Knotted Coherent States in Quantum Systems

• Knotted coherent states are quantum states featuring topologically knotted vortex cores or electromagnetic field lines, unifying macroscopic phase coherence with complex knot structures.
• In ultracold atomic systems, Raman coupling imprints knotted phase singularities onto Bose–Einstein condensates, enabling controlled realization and dynamic study of knot configurations.
• In electromagnetic settings, multi-mode coherent states realize knotted field-line geometries with measurable quantities such as energy and helicity directly linked to the underlying knot topology.

Knotted coherent states are quantum states in which coherence coexists with a nontrivial knot or link structure in a complex field, its singularity set, or its field-line geometry. In current arXiv usage, the expression has two distinct but related technical realizations. In ultracold matter, it denotes phase-coherent Bose–Einstein condensates whose order parameter hosts closed vortex cores forming knots or links, produced by coherent Raman transfer from structured optical fields (Maucher et al., 2015). In quantized electromagnetism, it denotes multi-mode coherent states of the electromagnetic field whose classical limit is a vacuum Maxwell solution with electric and magnetic field lines arranged as non-null torus knots (Silva et al., 14 May 2026). A broader mathematical backdrop is supplied by the theory of knotted zeros, according to which smooth complex-valued maps on $S^3$ can be constructed so that their inverse image of $0$ is any prescribed smooth knot or link (Kauffman et al., 2019).

1. Definition and conceptual scope

In the Bose–Einstein-condensate setting, a low-temperature condensate is described by a complex order parameter

$\psi(\mathbf r,t)=|\psi|e^{i\theta}.$

Vortices are phase singularities: lines where $|\psi|=0$ and the phase winds by an integer multiple of $2\pi$ around any loop encircling the line. The superfluid velocity $\mathbf v=(\hbar/m)\nabla\theta$ then gives circulation quantized in units of $h/m$. If the vortex cores form closed loops in three dimensions, those loops may be unknotted or may realize nontrivial knot or link types such as a Hopf link or a trefoil knot. In this usage, a knotted coherent state is a phase-coherent many-body state whose order parameter carries such a knotted vortex-line set (Maucher et al., 2015).

In the electromagnetic setting, a coherent state is the standard Glauber multi-mode coherent state defined by the eigenvalue property

$$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$

or equivalently by a displacement operator acting on the vacuum. The knotted aspect is not a nodal set of a scalar wavefunction but the topology of the classical field recovered from expectation values: electric and magnetic field lines lie on embedded tori and wrap with integer winding numbers, forming torus knots or links. The relevant family is parameterized by four positive integers $(n,m,l,s)$, with magnetic lines forming $(n,m)$ torus knots and electric lines forming $0$0 torus knots at $0$1 (Silva et al., 14 May 2026).

A third, more general formulation treats a complex wavefunction as a map

$0$2

or $0$3, with the knot represented by the nodal set $0$4. In that sense, knotted coherent states can be understood as coherent-state-like or mean-field states belonging to the much larger class of complex fields whose zeros, phase singularities, or field lines carry knot topology. This suggests a unifying viewpoint in which “knotted” refers to topology, while “coherent” refers either to Glauber coherence or to macroscopic phase coherence (Kauffman et al., 2019).

2. Matter-wave knotted coherent states in Bose–Einstein condensates

The matter-wave construction employs a $0$5-system in $0$6 with internal states

$0$7

and an excited manifold $0$8 used as a virtual intermediate level. The atoms are confined in a cylindrically symmetric harmonic trap

$0$9

A structured probe beam $\psi(\mathbf r,t)=|\psi|e^{i\theta}.$0 couples $\psi(\mathbf r,t)=|\psi|e^{i\theta}.$1, and a co-propagating plane-wave control beam $\psi(\mathbf r,t)=|\psi|e^{i\theta}.$2 couples $\psi(\mathbf r,t)=|\psi|e^{i\theta}.$3 with the same large detuning $\psi(\mathbf r,t)=|\psi|e^{i\theta}.$4. Although the direct $\psi(\mathbf r,t)=|\psi|e^{i\theta}.$5 transition is dipole forbidden, the two-photon Raman process yields an effective coherent coupling between the two ground states (Maucher et al., 2015).

After adiabatic elimination of $\psi(\mathbf r,t)=|\psi|e^{i\theta}.$6, the coupled Gross–Pitaevskii equations are

$\psi(\mathbf r,t)=|\psi|e^{i\theta}.$7

with $\psi(\mathbf r,t)=|\psi|e^{i\theta}.$8. For the specified $\psi(\mathbf r,t)=|\psi|e^{i\theta}.$9 states, $|\psi|=0$0 and $|\psi|=0$1.

The imprinting mechanism is transparent in the short-time limit. Starting from $|\psi|=0$2 and taking $|\psi|=0$3,

$|\psi|=0$4

Hence $|\psi|=0$5: the phase of $|\psi|=0$6 is initially the phase of $|\psi|=0$7, and the zeros of $|\psi|=0$8 become vortex cores in the target component. The knotted topology is therefore transferred from optical phase singularities to matter-wave vortex lines.

In this realization, coherence is macroscopic and many-body. The condensate remains a coherent matter wave, but the knot is a property of the phase singularity set of $|\psi|=0$9, not of individual atoms. Because reconnections are permitted in the scalar Gross–Pitaevskii framework, the resulting knots are generally metastable rather than topologically protected.

3. Optical synthesis of knotted vortex lines and condensate dynamics

The structured probe field is a paraxial beam with slowly varying envelope $2\pi$0 satisfying

$2\pi$1

and is expanded in Laguerre–Gaussian modes,

$2\pi$2

The coefficients $2\pi$3 are chosen so that the nodal lines of the field realize a prescribed knot or link. The construction follows Dennis and collaborators through Milnor polynomials. One begins with

$2\pi$4

uses inverse stereographic projection

$2\pi$5

and defines

$2\pi$6

For $2\pi$7, the zero set gives a Hopf link; for $2\pi$8, a trefoil knot. Matching $2\pi$9 to a finite Laguerre–Gaussian superposition at $\mathbf v=(\hbar/m)\nabla\theta$0 determines the required optical coefficients (Maucher et al., 2015).

The explicit examples used in the simulations are finite superpositions. For the Hopf link, the Raman coupling contains $\mathbf v=(\hbar/m)\nabla\theta$1, $\mathbf v=(\hbar/m)\nabla\theta$2, $\mathbf v=(\hbar/m)\nabla\theta$3, and $\mathbf v=(\hbar/m)\nabla\theta$4 terms. For the trefoil, the optimized superposition contains $\mathbf v=(\hbar/m)\nabla\theta$5, $\mathbf v=(\hbar/m)\nabla\theta$6, $\mathbf v=(\hbar/m)\nabla\theta$7, $\mathbf v=(\hbar/m)\nabla\theta$8, and $\mathbf v=(\hbar/m)\nabla\theta$9. In both cases the coupling is applied only during a finite pulse through a factor $h/m$0.

The subsequent condensate dynamics is nontrivial. For a Hopf-link example, the simulations use $h/m$1, $h/m$2, $h/m$3, $h/m$4, $h/m$5, $h/m$6, $h/m$7, $h/m$8, and $h/m$9. Shortly after imprinting, isodensity surfaces of $$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$0 show two linked loops with the expected phase winding. The structure exhibits center-of-mass motion predominantly in the $$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$1 direction and then undergoes reconnection events, eventually decaying into two separate unknotted vortex rings.

For a trefoil example, the parameters are $$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$2, $$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$3, $$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$4, $$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$5, $$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$6, $$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$7, $$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$8, and $$\hat a_{\mathbf k\lambda}|\{\alpha\}\rangle=\alpha_{\mathbf k\lambda}|\{\alpha\}\rangle,$$9. The trefoil is initially visible in $(n,m,l,s)$0 with the correct phase structure. It undergoes reconnections, but in the confined two-component geometry the dominant event is first the expulsion of a single vortex ring propagating rapidly in $(n,m,l,s)$1, after which the remaining part of the knot continues to evolve. These examples establish that matter-wave knotted coherent states are dynamically accessible yet metastable.

4. Electromagnetic knotted coherent states and their observables

The electromagnetic construction is formulated in Coulomb gauge with mode expansion

$(n,m,l,s)$2

with $(n,m,l,s)$3. A general classical solution $(n,m,l,s)$4 determines the coherent amplitudes through the Glauber mapping

$(n,m,l,s)$5

For the non-null torus-knot family, this becomes

$(n,m,l,s)$6

where $(n,m,l,s)$7 is linear in $(n,m,l,s)$8 and $(n,m,l,s)$9 supplies an exponential ultraviolet cutoff. The corresponding displacement operator $(n,m)$0 defines

$(n,m)$1

and the time-evolved state remains coherent with amplitudes $(n,m)$2 (Silva et al., 14 May 2026).

By construction,

$(n,m)$3

The state is therefore an ordinary multi-mode coherent state whose mean field is a knotted, non-null vacuum solution of Maxwell’s equations. The non-null character is specified by

$(n,m)$4

which do not vanish identically.

The principal expectation values are summarized below.

Observable	Expectation value	Dependence on knot data
Normal-ordered energy	$(n,m)$5	Sum of squares
Normal-ordered photon number	$(n,m)$6	Sum of squares
Optical helicity	$(n,m)$7, up to an overall sign depending on polarization conventions	Linking numbers
Magnetic helicity	$(n,m)$8	Linking numbers

The normal-ordered Poynting vector is

$(n,m)$9

so the energy flow has a toroidal structure, while the overall scale depends only on $0$00. For each mode, the quadratures satisfy

$0$01

and the normalized second-order correlation is

$0$02

Thus the state has the minimum-uncertainty and Poissonian-statistics properties of a standard coherent state, even though the spatially varying vector-field structure can make $0$03.

A distinguished special case is the Hopfion coherent state with $0$04. Then both electric and magnetic lines form Hopf links, and the general formulas reduce to explicit Hopfion values for energy, photon number, Poynting vector, and helicities.

5. Knotted zeros, classifying maps, and general wavefunction topology

The most general topological statement in this area is the theorem that for every smooth knot $0$05 there exists a differentiable map

$0$06

such that $0$07 is transverse to $0$08 and

$0$09

The same statement holds for links. Transversality implies that the zero set is a smooth codimension-$0$10 submanifold of $0$11, hence a knot or link. In this form, any smooth knot can be realized as the nodal set of a smooth complex-valued wavefunction on $0$12 (Kauffman et al., 2019).

The proof is constructive. Starting from a knot diagram, one uses the Wirtinger presentation of $0$13 to build a CW complex $0$14 with one circle for each generator and $0$15-cells attached along the Wirtinger relations. A map from the $0$16-skeleton to $0$17 is defined by sending each generator circle diffeomorphically to the target circle, and because the relations become null-homotopic this map extends over the $0$18-cells and then over the $0$19-cell. After passing from $0$20 to the knot complement, one obtains a map

$0$21

where $0$22 is a tubular neighborhood. On the boundary $0$23, the map is arranged to be projection to the second factor, and it is extended over the neighborhood by

$0$24

Consequently,

$0$25

This framework places earlier constructions in a broader context. Berry’s hydrogenic example, in which a time-independent wavefunction for hydrogen contains a trefoil in its zero set, becomes a special case of a more general classifying-map picture. The same section of the literature explicitly compares this viewpoint with the work of Dennis and collaborators, Peralta-Salas and collaborators, and Rudolph. What remains open in the formulation of Kauffman and Lomonaco is the characterization of when such a classifying map can be chosen to be a wavefunction for a specific physical quantum system.

A plausible implication is that knotted coherent states are best regarded not as a single unique object, but as a physically constrained subclass of complex fields with prescribed knot topology: coherent states in the strict Glauber sense when the field is quantized electromagnetism, and coherent macroscopic order parameters when the field is a condensate wavefunction.

6. Physical interpretation, limitations, and directions suggested in the literature

In the matter-wave setting, the required experimental scales are concrete. The species is $0$26 with $0$27, the trap frequency is typically $0$28, and the transverse oscillator length is $0$29. The optical wavelength is $0$30, beam waists are of order $0$31–$0$32, Rabi frequencies are of order $0$33, and $0$34–$0$35. Pulse durations such as $0$36 for a Hopf link and $0$37 for a trefoil are chosen to be shorter than the intrinsic condensate timescale $0$38, so the phase pattern is written before significant motion occurs. The proposed detection methods are state-selective imaging, in situ imaging, short time-of-flight expansion, tomography, and interferometry (Maucher et al., 2015).

The limitations are equally explicit. In the condensate protocol, spontaneous emission must remain small, laser noise and phase fluctuations can blur the imprinted pattern, finite temperature can damp vortex motion and accelerate decay, and extremely tight focusing may require non-paraxial corrections. Most importantly, the knot type is not dynamically protected: Gross–Pitaevskii evolution permits vortex reconnections, sound emission, deformation, and breakup into simpler loops.

In the electromagnetic construction, the limitations are of a different kind. The theory is formulated for free Maxwell fields in vacuum; coupling to sources would spoil exact knotted solutions. The exact mode distribution

$0$39

is derived analytically, but a concrete optical setup that generates it is not provided. Observable calculations are developed for quadratic quantities and standard Glauber correlations, and the work explicitly suggests extensions to squeezed knotted states, entangled knotted states, the interaction of knotted coherent fields with charged particles, and the study of decoherence in relation to topological features (Silva et al., 14 May 2026).

Across the three strands of the literature, several extensions are suggested or implied. In Bose–Einstein condensates, higher-order knots and links may be generated by replacing $0$40 with more general Milnor polynomials, and spinor BECs may allow knotted vortices combined with nontrivial spin textures. In quantum optics, non-null torus-knot coherent states provide a template for promoting classical topological light to fully quantized states with explicit formulas for energy, helicity, photon number, and correlations. In the general nodal-set framework, the existence of classifying maps for arbitrary knots suggests a broad design space for quantum states whose zeros or singularities realize prescribed knot types. The shared theme is that topology is encoded in a coherent field configuration, while the dynamical fate of that topology depends strongly on the underlying physical theory.