits an intriguing effect, where a horizontally-polarised interferometer with just one arm, can give rise to interference which is vertically-polarised. The interference appears to arise as the result of the phase difference between the physical arm and a \textit{phantom} arm. This peculiar effect arises from the interplay between the pre-selected and post-selected states, which characterises WVs. The QCC has not yet been unambiguously experimentally demonstrated. The QCC dual offers an alternative pathway to experimental realisation.

Characterizing non-deterministic noiseless linear amplifiers at the quantum limit

We address the characterisation of non-deterministic noiseless linear amplifiers (NLAs) and compare the performances of different estimation strategies. In particular, we focus on estimation of the gain, whose value may result from a combination of different experimental effects, as those governing the overall efficiency of the detector or the transmissivity of the involved optical components. At first, we show that, contrary to naive expectations, post-selecting only the amplified states does not offer the most accurate estimate. We then focus on minimal implementations of a NLA, i.e. those obtained by coupling the input state to a two-level system, and show that the maximal amount of information about the gain of the NLA is obtained by measuring the whole composite system. The quantum Fisher information (QFI) of this best-case scenario is analysed in some detail, and compared to the QFI of the post-selected states, both for successful and unsuccessful amplification. Eventually, we show that full extraction of the available information is achieved when the non-deterministic process is implemented by a Lüders instrument. We also analyse the precision attainable by probing NLAs by single-mode pure states and measuring the field or the number of quanta, and discuss in some detail the specific cases of squeezed vacuum and coherent states.

Optimized weak measurement of orbital angular momentum-induced beam shifts in optical reflection

Tiny but universal beam shifts occur when a polarized light beam is reflected upon a planar interface. Although the beam shifts of Gaussian beams have been measured by the weak measurement technique, the weak measurement for orbital angular momentum (OAM)-induced spatial shifts of vortex beams is still missing. Here, by elaborately choosing the preselection and postselection states, the tiny OAM-induced Goos–Hanchen and Imbert–Fedorov shifts are amplified at an air–prism interface. The maximum shifts along directions both parallel and perpendicular to the incident plane are theoretically predicted and experimentally verified with optimal preselection and postselection states. These maximum shifts can be used to determine the OAM of vortex beams.

Periodically manipulating the photonic spin Hall effect with an electric field

How to control the photonic spin Hall effect (PSHE) at will is a very interesting and significant task. In this letter, we propose an electrically driven method for modulating the spin-dependent splitting in the PSHE. When the light transmits through a liquid crystal variable retarder, the spin-dependent spatial and angular shifts can be periodically manipulated with an applied electric field. The weak measurement technique is introduced to experimentally observe the spatial and angular displacements by choosing different postselection states. These findings may have potential applications in spin-based nanophotonic devices.

Simulating Broken PT-Symmetric Hamiltonian Systems by Weak Measurement.

By embedding a PT-symmetric (pseudo-Hermitian) system into a large Hermitian one, we disclose the relations between PT-symmetric quantum theory and weak measurement theory. We show that the weak measurement can give rise to the inner product structure of PT-symmetric systems, with the preselected state and its postselected state resident in the dilated conventional system. Typically in quantum information theory, by projecting out the irrelevant degrees and projecting onto the subspace, even local broken PT-symmetric Hamiltonian systems can be effectively simulated by this weak measurement paradigm.

Weak value amplification for nonunitary evolution

We discuss interferometric parameter estimation of the amplitude, instead of the phase, via weak value amplification. The considered weak interaction introduces modulation on the amplitude of the wave function; therefore, the two-party state experiences a nonunitary evolution. With the same pre- and postselection states as those of original weak value amplification, a much larger anomalous amplification factor can be attained. The shift in the intensity profile at the dark port and the signal-to-noise ratio for the parameter of interest get further amplified by the weak value, compared to phase-based weak value amplification. Although the nonunitary evolution introduces loss, more information about the parameter of interest can be extracted. Thus the weak value amplification is extended to measurement of amplitude and nonunitary processes. We also discuss possible applications of this idea for precisely measuring tiny rotations or enhancing the image contrast of transparent objects.

Characterization of a quasistatic environment with a qubit

We consider a qubit initalized in a superposition of its pointer states, exposed to pure dephasing due to coupling to a quasi-static environment, and subjected to a sequence of single-shot measurements projecting it on chosen superpositions. We show how with a few of such measurements one can significantly diminish one's ignorance about the environmental state, and how this leads to increase of coherence of the qubit interacting with a properly post-selected environmental state. We give theoretical results for the case of a quasi-static environment that is a source of an effective field of Gaussian statistics acting on a qubit, and for a nitrogen-vacancy center qubit coupled to a nuclear spin bath, for which the Gaussian model applies qualitatively provided one excludes from the environment nuclei that are strongly coupled to the qubit. We discuss the reason for which the most probable sequences of measurement results are the ones consisting of identical outcomes, and in this way we shed light on recent experiment (D. D. Bhaktavatsala Rao et al., arXiv:1804.07111) on nitrogen-vacancy centers.

Quantum geometric phase under pre- and post-selection

We consider a quantum system subjected to a controlled phase transformation and interaction with the environment in between the acts of selection, which leads to the emergence of interference effects. It is shown that the shift of the dependence of the statistics of contacts’ information with the environement on the controlled phase shift can be interpreted as a geometric phase. This interpretation is consistent with the known operational approach to the geometric phase. As a result, we suggest generalising the operational approach to the realm of pre- and post-selected quantum states.

Experimental violation of Mermin steering inequality by three-photon entangled states with nontrivial GHZ-fidelity.

Einstein-Podolsky-Rosen steering is an intermediate relationship between entanglement and Bell nonlocality in the hierarchical structure of quantum nonlocality. To certify the steerability of the entangled state, Mermin steering inequality is supposed to be violated by exceeding the inequality bound of 2. We present an experimental generation of post-selected three-photon entangled states and witness a maximal violation of the inequality up to 3.50±0.05. In the context of observing the maximal violation of Mermin steering inequality which requires measuring on the GHZ state, we derive a tight lower bound on the GHZ-fidelity that can be certified from the Mermin steering inequality violation. From this bound, it follows that the violation of Mermin steering inequality by 3.5 certifies the GHZ-fidelity of 78.66% at least. On the other hand, the above maximal violation of Mermin steering inequality observed in our experimental setup is produced by a post-selected entangled state having the GHZ-fidelity of 87.25 ± 0.34% through quantum tomography.

Surface Plasmon Resonance (SPR) Sensor Based on Optimal Pre- and Post-Selection

A surface plasmon resonance (SPR) refractive index sensing technique based on the optimal pre-and post-selection (PPS) is put forward. The intensity contrast ratio (ICR) is introduced to estimate the precision of this sensing technique. Compared with the traditional sensing techniques, our method shows that the ICR can be improved by about two orders of magnitude with optimal pre-and post-selected states. It is also proved that our sensing technique has a large tolerance for the metal film thickness and the incident angle. Remarkably, the method greatly reduces the cost of experiments, which is significant for promoting the widespread applications of SPR sensors.

Pre- and post-selected measurements with coupling-strength-dependent modulation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11674234 and 11605205), the Fundamental Research Funds for the Central Universities, China (Grant No. 2012017yjsy143), the National Key Research and Development Program of China (Grant No. 2017YFA0305200), the Youth Innovation Promotion Association of Chinese Academy of Sciences (CAS) (Grant No. 2015317), the

Pre- and post-selected (PPS) measurement, especially the weak PPS measurement, has been proved to be a useful tool for measuring extremely tiny physical parameters. However, it is difficult to retain both the attainable highest measurement sensitivity and precision with the increase of the parameter to be measured. Here, a modulated PPS measurement scheme based on coupling-strength-dependent modulation is presented with the highest sensitivity and precision retained for an arbitrary coupling strength. This idea is demonstrated by comparing the modulated PPS measurement scheme with the standard PPS measurement scheme in the case of unbalanced input meter. By using the Fisher information metric, we derive the optimal pre- and post-selected states, as well as the optimal coupling-strength-dependent modulation without any restriction on the coupling strength. We also give the specific strategy of performing the modulated PPS measurement scheme, which may promote practical application of this scheme in precision metrology.

Universality of local weak interactions and its application for interferometric alignment

The modification of the effect of interactions of a particle as a function of its preselected and postselected states is analyzed theoretically and experimentally. The universality property of this modification in the case of local interactions of a spatially preselected and postselected particle has been found. It allowed us to define an operational approach for the characterization of the presence of a quantum particle in a particular place: the way it modifies the effect of local interactions. The experiment demonstrating this universality property provides an efficient interferometric alignment method, in which the position of the beam on a single detector throughout one phase scan yields all misalignment parameters.

Uncertainty relations for time-averaged weak values

Time averaging of weak values using the quantum transition path time probability distribution enables us to establish a general uncertainty principle for the weak values of two not necessarily Hermitian operators. This new principle is a weak value analog of the Heisenberg-Robertson strong value uncertainty principle. It leads to the conclusion that it is possible to determine with high accuracy the simultaneous mean weak values of non-commuting operators by judicious choice of the pre- and post-selected states. Generally, when the time fluctuations of the two weak values are proportional to each other there is no uncertainty limitation on their variances and, in principle, their means can be determined with arbitrary precision even though their corresponding operators do not commute. To exemplify these properties we consider specific weak value uncertainty relations for the time-energy, coordinate-momentum and coordinate-kinetic energy pairs. In addition we analyze spin operators and the Stern-Gerlach experiment in weak and strong inhomogeneous magnetic fields. This classic case leads to anomalous spin values when the field is weak. However, anomalous spin values are also associated with large variances implying that their measurement demands increased signal averaging. These examples establish the importance of considering the time dependence of weak values in time of flight scattering experiments.

Weak Measurement of Elliptical Dipole Moments by C-Point Splitting.

We investigate points of circular polarization in the far field of elliptically polarized dipoles and establish a relation between the angular position and helicity of these C points and the dipole moment. In the case of highly eccentric dipoles, the C points of opposite handedness exhibit only a small angular separation and occur in the low intensity region of the emission pattern. In this regard, we introduce an optical weak measurement approach that utilizes the transverse electric (azimuthal) and transverse magnetic (radial) far-field polarization basis. Projecting the far field onto a spatially varying postselected polarization state reveals the angular separation and the helicity of the C points. We demonstrate the applicability of this approach and determine the elliptical dipole moment of a particle sitting on an interface by measuring the C points in its far field.

Optimization of a quantum weak measurement system with its working areas.

Phase-sensitive weak measurement systems have been receiving an increasing amount of attention. In this paper, we introduce a series of weak measurement working areas. By adjusting the pre-selection and post-selection states and the total phase difference between vertically polarized light and horizontally polarized light, the measurement of the weak value is amplified by several times in one system. Its applicability is verified in a label-free total internal reflection system. The original sensitivity and resolution are improved at different working areas, reaching 1.85 um/refractive index unit (RIU) and 6.808 × 10-7 RIU, respectively.