Phase Space Tomography Overview

• Phase space tomography is the process of reconstructing multidimensional distributions from lower-dimensional projections, enabling detailed analysis of beam dynamics and quantum states.
• It employs reconstruction algorithms such as MLEM, SART, and machine learning methods to address inverse problems and improve resolution in noisy data.
• Practical applications include accelerator diagnostics, quantum state imaging, and biomedical electromagnetics, all leveraging inversion of Radon-type projections.

Phase space tomography is the reconstruction of a phase-space distribution or quasiprobability distribution from a set of lower-dimensional projections obtained after controlled evolution, transport, or measurement. In accelerator physics the reconstructed object is commonly a transverse or longitudinal beam density such as $f(x,x')$, $f(y,y')$, or $f(z,\delta)$; in quantum optics and related fields it is often a Wigner, $Q$, or $P$ function; and in some many-body settings the term is used more broadly for indirect mapping of a state’s support in classical phase space through fluctuation data (Ye et al., 8 Aug 2025, Koczor et al., 2017, Khripkov et al., 2012).

1. Conceptual scope

In beam physics, phase space encodes position and angle or momentum with respect to a reference trajectory. For one transverse plane this is typically $(x,x')$ or $(x,p_x)$, with $x' = dx/dz$; in the full transverse problem one has $(x,x',y,y')$, and in extensions that include longitudinal structure one may use $(x,x',y,y',t)$ or $f(y,y')$0 (Ye et al., 8 Aug 2025, Jaster-Merz et al., 2023, Hoover et al., 2024). The corresponding distribution function gives the density of particles in that space and carries the information from which emittances, Courant–Snyder parameters, halo, coupling, and slice structure are derived.

In quantum and wave settings, phase space denotes a representation over conjugate variables such as $f(y,y')$1, $f(y,y')$2, $f(y,y')$3, or $f(y,y')$4. The relevant objects are quasiprobability distributions, especially the Wigner function. For finite-dimensional quantum systems, continuous phase-space representations can be defined on the sphere $f(y,y')$5 for spin-$f(y,y')$6 systems, while discrete Wigner functions are defined on a $f(y,y')$7 grid for an $f(y,y')$8-dimensional Hilbert space (Koczor et al., 2017, Khushwani et al., 2023). In optomechanics, the phase-space distribution of a mechanical resonator is built from quadratures $f(y,y')$9 and $f(z,\delta)$0, and in optical coherence theory tomography may target a mutual intensity or ambiguity function rather than a particle density (Suchoi et al., 2014, Tian et al., 2011).

A recurring point across these domains is that the measured quantity is not the phase-space distribution itself but a projection or marginal of it. This distinguishes tomography from direct profile measurements, standard second-moment analysis, or ellipse fitting. In the Fermilab Linac case, the explicit motivation was that the beam did not occupy a strictly elliptical phase space, so conventional emittance methods were inadequate and tomography was required to retrieve the actual particle distribution in phase space (Garcia et al., 2013).

2. Projection geometry and forward models

The formal core of phase space tomography is a projection operator. In the simplest two-dimensional setting, if the unknown object is $f(z,\delta)$1, a projection at angle $f(z,\delta)$2 and coordinate $f(z,\delta)$3 is written as

$f(z,\delta)$4

This Radon-transform form appears explicitly in accelerator and optical formulations and makes clear that tomography reconstructs a multidimensional distribution from line integrals measured at different effective angles (Hermann et al., 2021, Funkner et al., 2019).

In quadrupole-based transverse beam tomography, the effective projection angle is generated by linear transport. For a quadrupole setting $f(z,\delta)$5, the screen coordinate obeys

$f(z,\delta)$6

and the FACET-II implementation defines

$f(z,\delta)$7

The measured screen profile is therefore a scaled projection of the upstream phase-space distribution at angle $f(z,\delta)$8, and scanning $f(z,\delta)$9 yields a sinogram (Ye et al., 8 Aug 2025).

Other experimental systems realize the same geometry differently. Nanofabricated wire scanners at SwissFEL obtain projections by scanning wires at different transverse angles and at several longitudinal positions around the waist, so that wire angle and phase advance jointly supply the projection diversity (Hermann et al., 2021). In storage-ring longitudinal tomography, successive bunch profiles become different projections because synchrotron motion rotates the longitudinal distribution in $Q$0 space; concatenating profiles over half a synchrotron period yields a sinogram (Funkner et al., 2019). In optical phase-space-time-frequency tomography, balanced heterodyne detection with a designed local oscillator yields a convolution between the unknown Wigner distribution and the local-oscillator Wigner distribution, turning the measurement into a tomographic sampling of $Q$1 space (1101.01323).

The quantum-optical formulation uses the Wigner function as the phase-space object. A representative form is

$Q$2

which exhibits phase space as a double Fourier transform of a two-point coherence function (1101.01323). In finite-dimensional quantum tomography, the same role is played by rotated parity operators or discrete phase-space point operators, so that $Q$3 becomes the tomographic target (Koczor et al., 2017, Khushwani et al., 2023).

3. Inverse problems and reconstruction algorithms

The reconstruction step converts measured projections into an estimate of the underlying distribution. One widely used method is Maximum Likelihood Expectation Maximization (MLEM), used at FACET-II for transverse phase-space reconstruction. After discretizing the unknown density into pixels $Q$4 and the measurements into bins $Q$5, the forward model is

$Q$6

with $Q$7 determined by transport and geometry. Under Poisson statistics, MLEM iteratively updates the estimate and is well suited to non-Gaussian distributions and structured features (Ye et al., 8 Aug 2025).

SART and related algebraic methods are common in higher-dimensional accelerator tomography. In the 5D reconstruction proposed for ARES and demonstrated experimentally at FLASHForward, SART is used for 3D $Q$8 reconstruction at the screen and for subsequent slice-by-slice transverse reconstruction, with positivity and charge invariance imposed as constraints; two iterations were reported as sufficient because one iteration already gave a good reconstruction and additional iterations enhanced sharp features but could increase noise (Jaster-Merz et al., 2023, Jaster-Merz et al., 19 May 2025).

Particle-based algorithms provide another route. The SwissFEL wire-scanner study represents the 4D transverse phase space by macro-particles convolved with Gaussian kernels rather than by a fixed grid, which avoids a direct voxel representation and builds the density from an ensemble of smoothed particles (Hermann et al., 2021). In optical coherence tomography of mutual intensity, the inverse problem can be written as a linear system $Q$9, after which low-rank matrix recovery and nuclear-norm minimization are used under a low-entropy assumption (Tian et al., 2011).

Maximum-entropy reconstruction addresses underdetermination by selecting the most conservative distribution consistent with the data. For phase-space density $P$0 and prior $P$1, the relative entropy functional is

$P$2

and the maximum-entropy solution has the multiplicative form

$P$3

This formulation is used explicitly in 6D tomography with normalizing flows and in 4D hadron-beam tomography from 1D wire-scanner data (Hoover et al., 2024, Hoover, 2024).

A central practical issue is that high-dimensional tomography is often severely underdetermined. This motivates priors, positivity constraints, covariance constraints, low-rank assumptions, or measurement-adapted representations. A plausible implication is that the choice of reconstruction space is itself part of the measurement model, not merely a numerical convenience.

4. Accelerator-beam implementations

Accelerator physics has produced some of the most operationally mature forms of phase space tomography. At FACET-II, transverse phase-space tomography was used to diagnose a two-bunch configuration in which drive and witness bunches were spatially superimposed on a screen but had different transverse phase spaces because they experienced different RF phases, energies, and focusing. The combined horizontal phase space exhibited an X-shaped pattern, and separate witness-only and drive-only measurements showed that the two arms of the X corresponded to the two individual bunch phase spaces. The same study reported charge-dependent stretching of the phase space and a horizontal flip when the bunch charge was halved in a zero-separation configuration, which was interpreted as evidence of unexpected space-charge-dominated dynamics (Ye et al., 8 Aug 2025).

At SwissFEL, nanofabricated gold wire scanners with $P$4 width and edge roughness of $P$5 were scanned through micrometer-scale beams. Measurements at six longitudinal positions around the waist, together with nine wire orientations, provided sufficient phase advance diversity to reconstruct the transverse phase-space density of micrometer-sized electron beams (Hermann et al., 2021). In the Fermilab Linac MTA line, a 10 m dispersion-free and magnet-free straight with upstream quadrupole-triplet control was designed explicitly so that varying waist conditions would rotate the phase space and enable deconvolution of multiwire profile data (Garcia et al., 2013).

Longitudinal beam tomography has comparable significance. At KARA, turn-by-turn electro-optical measurements of bunch profiles at MHz repetition rates were assembled into sinograms and inverted by filtered back-projection, revealing microstructuring and nonequilibrium longitudinal phase-space dynamics in the sawtooth bursting regime (Funkner et al., 2019). At CERN PSB and SPS, longitudinal phase-space tomography was used not primarily as a state-imaging tool but as a precision metrological diagnostic: the discrepancy between measured and reconstructed bunch profiles depended strongly on the assumed RF voltage and phase, allowing voltage calibration to the $P$6 level after correction factors were applied in the LLRF (Quartullo et al., 2023).

These examples show that tomography in accelerator settings is not restricted to visualization. It functions as a beam diagnostic for emittance analysis, bunch separation, slice characterization, collective-effects studies, and machine calibration.

5. High-dimensional tomography, machine learning, and resolution

The progression from 2D to 4D, 5D, and 6D tomography changes both the geometry and the inverse problem. A 5D reconstruction method was proposed for ARES by combining quadrupole-based transverse phase-space tomography with the adjustable streaking angle of a polarizable X-band transverse deflection structure, so that a 4D transverse tomography could be performed for each time slice and the slices stacked into $P$7 (Jaster-Merz et al., 2023). The first experimental demonstration at FLASHForward reconstructed the 5D phase-space distribution of a GeV-class electron bunch, used the measured phase space to generate a particle distribution for simulations, and extracted transverse 4D slice emittance (Jaster-Merz et al., 19 May 2025).

For hadron beams, the SNS study inferred a 4D transverse phase-space density from only 1D wire-scanner profiles by maximizing entropy subject to measurement constraints. The reconstructed distribution reproduced the measured profiles down to the noise level, and simulation studies indicated that the problem was reasonably well constrained (Hoover, 2024). For full 6D phase space, normalizing flows were introduced as invertible generative models that approximate the maximum-entropy solution while remaining computationally viable for large measurement sets (Hoover et al., 2024).

Machine learning has also been used to account for imperfect beamline knowledge. At CLARA, an autoencoder-based workflow reconstructed the 4D transverse phase space from a stack of screen images while simultaneously estimating unknown quadrupole-strength errors, using an extended latent space that encoded both the phase-space representation and magnet-error parameters (Wolski et al., 2024).

A distinct but related issue is resolution. The paper “Resolving the phase space” argues that not all fine structure visible in a reconstructed Wigner function is necessarily resolved by the measurement itself. The effective resolution is determined by a sampling operator linked to the Gram matrix of the measurement, which acts analogously to a transfer function in imaging and defines the experimentally accessible degrees of freedom and reconstruction bandwidth (Hradil et al., 28 May 2026). This provides an operational criterion for distinguishing resolved structure from artifacts induced by incomplete sampling or reconstruction assumptions.

6. Quantum, atomic, and hybrid formulations

Outside accelerator physics, phase space tomography is central to quantum-state reconstruction. For finite-dimensional spins, continuous $P$8-parametrized phase-space representations can be written as

$P$9

with $(x,x')$0 on the sphere and $(x,x')$1 a generalized parity operator. Tomography then reconstructs the phase-space function pointwise from rotated projective measurements (Koczor et al., 2017). In discrete quantum systems, discrete Wigner tomography estimates $(x,x')$2, and selective Wigner phase-space tomography exploits sparsity in the Wigner representation to infer dynamics from only a subset of phase-space points (Khushwani et al., 2023).

In optomechanics, phase-space tomography was used to reconstruct the phase-space distribution of a mechanical resonator coupled to microwave and optical cavities. Time-resolved tomography tracked the transition from optomechanical cooling to self-excited oscillation, and the reconstructed distributions were compared with a Fokker–Planck description (Suchoi et al., 2014). In cold-atom dynamics, harmonic evolution was used to rotate the phase space of trapped $(x,x')$3 atoms, and filtered backprojection reconstructed the phase-space distribution, revealing sensitivity to weak corrugations of the trapping potential through angular-velocity dispersion of isoenergetic trajectories (Zhou et al., 2014).

The meaning of “phase space tomography” is not identical in every field. In the bosonic Josephson junction literature, temporal fluctuations of one-body coherence were proposed as a probe for phase space tomography because the fluctuation variance factorizes into a participation-number term and a semiclassical function that reflects phase-space characteristics of the observable; this is a diagnostic mapping of phase-space structure rather than a full quasi-probability reconstruction (Khripkov et al., 2012). A plausible implication is that the term denotes a family of inverse and inference procedures unified by their phase-space target, not a single experimental protocol.

Recent proposals extend the idea to biomedical electromagnetics. Quantum Phase-Space Tomography for Electromagnetic Biomaterial Imaging proposes preparing a structured quantum electromagnetic probe, performing full quantum state tomography of the outgoing field, reconstructing a Wigner distribution, and then projecting the result onto an analytically derived tissue-response manifold based on multilayer Maxwell and Cole–Cole modeling (Settimi, 30 Aug 2025). This suggests a transfer of phase-space methods from quantum optics into metrological imaging.

Phase space tomography therefore encompasses a set of closely related practices: inversion of Radon-type projections, reconstruction of beam or field distributions under known transport, entropy-regularized inference in high dimensions, and resolution-aware interpretation of reconstructed structure. Across these variants, the common principle is that the experimentally accessible data are projections, while the scientific objective is the underlying distribution in phase space.