Orthogonal flexible Rydberg aggregates
We study the link between atomic motion and exciton transport in flexible Rydberg aggregates, assemblies of highly excited light alkali atoms, for which motion due to dipole-dipole interaction becomes relevant. In two one-dimensional atom chains crossing at a right angle adiabatic exciton transport is affected by a conical intersection of excitonic energy surfaces, which induces controllable non-adiabatic effects. A joint exciton/motion pulse that is initially governed by a single energy surface is coherently split into two modes after crossing the intersection. The modes induce strongly different atomic motion, leading to clear signatures of non-adiabatic effects in atomic density profiles. We have shown how this scenario can be exploited as an exciton switch, controlling direction and coherence properties of the joint pulse on the second of the chains [K. Leonhardt et al., Phys. Rev. Lett. 113 223001 (2014)]. In this article we discuss the underlying complex dynamics in detail, characterise the switch and derive our isotropic interaction model from a realistic anisotropic one with the addition of a magnetic bias field.
pacs:32.80.Ee, 82.20.Rp, 34.20.Cf, 31.50.Gh
Over the last decade fundamental phenomena in many-body Rydberg systems have been discovered, from dipole blockade Lukin et al. (2001); Urban et al. (2009); Gaëtan et al. (2009); Tong et al. (2004) and antiblockade Ates et al. (2007); Singer et al. (2004) over long-range molecules Greene et al. (2000); Liu et al. (2009) to excitonic dynamics due to resonant dipole-dipole interactions Ates et al. (2008); Wüster et al. (2010); Möbius et al. (2011); Wüster
et al. (2011a); Zoubi et al. (2014); Möbius
et al. (2013a); Barredo et al. (2015); Günter et al. (2013); Bettelli et al. (2013); Mülken
et al. (2007a).
Underlying all of these are strong long range interactions Park et al. (2011a, b); Li et al. (2005); Westermann et al. (2006); Ravets et al. (2014), which also make Rydberg atoms promising for quantum computing Saffman et al. (2010), quantum simulators Weimer et al. (2010); Schempp et al. (2014) and model systems for biological processes Mülken
et al. (2007b); Schönleber
et al. (2015).
It also became apparent, that the interplay of atomic motion and excitonic dynamics Frenkel (1931) yields entanglement transport Wüster et al. (2010); Möbius et al. (2011) and conical intersections (CIs) Wüster et al. (2011a). The latter play a crucial role in many quantum chemical processes, where they enable highly non-adiabatic dynamics on an ultrafast timescale Dantus and Zewail (2004). They may also protect the DNA structure from damage by UV radiation Perun et al. (2005). In Ref. Leonhardt et al. (2014) we have explored how conical intersections affect the wavepackets of atomic motion, which describe the entanglement transport and the associated energy and momentum transfer Wüster et al. (2010). We have shown that the CI can split those wavepackets, resulting in a superposition of two different excitation transfer processes. Since the relative weight of the processes after the splitting can be tuned through the system geometry and hence CI position, this gives rise to a sensitive switch for exciton transport properties.
Here we show the evolution of the underlying exciton spectra as a function of time in more detail. We also provide a parameter space survey for the operation of the conical intersection switch. Our results are derived from a model of isotropic dipole-dipole interactions. We finally show qualitatively similar features using the inherently anisotropic dipole-dipole interactions, where the anisotropy is suppressed through the application of an external magnetic field.
Dipole-dipole forces between atoms in two different Rydberg states and cause a strong interdependence of atomic motion and Rydberg state dynamics Ates et al. (2008). On a homogeneous chain of all but one atom in state , a single excitation will delocalize forming a Frenkel exciton Frenkel (1931). If the chain has a dislocation formed by two more closely spaced atoms, the exciton localises on these and in a repulsive state causes a combined pulse of chain dislocation, excitation and entanglement to propagate through the chain, akin to Newton’s cradle Wüster et al. (2010). In the scenario considered here, this combined pulse is directed towards a second chain orthogonal to the first, causing atoms to reach a configuration with a conical intersection in the exciton energy spectrum. The intersection causes non-adiabatic effects, triggering two different modes of pulse propagation on the second chain. The basic mechanism by which the conical intersection acts as a switch between these modes, can best be understood considering the essential subunit of two orthogonal atomic dimers.
The paper is organized as follows: In section II, we discuss Rydberg aggregates and their numerical treatment. We then proceed in section III.1 to describe a double Rydberg dimer, with two atoms freely moving on each of two orthogonal chains. We study the consequences of a conical intersection and characterise the dynamics on each of the two participating potential surfaces. Subsequently, in section III.2 we extend this setup to a seven atom system, with three and four atoms on the two chains. In section IV we highlight how the conical intersection can be functionalised as a switch controlling the transport dynamics on the second chain. Finally in section V we examine the feasibility to realise our isotropic interaction model experimentally. The appendix contains technical details regarding the engineering of isotropic interactions and removal of spin degrees of freedom using a magnetic field.
Ii Rydberg aggregates
We study a system of Rydberg atoms with masses , all with the same principal quantum number which we restrict for the sake of clarity to the two cases and . With of the atoms constrained on the -axis, and on the -axis their total number is , for a simple example see Fig. 1a. All atoms can move freely in only one dimension, with their positions described by the vector . The one-dimensional confinement could for example be realized by running laser fields and optical trapping of alkali Rydberg atoms Li et al. (2013), or earth alkali Rydberg atoms through their second valence electron Mukherjee et al. (2011). Furthermore, we assume atoms prepared such that only one atom is in an angular momentum state, all the other atoms are in angular momentum states. This allows us to expand the electronic wavefunction in the single excitation basis , where denotes a state with the th atom in the state Ates et al. (2008); Wüster et al. (2010).
ii.1 Rydberg-Rydberg interaction and the electronic Hamiltonian
Interaction potentials between Rydberg atoms can be determined by diagonalizing a dimer Hamiltonian in a restricted electronic state space, using the dipole-dipole approximation Gallagher (1994). Here we capture the essential features of these potentials into an effective model, including van-der-Waals (vdW) interactions between two atoms in the same state ( or ), and resonant dipole-dipole interactions between two atoms in different states (), leading to the electronic Hamiltonian
with resonant dipole-dipole interaction Hamiltonian
|and non-resonant van-der-Waals interaction|
where is the unit operator in the electronic space and the distance between atoms and . The interactions (2a) are isotropic, determined by the scaled radial matrix element . In section V and appendix A we discuss how this simplification can be realized using a magnetic field and isolating specific azimuthal angular momentum states. Using just one coefficient in (2b) ensures repulsive behavior at very short distances for all electronic states, assuming identical vdW interactions between Rydberg atoms in and states for simplicity. Their difference in reality can give rise to interesting effects at shorter distances Zoubi et al. (2014) that are, however, not relevant in our context.
Diagonalizing the electronic Hamiltonian for fixed nuclei,
gives the eigenstates called Frenkel excitons Frenkel (1931) and the eigenenergies which form Born-Oppenheimer surfaces (BO surfaces).
ii.2 Initial state
The initial state of the system can be written as a direct product
where is the initial electronic state and the initial spatial position state. We choose such that the initial position space representation for atoms,
is a product of Gaussians, centered on the atoms initial positions . We enforce for atoms on the horizonal chain () and for atoms on the vertical chain (), where are unit vectors in the , directions. This makes sure that the atoms have no initial position uncertainty in directions orthogonal to their chain. The longitudinal width models an experimental preparation in the lowest oscillator state of a harmonic trap.
ii.3 Propagation scheme for atomic motion
For larger numbers of atoms, solving the time dependent Schrödinger equation of the full Hamiltonian
for the motion in is not feasible in a reasonable time. Instead, we take the Wigner transform of the atomic Gaussian wave function (5a) as a weighting distribution for initial conditions to propagate classical trajectories fulfilling Newton’s equation
The mechanical force exerted on the atoms is derived from a single Born-Oppenheimer surface of the electronic Hamiltonian. However, we do allow transitions to other adiabatic surfaces during the time evolution according to Tully’s fewest switching algorithm Tully and Preston (1971); Hammes-Schiffer and Tully (1994); Barbatti (2011). These jumps, are random but occur with the probability for a transition between two surfaces and , which is proportional to the non-adiabatic coupling vector
Simultaneously, we propagate the electronic wave function of the Rydberg aggregate , expressed in the atomic (diabatic) basis , by solving the matrix equations for the coefficients ,
In addition to coefficients in the diabatic basis, we will also refer to those in the adiabatic basis defined by . Each trajectory is a self consistent solution of the coupled atomic and electronic equations of motion, (7) and (9) taking into account in addition the random jump sequence among the BO surfaces discussed above. With a recorded jump-sequence , each trajectory represents a realization. More precisely, the propagation sequence for a single trajectory with initial conditions at time starting on the BO surface consists of the following steps:
The electronic Hamiltonian is diagonalized and at any given time a single eigenenergy is the potential which exerts force on the atoms.
We determine whether the surface index undergoes a stochastic jump after this time step to (see Ates et al. (2008) for the precise prescription).
The new positions lead to new eigen states and -energies. Hence, we repeat the procedure from point 1 with updated time variable .
We use a single realization for each initial condition , which is in practice sufficient to converge the stochastic character of the jumps between surfaces if a large number of trajectories is propagated whose initial conditions randomly sample the initial Wigner distribution. Results obtained in this way agree very well with full quantum results for three atoms where quantum calculations can be done Wüster et al. (2010); Möbius et al. (2011); Möbius et al. (2013a); Leonhardt et al. (2014). Since Tully’s algorithm is expected to be an even better approximation for a larger number of dimension we confidently have adapted it to our present problem as described above.
Iii Nonadiabatic dynamics
Equipped with the tools to describe dynamics where the aggregate can make transitions between different adiabatic surfaces/ excitons, we would like to control these transitions in order to steer exciton transfer in a complex landscape of potential surfaces. It is well known that conical intersections – where adiabatic surfaces touch each other – lead to enhanced non-adiabatic transfer from on surface to another. Previously, we investigated transport of excitation and entanglement in linear flexible Rydberg chains without non-adiabatic transfer Wüster et al. (2010); Möbius et al. (2011), and the opposite, non-adiabatic processes near conical intersections without transport in ring configurations Wüster et al. (2011a).
A system exhibiting both features, transport and induced non-adiabatic transfer, is a T-shape configuration of two one-dimensional chains of atoms. We will present their dynamics with three and four atoms respectively on the two chains, using Li atoms with a mass of au. First, however, we investigate the simplest possible realization, a double dimer with two atoms on each of two orthogonal lines as sketched in Fig. 1.
iii.1 Two perpendicular dimers
We use Rydberg states in principal quantum number , leading to a transition dipole moment of au. For simplicity we set in this section. The parameters of the initial atomic configuration were set to , and . To sample the initial nuclear (Wigner) distribution with Gaussian width of about each atomic position (5a), we have propagated trajectories.
A single p-excitation in the system shared between atom 0 and 1 leads to their mutual repulsion as indicated by the blue arrows in Fig. 1a and enables atom 1 to move towards the vertical chain. On this way the three atoms 1-3 form a triangular sub-unit corresponding to the ring trimer studied in Wüster et al. (2011a). The CI of the trimer is realised for , in Fig. 1a, where the three atoms form an equilateral triangle. To illustrate the CI, we show in Fig. 1b the two intersecting energy surfaces as a function of two selected atomic position variables. The upper surface, shaded in red, will be hereafter referred to as the repulsive surface (with exciton state ), as it always entails repulsive interactions of nearby atoms. The lower surface at the intersection, shaded in green, will be referred to as the adjacent surface (with exciton state foo (a)). Further surfaces are not shown and play no significant role.
We will now systematically construct and interpret the atomic motion triggered by the initial excitation, firstly by analyzing typical trajectories and their energy spectra, then by constructing the atomic densities on the repulsive and adjacent adiabatic surface, which finally will enable us to understand the full evolution of the atomic density. We consider its evolution in time, spatially resolved, in terms of population of adiabatic surfaces and regarding the purity of the state.
iii.1.1 Evolution and energy spectra of typical trajectories
Since we neglect all uncertainties transverse to each of the trapping beams (see (5a)), atoms zero and one have well defined -coordinates. However, the atoms on the vertical line have a distribution in about their central position such that different trajectories have different asymmetry parameters . We distinguish the “symmetric” part of the nuclear wavepacket with (a typical trajectory is shown in Fig. 2e) from the “asymmetric” part which realizes other values (two trajectories that are nearly related through vertical mirroring are shown in Fig. 2a,c). One can see that initially, symmetric and asymmetric trajectories evolve very similarly. In particular, both of them jump from the repulsive surface (red) to the one we have called the adjacent surface of the trimer (green) a bit before one into the dynamics, see the energy spectra Fig. 2b,d,f. This point in parameter space represents a further conical intersection with only trivial consequences for our dynamics here, see dis (). During this initial time, atom 0 and 1 simply repel (not shown), until atom 1 sets 2 and 3 into motion around s. Shortly afterwards (during the grey shaded time interval) the dynamics of the trajectories is ruled by the essential conical intersection of the trimer.
Here, the symmetric part, coming near the degenerate point of the CI, is more likely to make a transition to the repulsive surface than the asymmetric part, which stays dominantly on the adjacent surface and passes the CI at a greater distance, see Fig. 1b). This is reflected in the energy spectrum of the trajectories by a very narrow avoided crossing (it appears actually as a crossing within the resolution of Fig. 1f) for the symmetric trajectory in the grey shaded area. Consequently, the symmetric trajectory follows a diabatic path, jumping from the adjacent surface to the repulsive one, while the two asymmetric trajectories that miss the CI move adiabatically in their energy landscape Fig. 2b,d), with a relevant avoided crossing in the grey shaded area that is wider allowing them to stay on the adjacent surface. A final transition to the repulsive surface leads to a significant separation of atoms 2 and 3 in the course of time, while the asymmetric trajectories that stay on the adjacent surface contain one atom remaining almost at rest, while the other one, initially at larger distance to the horizontal chain experiences the stronger force (Fig. 1c) and moves away foo (b).
iii.1.2 Atomic densities on the adiabatic surfaces forming the conical intersection
To evolve the Rydberg aggregate from its initial (Wigner)-distribution, we have calculated trajectories, the surface index of which at each instant of time allows us to show the atomic density foo (c) on the adjacent and repulsive surface separately as done in Fig. 3 for atoms 2 and 3. One directly recognises that the wave packet enters the CI on the adjacent surface (no density in Fig. 3b for short times) and that through the CI roughly half of the density is transferred to the repulsive surface where the two repelling branches show a wide distribution. On the adjacent surface, on the other hand, the four-fold branching is due to the asymmetric character of its individual trajectories as illustrated in Fig. 2a,c).
iii.1.3 Time dependent observables of the double dimer
We finally present the full time evolution of the double dimer in Fig. 4, where we clearly recognize in (a) the initial repulsion of atoms 0 and 1 of the horizontal chain. Furthermore, we can appreciate and understand the complicated fanning out of density in the vertical chain after the CI as a superposition of the wide two-branch distribution on the repulsive surface and the narrower four finger double fork on the adjacent surface, Fig. 4b. The population on both surfaces as a function of time in Fig. 4c confirms that the wave packet starts out on the repulsive surface (red) and continues after the first crossing on the adjacent surface (green), while it is split at the conical intersection to finally populate both surfaces roughly equally. As a consequence, the purity foo (d) of the wave packet decreases from unity to one half in Fig. 4d.
iii.2 The seven atom T-shape system
The double dimer was a necessary prerequisite to create a coherent exciton pulse along a path defined by atoms positioned in space: We now know that the atoms on the vertical chain will not attract each other after the atomic wave packet has passed the CI crossing, which is the first condition to continue excitation transport from the horizontal onto the vertical chain. The transport from its initiation, by creating a repulsive exciton localised on the first two atoms, to the CI on the horizontal chain follows the dynamics we have already established with “Newton’s cradle” in Ref. Wüster et al. (2010). However, for more atoms in the vertical chain, the CI may induce features which qualitatively go beyond what we have seen in the double dimer. Hence we extend our system to three atoms in the horizontal and four atoms in the vertical chain, keeping the configuration of two orthogonal lines as shown in Fig. 5. The resulting setup corresponds to two linear atomic ”Newton’s Cradle” configurations as in Ref. Wüster et al. (2010) that intersect at a right angle, such that transport involves a CI.
iii.2.1 Atomic configuration
We place the atoms initially as shown in Fig. 5 and prepare the system in the repulsive exciton localized on atoms 0 and 1. As before, this leads to a repulsive motion, such that atom 1 kicks atom 2, which eventually transfers its energy and electronic excitation to atom 2 as familiar from Newton’s cradle. Once atom 2 reaches the system can again be understood by considering the atomic trimer now formed by atoms 2, 4 and 5.
iii.2.2 Fully coherent exciton transport to the vertical chain
The simplest case essentially involves only a single BO surface. For sufficient, negative , atom 2 misses the CI and traverses the y-axis much closer to atom 5 than to atom 4. In this case Newton’s cradle like binary exciton transfer from atom 0 to 1 over 1 to 2 on the horizontal chain will carry over to upward transport on the vertical chain via transfer from atom 2 to 5 and ultimately 5 to 6, realising fully adiabatic dynamics with respect to the energy surfaces. Analogously one gets for positive ultimate exciton transfer to atoms 4 and 3. Hence changes in switch between downwards or upwards exciton transport, as discussed further in section IV.
Less clear is what happens with the excitation and entanglement transport if atom 2 approaches the vertical chain with . As we know from the double dimer, motion of atoms 4 and 5 is then affected by the CI through dynamics on its constituting energy surfaces, the adjacent and the repulsive one. To elucidate what this means for the ensuing excitation transport on the vertical chain we present calculations with Rydberg excitation , which gives a transition dipole moment of of the lithium atoms. The parameters of the perpendicular chains are , , and we use the same initial Gaussian distributions (5a) for the atomic positions.
iii.2.3 Simultaneous dynamics on several surfaces
We first discuss the case implying that the incoming atomic wave packet directly hits the CI.
The total atomic density on the horizontal chain Fig. 6a clearly shows the repulsive behavior of the atoms transferring the initial momentum and excitation shared by atoms 0 and 1 from atom 1 to atom 2. Once atom 2 approaches the vertical chain it enters the region of the CI which leads, as for the double dimer, to a distribution of the atomic density on the vertical chain over the repulsive and adjacent surface (Fig. 6b). The exciton spectra of selected single trajectories on these surfaces shown in Fig. 7 behave quite similarly as for the double dimer. In the beginning on the horizontal chain the trajectories switch diabatically a couple of times between the adjacent and repulsive surface before their paths finally become qualitatively different close to the CI (grey shaded region) when the two asymmetric trajectories shown in Fig. 7a,b,c,d stay on the adjacent surface, which then rules their subsequent dynamics on the vertical chain, while the more symmetric trajectory of Fig. 7e,f jumps to the repulsive surface. The main difference compared to Fig. 4 that one recognises is the transfer of momentum to the two new outer atoms 3 and 6 on the vertical chain.
iii.2.4 Inversion of excitation transport direction on the adjacent surface
Based on the understanding gained in section III.1.1, we can now understand an essential qualitative difference between transport dynamics on the repulsive and adjacent surfaces. To this end, we visualize in Fig. 8 the allocation of excitation transport onto the different atoms of the vertical chain, for two cases that differ only in the parameter . While in both cases, atom passes most closely to atom , the repulsive surface realizes the intuitively expected transport continuing with atom , while the adjacent surface instead leads to transport continued with atom in the opposite direction. As explained in section III.1.1, this can be directly traced back to differences between the repulsive and adjacent exciton states near the essential conical intersection.
Iv Exciton switch
Depending on how the passage through or near the CI distributes population onto the repulsive or adjacent energy surface, either of the transport mechanisms discussed above can be made dominant. This population distribution depends on the width and position of the multi-dimensional atomic wavepacket describing all co-ordinates , as well as the effective size of the CI which is essentially controlled by the corresponding multi-dimensional velocity Wüster et al. (2011a). All these features can be varied as a function of our configuration parameters , , and .
Extending the work of Leonhardt et al. (2014) we show a parameter space survey of the ensuing ”exciton switch” in Fig. 9. To characterize the exciton transport in a given configuration, we use the binary entanglement of the final atoms on the vertical chain, at the moment when the first of them reaches its ”entanglement readout” location, described in Fig. 5. For a definition of the binary electronic entanglement of two neighboring atoms , see Wüster et al. (2010). A large value of  corresponds to a coherent exciton-motion pulse travelling in the downward (upward) direction.
For various combinations of , , and , we plot and in Fig. 9, determined numerically from simulations as in Fig. 6 with a reduced number of trajectories. The three cases selected in Leonhardt et al. (2014) are shown as symbols (, and ). To understand the inversion of the entanglement transport direction for varying at fixed , please refer to our discussion in section III.2.4.
V Isotropic versus anisotropic dipole-dipole model
The model of interactions (1) employed so far was isotropic, neglecting the angular dependence of dipole-dipole interactions Gallagher (1994); Robicheaux et al. (2004), and the presence of several azimuthal sub-levels of the Rydberg state. In general we have to expand our basis to , where is the magnetic quantum number of atom in the state. For a one-dimensional atomic chain, the model (1) with basis can be recovered by restricting the initial state to , when the quantisation axis is parallel to the atomic chain Möbius et al. (2011). Dipole-dipole interactions then do not populate any other azimuthal states.
For the phenomena discussed in this article, we require at all , for the system to display the conical intersection between the repulsive and adjacent surface. Since additionally atomic distances here span two spatial dimensions, the procedure to eliminate the azimuthal degree of freedom described above is no longer possible. To ensure the correct sign of interactions in this case, we concentrate on the azimuthal sub-levels with quantisation axis perpendicular to the - plane. To prevent undesired coupling to the states, we additionally impose a uniform external magnetic field , where is a unit vector in the -direction. This creates an energy offset for all electronic states, effectively decoupling those with from . We discuss formally in appendix A how this results in our isotropic model (1) in the limit of an infinitely strong magnetic field, and how corrections to this model for realistic field strengths can be calculated. These corrections leave the qualitative results unchanged as we will show.
To this end we now repeat the simulations of Fig. 4 using a model that allows all magnetic states for all four atoms in the presence of an external magnetic field, described in appendix A and further in Leonhardt et al. (2015). The motion of the atoms is still constrained onto two orthogonal lines, as in the rest of the article. We demonstrate in Fig. 10 that qualitatively the same main features are found as for the isotropic model. However, for the greatest resemblance the parameters of both models have to be chosen slightly different due to the quantitative difference of potentials, which affect most importantly the initial acceleration of atoms and , in turn controlling the relative population of the two energy surfaces after CI crossing, seen in panel (e). For Fig. 10 we have adjusted in both models separately to achieve a rough splitting on the two surfaces. Both variants then qualitatively agree, in particular regarding clear signatures of multiple populated Born Oppenheimer surfaces in the snapshot shown in panel (f).
We have considered assemblies of dipole-dipole interacting Rydberg systems, whose motion and quantum state dynamics is affected by a conical intersection. For atoms free to move on either of two orthogonal chains, we have shown that the CI in the spectrum acts as a selector, through which parts of the wave packet where atoms are spaced symmetrically around one of the chains tend to be transferred to the repulsive surface, while the rest remains on the adjacent surface. As a result of the splitting onto two surfaces, the total atomic density shows several characteristic features, clearly signaling the crossing of a conical intersection.
The CI subunit of the two chains, with four atoms in two orthogonal dimers, can be understood as a building block for more complex systems. The block allows branching and switching of dislocation pulses within the system. Since the fraction of total population on either of the surfaces sensitively depends on the motion velocity and the width of the nuclear wave packet, the switching can be tuned or externally controlled. We have shown that this can be achieved through small changes of the confinement geometry, extending the work of Leonhardt et al. (2014) by a detailed parameter space survey of the ensuing exciton switch. Possible future extensions could be velocity adjustments of Rydberg atoms via external fields, e.g. Breeden and Metcalf (1981).
In this article we have employed an isotropic interaction model with just one participating azimuthal quantum number, which results in numerical simplifications and easier interpretation of quantum dynamics. We have demonstrated how this model can experimentally be realized using a magnetic field. However, the essential physics reported, the splitting of a Rydberg atom wave-packet onto multiple Born-Oppenheimer surfaces through acceleration by resonant dipole-dipole interactions is a generic feature of higher dimensional Rydberg aggregates. It neither relies on any of the above simplifications (as shown here) nor on the confinement of atomic motion on one-dimensional lines (as we will report in Leonhardt et al. (2015)).
Finally, note that similar transport processes on different energy scales could be studied through Rydberg dressed dipole-dipole interactions Wüster et al. (2011b); Genkin et al. (2014); Möbius et al. (2013b), or could rely on Rydberg atoms immersed in host cold atom clouds Möbius et al. (2013a); Wüster et al. (2013) instead of individual atoms.
Acknowledgements.We gratefully acknowledge fruitful discussions with Alexander Eisfeld and Sebastian Möbius.
Appendix A Isotropic dipole-dipole interactions
As demonstrated in section V, an external magnetic field can be used to reduce the number of electronic angular momentum states participating in the dynamics of Rydberg aggregates and thus render the interactions predominantly isotropic. Here we describe the underlying model and analytically derive an expression for the resulting effective interactions, including anisotropic correction terms. We additionally discuss spin of the Rydberg atoms, and show that the same mechanism that renders interactions isotropic also removes spin as a degree of freedom.
Besides including the azimuthal quantum number in our extended single excitation basis , see section V, we also have to describe the spin of the atoms, where it suffices to label the z-component for each atom by . We start by defining the Hamiltonian for the interaction of the magnetic field with atoms, given by
The magnetic field points in z-direction and , denote the operators of angular momentum and spin in z-direction for the th atom, and is the Bohr magneton. A spin configuration for the aggregate is uniquely defined by the tuple . We denote the corresponding state with , which is the product state of all single atom spin states, labeled for the th atom with . Introducing the quantum number for the z-component of the aggregate spin
which is the sum over all individual spin quantum numbers, the energy shift of the magnetic field is for states given by
The detuning between the and states inside a single -manifold is . This has to be larger than the anisotropic matrix elements of dipole-dipole interactions, which for Lithium atoms and our typical parameters also makes it larger than the finestructure-splitting. This implies we are in a strong field regime, where spin and angular momentum couple separately to the magnetic field, which effectively removes the finestructure and gives an energy level structure sketched in Fig. 11 (b). The energy spacing in this strong field regime is , which is of the order of MHz.
The finestructure furthermore yields doublets for neighboring states with , sketched as red and blue lines in Fig. 11 (b). The only singlet states with are the manifolds and . Hence we concentrate on the manifold, which can be well addressed during the Rydberg excitation process. The magnetic field yields increasing decoupling of the manifold from the manifold with increasing magnetic field strength. We study this decoupling in detail, giving the Hamiltonian structure in appendix A.1 and derive effective interactions in appendix A.2. The only coupling of the state to other manifolds than is through spin-orbit interactions and can thus be neglected.
a.1 Planar Rydberg aggregates in strong B-fields
Here we derive the Hamiltonian for a Rydberg aggregate in an external magnetic field pointing in the z-direction, where the magnetic field shift is much larger than the finestructure. This strong field regime allows a reduction of the electronic Hilbert space to a single spin manifold. All atoms are assumed to be located within a plane, as in all cases considered here, with quantization axis perpendicular to that plane. Then, the manifold is already completely decoupled Möbius et al. (2011). We are interested in and thus neglect the manifold in the following. We use the aggregate states as in the main article and denote with the states of angular momentum magnetic quantum number . The combined basis is . We consider two Hilbert spaces: The “pure” aggregate space, , spanned by the basis and the space including the angular momentum magnetic quantum numbers, , spanned by .
The dipole-dipole interaction Hamiltonian with magnetic field shift for fixed is given by
with the Hamiltonian in (1) and the identity operator, acting on states of .
The dipole-dipole transitions from to are described by
where is the radial dipole matrix element between and for principal quantum number , and and are the modulus and azimuthal angle of the separation between atoms and , within the co-ordinate system defining our quantization axes.
We treat now as unperturbed system and as perturbation and set up extended Hamiltonians:
The Hamiltonian (18) describes Rydberg aggregates with magnetic field shifts for the states, but neglecting fine structure shifts.
a.2 Effective interactions from block-diagonalization
To see how the mixing of the manifolds due to the interaction described by is weakened with increasing magnetic field, we block-diagonalize , according to van Vleck perturbation theory J. H. Van Vleck (1929) in a canonical form as outlined by Shavitt et al Shavitt and Redmon (1980). We define the projectors
and denote for every operator that acts on states in the diagonal blocks with
and the offdiagonal blocks with
The goal is to find a unitary transformation, , such that is blockdiagonal, i.e. . In the following we omit the dependency of the operators on for better readability.
In canonical van Vleck perturbation theory, the transformation is rewritten as with the property and . These conditions lead to the following form of :
with acting on states in . An expansion in orders of the perturbation Shavitt and Redmon (1980) leads to equations for , labeling different orders:
with constants and operators . The solution to this equation is
with and . The expansion of the block-diagonalization is given by
The first correction of the blockdiagonalization procedure is (34). Using solution (31) for equation (28), we see that and so . The second contribution is thus from , the leading order of which is , thus we do not consider it further. We now concentrate on the Hamiltonian for states with , . Using (17), (22) and (31) in (34), we can evaluate the corresponding first correction term
Let us define the maximum dipole-dipole interaction element of , , with . To assess if (37) is small compared to , as required for , we introduce rescaled quantities, , with . This yields and . Introducing further a decoupling number, and setting as our zero of energy, we find up to the following effective interaction Hamiltonian for :
Note that is measured in units of . The leading order of the effective Hamiltonian (38) corresponds to our original isotropic model (1) and the additional terms can be explicitly constructed to allow for small anisotropic corrections.
a.3 Comparison between isotropic, effective and complete Hamiltonian
In this section we assess how good the isotropic or the effective Hamiltonian (both without spin degrees of freedom) approximate the complete Hamiltonian, which includes spin-orbit coupling in addition to the magnetic field.
We denote the space of the electron spin states of the th-atom with , the basis of which is spanned by . The states denote downwards oriented electron spin and upwards oriented electron spin for the th-atom.
The space of all electron spins is then given by , with the product basis .
The spin-orbit interaction destroys the decoupling of the states, such that we have to redefine some quantities of appendix A.1. The space is now spanned by , with the states of the quantum number . The Hamiltonian for the states is given by
with the resonant dipole-dipole Hamiltonian in (2a) and the non-resonant van-der-Waals Hamiltonian in (2b). Note that experiences no magnetic field shift. We redefine from (16), such that it includes the states:
The Hamiltonian in (18), , is now calculated with the redefined .
With these definitions, we can span the complete space , describing the orientation of the p-states and the spins of the electrons together. The product basis, where spin and orbital angular momentum of the p-states are not combined to a total angular momentum, is then given by .
The Hamiltonian in (18) includes the magnetic field shifts for the orbital angular momentum only. The magnetic field shift for the spins is described by
The dipole-dipole interactions together with the total magnetic field shift is then given by
To set up the spin-orbit interaction Hamiltonian in a simple way, it is useful to employ yet another basis. First we define the spin spaces , which describe all spins except those of the the th-atom, . Their product basis is spanned by . The spin-orbit coupling yields a total angular momentum, per atom. The pair are the quantum numbers of , with and . The result of the spin-orbit coupling is the finestructure splitting , between p-states with and p-states with . To write down the spin-orbit Hamiltonian in its eigenbasis, we first introduce aggregate states which include the spin of the p-states, . We now define spaces , which we span with the basis . The orthogonal sum of both ’-spaces’ spans the complete space, . This yields the eigenbasis of the spin-orbit Hamiltonian, . Introducing the unitary transformation , which performs the basis transformation from to , the spin-orbit Hamiltonian in the basis is given by
where is the null-operator acting on elements in and transforming them into its zero. Note that we thus shift the origin of energy to the manifold. The complete Hamiltonian is given by
We compare the three different Hamiltonians in (1), (38) and (44) by using them to calculate the eigenenergies for a four atom system with a symmetric configuration (), as sketched in Fig. 1 (a). We show for (44) only the manifold, which is the one, we propose to work with. The positions of atoms (2,3) are fixed, whereas the positions of atoms 0 and 1 are parameterized as and . The eigenenergies of all three Hamiltonians are plotted as a function of the co-ordinate in Fig. 12. The isotropic model in (1) approximates (44) well for all locations crucial in our simulations, as shown in Fig. 12 (a-c). Crucial for the simulations are configurations with small values, where the atoms are accelerated due to the interactions, and in the neighborhood of the conical intersection. The agreement is not good for the equidistant linear trimer configuration . The excitation is there mostly delocalized and the phase of the dipole-dipole interaction plays a role. The effective Hamiltonian in (38) approximates the complete one for this configuration very well, as shown in Fig.