INTRODUCTION

Excitation of waves by a moving object is ubiquitous in many areas of physics, such as electrodynamics (1), acoustics (2), and hydrodynamics (3)examples are the Cherenkov effect, sound waves, and ship wakes. These processes are thought to be successfully explained by classical physics, wherein wave interference is often critical for describing the phenomena. For example, in electrodynamics (1), radiation emission patterns are predicted by Maxwells equations.

Such is the case for Cherenkov radiation (CR): the emission of light by free charged particles moving faster than the phase velocity of light in a medium (4). Since its discovery in 1934, a long-standing hallmark of CR is its manifestation as a shock wave of light (1, 59), resulting from the coherent temporal interference of radiation at a wide spectral range. Despite the wide applicability of this effect, no experiment has ever directly observed the shock wave dynamics emitted by a single particle. As one of the implications of this work, we shall see that the underlying quantum nature of CR fundamentally limits the shock wave duration in many existing experimental settings, which can be understood in terms of the entanglement of the light with the emitting particle.

Looking at the bigger picture in electromagnetism, light emission by free charged particles constitutes a family of effects (10, 11) including, for example, transition radiation (12), Smith-Purcell radiation (13), undulator radiation (14), and CR. Often called coherent cathodoluminescence (10) (CL), these phenomena are used in many areas of physics and engineering, from electron microscopes (10, 15), particle detectors (16, 17), free-electron lasers (18), engineerable light sources (19, 20), and medical imaging (21). As the spectral range of emitted light can be straightforwardly tuned by varying the particle energy, coherent CL is a promising platform for light generation in otherwise inaccessible regimes (18, 22), such as at terahertz, ultraviolet, and x-ray frequencies.

The broad tunability of coherent CL, alongside recent advances in shaping (23, 24), coherent control (2527), and entanglement (28, 29) of free electrons, makes it a probe of fundamental light-matter interaction (15, 30) and a prominent candidate for quantum measurement (15). These advancements brought about fundamental questions regarding the role of the particle wave function (3034) in coherent CL. However, in all relevant experimental settings, coherent CL is still considered as classical (3539) or semiclassical (31, 34, 40, 41). The general expectation from a quantum theory is that when the emitting particle is not directly measured (42), the quantum features of its wave function (4345) cannot leave a detectable mark on the emitted light. A milestone of fundamental importance would be, therefore, to identify observables of coherent CL radiation that are both detectable in practical settings and directly depend on the quantum state of the emitting particles. This observation has implications also for general wave phenomena, such as any mechanical waves excited by free moving objects. Can fundamental quantum aspects of a particle affect the patterns of waves in seemingly classical regimes?

Here, we introduce the quantum optical paradigm to describe coherent CL and identify the specific measurements that depend on the quantum wave nature of the emitter. By formulating a general quantum theory of spontaneous light emission by charged particles, we show that already in what are generally assumed to be classical regimes, coherent CL can be dominated by quantum features such as wave function uncertainty, quantum correlations, and decoherence. These effects can be exposed in quantum optical measurements, such as first-order correlation measurements, even in seemingly classical features such as the emitted pulse duration. Although the concept of coherence transfer was thoroughly studied in quantum optics for nonrelativistic bound-electron systems [for example, in effects such as quantum beats (46)], it was never applied to light emission from relativistic free charged particles, still commonly described in classical or semiclassical terms. Hence, new insight is gained by analyzing the optical coherence of such system through the prism of quantum optics.

As an unexpected implication for the Cherenkov effect, we find that quantum decoherence imposes a fundamental lower bound for the Cherenkov shock wave duration, predicting an uncertainty principle that connects it to the particle momentum uncertainty. Quantum coherence is the ability of a quantum system to demonstrate interference. The coherence between different parts of a wave function (in momentum or real space) allows for the famous double-slit interference and the formation of short quantum wave packets propagating in space. Quantum decoherence, as its name suggests, is the loss of quantum coherence, hindering the visibility of interference. Most commonly, this process happens when an open quantum system interacts with its surrounding environment (47). The underpinning mechanism for decoherence is the entanglement of the observed subsystem (for example, an emitted photon) with another, unobserved subsystem (for example, a charged particle). In our context, we identify many practical scenarios in which CR is not a shock wave, owing to the underlying quantum decoherence of the emitted light.

Our quantum theory of coherent CL has new applications, such as detecting the shape, size, and coherence of the emitters wave function by measuring the spectral autocorrelations of the light it emitsthereby gaining information on the wave function uncertainty. Our findings can resolve a question, which, with the advent of ultrafast electron microscopes, has been frequently asked: What part of the measured energy spread of an electron beam is due to coherent energy uncertainty, and what part is due to incoherent uncertainty? Moreover, our work sheds light on fundamentally new capabilities to measure quantum properties of charged particles that can serve as an alternative to matter wave holography, which is especially important for many high-energy particles observed in Cherenkov detectors, where holographic techniques do not exist. The results presented in this work pave the way toward novel tunable light sources and measurements sensitive to the wave function of free charged particles.

In classical physics, waves interfere coherently when they are generated from different point particles constituting an emitter (48), so long as the different emission points are perfectly correlated with each other (Fig. 1A). In particular, the emission from each individual particle is considered to always be coherent with itself. In quantum mechanics, an emitter is described by a spatially varying wave function. Following the emission of wave quanta, the particles and waves are in an entangled state, known to cause quantum decoherence (Fig. 1B) (47) if one of the constituents of the bipartite system is not measured. As spontaneous emission of light by free charged particles is usually described classically (3539), it is generally assumed that the abovementioned effect is negligible, on the grounds that the correspondence principle (49) is always valid. This assumption is backed by the small quantum recoil (10) exerted by the photon, amounting to only minor corrections (4345). It is the purpose of the following analysis to show that under certain common conditions, quantum mechanics fundamentally modifies light emission, even in regimes that are traditionally seen as classical.

(A) Classical wave dynamics. A point particle with velocity v passes through an optical medium and emits waves that may interfere coherently. The classical emitter current density J(r, t) = ev(r vt) emits a temporally coherent shock wave. (B) Quantum description. A quantum particle is described by a delocalized wave function (r, t). A current operator J(r,t) is then associated with the particle. Even when the initial particle is only described by a single momentum ki, it may spontaneously emit many wave quanta (momenta q, q, ). The waves are then entangled with the particle because of momentum conservation (leaving the final particle having momenta kf, kf, respectively). When only the emitted waves are observed, this entanglement can lead to quantum decoherence and lack of interference visibility, resulting in the emission of incoherent radiation.

Recent works considered in depth the effect of the wave function size and shape on the spontaneously emitted radiation by free charged particles. Investigations based on semiclassical analysis (31, 34, 40, 41) imply size- and shape-dependent effects on the emitted power spectrum, while a quantum analysis (32, 33) suggested no such effects. Importantly, experiments have demonstrated wave function dependence upon postselection of a final electron state (30), while no such effects were observed when only the light was measured (no postselection) (33). The findings detailed below determine between the contradicting results, showing explicitly that without postselection, the wave function does not affect the power spectrum of spontaneous emission, while suggesting a new observablespectral coherencewhich explicitly depends on the wave function, and how the size and shape of the latter could be extracted from it. In this context, our findings can help promote the fast-growing field of free-electron quantum optics (50, 51) and emphasize the effect of the electron wave function in ultrafast electron beam spectroscopy experiments (15).

Without loss of generality, consider the emitting charged particles to be free electrons. We also consider the emitted electromagnetic field to be in a general optical environment. The initial state is described by a density matrix i, where the electrons have a reduced density matrix e, and the radiation field is found in the vacuum state 0, such that the initial state is separable i = e 00. The interactions between the electrons and the electromagnetic field are governed by the Dirac Hamiltonian: Hint = ec A, where e is the electron charge, c the speed of light, i=0i are the Dirac matrices, and A is the electromagnetic vector potential operator. Considering a weak coupling between the electrons and photons, the final quantum state of the system, f, is found by first-order time-dependent perturbation theory (see section S1).

In general, after the interaction, the electrons are entangled to many photonic modes because emission is allowed for different directions and at many different frequencies. For example, starting from an arbitrary initial wave function of a single electron and zero photons, i = kikiki0, and if momentum is conservedas in CRthe photon can be emitted with different momenta q = ki kf, giving an entangled final statef=kikikfMkikf;qeiEft/eiqtkfq=kikf(1)where Mki kf;q is the transition amplitude. Information regarding the electron initial state ki can be extracted by measuring the photon momentum q = ki kf in coincidence with (or postselection of) an electron momentum kf. However, this is not the experimental situation of CL, where only the light is measured, and the electron degrees of freedom are traced out. In this case, both experimental and theoretical evidence suggest that the initial electron wave function has no influence on observables of the emitted radiation (32, 33, 44), such as the power spectrum. Below, we will examine this situation carefully and show how the emitted light autocorrelations can be strongly influenced by the single electron wave functionalthough the power spectrum is not, suggesting a ubiquitous, hidden quantumness to the radiation by free electrons.

To describe the photonic final state in the experimental scenario of coherent CL, we calculate the reduced density matrix of the electromagnetic field, ph = Tre{f}, with Tre denoting the partial trace over the electronic state. The electric field autocorrelation is determined by the final photonic state, ph, via the quantum mechanical expectation value E()(r, t)E(+)(r, t) = Tr{E()(r, t)E(+)(r, t)ph}, where E(+)(r, t) and E()(r, t) = (E(+)(r, t)) are, respectively, the positive and negative frequency parts of the electric field operator. Instead of the simplified momentum-space picture of Eq. 1, which strictly holds only for CR (see section S8), we use a more general formalism. On the basis of quantum electrodynamical perturbation theory, the formalism holds for all coherent CL processes and for an arbitrary number of electrons (see section S1 for derivation), yieldingE(r,)E(r,)=02d3Rd3RG(r,R,)G(r,R,)j(R,)j(R,)e(2)where G(r, r, ) is the Dyadic Greens function of Maxwells equations for the dielectric medium (52), and where E(+)(r,t)=0deitE(r,). The quantity j(r, )j(r, )e = Tr{ejj} is the expectation value, with respect to the emitter initial state, of the correlations in the current density operator j(r, t) = ec, where (r, t) is the emitter spinor field operator described in second quantization (see sections S1 and S2). From here onward, we assume that the particles propagate as wave packets with a well-defined carrier velocity v0 (the paraxial approximation, where the particle dispersion is linearized about its mean momentum/energy).

Now, let us constrain the discussion to the seemingly classical regime, where photon recoils q are much smaller than electron momenta pe. This constraint is applicable to a vast number of effects, including all cases in which the emitter is relativistic, all current free-electron nanophotonic light sources, and all free-electron sources in the microwave and radio frequency ranges. In general, this derivation applies to both the single- and many-particle emitter states, described via second quantization of the emitter. The current correlations in Eq. 2 can then be written as (see section S2 for derivation)j(x)j(x)=e2v0v0[Ge(2)(x,x)+(xx)Ge(1)(x,x)](3)where x = r v0t and x = r v0t. In Eq. 3, we define the first- and second-order correlation functions of the emitter Ge(1)(x,x)=Tr{e(x)(x)} and Ge(2)(x,x)=Tr{e(x)(x)(x)(x)}, respectively, where (x) are operators corresponding to the particle spin components = , . Equation 3 is valid for both fermionic and bosonic statistics, under the approximations detailed above.

The current correlations comprise two terms: a pair correlation term proportional to Ge(2)(x,x), giving rise to spatially and spectrally coherent spontaneous radiation (henceforth called coherent radiation) when substituted into Eq. 2, and a term proportional to the probability density Ge(1)(x,x), contributing a spatially and spectrally incoherent spontaneous radiation (33) (which we refer to as incoherent radiation). In this work, we focus on the case of a single particle, wherein Ge(2)(x,x)=0, and discuss the nature of quantum decoherence of the light it emits. A derivation of the effects of many-body quantum correlations [Ge(2)(x,x)0] on the radiation will be reported in a separate work (53).

CR is characterized by a directional, polarized, cone-shaped radiation pattern with opening semi-angle c satisfying cos c = 1/n(), where = v/c is the speed of the particles normalized by the speed of light and n() is the refractive index of the medium. We assume that the emission is detected with a far-field detector located at a specific azimuthal angle on the cones rim, providing broadband detection of all frequency components [note that in certain practical situations, the entire emission ring (over all azimuthal angles) could be collected using special optics (54)thereby increasing the signal level]. Using the far-field expression for the dyadic Green tensor of a uniform dielectric medium (52), G(r,r,)=eiqr4r(Irr)eiqr, and assuming weak material dispersion, we find from Eqs. 2 and 3 that the radiation field projected on the detector is described by the following frequency-domain quantum autocorrelation (see section S3 for derivation)E()(r,)E(+)(r,)=U02ei(qq)r2n0cr2d3xei(qq)xGe(1)(x,x)(4)where we denote q=rcq, with rc being the observation direction on the Cherenkov cone, q = n()/c, and with U0 = sin2 c. Equation 4 implies that for a single emitting particle, the first-order autocorrelation of CR is intimately relatedthrough a Fourier transformto the probability density of the particle wave function. The same conclusionyet with more complex expressionsapplies to all coherent CL processes, such as Smith-Purcell and transition radiation. Smith-Purcell radiation (13) occurs when an electron passes near a periodic grating. The grating introduces a boundary condition defining periodic photonic Bloch modes u(r) ( standing for all relevant indices such as Bloch vector, band number, and polarization). The periodicity of the photonic near field allows simultaneous energy and momentum conservation in the emission process, which results in the emission to the far field. A possible way to obtain the dyadic Green function of Eq. 2 is via mode expansion (52) G(r,r,)=c2u(r)u*(r)/(22), and a result similar to Eq. 4 could be derived.

The emission of classical shocks from a point charge (9), for which j(r, t) = ev(r vt), is optically coherent over an arbitrarily wide spectral range only limited by the optical response of the medium. As such, the measured duration of the shock wave intensity envelope E(t)2 is only limited by the material dispersion and/or the detection bandwidth, theoretically enabling shock waves on the scale of femtoseconds and below (8, 9). The quantum description, however, incorporates the finite-sized single-particle wave function through Eq. 4. The incoherent emission from different points on the wave function (resulting from the delta-function term in Eq. 3) is a manifestation of quantum decoherence of the emitted light, expected to inhibit interference visibility and stretch the shock duration. Here, the photon acts as the observed subsystem, and the electron it was emitted from acts as the unobserved subsystem (47). As these two subsystems are entangled in momentum and the electron degrees of freedom are now traced out, the different photon momenta tend to a classical mixture instead of a pure quantum superposition. As a result, the quantum coherence of that photon is hindered, and the interference visibility can be greatly reduced.

For CR, this observation manifests itself in a rather straightforward manner. Considering weakly dispersive media and wide detection bandwidths, the shock wave power envelope P(t) = 2r20ncE()(t)E(+)(t) travelling at a group velocity vg is given by the equal-time temporal Fourier transform of Eq. 4. The probability cloud Ge(1)(x,x) is projected along the direction of observation rc on the Cherenkov cone. From this relation, we find that if the emitting particle wave function has a momentum uncertainty pe in the direction of CR, then the position uncertainty of the shock wave, xshw, satisfies a generalized uncertainty principle (see section S3 for derivation)xshwpe2(5)where the inequality becomes a strict equality for a minimum-uncertainty particle (satisfying xepe = /2) and for broadband detection. The intuition behind Eq. 5 is the following: If the particle has a position uncertainty along the emission direction, the shock wave emitted from it will demonstrate this same position uncertainty. For a classical particle, this uncertainty approaches zero, giving a classical shock. However, for a quantum particle, the Heisenberg uncertainty principle (55) defines a lower bound to the position uncertainty, thereby affecting the presumably classical light.

The seemingly elementary result in Eq. 5 represents a rather deep conclusion: It demonstrates how the well-known classical wave interference can only be generated by a quantum particle that has a certain momentum uncertainty. This result also provides a fundamental quantum lower bound on the interference (the shock wave duration) that cannot be captured within a classical theory considering point particles or with a semiclassical theory treating the wave function as a coherent spread-out charge density (33, 41) (see discussion below). As a concrete example, Fig. 2 shows how the momentum coherence pe pe (or coherent momentum uncertainty) determines the tight lower bound on the shock duration. For example, particles in a mixed quantum state in momentum space, with low coherent uncertainty pe pe, emit temporally incoherent light and, consequently, a longer shock wave. These kinds of considerations also show why low-frequency radiation (radio frequency, microwave, etc.) will generally be classical.

(A) A quantum particle with a coherent momentum uncertainty pe that equals its total momentum uncertainty pe displays a broad quantum coherence between its initial momenta pi (yellow glow). When the particle transitions to any final momentum pf, the emitted wave inherits this initial coherence because of the which path interference between the initial particle states. Hence, different wave vector components of the wave are coherent (red glow). (B) A quantum particle in a mixture of momenta (total uncertainty pe) with low coherent uncertainty pe pe emits temporally incoherent waves. The limited interference inhibits the pulse formation, and its length exceeds the classical prediction. (C and D) The temporal field autocorrelations, 2r20ncE()(t)E(+)(t) (in W), for 1-MeV electrons in silica in the visible range. The electrons are modeled as spherical Gaussian wave packets with coherent energy uncertainty (A) e = 3.72 eV (wave packet radius ~50 nm) and (B) e = 0.19 eV (wave packet radius ~1 m). The diagonal (t = t) indicates the temporal power envelope, P(t), being transform-limited in (A) and incoherent in (B). Insets show a scaled comparison between P(t) and the degree of first-order coherence of the light, g(1)(). For both (A) and (B), the classically expected shock wave full width at half maximum is 1.4 fs.

Experimentally, the decoherence effect best manifests itself in the temporal (or spectral) autocorrelations E()(t)E(+)(t) [or E()()E(+)()], where the off-diagonal (t t or ) terms relate to the coherence. Temporally coherent CR results in a transform-limited shock wave, as the classical theory suggests. However, the quantum corrections may alter the temporal behavior: Coherent (incoherent) shock waves exhibit g(1)() wider (narrower) than the pulse envelope. Figure 2 (C and D) demonstrates this behavior by simulating CR emission from 1-MeV electrons in silica for varying uncertainties. We note that this energy was chosen because it is closer to values often considered in high-energy physics to quantify Cherenkov detectors (with a relativistic particle velocity = 0.94 close to 1), while being of a similar order of magnitude to what one finds in a transmission electron microscope (TEM). Lower electron energies (such as those useable in TEMs) could readily be considered, keeping in mind the experimental limitations detailed below in the Experimental considerations section.

In this context, it is noteworthy to mention that classical and semiclassical theories predict that the emitted radiation is always perfectly coherent, both temporally and spectrally. The reason for this lies in the treatment of the electron probability density Ge(1)(x,x) as a classical charge density that emits light coherently from different points. While this approximation holds in the limit of a point particle, it fails when the electron wave function is delocalized such that it exceeds the photon wavelength. Subsequently, it can be shown that these theories do not satisfy the quantum uncertainty principle (Eq. 5) in practical experimental situations of CR (e.g., in standard electron microscopes)as they always predict a larger optical coherence in the semiclassical picture (because quantum decoherence is ignored). One is able to amend the semiclassical picture by adopting an ad hoc probabilistic approach (33) demanding that the electron emits light incoherently from different points, although a fully quantum treatment is necessary to unveil other important aspects such as quantum correlations (53, 56). An elaborate comparison between these theories can be found in section S7 and other works (32, 33, 41).

Quantum optical measurement of the spectral autocorrelations may unveil information about the emitter wave function itself and provide an unprecedented analytical tool for particle identification. Equation 4 provides a direct relation between the frequency-domain autocorrelation E()()E(+)() and the spatial Fourier transform (or structure factor) of the emitter probability density Ge(1)(x,x). This structure factor is equivalent to a momentum coherence function of the particle e(q q) = d3k e(k + q, k + q), namelyE()()E(+)()d3xei(qq)xGe(1)(x,x)=e(qq)(6).

Equation 6 implies that a spontaneously emitted photon is only as spectrally coherent as the emitting particle it originated from (see Fig. 2, A and B). Spontaneous CR can, therefore, be used to map the structure of the emitter wave function and its momentum coherence by analyzing the correlations of emitted photons. Note that only the spectral coherence of the photons plays a role here, namely, the width of the off-diagonal part of the autocorrelations (see Fig. 3C). The diagonal part (optical power spectrum), E()()E(+)(), is wave function independent.

(A) A charged particle wave packet (r, t) of finite size and carrier velocity v0 impinges on a Cherenkov detector with material dispersion n(). The particle spontaneously emits quantum shock waves of light into a cone with opening half-angle c() = acos[1/n()]. Collection optics is situated along the cone in the direction rc in the far field. (B) Detection scheme for measuring the spectral field autocorrelations E()E() using an interference between spectrally/temporally sheared fields (57). (C) The reconstructed photon density matrix determines the spatial probability distribution (r)2. (D and E) Simulation of particle wave function size reconstruction from the photon density matrix. A single 1-MeV electron ( = 0.94) in a silica Cherenkov detector [dispersion taken from (77)] emits CR that is collected within the visible range ( = 400 to 700 nm, centered at 0 = 550 nm). The electron wave function envelope is Gaussian and spherically symmetric, with position uncertainty of (d) xe = 254 nm and (E) xe = 1016 nm (bottom insets). In both (D) and (E), the measured photon density matrix, ph(, ), is plotted. The wave functionindependent diagonal ph(, ) that denotes the photodetection probability is the same for both cases. However, the off-diagonal spectral coherence ph( ) is strongly dependent on the wave function. Measuring its width coh (top insets) and using the approximate Eq. 7 provide the estimates (D) xe=290nm and (E) xe=1006nm.

The wave function size and shape can be estimated, for example, by assuming a spatial variance matrix for the emitter probability cloud given as ij2=Tr{rirje}. The photons are collected at an observation direction rc on the Cherenkov cone, and the width of their spectral coherence is measured (see Fig. 3, A to C). It then gives an estimate for the wave function dimensions along the observation direction (see section S4 for the derivation)rcT2rc=vg22(7)where vg denotes the shock wave group velocity. If the electron wave function is not spherical, we can further reconstruct the three-dimensional by measuring the spectral coherence along different Cherenkov cones rc [which can be done by measurements of multiple particles with the same wave function moving through media of different refractive indices n, as done in threshold detection (16)]. At least two such measurements are necessary to find both the longitudinal and transverse sizes of the wave function.

The quantum optical measurements necessary for the reconstruction of the photon density matrix in the frequency domain have been demonstrated experimentally for single photons (5759). Combining these quantum optical reconstruction techniques with Cherenkov detectors may allow for completely new and exciting capabilities. Currently available techniques for particle identification in Cherenkov detectors are limited to measuring velocity or mass (16, 17). Our proposed scheme further enables the measurement of the wave function dimensions and coherences of naturally occurring particles such as in cosmic radiation and beta decay (16), as well as the characterization of charged particle beams (for example in microscopy). Figure 3 (D and E) shows an example for such measurement scheme for the case of 1-MeV electrons. This method can provide an alternative to matter wave holography [used in electron microscopes (60) to measure the transverse wave function], which is currently unavailable for high-energy charged particles, such as muons, protons, kaons, and pions. In contrast, the measurement we propose is relevant for these particles and can be used as part of Cherenkov detectors, which also have the advantage of being a nondestructive measurement.

Beyond the capability of reconstructing the wave function size, our technique can be used to detect the signature of non-Gaussian wave packets (in energy-time space), such as coherent electron energy combs produced in photon-induced near-field electron microscopy (PINEM) (61), ultrafast TEM (27), and other methods (see Fig. 4) (6264). In PINEM, a free electron traverses a near-field optical structure and interacts with a coincident laser pulse. As a result, the electron wave function is modulated and given by a coherent superposition of energy levels. Following free-space propagation, the electron wave function takes the form of a pulse train (27). When this electron emits CR, notice how the interference fringes due to its shaped wave function appear only in the photon spectral autocorrelations (off-diagonal) and not in the radiation spectrum (diagonal).

(A) A free electron wave function is shaped by the interaction with a strong laser field of frequency (here, = 2 200 THz), as done in photon-induced near-field electron microscopy (61). The result is a coherent electron energy ladder, manifested as a temporal pulse train. (B) Cherenkov photon autocorrelations reveal the electron wave function spectral interference pattern, matching the laser frequency. The measurement scheme is the same as in Fig. 3 (A to C).

Here, we briefly discuss some important considerations for realizing our predictions in an experiment. For the analysis discussed in the previous section, the electrons coherent interaction length Lint must be in the range /n Lint (n/n)(/), where n = n ng is the difference between the refractive and group indices of the material, and denotes the wavelength band collected by the detection system (see section S5). The lower limit ensures that the Cherenkov angle is sharply defined, while the upper limit ensures that the material dispersion has a weak effect on the correlation between different frequencies. For standard materials and optical wavelengths, Lint is in the order of a few micrometers.

For CR in bulk media, other scattering processes with mean free paths smaller than Lint can readily broaden the particle spectrum e(k, k). In section S6, we show that for a general uniform medium of optical response function Im G(q, ) (which encompasses all types of inelastic processes, such as scattering by phonons and plasmons, and excitations of electron-hole pairs), the momentum coherence function e(q q) of Eq. 6 remains unchanged. As the latter quantity is the one responsible for the Cherenkov autocorrelation through Eq. 6, we expect the signature of the wave function to persist. In electron microscopy, one can avoid these scattering processes by using an aloof beam geometry having electrons that propagate in vacuum near an optical structure, such as in Smith-Purcell experiments or in emission of Cherenkov photons near dielectric boundaries (65).

Here, we investigated light emission by free charged particles from a quantum-optical viewpoint, by using a fully quantum formalism of light-matter interaction. Our conclusions take into account the experimental situation that the emitting particle itself is not measured. In this situation, recent studies show that the particle wave function has no influence on the emitted spectrum. We complement this realization by showing that quantum optical observables such as the emitted pulse duration and optical autocorrelations are all strongly influenced by the particle wave function. Moreover, all the quantum features of the particle such as coherence, uncertainty, and correlations embedded in the emitter wave function play an important role in determining the properties of the emitted light.

As an example, we considered the Cherenkov effect, and its characteristic optical shock wave, envisioned classically for almost a century as a coherent, transform-limited pulse of light. Instead, we found that it is fundamentally limited by the particle quantum uncertainty, satisfying a generalized uncertainty principle. The smaller the coherent momentum uncertainty of the particle is, the longer (and less coherent) the shock wave becomes. We further showed how this uncertainty relation can be harnessed to unveil information about the particle wave function, allowing unprecedented capabilities for particle detection. For example, Cherenkov detectors together with a quantum-optical measurement of the emitted light can be used to reconstruct the particle wave function size, shape, coherence, and quantum correlations.

Our findings can be used to resolve an important fundamental question of practical importance: What part of the energy uncertainty of a free electron is coherent, and what part is incoherent? This property can be measured from the radiation autocorrelations and spectrum. With the advent of laser-driven electron sources, for example, in ultrafast electron microscopes (2527, 63, 66, 67), the particle coherent energy uncertainty is believed to be dictated by the laser linewidth (26, 68), e.g., spanning tens of millielectron volts for excitations with femtosecond lasers. With the ability to coherently control the spatial electron wave function (23, 24), the transverse momentum uncertainty can be further lowered. Such conditions allow for the predictions of our work to be tested experimentally under controllable settings.

Considering the outlook for using free electrons as quantum probes (15, 51, 69), our work paves the way toward quantum measurement of free electrons and other charged particles based on spontaneous emission. One interesting direction for extending the research is to consider light emission from low-energy (tens to hundreds of electron volts) coherent electrons (70, 71), for which the zero-recoil approximation is no longer valid. In addition to recoil-induced quantum corrections in the emitted light (45), we expect the coherence of such light to be limited by the high spatial coherence of the electrons. Furthermore, our results may readily be generalized to other physical mechanisms of wave emission, for example, analogs of the Cherenkov effect (72, 73), as in Bose-Einstein condensates. Similar effects can be explored with any photonic quasiparticle (74), and even with sound waves, and phonon waves in solids (75), which all have the same underlying quantum nature and must have exact analogous phenomena.

Another intriguing question is the effect of many-body correlations (as manifested by the second term in Eq. 2) on such radiation phenomena, giving rise to yet unexplored quantum super- and subradiance regimes of coherent CL. These arise from coherent interference of multiparticle wave functions, which will be discussed in forthcoming work (53).

Note added in proof: This work was first presented in the Conference on Lasers and Electro-Optics in May 2020 as a conference presentation (76). A related paper (doi: 10.1126/sciadv.abf6380) appears in Science Advances.

Acknowledgments: Funding: This work was supported by the ERC starting grant NanoEP 851780 and the Israel Science Foundation grants 3334/19, 831/19, and 1415/17. A.K. acknowledges support by the Adams Fellowship of the Israeli Academy of Sciences and Humanities. N.R. was supported by the Department of Energy Fellowship DE-FG02-97ER25308 and by a Deans Fellowship by the MIT School of Science. Author contributions: A.K., N.R., A.A., and I.K. conceived the idea and contributed to writing the paper. A.K. performed the theoretical derivations. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

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The coherence of light is fundamentally tied to the quantum coherence of the emitting particle - Science Advances

This book explains quantum computing in terms of elementary linear algebra, emphasizing computation and algorithms and requiring no background in physics. Richard J. Lipton and Kenneth W. Regan's book is concise but comprehensive, covering many key algorithms. It is mathematically rigorous but requires minimal background and assumes no knowledge of quantum theory or quantum mechanics.The book explains quantum computation in terms of elementary linear algebra; it assumes the reader will have some familiarity with vectors, matrices, and their basic properties, but offers a review of the relevant material from linear algebra.

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LOS ANGELES, CA - JANUARY 15: Soleil Moon Frye, Linda Perry, at 10th Anniversary Gala Benefiting ... [+] CORE at The Wiltern Theatre in Los Angeles, California on January 15, 2020. Credit: Faye Sadou/MediaPunch /IPX

For actress Soleil Moon Frye, music is inextricably linked with her experiences growing up in the 1990s.

Early on, Frye took stock of her upbringing as a child actress, keeping a diary from the age of just 5, later utilizing audio recording and video cameras to document the lives of her and her famous friends at a time prior to the advent of camera phones when recording every moment wasnt ubiquitous in the way it is today in the time of social media.

In the new Hulu coming of age documentary Kid 90, Frye utilizes all of those materials in the telling of a story that defines both time (the 90s) and place (Los Angeles and New York), directing a uniquely rare look back defined by the films unflinching honesty.

The documentary explores relatable topics like friendship, love, loss, death, acceptance and moving on, addressing the importance of having an open and honest conversation about mental health thats free of stigma.

Music plays a major role. Artists like Janes Addiction frontman Perry Farrell and House of Pain rapper Danny Boy appear in a film that also features era appropriate tracks from Eddie Vedder, Nine Inch Nails, Liz Phair and more.

At the heart of the film is an original score (now available on streaming platforms), created by singer, songwriter, producer, manager and label executive Linda Perry, who navigated the first half of the 90s as vocalist of platinum selling alternative group 4 Non Blondes, which tallied a top 20 single in 1992 thanks to the inescapable Whats Up?

While Perry has worked with artists like Christina Aguilera, Pink and Adelle, Kid 90 marks her own first new music in over 15 years. On the soundtrack, she works with stunning 16 year old singer Willa Amai on The Show and recently released a video for The Letter, a heartfelt song inspired by one of the films most emotional scenes, one which sees Frye, 44, revisiting a letter written to her adult self when she was just 16.

Motivated by the success of a partnership years in the making with Kid 90, Frye and Perry collaborated further on the Peacock reboot of the hit 80s sitcom Punky Brewster.

Perry scores the new continuation of a series which sees Frye nodding in the direction of nostalgia, revisiting not just her most famous role but one she continues to embrace both on and off the screen.

Punky is really my inner superhero. In really, truly discovering my Punky power, Ive also rediscovered my Soleil and Punky power. That inner spark that I really associate with youth, I feel it coming alive so strongly through the process of Punky and the documentary. And I feel so grateful for that, said Frye. Punky was always such a survivor. And I remember from the earliest age saying, Punky and I are so much alike! That heart and that perseverance and that strength - I really lean into her so much. Our life journeys - whether its life imitating art or art imitating life, we definitely have so many shared experiences. And I will always hold Punky close to my heart.

I spoke with Soleil Moon Frye and Linda Perry about the art of collaboration and the role of music in both Punky Brewster and Kid 90. Highlights of two separate conversations, lightly edited for length and clarity, follow below.

Freddie Prinze Jr.s character in Punky Brewster is a musician. Theres an 80s episode where music plays a big role. Generally speaking, how important is music to the Punky Brewster reboot?

SOLEIL MOON FRYE: Music is so important in life. And I feel as though with Punky, weve really always tried to be authentic in the storytelling. The original Punky had so much heart and spirit and soul. And it was so important to carry that on in the continuation. I like to call it a continuation - because to me it feels like a continuation more than a reboot, you know?

I think because music is such a part of life that it makes sense that its such an important part of the show. I love that our creators are so driven by music. We all love music and are so inspired by it. We had talked about it from the very beginning that that would be such a big part of the DNA of the show.

The show runners were really passionate about the music all along. Linda and I have spent years now working together on the documentary and it just felt so natural to continue that. Because I think, although theyre so different in so many ways, theyre both about coming of age at any stage of our lives. And Linda really does such a beautiful job of capturing that essence. I was just so grateful when she came on.

American child actor Meeno Peluce with his sister, American child actress Soleil Moon Frye, attend ... [+] the 37th Annual Primetime Emmy Awards, held at the Pasadena Civic Auditorium in Pasadena, California, 22nd September 1985. (Photo by Vinnie Zuffante/Michael Ochs Archives/Getty Images)

Certainly, nostalgia is part of a continuation like this. But in presenting the Punky character to a new generation, how important was it to kind of embrace nostalgia while still keeping the show contemporary and relevant too?

SOLEIL: Its so amazing. Because Im such a nostalgic person. And I love the 80s and I love the 90s. I love music from all decades. And I think thats something that really is part of the connective tissue of the show - that it does have a nostalgic feel and also feels very much like what we would listen to today, you know?

How did you go about creating the sound for this show?

LINDA PERRY: For me, I was just trying to put music that didnt sound like bad, cheesy sitcom music. I was just trying to do my best version of what songs would sound like and what the theme was to me. They didnt really give me a direction. I just kind of went with what felt like the right thing to do.

I didnt go 80s whatsoever with it. They had an 80s episode and I may have geared that music on that particular episode more 80s - because I thought it was fun. But it was very minimal. Its just quirky.

As far as the songs that Freddie Prinze sang, I had nothing to do with those. Those were already written - because they were to camera, to film. They shot it that way. So I wouldve loved to have gotten in on those. But those were already cleared.

NEW YORK, NEW YORK - NOVEMBER 19: Linda Perry performs on stage during the annual Make Equality ... [+] Reality Gala hosted by Equality Now on November 19, 2019 in New York City. (Photo by Dia Dipasupil/Getty Images)

Whats the goal when youre creating the quick, cohesive little guitar blasts that kind of establish a consistent tone while ending scenes and advancing narrative?

LINDA: I thought the characters were great and the kids were great. You cant go too deep, man. Its Punky Brewster. You kind of have to find a balance of whats going to be realistic. I feel like I somehow stumbled on a balance that was very guitar driven - but with a quirk to it, you know?

Theres some quirky cues in there that I was surprised they actually liked. I had an alternative feel - but it had my rock sensibility. You hit these moments where they just want like two seconds [of music]. So I tried to be as clever as I could in every moment.

My goal was just to remain authentic. I dont do series. This was the first time Id ever done anything like that. Im a deep thinker. I get really into things. Not that this wasnt deep - but it was just a different part of my brain. Because youre not writing a full song and youre not trying to get on the radio. Youre just simply trying to help narrate the story and push it along. Ego has to go. My personal tastes have to go. I have to think about what is best for the situation. I was given a job. My job was to help move the story along and still remain kind of fun. And I feel like I did that job.

I really have a good sense of emotion and finding the listening - like if someones talking. If Soleil is talking or Punky is talking, I can hear their tone and I base whatever Im going to do on that tone of the voice. Sometimes I shut off the volume and I just watch. I try to understand the body language and whats happening. And then Ill base something simply on that.

What was it like working with Soleil to create a score for Kid 90?

LINDA: That movie is so beautiful to me. I love the music created. Its an emotional documentary - a coming of age story thats very real. Theres a lot of emotional pieces in it. Its Soleils story. And her story is a very relatable story. She originally wanted 90s music. And when I started watching it, it was like, No. This is timeless. We dont want to put a date on it. So, I went with the emotions of what was happening and thats where the music came from.

Theres a couple of songs in there that were songs that actually were very prominent for her in her story that were playing on the radio or something that was left in because they were important. But the majority of it [is original new music].

NEW YORK, NEW YORK - NOVEMBER 20: (EXCLUSIVE COVERAGE) Soleil Moon Frye visits SiriusXM Studios on ... [+] November 20, 2019 in New York City. (Photo by Dia Dipasupil/Getty Images)

Music certainly plays a pretty key role in the film. As a director, how did music kind of work in tandem for you with the story to kind of drive the narrative?

SOLEIL: I am so driven by music. And music is such an important part of my process.

This documentary has been in the making since Well, I started carrying my diary at age 5 and my audio recorder at 12 and then a video camera as a teen. And music has really always been such a part of my life. And you can see it in so much of the found footage where music is playing.

Spending the last more than four years living with these tapes and the audio recordings and the voicemails and the diaries - music is always such a big part of that. So I was so grateful when I have this beautiful experience with someone [like Linda] who has been such a huge inspiration to me throughout this process. She just put her entire heart into it. She would send me music and send me more music. It was just a beautiful process. She really was such a muse for me.

Truly, Lindas score, and what she has created, is really the soundtrack and score to so many of our lives. Im so forever in the deepest gratitude.

Theres such a diverse array of 90s music in the film. Janes Addictions Perry Farrell and Danny Boy from House of Pain both appear in it. We also hear songs from artists like Eddie Vedder, Nine Inch Nails and Liz Phair. What is it about that 90s music that you still connect with so deeply today?

SOLEIL: It brings me back to a time. I love how music transports us and moves us through the decades and the times. Im a big believer in quantum physics. I love quantum physics and I feel time has folded over in some way where the teen journalist in me and the adult journalist in me come together. And it was really important to me in this documentary to allow those moments to play and authentically be themselves.

And so the music, whether its my best friend from childhood and I sitting in a car with Cranberries playing - thats what was on the radio at that time. Nine Inch Nails. Thats what we were listening to. Liz Phair! I think I mustve listened to her music every day of my life in the 90s. So I love the way that we were able to allow that music to play naturally and fit in. And then Linda did this beautiful score throughout that really is timeless.

I think that, in a lot of ways, we so often see things that are manufactured to the times. But with this, I love that, to me, it feels very much of the time and of those moments. I feel that theres a rawness to it that allows it to breathe and feel of that time.

My dream was that when people would watch it, they would watch it through their own lens and be able to live their own stories through this story. And the response of other people responding to it so beautifully and to their own life experiences and to the music, to that decade that I think is so important to so many of us - and also to those that are growing up today that wonder what that decade was like - thats been really incredible.

I cant possibly imagine looking back at my childhood in the depth that you did while making Kid 90. I would cringe. What would you say is the most important thing you learned during that process?

SOLEIL: Its really been a coming of age for both the teen me and the adult me. And that inner spark that I really associated with youth. I really thought that that inner spark was something I felt when I moved to New York City and was living in the rawness of it all. To rediscover it felt like my teenage self had left this chronological blueprint to come home to and rediscover myself once again. And that has been so illuminating, transformational and really life changing.

I will also say, I hope you get to see Linda Perrys The Letter - her incredible song and music video. Its brilliant. The fact that my teen self was asking these questions to my adult self and I really had to look within and go, Am I living with my full purpose? Have I made my life? Asking those questions and wanting to make that teen girl proud of me. And I think she would be.

At the same time, my grown up arms were able to hug the little girl who felt insecure in so many ways and felt shame in so many ways at some of her experiences - felt maybe responsible for. It was like being able to go back and say, Everything that you go through - all of the messiness, all of the love, all of the heartache and pain - every step of that is going to bring you right where you want to be.

Im so grateful for the healing.

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Linda Perry And Soleil Moon Frye On Role Of Music In Punky Brewster Reboot, Kid 90 Documentary - Forbes

Even when you understand the science, its quite hard to accept that rainbows are not real. You can think you can see one, and the person standing next to you agrees. But you are both being fooled. A rainbow is the pattern of coloured light that you perceive when you look out on a very specific set of atmospheric conditions. Because each of your eyes looks out at a slightly different angle, you actually perceive two different rainbows, one in each eye. And the person next to you sees a different rainbow again. None of those rainbows exist out there, outside of a mind.

If you can accept this, maybe you can accept the Italian physicist Carlo Rovellis perception of the universe. His new book, Helgoland, is an argument that nothing we see and experience actually exists. Just as the rainbow is a manifestation of the angle between you, some water droplets in the sky, and the sun, Rovelli tells us that the atoms, electrons, photons of light and other stuff of the universe manifest only in their interactions with each other. Individual objects are the way in which they interact, he says. Reality is a vast web of interacting entities, of which we are a part.

This relational reality is Rovellis favoured way of interpreting quantum theory, physicists best mathematical description of how the universe behaves at its most fundamental level. Quantum physics invites such interpretations because it doesnt actually have anything to say about the nature of reality. It was cobbled together, a somewhat Heath Robinson affair, on the back of late 19th-century attempts to make better electric light bulbs.

First came the assertion from the German physicist Max Planck that energy is emitted by atoms in lumps: it pours out like cereal from a packet, not like milk from a bottle. Planck could not justify his idea he called it an act of desperation. Nonetheless, it enabled him to explain the ratio of heat to visible light given out by electric light bulb filaments. After this problem was solved, one ingenious hack was forced on top of another until we ended up with a theory that could accurately describe the outcomes of any experiment involving atoms and their ilk.

Its been a huge success; developments of quantum theory have given us innumerable technological and scientific breakthroughs. At the same time, though, the theory has never been able to tell us what the constituents of the universe actually are.

This wouldnt have been a problem if a small cadre of physicists didnt insist that it should be. Early in the quantum story the 1920s and 1930s the Danish physicist Niels Bohr tried his best to stem this tide. When Einstein objected to the apparent randomness at the heart of quantum theory, Bohr allegedly told him to, Stop telling God what to do. In the face of efforts to describe what atoms were, Bohr warned that, when it comes to atoms, language can be used only as in poetry.

Bohr might as well have saved his breath. We now have myriad interpretations of quantum theory, each one an attempt to describe an underlying reality that gives rise to the results we obtain in quantum experiments. And they are, essentially, guesswork.

[see also:The moonshot delusion]

You might be familiar with some of the guesses. Theres the many worlds interpretation, for instance, which claims that you are reading this in one of a near-infinite number of alternate universes. Each one is the host of a different outcome of a single event in the quantum world. Another famous interpretation is the hidden variables idea favoured by Einstein, where an atom is not a particle as we tend to think of it, but consists of a particle and an invisible, undetectable quantum wave that guides the particles behaviour.

Rovellis relational interpretation, the central subject of his book, has its roots in the work of a young physicist called Werner Heisenberg. In the summer of 1925, Heisenberg took himself on a retreat to the small, near-treeless North Sea island of Helgoland. Here, he dedicated himself to creating a new mathematical approach to quantum theory: matrix mechanics.

You probably wont have heard of matrix mechanics because it has been suffocated by the popularity of an alternative: Erwin Schrdingers wave equation. Schrdinger treated isolated quantum entities, such as atoms, as if they were waves. When different quantum waves come together, they create what seem like otherworldly influences between quantum stuff, and strange behaviours such as one entity simultaneously existing in multiple places, or simultaneously moving in multiple directions.

In the commonly accepted view, Schrdingers waves eventually crash on the shores of their environment (such as our laboratory measuring apparatus), leaving imprints that we have generally taken as evidence for the existence of quantum particles. However, these imprints often reveal a wave-like past to these particles existence, confounding our understanding of what these things actually are. Hence the American physicist Richard Feynmans evergreen assertion that quantum physics is not actually comprehensible.

Heisenberg appreciated this far earlier than most, declaring Schrdingers quantum waves to be repulsive and crap, and offering his matrix mechanics the pure, unadorned mathematics of how one state of an atom relates to another state in the moment of a measurement of its properties as an alternative. Even Schrdinger conceded that this was more accurate. It is better, he admitted, to consider a particle not as a permanent entity but rather as an instantaneous event.

And according to Rovelli, this series of fortunate events is all there is. The properties of a quantum object are only real with respect to some other object at some moment, just as a rainbow is only real in the mind of an observer at the moment of its observation. Whats more, a third quantum object might not perceive those same properties at all. Putting it another way, reality is relative and truth is subjective. Is it possible that a fact might be real with respect to you and not real with respect to me? Rovelli asks. Quantum theory, I believe, is the discovery that the answer is yes.

Rovelli doesnt really push things much further than that; you wont come away from Helgoland with a sense that you finally understand the true nature of reality. He doesnt explain, for instance, what it is that is doing the interacting, if the entities are nothing but their interactions. But it is a pleasure to travel in his company regardless.

Thats partly because, as with Rovellis previous books, the prose is translated from his Italian by the writer Erica Segre and Simon Carnell, a poet, who have made it a delight to read. They describe reality as a luxuriant stratification: snow-covered mountains and forests, the smile of friends, the rumble of the underground on dirty winter mornings With phrasing like this, who cares if there are no real answers?

And lets not pretend that any books on quantum physics can contain satisfying answers about what reality is. How could they, when the theory is not designed to give any? Using it as a guide to the nature of reality leaves us stranded in the mist, like Heisenberg lost in his thoughts on Helgoland.

Aware of the inadequacy of the science, Rovelli offers a second source for his intuitions. In the last third of the book we are seated with him at the feet of the Buddhist philosopher Nagarjuna, who teaches that there is nothing that exists in itself, independently from something else. For me as a human being, Rovelli says, Nagarjuna teaches the serenity, the lightness and the shining beauty of the world: we are nothing but images of images. Reality, including our selves, is nothing but a thin and fragile veil, beyond which there is nothing. Much like that damned rainbow.

Michael Brookss books include The Quantum Astrologers Handbook (Scribe)

HelgolandCarlo RovelliAllen Lane, 208pp, 20

[see also:This risks creating an arms race: inside Europes battle over the future of quantum computing]

Read more here:

Carlo Rovelli's Helgoland argues that all reality is relative - New Statesman

The Muon g-2 ring sits in its detector hall amidst electronics racks, the muon beamline, and other equipment. Credit: Reidar Hahn, Fermilab

The first results from the Muon g-2 experiment at the U.S. Department of Energys Fermi National Accelerator Laboratory have revealed that fundamental particles called muons behave in a way that is not predicted by scientists best theory to date, the Standard Model of particle physics. This landmark result, published recently in Physical Review Letters, confirms a discrepancy that has been gnawing at researchers for decades.

The strong evidence that muons deviate from the Standard Model calculation might hint at exciting new physics. The muons in this experiment act as a window into the subatomic world and could be interacting with yet-undiscovered particles or forces.

This experiment is a bit like a detective story, said team member David Hertzog, a University of Washington professor of physics and a founding spokesperson of the experiment. We have analyzed data from the Muon g-2s inaugural run at Fermilab, and discovered that the Standard Model alone cannot explain what weve found. Something else, perhaps beyond the Standard Model, may be required.

The Muon g-2 experiment is an international collaboration between Fermilab in Illinois and more than 200 scientists from 35 institutions in seven countries. UW scientists have been an integral part of the team through the Precision Muon Physics Group constructing sensitive instruments and sensors for the experiment, and leading data analysis endeavors. In addition to Hertzog, current UW faculty and lead scientists involved include Peter Kammel, research professor of physics; Erik Swanson, a research engineer with the UWs Center for Experimental Nuclear Physics and Astrophysics, or CENPA; Jarek Kaspar, a research scientist; and Alejandro Garcia, a professor of physics.

Lead fluoride crystals, which are used in detectors designed and constructed at the UW that measure muon decay products for the Muon g-2 experiment. Credit: University of Washington

The UW custom-built instrumentation would not have been possible without the extraordinary dedication and expertise of our CENPA technical staff, who work closely with our postdocs and graduate students, said Hertzog.

A muon is about 200 times as massive as its cousin, the electron. They occur naturally when cosmic rays strike Earths atmosphere. Particle accelerators at Fermilab can produce them in large numbers. Like electrons, muons act as if they have a tiny internal magnet. In a strong magnetic field, the direction of the muons magnet precesses, or wobbles, much like the axis of a spinning top. The strength of the internal magnet determines the rate that the muon precesses in an external magnetic field and is described by a number known as the g-factor. This number can be calculated with ultra-high precision.

As the muons circulate in the Muon g-2 magnet, they also interact with a quantum foam of subatomic particles popping in and out of existence. Interactions with these short-lived particles affect the value of the g-factor, causing the muons precession to speed up or slow down slightly. The Standard Model predicts with high precision what the value of this so-called anomalous magnetic moment should be. But if the quantum foam contains additional forces or particles not accounted for by the Standard Model, that would tweak the muon g-factor further.

Hertzog, then at the University of Illinois, was one of the lead scientists on the predecessor experiment at Brookhaven National Laboratory. That endeavor concluded in 2001 and offered hints that the muons behavior disagreed with the Standard Model. The new measurement from the Muon g-2 experiment at Fermilab strongly agrees with the value found at Brookhaven and diverges from theory with the most precise measurement to date.

The accepted theoretical values for the muon are:

The new experimental world-average results announced by the Muon g-2 collaboration today are:

The combined results from Fermilab and Brookhaven show a difference with theoretical predictions at a significance of 4.2 sigma, a little shy of the 5 sigma or 5 standard deviations that scientists prefer as a claim of discovery. But it is still compelling evidence of new physics. The chance that the results are a statistical fluctuation is about 1 in 40,000.

This result from the first run of the Fermilab Muon g-2 experiment is arguably the most highly anticipated result in particle physics over the last years, said Martin Hoferichter, an assistant professor at the University of Bern and member of the theory collaboration that predicted the Standard Model value. After almost a decade, it is great to see this huge effort finally coming to fruition.

UW research engineer Erik Swanson with equipment used to measure magnetic fields in the Muon g-2 experiment. Credit: University of Washington

The Fermilab experiment, which is ongoing, reuses the main component from the Brookhaven experiment, a 50-foot-diameter superconducting magnetic storage ring. In 2013, it was transported 3,200 miles by land and sea from Long Island to the Chicago suburbs, where scientists could take advantage of Fermilabs particle accelerator and produce the most intense beam of muons in the United States. Over the next four years, researchers assembled the experiment; tuned and calibrated an incredibly uniform magnetic field; developed new techniques, instrumentation, and simulations; and thoroughly tested the entire system.

The Muon g-2 experiment sends a beam of muons into the storage ring, where they circulate thousands of times at nearly the speed of light. Detectors lining the ring allow scientists to determine how fast the muons are wobbling.

Many of the sensors and detectors at Fermilab were constructed at the UW, such as instruments to measure the muon beam as it enters the storage ring and to detect the telltale particles that arise when muons decay. Dozens of scientists including faculty, postdoctoral researchers, technicians, graduate students and undergraduate students have worked to assemble these sensitive instruments at the UW and then install and monitor them at Fermilab.

UW scientists have also been involved in theoretical work around the Muon g-2 collaboration.

The prospects of the new result triggered a coordinated theory effort to provide our experimental colleagues with a robust, consensus Standard-Model prediction, said Hoferichter, who was a UW research assistant professor from 2015 to 2019. Future runs will motivate further improvements, to allow for a conclusive statement if physics beyond the Standard Model is lurking in the anomalous magnetic moment of the muon.

In its first year of operation, in 2018, the Fermilab experiment collected more data than all prior muon g-factor experiments combined. The Muon g-2 collaboration has now finished analyzing the motion of more than 8 billion muons from that first run. The UW team was central to this effort, leading to four doctoral theses to date.

Data analysis on the second and third runs of the experiment is under way; the fourth run is ongoing, and a fifth run is planned. Combining the results from all five runs will give scientists an even more precise measurement of the muons wobble, revealing with greater certainty whether new physics is hiding within the quantum foam.

So far we have analyzed less than 6% of the data that the experiment will eventually collect, said Fermilab scientist Chris Polly, who is a co-spokesperson for the current experiment and was a lead University of Illinois graduate student under Hertzog during the Brookhaven experiment. Although these first results are telling us that there is an intriguing difference with the Standard Model, we will learn much more in the next couple of years.

With these exciting results our team, in particular our students, is enthusiastic to push hard on the remaining data analysis and future data-taking in order to realize our ultimate precision goal, said Kammel.

For more on this research:

Reference: Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm by B. Abi et al. (Muon g2 Collaboration), 7 April 2021, Physical Review Letters.DOI: 10.1103/PhysRevLett.126.141801

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Muon g-2 Particle Accelerator Experiment Results Are Not Explained by Our Current Theories of Physics - SciTechDaily