Helical Optical Tube (HOT): 3D Atom Trapping

• Helical Optical Tube (HOT) is a 3D optical trapping potential formed by interfering Laguerre–Gaussian beams with opposite winding numbers, creating helical channels for atom guidance.
• It combines a ring-shaped radial envelope with a helical interference pattern to yield quantized transverse modes and Mathieu-type longitudinal motion for precise matter-wave control.
• HOT applications include coherent atom beam shaping, matter-wave interferometry, and concepts like twisted atom lasers, enabling dynamic studies of quantum transport and interference.

A Helical Optical Tube (HOT) is a three-dimensional helical optical trapping potential generated by the interference of two counter-propagating Laguerre–Gaussian (LG) beams with opposite winding numbers, such that the optical dipole minima form one or more helical guiding channels for atoms. In the cold-atom literature, it is treated as a helicoidal potential, a three-dimensional helical lattice structure, and a waveguide for atomic matter waves; the resulting confinement geometry has been used to analyze stationary quantum states, rotating transport, and the free release of single atoms and Bose–Einstein condensates (BECs), including proposals framed as routes toward twisted BECs and directed atom lasers (Lembessis et al., 2022, Jaouadi et al., 19 Aug 2025).

1. Definition and optical construction

In the HOT literature, the basic optical arrangement consists of two counter-propagating LG beams with opposite topological charges $+\ell$ and $-\ell$. Each beam carries photon orbital angular momentum $\ell\hbar$, and their interference produces a helicoidal intensity pattern whose minima wind around the optical axis. The 2025 release-dynamics treatment describes the HOT as a structured-light optical dipole trap whose minima form one or more helical guiding channels for atoms and explicitly states that it “serves as a waveguide for atomic matter waves” (Jaouadi et al., 19 Aug 2025).

The central physical picture is that atoms are trapped along a twisted trough in space rather than in a straight guide or a simple ring. In the far-detuned dipole-trap regime, the atoms experience a time-averaged optical dipole potential proportional to intensity. The essential ingredient in this construction is the OAM structure and interference of the LG beams; the 2025 treatment states that polarization is not central in the HOT construction, whereas the 2016 rotating-HOT study specifies equal polarization, equal power, and equal beam waist for the two beams (Jaouadi et al., 19 Aug 2025, Rsheed et al., 2016).

A rotating HOT is obtained by introducing a small frequency difference between the two counter-propagating beams. In that case, the helical interference pattern rotates with angular velocity

$\Omega_R=\frac{\Delta\omega}{2l},$

where $\Delta\omega=\omega_1-\omega_2$. This converts the HOT from a static helical dipole guide into a rotating helical conveyor for cold atoms (Rsheed et al., 2016).

2. Optical potential, geometry, and local mode structure

The principal HOT potential used in the 2025 quantum-release study is

$$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$

with

$w(z) = w_0\sqrt{1 + (z/z_R)^2}.$

Here $w_0$ is the beam waist, $z_R$ the Rayleigh range, $k$ the optical wave number, $-\ell$0 the trap-depth scale, and $-\ell$1 the LG winding number (Jaouadi et al., 19 Aug 2025).

This expression combines a ring-shaped LG radial envelope with a helical interference factor $-\ell$2. The helical geometry follows directly from the phase-locking condition

$-\ell$3

which yields

$-\ell$4

Thus the HOT is neither merely a standing wave in $-\ell$5 nor merely a transverse ring trap: it combines radial localization near the LG intensity ring with axial–azimuthal coupling that defines a helical guide (Jaouadi et al., 19 Aug 2025).

The same study identifies several basic geometric characteristics. The centerline radius is approximately

$-\ell$6

which becomes nearly constant in the paraxial region $-\ell$7,

$-\ell$8

The pitch magnitude is

$-\ell$9

The potential landscape contains $\ell\hbar$0 intertwined minima winding around the optical axis, and the sign of $\ell\hbar$1 fixes the sense of azimuthal winding for a given propagation convention (Jaouadi et al., 19 Aug 2025).

For local analysis, the HOT is recast in adapted helical coordinates. In that form the potential is written as

$\ell\hbar$2

which makes the helicoidal symmetry explicit. Near the troughs, the stationary-state analysis yields a separated wavefunction

$\ell\hbar$3

with harmonic confinement in the two transverse directions and Mathieu-type structure along the helicoidal trough. The corresponding energy spectrum is

$\ell\hbar$4

which formalizes the HOT as a waveguide-like tube with quantized transverse modes and structured longitudinal motion (Jaouadi et al., 19 Aug 2025).

3. Stationary quantum states in a HOT

The stationary quantum theory of atoms in a HOT is developed explicitly for a two-level Rb atom in the far-detuned dipole regime. In that treatment, the HOT is again formed by two counter-propagating LG beams with opposite winding numbers, and the atomic center-of-mass motion separates into three quantized sectors labeled by a triplet of quantum numbers. The paper states that the corresponding wavefunctions exhibit a distinctly twisted, spiral-like 3D spatial shape (Lembessis et al., 2022).

In the 2022 formulation, the wavefunction separates into radial, binormal/helical-transverse, and helical-longitudinal factors. Two sectors reduce to harmonic-oscillator equations, while the helical coordinate yields a position-dependent-mass problem that is transformed to a Mathieu equation. The helical-sector eigenfunctions are therefore expressed in terms of Mathieu functions, and the total stationary states take the form

$\ell\hbar$5

The full probability density wraps around the $\ell\hbar$6-axis and follows the spiral geometry of the tube (Lembessis et al., 2022).

The 2025 release-dynamics study adopts a closely related separation and emphasizes that the trapped ground state $\ell\hbar$7 has a helical density distribution centered around the optical intensity ring. The initial atomic probability density is described as ring-shaped in cross-section, helically arranged in three dimensions, and localized near the helical trough. The same work is careful to note that it does not explicitly write an atomic wavefunction with a factor $\ell\hbar$8 or claim a sharply quantized atomic OAM eigenstate before release; what is directly supported is a helically localized matter wave with geometry-imprinted phase coherence (Jaouadi et al., 19 Aug 2025).

For the BEC case, the initial state is treated in the Thomas–Fermi approximation,

$\ell\hbar$9

The condensate therefore occupies the region where $\Omega_R=\frac{\Delta\omega}{2l},$0, so its density follows the inverse HOT geometry and inherits a helicoidal density pattern (Jaouadi et al., 19 Aug 2025).

Representative bound-state scales have also been reported. For the $\Omega_R=\frac{\Delta\omega}{2l},$1 transition in Rb with

$\Omega_R=\frac{\Delta\omega}{2l},$2

the 2022 paper gives

$\Omega_R=\frac{\Delta\omega}{2l},$3

indicating that the HOT can support a large manifold of bound center-of-mass states in that regime (Lembessis et al., 2022).

4. Free release, matter-wave evolution, and the twisted atom-laser proposal

When the HOT is switched off, the released atomic wavefunction evolves under gravity according to

$\Omega_R=\frac{\Delta\omega}{2l},$4

The 2025 study transforms this to a freely falling frame,

$\Omega_R=\frac{\Delta\omega}{2l},$5

so that the evolution becomes free expansion in the comoving frame: $\Omega_R=\frac{\Delta\omega}{2l},$6 The time-dependent Schrödinger equation and, when relevant, the Gross–Pitaevskii equation are solved numerically with the split-step Fourier method (Jaouadi et al., 19 Aug 2025).

The central HOT-specific result is that release from a helical trap does not generate a generic Gaussian-like falling packet. Instead, the initial ring/helical geometry produces rapid radial expansion, concentric self-interference fringes, temporary refocusing/revivals, and preservation of structured, twisted spatial features. The study states that interference fringes appear between $\Omega_R=\frac{\Delta\omega}{2l},$7 and $\Omega_R=\frac{\Delta\omega}{2l},$8 ms, and that around $\Omega_R=\frac{\Delta\omega}{2l},$9 ms a temporary refocusing occurs, interpreted as constructive self-interference enabled by the curved ring/helical geometry (Jaouadi et al., 19 Aug 2025).

The same work identifies internal time scales inherited from the HOT: $\Delta\omega=\omega_1-\omega_2$0 with reported values

$\Delta\omega=\omega_1-\omega_2$1

The revivals are attributed mainly to the $\Delta\omega=\omega_1-\omega_2$2-coordinate dynamics along the helicoidal trough, where the effective potential has a Mathieu spectrum and acts as an internal clock (Jaouadi et al., 19 Aug 2025).

For released BECs, the reported dynamics include helical unwrapping or partial washing out of helicity, interaction-driven self-focusing, axial density redistribution, interference fringes, and formation of narrow high-density output channels under tighter confinement or higher helicity. For $\Delta\omega=\omega_1-\omega_2$3, tighter waist $\Delta\omega=\omega_1-\omega_2$4 produces a stronger central density peak and more pronounced nonlinear dynamics than $\Delta\omega=\omega_1-\omega_2$5; for higher $\Delta\omega=\omega_1-\omega_2$6, the initial condensate contains multiple intertwined strands, and larger $\Delta\omega=\omega_1-\omega_2$7 leads to stronger localization and stronger nonlinear focusing (Jaouadi et al., 19 Aug 2025).

These release calculations motivate the paper’s proposal of a twisted atom laser. The modeling, however, is limited to HOT confinement followed by free release under gravity. The work explicitly does not model a full atom-laser output-coupling mechanism in the conventional sense. The “twisted atom laser” is therefore a conceptual and dynamical pathway, not a complete lasing-device model (Jaouadi et al., 19 Aug 2025).

5. Rotating HOTs and the optical Archimedes’ screw

The 2016 rotating-HOT study shifts attention from quantum stationary states to the classical or semiclassical center-of-mass dynamics of a cold atom inside a vertical rotating HOT in gravity. In the rotating frame,

$\Delta\omega=\omega_1-\omega_2$8

the optical potential becomes time independent, while the Lagrangian acquires the usual inertial terms of a rotating frame (Rsheed et al., 2016).

The resulting equations of motion contain both centrifugal and Coriolis contributions. The paper emphasizes a specific mechanism: the centrifugal force first pushes the atom radially, the resulting radial motion generates a Coriolis force, that Coriolis force drives azimuthal motion opposite to the HOT rotation, and because the trap itself is helical, the induced azimuthal drift translates into vertical motion along the tube (Rsheed et al., 2016).

For a left-handed HOT, the reported directional rule is:

• counter-clockwise HOT rotation ($\Delta\omega=\omega_1-\omega_2$9) drives the atom upward;
• clockwise HOT rotation ($$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$0) drives the atom downward.

At sufficiently small $$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$1, the atom remains globally trapped and performs axial oscillations between turning points. As $$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$2 increases, the effective global oscillation frequency decreases, and when the global axial frequency vanishes while radial confinement remains, the rotating HOT operates as an Optical Archimedes’ Screw (OAS) (Rsheed et al., 2016).

For one numerical parameter set based on $$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$3, $$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$4, $$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$5, and initial velocity $$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$6, the study reports: $$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$7 for upward transport,

$$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$8

for downward transport, and radial escape thresholds

$$U(\rho,\phi,z) = V_0 \left( \frac{\sqrt{2}\rho}{w(z)} \right)^{2|\ell|} \exp\left(-\frac{2\rho^2}{w(z)^2}\right)\cos^2(kz + \ell\phi),$$9

The OAS operating regime is therefore

$w(z) = w_0\sqrt{1 + (z/z_R)^2}.$0

where $w(z) = w_0\sqrt{1 + (z/z_R)^2}.$1 is the axial-release threshold and $w(z) = w_0\sqrt{1 + (z/z_R)^2}.$2 the radial-escape threshold (Rsheed et al., 2016).

The paper further states that the transport window can be controlled by adjusting the orbital angular momentum number $w(z) = w_0\sqrt{1 + (z/z_R)^2}.$3, beam waist $w(z) = w_0\sqrt{1 + (z/z_R)^2}.$4, power $w(z) = w_0\sqrt{1 + (z/z_R)^2}.$5, and detuning $w(z) = w_0\sqrt{1 + (z/z_R)^2}.$6, and that stronger dipole potentials yield larger threshold values and a wider OAS operating window (Rsheed et al., 2016).

6. Related helical-guiding frameworks and analogs

HOT research sits within a broader class of helical and tubular wave-guiding structures. The 2025 HOT paper itself places the HOT relative to optical-vortex ring traps, Ferris-wheel lattices from co-propagating opposite-OAM beams, and more conventional cylindrically symmetric dipole traps and lattices, identifying the HOT as a three-dimensional helical extension of the optical Ferris wheel in which counter-propagation converts transverse petal structure into a helicoidal guide (Jaouadi et al., 19 Aug 2025).

A rigorous electromagnetic analog is provided by the theory of helical Bloch modes in $w(z) = w_0\sqrt{1 + (z/z_R)^2}.$7-fold rotationally symmetric rings of coupled spiralling optical waveguides. That work develops a vector coupled-mode description in local Frenet–Serret frames for a helically twisted, azimuthally periodic guiding structure. A plausible implication is that segmented or discretized HOT-like guides should inherit twist-induced frame rotation, azimuthal Bloch phase structure, and polarization-dependent propagation-constant shifts (Chen et al., 2020).

Another related direction is the use of flat helical nanosieves to generate nondiffracting hollow vortex beams, helical axicons, and helicon waves. These are not cold-atom HOTs, but they realize elongated hollow-core vortex propagation and rotating longitudinal intensity patterns in free space. This suggests a phase-engineering route to optical fields that are geometrically close to tube-like or helically modulated guides (Mei et al., 2016).

The literature also contains more indirect analogs. A $w(z) = w_0\sqrt{1 + (z/z_R)^2}.$8-symmetric non-Hermitian square-lattice tube with an embedded SSH ring is described explicitly as not a HOT in the usual continuum-waveguide sense, but as a lattice/photonic analog of a HOT because it supports helicity-selective transport, mode conversion, and directional amplification in a tube-like geometry (Zhang et al., 2019). Likewise, ultraslow helical optical bullets in an EIT medium are stated explicitly not to be HOTs, yet they provide a Maxwell–Bloch-based framework for ring confinement, controlled azimuthal motion, and a helical trajectory of a localized optical packet in a coherent atomic medium (Hang et al., 2014).

7. Applications, experimental relevance, and interpretive cautions

The applications emphasized in the HOT literature are coherent atom beam shaping, matter-wave interferometry, guided transport of quantum matter, precision inertial sensing, rotational sensing, and, more generally, quantum technologies based on structured matter-wave modes. The argument is that a HOT provides coherent guided initial modes, geometry-imprinted phase and density structure, and access to linear momentum, angular momentum, and helicity or topology as matter-wave control parameters (Jaouadi et al., 19 Aug 2025).

The 2025 release calculations use $w(z) = w_0\sqrt{1 + (z/z_R)^2}.$9 and quote representative parameters such as $w_0$0 mW, detuning $w_0$1 Hz or $w_0$2 Hz, beam waists $w_0$3, winding number $w_0$4 in baseline simulations, and trap depth $w_0$5. The paper presents these as typical experimental conditions and notes that the HOT relies on established structured-light atom-optics প্রযুক্তures: LG-beam generation, topological-charge control, counter-propagating alignment, and far-detuned dipole trapping (Jaouadi et al., 19 Aug 2025).

Several interpretive cautions are explicit in the cited work. A common misconception is to equate a “twisted” trapped state with a definite atomic OAM eigenstate; the 2025 paper does not support that stronger statement and instead supports helically localized matter waves with geometry-imprinted phase coherence (Jaouadi et al., 19 Aug 2025). Another misconception is to treat the HOT as simply a ring trap or a longitudinal standing wave; the defining feature is the coupled phase factor $w_0$6, which creates a genuine helical guide (Jaouadi et al., 19 Aug 2025).

The literature also contains manuscript-level inconsistencies that should be read cautiously rather than overinterpreted. The 2025 HOT paper notes oscillation between the terms “Helical Optical Tube” and “Helical Optical Trap” in figure captions and identifies visibly malformed printed expressions for the characteristic well depth. The 2016 rotating-HOT study likewise contains unit inconsistencies in parts of the text, even though its dynamical interpretation is clear (Jaouadi et al., 19 Aug 2025, Rsheed et al., 2016).

Taken together, the HOT literature defines a structured-light helical dipole guide in which geometry is not a secondary detail but the central dynamical resource. In the stationary problem, that geometry yields quantized transverse modes and Mathieu-type motion along a helicoidal trough; in the dynamical problem, it imprints structured coherence that survives release, producing self-interference, revivals, nonlinear focusing, and guided transport; and in the rotating problem, it converts inertial-frame effects into controllable vertical transport. This suggests that the HOT is best understood as a platform for topological mode imprinting on matter waves, with the qualification that several proposed applications remain at the level of modeled pathways rather than complete device realizations (Jaouadi et al., 19 Aug 2025).