Tying Quantum Knots*
Knots are familiar entities that appear at a captivating nexus of art, technology, mathematics, and science Adams1994 (). As topologically stable objects within field theories, they have been speculatively proposed as explanations for diverse persistent phenomena, from atoms and molecules Thomson1867 () to ball lightning Ranada1996 () and cosmic textures in the universe Cruz2007 (). Recent experiments have observed knots in a variety of classical contexts, including nematic liquid crystals Smalyukh2009 (); Tkalec2011 (); Sec2014 (), DNA Han2010 (), optical beams Leach2005 (); Dennis2010 (), and water Kleckner2013 (). However, no experimental observations of knots have yet been reported in quantum matter. We demonstrate here the controlled creation Kawaguchi2008 () and detection of knot solitons Faddeev1997 (); Battye1998 () in the order parameter of a spinor Bose–Einstein condensate. The experimentally obtained images of the superfluid directly reveal the circular shape of the soliton core and its accompanying linked rings. Importantly, the observed texture corresponds to a topologically non-trivial element of the third homotopy group Nakahara2003 () and demonstrates the celebrated Hopf fibration Hopf1931 (), which unites many seemingly unrelated physical contexts Urbantke2003 (); Moore2008 (). Our observations of the knot soliton establish an experimental foundation for future studies of their stability and dynamics within quantum systems Hietarinta2004 ().
Knots are defined mathematically as closed curves in three-dimensional space Adams1994 (). A trivial example is a circle, which is also known as an unknot. More complicated knots have been extensively classified and tabulated by determining whether they can be continuously deformed, one into another, without permitting the curve to pass through itself. Although nontrivial knots are commonly associated with physical strings, they can also appear in line-like vortices, the cores of which trace the closed curves. A celebrated example is Kelvin’s early atomic theory, which is linked to the existence and dynamics of knotted vortex rings in an ethereal fluid Thomson1867 (). More recently, nontrivial vortex knots have been created and identified experimentally in water Kleckner2013 () and optical beams Leach2005 (); Dennis2010 (), and discussed theoretically in the context of superfluid turbulence Barenghi2007 ().
Knots can also appear as particle-like solitons in classical and quantum fields Manton2004 (), the nature of which has been a subject of intense mathematical interest for more than eighty years Hopf1931 (). In this case, the closed curve is a ring that describes the core of the soliton. This ring is surrounded by an infinite number of similar rings, each linked with all of the others to generate a knotted field structure Radu2008 (). For example, the classical Maxwell’s equations admit solutions that involve knot solitons in which the rings are electric and magnetic field lines Ranada1990 (). Quantum-mechanical examples have been theoretically proposed Faddeev1997 (); Battye1998 () for the Faddeev–Skyrme model, in which each of the linked rings consists of the points in space sharing a particular direction of the field.
In general, knot solitons are non-singular topological excitations Nakahara2003 () that change smoothly and non-trivially in all three spatial dimensions. They are therefore described by the third homotopy group, , which classifies such textures according to whether or not they can be continuously transformed into one another. One-dimensional solitons and singular vortex lines, both belonging to the fundamental group , have been identified experimentally in superfluids Denschlag2000 (); Burger1999 (); Vinen1961 (), as have two-dimensional skyrmions Choi2012 () and singular monopoles Ray2015 () belonging to the second homotopy group, . Since singular defects within are unrealisable monopoles in four spatial dimensions, the knot soliton is an instance of the only general texture type that has not yet been identified experimentally within a medium described by a quantum-mechanical order parameter.
In this Letter we demonstrate the creation and observation of knot solitons in the polar phase of a spinor Bose–Einstein condensate. We adopt the theoretical method proposed in ref. dccclxxx(12) and implement it using experimental techniques that have recently been used to create Dirac monopoles Ray2014 () and isolated monopoles Ray2015 (). An overview of the experiment is given in Fig. 1. The core of the knot soliton is first observed as a ring of enhanced particle density at the periphery of the condensate that shrinks inwards as time advances (Fig. 2). The presence of the soliton is confirmed by its excellent agreement with analytical theory and with numerical simulations (Fig. 3). Strikingly, we directly image the linked structure of the knot soliton, as shown in Fig. 4.
The order parameter describing a spin-1 Bose–Einstein condensate may be written as
where is the atomic density, is a scalar phase, and is a three-component -quantized spinor with . We restrict our attention here to the polar phase of the condensate, which is obtained by spin rotations
for angles and about the and axes, respectively. The polar order parameter may therefore be expressed as
in terms of the nematic vector, , defined by equation (2).
The nematic vector field maps points in real space to points on the surface of the unit sphere . In our case, the nematic vector assumes a constant value, , at the boundary of a certain volume . We restrict our studies to textures inside the volume , which as a result can be identified with , the surface of a four-dimensional ball. Nontrivial mappings from to lead to knotted field configurations characterised by integer topological charges or Hopf invariants Manton2004 (), , as determined by the third homotopy group . Field configurations with different Hopf invariants cannot be continuously deformed into one another and are therefore topologically distinct.
Taken together, the points in at which assumes the same value, , define a closed curve known as the preimage of . Each of these preimages is linked with all of the others, which are associated with different , exactly times. Thus the linking number is equivalent to the Hopf invariant Manton2004 (), and provides an alternative perspective on its physical significance.
In this experiment we consider the Hopf map Hopf1931 (), which has and is generated physically in our system by spin rotations in an inhomogeneous magnetic field. We begin with an optically trapped Rb condensate (see Methods) described by the spinor , corresponding to . The inhomogeneous magnetic field is given by
where the condensate is taken to be at the origin of the rescaled coordinate system , , and . In the gradient G/cm and effective bias field mG, the zero point of the magnetic field is initially 33 m away from the centre of the condensate. The creation of the knot begins with a sudden change of that places the field zero at the centre of the condensate, ideally leaving its state unchanged (see Fig. 1a,b and Extended Data Fig. 1). The nematic vectors then precess about the direction of the local magnetic field at their spatially-dependent Larmor frequencies
where is the atomic Landé -factor, is the Bohr magneton, and . The optimal result is the time-dependent nematic vector field
where is the vector of dimensionless spin-1 matrices in the Cartesian basis. Importantly, for all points satisfying , and hence we may choose the volume to be a ball of radius
Figure 1d–e illustrates how the nematic vector assumes its knot soliton configuration as a result of the spatially dependent Larmor precession. The core of the soliton is identified with the preimage of the south pole of , i.e., , which lies in the -plane. This ring defines a circle (Fig. 1c–e) in which the condensate is entirely in the spinor component [equation (2)]. The comparable preimage of the north pole, , includes the -axis and all points on the boundary of . Because antipodal points on correspond to the same spinor up to a sign [equation (3)], this preimage is also entirely in the component. The preimages of the equatorial points on the two-sphere consist of linked rings that define a toroidal tube enclosing the core, as shown in Fig. 1c. Since at the equator of , this torus consists of overlapping components [equation (2)]. Elsewhere, smoothly interpolates between these values.
After an evolution time we apply a projection ramp in which the bias field is rapidly changed to move the field zero far from the centre of the condensate Ray2014 (); Ray2015 (). The condensate is then released from the optical trap, whereupon it expands and falls under the influence of gravity. Subsequently, its spinor components are separated and imaged simultaneously along both the vertical () and horizontal () axes.
The temporal evolution of the particle column densities in the component, , is shown in Fig. 2a,b. The pictures show the combined preimages of the poles of (), revealing in one picture both the core of the knot and the boundary of . The preceding analytical result for the radius of the core, from equation (7), agrees well with the experimental observations.
Figure 3 provides a detailed comparison of the experimentally obtained knot soliton with numerical simulations of the corresponding Gross–Pitaevskii equation (see Methods) with no free parameters. The very good correspondence between the experiment and the simulation, together with the qualitatively correct behaviour of the spinor components that jointly accumulate in the vicinity of the intensity minima of the component, provide further evidence that the observed texture is that of a knot soliton. Note that the components do not overlap as a result of the time-of-flight imaging technique (see Extended Data Fig. 2 for simulated images of the spinor components prior to expansion).
By definition the nematic vector is aligned with the local spin quantization axis, along which the condensate is fully in the component [see equation (2)]. Remarkably, a projection ramp taken along an arbitrary axis, , populates the component with the preimages of the antipodal points in corresponding to . By performing the projection ramp along and , for example, we can observe the linked preimages of and , respectively, in the component (Fig. 4). These images explicitly demonstrate the linked rings of the Hopf fibration and provide conclusive evidence of the existence of the knot soliton.
Our observations suggest future experiments on the dynamics, stability, and interactions of knot solitons Hietarinta2004 (). Experimental creation of multiply-charged and knotted-core solitons in quantum fields stands as another promising research direction. Furthermore, stabilising the knot soliton against dissipation, a feature associated with textures in the Faddeev–Skyrme model Faddeev1997 (); Battye1998 (), remains an important experimental milestone.
Experiment. The condensate initialisation, trapping, and imaging techniques are essentially identical to those of ref. Ray2015, . The key technical difference in the present experiments is that we bring the magnetic field zero suddenly into the condensate centre, in contrast to the adiabatic creation ramp in ref. Ray2015, . Extended Data Fig. 1 shows the measured temporal evolution of the electric current controlling during its excursion, expressed in units of the magnetic field. We define to be the moment at which the field zero has traversed 90% of the distance towards its final location at the centre of the condensate. The strength of the quadrupole gradient field is estimated by repeating the knot creation experiment with a small bias field offset applied along the -axis, which introduces a fringe pattern that winds at a rate proportional to the strength of the gradient.
The crossed-beam optical dipole trap operates at 1064 nm with frequencies Hz and Hz in the radial and axial directions, respectively. The total number of particles in the condensate at the moment of imaging is typically .
Data. The experimentally obtained images of knot solitons shown in this manuscript represent typical results selected from among several hundred successful realisations taken under similar conditions over the course of more than a year. Remarkably, almost identical knot solitons have been created with several minutes of time elapsed between the realisations while not changing the applied control sequences.
Simulation. We theoretically describe the low-temperature dynamics of the condensate using the full three-dimensional spin-1 Gross–Pitaevskii equation
where we denote the single-particle Hamiltonian by , the spin vector by , and the density–density and spin–spin coupling constants by and , respectively. We employ the literature values for the three-body recombination rate cm/s, the Rb mass kg, and the -wave scattering lengths nm and nm. The single-particle Hamiltonian assumes the form where the strength of the quadratic Zeeman effect is given by MHz/T and the optical trapping potential is approximated by The Gross–Pitaevskii equation is integrated using a split-operator method and fast Fourier transforms on a discrete grid of size . The computations are carried out using state-of-the-art graphics processing units. The simulations reproduce the experimental results with no free parameters: Only literature values for constants and independently measured parameters, such as the temporal dependence of the magnetic field, are employed. The magnetic field gradient that is briefly applied to separate the different spinor components during the time-of-flight imaging is not included in the simulations.
Acknowledgements We acknowledge funding by the National Science Foundation (grant PHY–1205822), by the Academy of Finland through its Centres of Excellence Program (grant no. 251748) and grants (nos 135794 and 272806), Finnish Doctoral Programme in Computational Sciences, and the Magnus Ehrnrooth Foundation. CSC - IT Center for Science Ltd. (Project No. ay2090) and Aalto Science-IT project are acknowledged for computational resources. We thank N. Johnson for making public his Hopf fibration code, A. Li for assistance with figures, and W. Lee and S.J. Vickery for experimental assistance.
Author Contributions M.W.R., A.H.G, and D.S.H. developed and conducted the experiments and analysed the data. K.T. and E.R. performed the numerical simulations under the guidance of M.M. who provided the initial suggestions for the experiment. M.M. and D.S.H. developed the analytical interpretation of the data as preimages. All authors discussed both experimental and theoretical results and commented on the manuscript.
Author Information The authors declare that they have no competing financial interests. Correspondence and requests for materials should be addressed to D.S.H. (email@example.com).
*PLEASE NOTE This is the version originally submitted to the journal and contains some minor typos and errors. The published and corrected version will be posted to arXiv six months after publication in the journal.
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