Compensation of the trap-induced quadrupole interaction in trapped Rydberg ions

The quadrupole interaction between the Rydberg electronic states of a Rydberg ion and the radio frequency electric field of the ion trap is analyzed. Such a coupling is negligible for the lowest energy levels of a trapped ion but it is important for a trapped Rydberg ion due to its large electric quadrupole moment. This coupling cannot be neglected by the standard rotating-wave approximation because it is comparable to the frequency of the trapping electric field. We investigate the effect of the quadrupole coupling by performing a suitable effective representation of the Hamiltonian. For a single ion we show that in this effective picture the quadrupole interaction is replaced by rescaled laser intensities and additional Stark shifts of the Rydberg levels. Hence this detrimental quadrupole coupling can be efficiently compensated by an appropriate increase of the Rabi frequencies. Moreover, we consider the strong dipole-dipole interaction between a pair of Rydberg ions in the presence of the quadrupole coupling. In the effective representation we observe reducing of the dipole-dipole coupling as well as additional spin-spin interaction.

Ion trap system is one of the leading platforms in quantum information technologies [1][2][3] . The ability to control and read out the external and internal degrees of freedom of the trapped ions with high accuracy leads to experimental implementation of various entangled states [4][5][6][7] and quantum gates with high fidelity [8][9][10] . However, the control of trapped ions becomes more difficult for multiple ions. Indeed, the phonon mode structure becomes too complicated for entanglement operations and the storage capacity is rendered limited. One way to overcome this limit is to use array of ion traps which store a small number of ions [11][12][13] . Another approach is based on using trapped Rydberg ions. In this approach instead of using the common phonon mode for entanglement, the strong dipole-dipole interaction may be used for implementation of entangled states and quantum gates 14,15 , as well as for quantum simulation 16 .
The strongly interacting Rydberg atoms offer a promising platform for quantum computation and simulation [17][18][19][20] . The long-range Rydberg-Rydberg interaction between the neutral atoms can be controlled and enhanced by applying radio-frequency electric field 21,22 . One hope that one may use the advantages of both trapped ions (individual addressability, entanglement operations with small errors, etc.) and the strong long range interaction of Rydberg ions. However this novel system suffers from some disadvantages. For example, stray electric and magnetic fields due to the trap could induce electric dipole moment of the Rydberg ion. Despite that, trapped Rydberg ions have been recently experimentally accomplished 23 . The Floquet sidebands due to quadrupole interaction as well as modification of the trapping potential due to the strong polarisability of the 88 Sr + Rydberg ion have been observed 24 .
In refs 23,25 the Rydberg levels are excited using a single-photon excitation with vacuum ultraviolet laser light at 122 nm. However, this is quite difficult to handle experimentally. Another experimental approach is to use 88 Sr + Rydberg ions 24 . In that case the Rydberg ions are excited by two-photon transitions at 243 and 309 nm respectively. Though the 88 Sr + ions are excited more easily to Rydberg levels, the nD 3/2 Rydberg states are coupled by the quadrupole field of the trap. These undesirable transitions may transfer population out of the Rydberg state.
In this paper we consider the effect of the quadrupole coupling on the coherent dynamics of Rydberg trapped ions. Such an effect arises due to the huge electric dipole moment of the Rydberg ion and causes undesired excitation of electronic transitions driven by the radio-frequency electric field of the Paul trap, see Fig. 1. We consider single Rydberg ion with one and two Rydberg state manifolds subject to the quadrupole coupling. We show that as long as the radio trap frequency ω of these quadrupole transitions is sufficiently large, the negative effect is (2019) 9:7340 | https://doi.org/10.1038/s41598-019-43865-5 www.nature.com/scientificreports www.nature.com/scientificreports/ averaged and traced out. To see that, we perform a suitable unitary transformation and investigate the system into a different picture. We show that the effect of the quadrupole coupling is merely to rescale the Rabi frequencies Ω i which drive the transition between the Rydberg levels. The quandrupole interaction also induces an energy shift of the respective Rydberg levels. Moreover, we consider the strong dipole-dipole interaction between the Rydberg ions in the presence of quadrupole coupling. We show that the effect of the coupling is to reduce the strength of the dipole-dipole interaction. We also find that the quadrupole coupling induces residual dipole-dipole interaction which can be neglected only in the rotating-wave approximation.
The paper is organized as follows. In Section II we provide discussion of the effect of the radio-frequency electric field on the electronic Rydberg transition. In Sec. III we introduce the suitable unitary transformation which allows to treat the effect of the oscillating quadrupole coupling. In Sec. IV we discuss the single trap ion with one and two Rydberg state manifolds sensitive to the quadrupole coupling. In Sec. V we consider two trapped Rydberg ions interacting via strong dipole-dipole coupling. Finally, in Sec. VI we summarize our findings.
the Level system of 88 sr + trapped Ion Our quantum system consists of a single trapped Rydberg ion. Although the method is applicable for any Rydberg ion we consider for concreteness Rydberg 88 Sr + ion with the level structure shown on Fig. 2. The Rabi frequency Ω 1 drives the two-photon transition between the states |1〉 and |2〉. State |2〉 belongs to a Rydberg nD 3/2 manifold with detuning Δ 2 . We apply an additional laser field with Rabi frequency Ω 2 which couples level |2〉 and level |3〉 with detuning Δ 3 . The latter is part of n′P 1/2 manifold. The interaction Hamiltonian becomes ( 1 = )   Two laser fields are applied which drive the transitions |1〉 ↔ |2〉 and |2〉 ↔ |3〉 with Rabi frequencies Ω 1,2 and detunings Δ 2,3 . The quadrupole field couples the levels |2〉 and |4〉 with peak Rabi frequency Ω. This transition oscillates with radio trap frequency ω. The level |4〉 has an energy shift Δ 4 . (b) Using the effective picture, the system is reduced to three state ladder system with rescaled Rabi frequencies 1,2 Ω′ and detunings Ω′ 2 .
www.nature.com/scientificreports www.nature.com/scientificreports/ a Ry ~ 100 nm. Thus, it follows that  a a Ry o . Therefore the Rydberg ion can no longer be considered as a point-like particle but rather as a composite object 14 and its internal structure must be taken into account. Indeed, as shown in refs 14,24,26 , the electric field of the Paul trap may excite internal electronic transitions which are no longer negligible contrary to the ordinary trapped ions.
The Paul trap electric field can be written as where α and β are electric field gradients and ω is the radio-frequency of the Paul trap 27 . In the customary ion traps, this electric field does not couple internal electronic states. The ion in that case can be considered as a point particle. However, in the case of Rydberg ions, it will couple electronic transitions. The coupling Ĥ e of the above electric field is given by , where e is the electronic charge. Generally, this quadrupole coupling cannot couple (to first order) states in the manifold nX J for J = 1/2 for any X = S, P, D, … Such transitions are only allowed for J > 1/2 due to selection rules. However, states in the manifold nD J (J = 3/2 or J = 5/2) are coupled even to first order by the quadrupole field. It turns out, that the time dependent interaction with the quadrupole is 24 where Ω is the effective Rabi frequency for the quadruple coupling which oscillates with the trap frequency ω. This transition may lead to a leak of population to an undesirable state |4〉 as is shown in Fig. 2. Unfortunately rotating wave approximation is not applicable because the effective Rabi frequency Ω is comparable with the trap frequency ω 24 . In the next section we shall propose solution to this problem.

General Theory of the Effective Picture
First, let us rewrite Eq. (3) for the 88  becomes time-independent to any desired order of ω −1 . Method for averaging of the rapidly oscillating terms was proposed in 28 , which however is not suitable for our case since it requires knowledge of the spectrum of Ĥ 0 . We derive K t ( ) following the method presented in 29,30 (see the Supplement for an overview of the derivation). Here we simply state the resultˆω

Including the quadrupole interaction the total Hamiltonian becomes
We find that the effective Hamiltonian becomeŝˆˆˆω  www.nature.com/scientificreports www.nature.com/scientificreports/ Interestingly, we observe that the quadrupole interaction between states |2〉 and |4〉 is removed. However the new Rabi frequencies in the effective picture are rescaled (renormalized) with the same factor (1 − Ω 2 /(4ω 2 )). Therefore in order to compensate the quadrupole interaction one needs to merely increase the laser intensities with the factor (1 − Ω 2 /(4ω 2 )) −1 . Additionally, we find that the quadrupole interaction caused an energy shift of the states |2〉 and |4〉 such that the laser detuning becomes In Fig. 3 we compare the exact dynamics governed by the full Hamiltonian (6) and the effective Hamiltonian (10). As can be seen very good agreement is observed. For comparison, we also show the effective dynamics which is obtained by standard rotating-wave approximation (RWA), where the effect of the fast oscillating terms is neglected, ≈ ω ve 0 i t . As expected, RWA significantly deviates from the exact solution. This is due to the fact that Ω and ω are of the same order of magnitude, namely Ω = 0.6ω. In Fig. 3(b) we plot the population of the level |2〉 which is subject of the strong quadrupole interaction. Because of that the time evolution of the population contains fast and slow components where the latter can be described within the effective picture. Figure 4 shows the frequency scan of the populations P 1,3 at fixed interaction time. The exact and the effective solutions are almost indiscernible. Lastly, we point out that our analysis is correct as long as the trap frequency is larger than the effective quadrupole coupling, ω ≳ Ω. When the coupling increases more than the driving trap frequency one observes that it cannot be neglected in a sense that the level structure is no longer decoupled.
After a lengthy calculation it can be shown that the next correction to the effective Hamiltonian Eq.
. This explains why the agreement in Fig. 3 is quite accurate. two Rydberg manifolds coupled by the quadrupole coupling. We extend the discussion by including higher angular momentum Rydberg states such as n′P 3/2 states, see Fig. 5(a). In tha explains why the agreeme couples not only states |2〉 and |4〉 but also states |3〉 and |5〉. We shall show that in the effective picture the quadrupole coupling is again removed. In with Ω and Ω being the effective Rabi frequencies for the quadrupole interaction. The expression Eq. (9) for the effective Hamiltonian as well as Eqs (7) and (8) for K remain valid. Thus, we obtain This result means that the initial five-level coupled system is reduced to two uncoupled ladders, see Fig. 5(b). The first ladder is a two level system consisting of states |4〉 and |5〉 driven by effective Rabi frequency Ω′ = as is shown in Fig. 5(b). As long as the initial population is in state |1〉, the population will remain in the second ladder, described by H 2 .
In Fig. 6 we show the resonance oscillations of the probability P 3 (t). We observe that the initial prepared population in state |2〉 exhibits Rabi oscillations where the exact solution is very closed to the effective picture. Although the quadrupole coupling between the states |2〉 and |4〉 is very strong and comparable with the radio trap frequency the corresponding probability is slightly affected.

two Rydberg Ions Interacting with Dipole-Dipole Interaction
In this section, we extend the discussion including the dipole-dipole interaction. We consider an ion chain consisting of two Rydberg ions. The generalization for chain with N ions is straightforward. The full Hamiltonian is quite complicated, see for example ref. 14 . However, under certain rather plausible approximations the Hamiltonian can be reduced to 14ˆˆˆĤ Here   = Ω | 〉 〈 | + | 〉 〈 | + . . .

Conclusion
In this paper we have shown that the quadrupole interaction which causes a reduction in the dipole moment in a trapped Rydberg ion can be dealt with by increasing the Rabi frequencies. To show that we have applied an unitary transformation. In this new picture, dubbed 'effective picture' , see Fig. 2, the Rabi frequencies are renormalized with the same factor [1 − Ω 2 /(4ω 2 )]. Therefore by increasing the laser intensities with the factor [1 − Ω 2 / (4ω 2 )] −1 , the negative effect of the quadrupole interaction Ωcosωt can be removed. In addition, we have extended the discussion, see Fig. 5, when both Rydberg manifolds are coupled by the quadrupole part of the trapping electric field. On the right side of Fig. 5 one observes that the effective five level system is decoupled and the Rabi frequencies are altered by different factors. Therefore even in that case, the negative effect of the quadrupole interaction can be removed. One merely has to rescale the laser intensities by different magnitudes. We have extended the discussion to an ion chain of two ions and we have shown that the Rabi frequencies are renormalized as well as the dipole-dipole coupling is modified. The latter is out of experimental control. However the reduction of the dipole-dipole coupling is only a few per cent for reasonable experimental parameters 24 , while the renormalization can be dealt with by increasing the Rabi frequency as was shown in the single ion case.