Photoionization of Rydberg Atoms in Optical Lattices
Ryan Cardman, Jamie MacLennan, Sarah E. Anderson, Yun-Jhih Chen, Georg Raithel
PPhotoionization of Rydberg Atoms in Optical Lattices
R. Cardman, ∗ J. L. MacLennan, S. E. Anderson, Y.-J. Chen, and G. Raithel
Department of Physics, University of Michigan, Ann Arbor, MI 48109 (Dated: February 22, 2021)We develop a formalism for photoionization (PI) and potential energy curves (PECs) of Rydbergatoms in ponderomotive optical lattices and apply it to examples covering several regimes of theoptical-lattice depth. The effect of lattice-induced PI on Rydberg-atom lifetime ranges from notice-able to highly dominant when compared with natural decay. The PI behavior is governed by thegenerally rapid decrease of the PI cross sections as a function of angular-momentum ( (cid:96) ), and bylattice-induced (cid:96) -mixing across the optical-lattice PECs. In GHz-deep lattices, (cid:96) -mixing leads to arich PEC structure, and the significant low- (cid:96)
PI cross sections are distributed over many lattice-mixed Rydberg states. In lattices less than several tens-of-MHz deep, atoms on low- (cid:96)
PECs areessentially (cid:96) -mixing-free and maintain large PI cross sections, while atoms on high- (cid:96)
PECs trendtowards being PI-free. Characterization of PI in GHz-deep Rydberg-atom lattices may be beneficialfor optical control and quantum-state manipulation of Rydberg atoms, while data on PI in shallowerlattices are potentially useful in high-precision spectroscopy and quantum-computing applicationsof lattice-confined Rydberg atoms.
I. INTRODUCTION
Rydberg atoms in optical lattices and traps havegained interest in the fields of quantum computing andsimulations [1–4], quantum control [5], and high-precisionspectroscopy [6–8], as the lattice confines the atoms andextends interaction times. However, the binding energyof Rydberg atoms is several orders of magnitude belowthe photon energy (cid:126) ω of commonly used optical-latticefields. Optical photoionization (PI) of the Rydberg va-lence electron leads to lifetime reduction and decoher-ence. Lattice-induced PI can broaden radio-frequency(RF) transitions between Rydberg states and limit the fi-delity of Rydberg-atom quantum-control and -simulationschemes that involve coherences in the RF domain. ThePI can also degrade optical coherences between groundand Rydberg states that can be induced by (cid:46) e A ( r ) / (2 m e ), and the e A ( r ) · p /m e inter-actions, with e , m e , r , p , and A ( r ) denoting the mag-nitude of the fundamental charge, electron mass, elec-tron position and momentum in the laboratory frame,and the position-dependent vector potential of the field,respectively [12]. Interplay between these two interac-tions has previously been discussed in Refs. [13–15] inthe context of above-threshold ionization. In an inhomo-geneous light field, such as an optical lattice, the pon-deromotive A term generates an optical force on theRydberg electron that depends on the intensity gradient ∗ [email protected] of the optical-lattice interference pattern and its over-lap with the spatial distribution of the Rydberg-electronwavefunction. Effects of the ponderomotive force on freeelectrons in a standing-wave laser field were studied be-fore in Refs. [16, 17]. The Rydberg electron is quasi-free,allowing the ponderomotive force to enable optical-latticetraps for Rydberg atoms [18, 19]. The spatial period ofRydberg-atom optical lattices, which is on the order ofthe laser wavelength λ , is similar to the diameter of thetrapped atoms, a situation that differs from most op-tical lattices, in which the atoms are point-like relativeto λ . A ponderomotive optical lattice couples Rydbergstates over a wide range of electronic angular momenta, (cid:96) , [20, 21], affording capabilities in high- (cid:96) Rydberg-stateinitialization [5, 21] and Rydberg-atom spectroscopy freeof selection rules for (cid:96) [20, 22].Optical and black-body-radiation-induced PI resultfrom the A · p -term [12]. Here we investigate laser-induced PI of Rydberg atoms trapped in an optical lat-tice. In Sec. II, we derive PI cross sections and ratesfor Rydberg atoms in plane-wave light fields and ex-tend the results to Rydberg atoms in optical lattices.In Sec. III, we obtain equations for the potential energycurves (PECs), the adiabatic Rydberg states, and theirPI-induced decay rates in the lattice. In the examples inSec. IV, we focus on rubidium Rydberg atoms in a one-dimensional lattice formed by counter-propagating laserbeams of 1064 nm wavelength. The lattice strength ischaracterized by the magnitude of the ponderomotive in-teraction relative to the unperturbed Rydberg-level sepa-rations. We present results for PECs and lattice-inducedPI of (cid:96) -mixed Rydberg atoms in a strong optical lattice,and of Rb 50 F atoms in a weaker, (cid:96) -mixing-free opticallattice. In the Appendix, we discuss fundamental aspectsof optical PI of Rydberg atoms. a r X i v : . [ phy s i c s . a t o m - ph ] F e b II. PI OF RYDBERG ATOMSA. PI cross sections and rates
The lowest-order transition rate between atomic statesdue to an interaction ˆ H int is given by Fermi’s golden rule,Γ = π (cid:126) |(cid:104) f | ˆ H int | i (cid:105)| ρ ( (cid:15) ), with final-state energy (cid:15) and den-sity of final states ρ ( (cid:15) ) [12]. For PI in a plane-wave field,it is ˆ H int = e ˆ A · ˆ p /m e , and the final state | f (cid:105) is a free-electron state. We normalize the free-electron states perunit energy, i. e. (cid:104) f (cid:48) | f (cid:105) = δ ( (cid:15) (cid:48) − (cid:15) ) δ η (cid:48) ,η , with η de-noting the angular-momentum quantum numbers ( (cid:96), m (cid:96) )and ρ ( (cid:15) ) being equal to 1 per unit energy. The PI crosssection σ PI is determined by dividing the PI rate by thephoton flux density, I/ ( (cid:126) ω ), where I is the field intensityand ω its angular frequency. In SI units, for a linearly po-larized field (polarization unit vector ˆ n ) with wave vector k , the PI cross section is (see Appendix A) σ PI = πe (cid:126) (cid:15) m ωc (cid:12)(cid:12)(cid:12)(cid:12) ˆ n · (cid:90) ψ ∗ f e i k · r e ∇ e ψ i d r e (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) E H a (cid:19) , (1)where r e denotes the relative Rydberg-electron coordi-nate, E H the atomic energy unit, and a the Bohr ra-dius. The last term converts the squared matrix element,which is in atomic units, into SI units. For PI of Rydbergatoms the electric-dipole approximation (EDA) typicallyis valid, as shown in [23] and discussed in greater de-tail in Appendix B. The EDA is implemented by setting e i k · r e = 1 in Eq. 1. The resultant expression for thematrix element is referred to as “velocity form”, usedthroughout this paper to compute the PI cross sections.For ˆ p -independent atomic potentials, the matrix ele-ment in Eq. 1, with the EDA applied, can be transformedinto “length form”, leading to σ PI , L = πe ω(cid:15) c (cid:12)(cid:12)(cid:12)(cid:12) ˆ n · (cid:90) ψ ∗ f r e ψ i d r e (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) a E H (cid:19) . (2)This length-form expression for the PI cross section isnot accurate if the atomic potential is (cid:96) -dependent, as inthe present work on Rb.In the following, n denotes the bound-state principalquantum number, (cid:15) (cid:48) the free-electron energy, and (cid:96) > thelarger of the bound- and free-electron angular momenta, (cid:96) and (cid:96) (cid:48) . The shell-averaged PI cross section, given bythe average of the PI cross sections of the m (cid:96) -sublevelsof the Rydberg state, is¯ σ (cid:15) (cid:48) ,(cid:96) (cid:48) n,(cid:96) = πe (cid:126) (cid:15) m ωc (cid:96) > (2 (cid:96) + 1) | M | (cid:18) E H a (cid:19) , (3)where M is the radial part of the matrix element fromEq. 1 in atomic units, which is, with the EDA applied, M = (cid:90) ∞ u (cid:15) (cid:48) ,(cid:96) (cid:48) ( r e ) (cid:20) u (cid:48) n,(cid:96) ( r e ) ∓ u n,(cid:96) ( r e ) r e (cid:96) > (cid:21) dr e . (4) FIG. 1. Total shell-averaged PI cross sections ¯ σ n,(cid:96) for Ryd-berg n and (cid:96) states of Rb for λ = 1064 nm, obtained bysumming Eq. 3 over (cid:96) (cid:48) . For each (cid:96) , the n values range from20 to 90 in steps of 5. The dashed line represents the Thomsoncross section, σ T =0.665 barn. There, the upper sign is for (cid:96) > = (cid:96) (cid:48) , the lower sign for (cid:96) > = (cid:96) , and (cid:96) (cid:48) = (cid:96) ±
1. The functions u ∗ ,(cid:96) ( r e ) are givenby u ∗ ,(cid:96) ( r e ) = r e R ∗ ,(cid:96) ( r e ), where R ∗ ,(cid:96) ( r e ) is the usual ra-dial wavefunction, and ∗ = n or (cid:15) (cid:48) for bound- and free-electron states, respectively. For light polarized in the z -direction, the PI cross section for an atom in a well-defined m (cid:96) -state is σ (cid:15) (cid:48) ,(cid:96) (cid:48) z,n,(cid:96),m (cid:96) = 3( (cid:96) > − m (cid:96) )(2 (cid:96) > + 1)(2 (cid:96) > −
1) (2 (cid:96) + 1) (cid:96) > ¯ σ (cid:15) (cid:48) ,(cid:96) (cid:48) n,(cid:96) , (5)with ¯ σ from Eq. 3. In the applications discussed below,it is convenient to assume a light field that is propagatingalong the atom’s quantization axis, ˆ z , and that is linearlypolarized along a transverse direction, which we may de-fine as ˆ x . The PI cross section for linear polarizationtransverse to the atom’s quantization axis is σ (cid:15) (cid:48) ,(cid:96) (cid:48) x,n,(cid:96),m (cid:96) = 32 ( (cid:96) (cid:48) ( (cid:96) (cid:48) + 1) + m (cid:96) )(2 (cid:96) > + 1)(2 (cid:96) > −
1) (2 (cid:96) + 1) (cid:96) > ¯ σ (cid:15) (cid:48) ,(cid:96) (cid:48) n,(cid:96) . (6)We calculate ¯ σ (cid:15) (cid:48) ,(cid:96) (cid:48) n,(cid:96) , required in Eqs. 5 and 6, for a widerange of bound states ( n, (cid:96) ) and both PI channels (cid:96) (cid:48) = (cid:96) ±
1. The free-electron energy in atomic units is (cid:15) (cid:48) = 2 π a αλ − n ∗ , with the laser wavelength λ in meters, the fine struc-ture constant α , and the effective quantum number of theRydberg state, n ∗ . For the calculation of the bound-stateand free-electron wavefunctions [24], we use model po-tentials from [25], which have previously been employedto compute polarizabilities [26] and two-photon excita-tion rates [27] in Rb. A table of the calculated ¯ σ (cid:15) (cid:48) ,(cid:96) (cid:48) n,(cid:96) , for λ = 1064 nm, is provided as Supplementary Material.To illustrate the general behavior of PI cross sectionsfor different Rydberg states, in Fig. 1 we show results forRb in a λ = 1064-nm field as a function of n and (cid:96) . Thecross sections are generally quite large for low (cid:96) , withan exception for the S -states that is caused by a Cooperminimum [28, 29]. The calculated PI cross sections de-crease rapidly as (cid:96) increases. For (cid:96) (cid:38)
10, they drop belowthe elastic photon scattering cross section, given by theThomson cross section, σ T = 0 .
665 barn. PI cross sec-tions (cid:46) σ T are likely too small to cause observable effectsin applications. B. Fine structure effects
So far, the fine structure has been neglected becauseit is very small on the scale of the Rydberg-atom bind-ing energy and, even more so, on the energy scale of theliberated photo-electron in the final state | f (cid:105) . In theapplications described in the following sections, the finestructure must be included, however, because it can beon the order of or larger than the optical-lattice trapdepth, and it therefore does affect the initial state | i (cid:105) .The PI cross sections of the fine-structure-coupled un-perturbed Rydberg states, | n, (cid:96), j, m j (cid:105) , follow from thecross sections in Eqs. 5 and 6 via σ (cid:15) (cid:48) ,(cid:96) (cid:48) ∗ ,n,(cid:96),j,m j = | c ↑ | σ (cid:15) (cid:48) ,(cid:96) (cid:48) ∗ ,n,(cid:96),m j − / + | c ↓ | σ (cid:15) (cid:48) ,(cid:96) (cid:48) ∗ ,n,(cid:96),m j +1 / (7)with Clebsch-Gordon coefficients c ↑ = (cid:104) j, m j | m (cid:96) = m j − / , m s = 1 / (cid:105) and c ↓ = (cid:104) j, m j | m (cid:96) = m j + 1 / , m s = − / (cid:105) , and coupled, orbital and spin magnetic quantumnumbers m j , m (cid:96) and m s , respectively. The identifier ∗ = z or x denotes the respective laser polarization directionfrom Eq. 5 or 6. The PI rates of the Rydberg states inthe “( j, m j )”-basis, {| n, (cid:96), j, m j (cid:105)} , are then given byΓ ∗ ,n,(cid:96),j,m j = ( σ (cid:15) (cid:48) ,(cid:96) − ∗ ,n,(cid:96),j,m j + σ (cid:15) (cid:48) ,(cid:96) +1 ∗ ,n,(cid:96),j,m j ) I/ ( (cid:126) ω ) , (8)with the light intensity at the atom’s center-of-mass(CM) location denoted by I , and σ -values from Eq. 7.In cases where the fine-structure is absent or decou-pled by an auxiliary field, the time-independent Rydbergstates are {| n, (cid:96), m (cid:96) , m s (cid:105)} . The PI rates of states in the“( m (cid:96) , m s )”-basis are m s -independent and followΓ ∗ ,n,(cid:96),m (cid:96) = ( σ (cid:15) (cid:48) ,(cid:96) − ∗ ,n,(cid:96),m (cid:96) + σ (cid:15) (cid:48) ,(cid:96) +1 ∗ ,n,(cid:96),m (cid:96) ) I/ ( (cid:126) ω ) , (9)with σ -values from Eqs. 5 or 6. C. PI in an optical lattice
In an optical lattice, intensity and polarization mayvary within the volume of a Rydberg atom. In fact, theatomic volume can extend over several nodes and anti-nodes of the light field [1, 18, 30]. The lattice-intensityvariation within the atomic volume is important for the PECs and state-mixing in the lattice, as discussed in thenext section. For PI, it must be considered what exactintensity value determines the PI rate of the atoms inthe lattice. Our analysis given in the Appendix showsthat the PI rates of Rydberg states are determined bythe intensity at the exact CM location of the Rydbergatom, I ( R ). We enter I ( R ) into Eqs. 8 or 9 to obtainthe PI rates of the Rydberg basis states. It is irrelevanthow the field varies over the atomic volume. Especiallynoteworthy is the fact that the light intensity within themain lobes of the Rydberg electron wavefunction is notimportant. This finding is a consequence of the valid-ity of the EDA for PI of Rydberg atoms, which is dis-cussed in the Appendix. Laser-induced Rydberg-atomPI was previously measured in plane waves [31] and, ina spatially-sensitive manner, in an optical lattice [23]. III. POTENTIAL ENERGY CURVESA. Strong optical-lattice regime
Rydberg atoms in an optical lattice are subject to boththe A · p and the ponderomotive ( A ) interactions, giv-ing rise to lattice-induced PI and the atom trapping po-tentials (PECs) at the same time. In the following wedescribe our comprehensive formalism for both PI andPECs. In a one-dimensional optical lattice along the z -direction, the PECs are calculated by finding and diago-nalizing the Hamiltonianˆ H lat = ˆ H + V P (ˆ z e + Z ) (10)on a grid of fixed CM positions Z of the atoms in the lat-tice. There, ˆ H is the field-free atomic Hamiltonian, andthe operator ˆ z e represents the relative z -coordinate of theRydberg electron. Further, V P ( z ) = e E ( z ) / (4 m e ω )is the free-electron ponderomotive potential that followsfrom the A -interaction, E ( z ) the total lattice electric-field amplitude, and z = z e + Z the z -coordinate of theRydberg electron in the laboratory frame. Classically,the A -term may be thought of as the time-averaged ki-netic energy of the electron quiver in the lattice electricfield at the optical frequency [30]. In a one-dimensionallattice along z , V P ( z ) = V (1 + cos(2 kz )) , (11)with the full free-electron potential depth 2 V and k =2 π/λ = ω/c . For a pair of lattice beams with equalsingle-beam electric-field amplitude E and equal linearpolarization, it is 2 V = e E / ( m e ω ). The potential V P ( z ) introduces couplings that are free of selection rulesfor (cid:96) [21, 32]. From a perturbation-theory viewpoint,the Rydberg-atom lattice is strong if the lattice depthapproaches the characteristic energy scale of the unper-turbed Rydberg atom, i.e., if 2 V (cid:38) sE H /n , with scalingparameter s ∼ . (cid:96) states, causing mix-ing among such states.To include the effect of PI, we add imaginary contri-butions to the energy eigenvalues of the basis states thataccount for the PI-induced decay. The resultant effectiveHamiltonian isˆ H ( Z ) = ˆ H lat ( Z )+ (cid:88) n(cid:96)jm j | n(cid:96)jm j (cid:105)(cid:104) n(cid:96)jm j | (cid:18) − i (cid:126) Γ x,n(cid:96)jm j ( Z )2 (cid:19) , (12)with ˆ H lat defined in Eq. 10. The PI rates Γ x,n(cid:96)jm j , whichtrace back to the A · p interaction, are obtained fromEq. 8, using the intensity at the CM location of the atom, I ( Z ), and the polarization identifier ∗ = x . While orig-inating in different terms of the atom-field interaction,the ionization rates and the free-electron ponderomotiveenergy at the atom’s CM location are both proportionalto I ( Z ), Γ x,n(cid:96)jm j ( Z ) = I ( Z ) σ x,n(cid:96)jm j (cid:126) ωV P ( Z ) = I ( Z ) e c(cid:15) m e ω . (13)We diagonalize the Hamiltonian in Eq. 12 in subspacesof fixed m j , which is conserved due to the assumed az-imuthal symmetry, on a grid of Z -values [21]. This yieldsˆ H ( Z ) | ψ k ( Z ) (cid:105) = (cid:20) W k ( Z ) + i (cid:126) Γ k ( Z )2 (cid:21) | ψ k ( Z ) (cid:105) , (14)with the PECs given by the real-valued W k ( Z ), the adi-abatic Rydberg states | ψ k ( Z ) (cid:105) , their PI rates Γ k ( Z ),and a PEC index k . Due to lattice-induced state mixing,the | ψ k ( Z ) (cid:105) are Z -dependent superpositions of field-freeatomic states | n, l, j, m j (cid:105) .We note that the PECs W k ( Z ) satisfy W k ( Z ) = (cid:90) V P ( z e + Z ) | ψ k ( r e ; Z ) | d r e , (15)which represents a spatial average of V P , weighted bythe wavefunction densities | ψ k ( r e ; Z ) | of the adiabaticstates | ψ k ( Z ) (cid:105) . The wavefunction density is traced overthe electron spin. Since the | ψ k ( Z ) (cid:105) are not known be-fore diagonalization of the Hamiltonian in Eq. 12, Eq. 15generally cannot be used to calculate PECs (exceptionsare discussed in Sec. III B). Instead, the Hamiltonian inEq. 12 must be diagonalized to simultaneously yield boththe PECs, W k ( Z ), and the | ψ k ( Z ) (cid:105) . B. Weak optical-lattice regime
If the Rydberg-atom lattice is weak, 2 V < sE H /n ,there are cases in which the ponderomotive potential V P ( z ) does not cause lattice-induced state mixing of theunperturbed Rydberg levels. These cases include nS / Rydberg levels, and nP j and nD j levels if 2 V is also lessthan the fine structure splitting. For Rydberg states thatare known to be mixing-free, the PECs can be obtainedfrom first-order non-degenerate perturbation theory, W k ( Z ) = (cid:90) V P ( z e + Z ) | ψ k, ( r e ) | d r e . (16)This expression amounts to a spatial average of V P ,weighted by the wavefunction density of the unperturbed, Z -independent state | ψ k, (cid:105) = | n, (cid:96), j, m j (cid:105) , | ψ k, ( r e ) | = | R n,(cid:96),j ( r e ) | (cid:104) | c ↑ Y m j − / (cid:96) ( θ e , φ e ) | + | c ↓ Y m j +1 / (cid:96) ( θ e , φ e ) | (cid:105) , with Clebsch-Gordon coefficients c ∗ defined as af-ter Eq. 7, and spherical Rydberg-electron coordinates( r e , θ e , φ e ). The PEC index k now merely is a short-hand label for the mixing-free state | n, (cid:96), j, m j (cid:105) . PECsin weak lattices have been investigated in Refs. [33, 34].Also, the PI rate of the state | n, (cid:96), j, m j (cid:105) in the lattice issimply given by Eq. 13, taken at the CM location Z .In certain scenarios, one can force applicability of non-degenerate perturbation theory by lifting degeneraciesvia application of an auxiliary DC electric or magneticfield, or a microwave field. If the auxiliary field sup-presses lattice-induced state mixing, the adiabatic Ryd-berg states in the lattice become independent of Z ,allowing a perturbative calculation of the PECs as inEq. 16 [7, 30]. In some of the cases, the fine-structurecoupling can be lifted by the DC field, and the time-and Z -independent states become | n, (cid:96), m (cid:96) (cid:105) ⊗ | m s (cid:105) . Inthose cases, the wavefunctions to be used in Eq. 16 are ψ k, ( r e ) = (cid:104) r e | n, (cid:96), m (cid:96) (cid:105) , and their PI rates follow fromEq. 9. One such example is the weak one-dimensional lat-tice of Rb 50 F -states with an external DC electric field,discussed in Sec. IV B. IV. RESULTSA. An implementation of a strong optical lattice
In strong Rydberg-atom optical lattices, lattice-induced state mixing gives rise to a rich structure ofPECs. This is illustrated in Fig. 2 for n = 50, m j = 1 / V = h × . × E rec , with the single-photon recoil energy ofRb for λ = 1064-nm, E rec = h × .
027 kHz. For this lat-tice it is 2 V ∼ . E H /n , placing it in the strong-latticeregime as defined in Sec. III. Fine structure and quantum - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2- 4 3 . 8 8- 4 3 . 8 6- 4 3 . 8 4- 4 3 . 8 2- 4 3 . 8 0 Wk (cm-1) Z ( (cid:1) )( a ) ( b ) ( c ) G k = 1 . 6 M s - 1 G k ( 1 0 s - 1 ) - 0 . 2 5 0 - 0 . 2 2 5 - 0 . 2 0 0 - 0 . 1 7 5 - 0 . 1 5 0- 4 3 . 8 7- 4 3 . 8 6- 4 3 . 8 5- 4 3 . 8 4- 4 3 . 8 3- 4 3 . 8 2- 4 3 . 8 1- 4 3 . 8 0 Wk (cm-1) Z ( (cid:1) )( b ) G k ( 1 0 s - 1 ) - 0 . 1 0 0 - 0 . 0 7 5 - 0 . 0 5 0 - 0 . 0 2 5 0 . 0 0 0 0 . 0 2 5 0 . 0 5 0 0 . 0 7 5 0 . 1 0 0- 4 3 . 8 7- 4 3 . 8 6- 4 3 . 8 5- 4 3 . 8 4- 4 3 . 8 3- 4 3 . 8 2- 4 3 . 8 1- 4 3 . 8 0 Wk (cm-1) Z ( (cid:1) ) ( d ) ( c ) G k ( 1 0 s - 1 ) - 0 . 0 3 - 0 . 0 2 - 0 . 0 1 0 . 0 0 0 . 0 1 0 . 0 2 0 . 0 3- 4 3 . 8 7 4- 4 3 . 8 7 2- 4 3 . 8 7 0- 4 3 . 8 6 8- 4 3 . 8 6 6- 4 3 . 8 6 4- 4 3 . 8 6 2- 4 3 . 8 6 0 Wk (cm-1) Z ( (cid:1) )( d ) G k = 0 . 0 8 8 M s - 1 G k ( 1 0 s - 1 ) FIG. 2. PECs in a one-dimensional ponderomotive optical lattice of Rb Rydberg atoms for n = 50 and m j = 1 / λ = 1064 nm,and lattice depth 2 V = h × . × E rec . PEC energies are in cm − relative to the ionization threshold, and CMpositions Z in units λ . The boxed regions in (a) correspond to the full regions displayed in panels (b) and (c), while the boxed region in(c) corresponds to the full region displayed in panel (d). The color of the dots on the PECs shows PI rate Γ k ( Z ),on the color scales provided, and the dot diameter is proportional to Γ k ( Z ). For clarity, the dot diameters in (b)are enhanced by a factor of 50 relative to those in (a), and those in (c) and (d) by a factor of 10. The close-up viewin (d) shows ≈ k ( Z ), of the PECs.The lattice primarily mixes states of small quantum de-fects, which covers the vast majority of Rydberg states.The adiabatic states of the PECs, | ψ k ( Z ) (cid:105) , are coher-ent superpositions of a wide range of low- (cid:96) and high- (cid:96) states, including circular Rydberg states. The lowest-energy curves in Fig. 2(a) are substantially perturbed50 F -states, which are lowered in energy due to theirquantum defect and are not entirely mixed into the man-ifold of high- (cid:96) states, which have near-zero quantum de-fect (states with (cid:96) ≥ F -character into the high- (cid:96) states is efficient enough to make the latter laser-excitable from alow-lying D -level. For instance, the three-step excitationsequence 5 S / → P / → D / → nF / using 795 nm,762 nm, and ∼ Z = ± λ/ V P ( z ) is linearin these regions, the analogy with the DC Stark effectis expected [21]. Near the nodes and anti-nodes of thelattice [ Z = 0 , ± λ/ k ( Z ), overall scale withthe lattice intensity at the atomic CM location, whichis proportional to (1 + cos(2 kZ )). The maximum Γ k -values in Fig. 2(a) are Γ k ≈ . × s − for the 50 F -like states at Z = 0, where the lattice intensity is max-imal. For the high- (cid:96) states within the range displayedin Fig. 2(b), which is near a lattice-intensity minimum,the Γ k range between 2 × s − and zero (at the ex-act anti-node positions). For the high- (cid:96) states withinthe range of Fig. 2(c), near an intensity maximum, theΓ k -values peak at about 10 s − . Since radiative decayrates and black-body-radiation-induced transition ratesof Rydberg levels around n = 50 are only on the orderof 10 s − , PI-induced decay in the lattice will be quitenoticeable for the high- (cid:96) states. For the 50 F -like states,it will greatly exceed natural decay, for conditions as inFig. 2.In possible future experimental work, an ultra-deepRydberg-atom lattice with a depth of 2 V = h × w = 20 µ m.Such a lattice can be prepared, for instance, by using anear-concentric field enhancement cavity [40], with theRydberg atoms loaded into the focal spot of the cavity.The PI-induced spectroscopic level widths in Fig. 2,which are Γ k / (2 π ) (cid:46)
250 kHz, should be large enoughto become visible in spectroscopic measurement of PECswith narrow-linewidth lasers (linewidth (cid:46)
100 kHz). An-other possible measurement method for PEC curves andlevel widths would be microwave spectroscopy from asuitable low- (cid:96) launch Rydberg state. This method wouldessentially be Doppler-effect-free and benefit from theHz-level linewidth of typical microwave sources, result-ing in higher spectral resolution. However, it would addexperimental complexity due to the need to account forthe PI and level shifts of the Rydberg launch state withinthe optical lattice.We note that near Z = 0 and ± λ/ ≈
10 nm and a depthin the range of h ×
10 to 100 MHz. The periodicity isabout a factor of 50 smaller than the fundamental λ/ B. An implementation of a weak optical lattice
In weak Rydberg-atom lattices it is 2 V (cid:28) . E H /n , (cid:96) -mixing plays no significant role for states with (cid:96) < (cid:96) PECs that have large PI cross sections (seeFig. 1). Hence, while the PI rate averaged over all PECsdrops in proportion with lattice intensity, atoms on low- (cid:96)
PECs may still photo-ionize at high rates.Examples of PECs for 50 F j in a weak lattice with adepth of 2 V = h ×
20 MHz= 9867 E rec are shown inFig. 3. The 50 F -levels split into seven resolved compo-nents of conserved m j . With the exception of | m j | = 7 / j = 5 / / {| F / , m j (cid:105) , | F / , m j (cid:105)} yields the PECsand their PI rates. As seen in Fig. 3(a), the modula-tion depth of the PECs varies from strongly modulatedat | m j | = 7 / | m j | = 1 /
2. Thevariation in PEC modulation depth arises from the differ-ing extent of the Rydberg-electron wavefunctions alongthe axis of the lattice, which results in varying amountsof averaging in Eq. 15 [34]. Generally, the sublevels withlesser values of | m j | have wavefunctions that extend morein the direction of the lattice axis, resulting in less deeplymodulated PECs. The j -mixing causes pairs of states ofsame m j to repel each other near the lattice inflectionpoints at Z = ± λ/
8. The j -mixing is illustrated inFig. 3(b), where the expectation value j on some PECsvaries considerably as a function of Z , while maintain-ing an average of 3 over pairs of coupled PECs with same m j . The level repulsion is seen best in Fig. 3(c), wherewe show a detailed view of the level pair with m j = 5 / k ( Z ) generally scalewith the lattice intensity, which is ∝ (1 + cos(2 kZ )).Further, according to Eqs. 6-9, the Γ k -values at fixed Z should increase with m (cid:96) , and by continuation, with m j .This trend is obvious in Fig. 3(a). To exhibit this be-havior more clearly, in Fig. 3(d) we show PECs and PIrates, Γ k , with an additional longitudinal electric fieldalong the z -direction. The field is sufficiently strong todecouple the fine structure, but weak enough to not causesignificant (cid:96) -mixing with nearby D and G Rydberg states.The adiabatic states | ψ k (cid:105) associated with the PECs thenapproximately are | F, m (cid:96) , m s (cid:105) , with strictly conserved m j = m (cid:96) + m s . With all degeneracies lifted, the PECs fol-low from Eq. 16 with ψ ( r e ) = (cid:104) r e | F, m (cid:96) (cid:105) . There still isa small fine-structure splitting between PECs with same m (cid:96) and different m s , with the exception of m (cid:96) = 0, where - 0 . 2 0 - 0 . 1 5 - 0 . 1 0 - 0 . 0 50 . 0 0 40 . 0 0 60 . 0 0 80 . 0 1 00 . 0 1 20 . 0 1 40 . 0 1 6 < j > D W k (GHz) Z ( l )| m j | = 5 / 2 ( c ) F i n e s t r u c t u r e m i x i n g i n E = 0 f o r | m j | = 5 / 2 [ Z o o m ] ( d ) P I r a t e s i n E z = 0 . 1 V / c m - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2- 0 . 0 8- 0 . 0 7- 0 . 0 6- 0 . 0 5- 0 . 0 4- 0 . 0 3- 0 . 0 2 | m l , m s > = | 0 , + / - 1 / 2 >| 3 , 1 / 2 > D W k (GHz) Z ( l ) G k ( 1 0 / s ) | 3 , - 1 / 2 >| 2 , - 1 / 2 > | 2 , 1 / 2 >| 1 , - 1 / 2 > | 1 , 1 / 2 > - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 20 . 0 0 20 . 0 0 40 . 0 0 60 . 0 0 80 . 0 1 00 . 0 1 20 . 0 1 40 . 0 1 60 . 0 1 8 | m j | = D W k (GHz) Z ( l ) < j > - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 20 . 0 0 20 . 0 0 40 . 0 0 60 . 0 0 80 . 0 1 00 . 0 1 20 . 0 1 40 . 0 1 60 . 0 1 8 | m j | = Z ( l ) G k ( 1 0 / s ) D W k (GHz) ( a ) P I r a t e s i n E = 0 ( b ) F i n e s t r u c t u r e m i x i n g i n E = 0 FIG. 3. (a) and (b): PECs of Rb 50 F in an optical lattice with λ = 1064 nm and a depth 2 V = h ×
20 MHz= 9867 E rec . Thedeviations of the PECs from the field-free Rb 50 F / state, ∆ W , in GHz are plotted vs CM position, Z , in units λ . The PEClabels show | m j | . Symbol sizes and colors in (a) and (b) show PI rate and expectation value of j , respectively, on the given colorscales. (c) Magnified view of a | m j | = 5 / j -mixing and level repulsion. (d)PECs for the same conditions as in (a-c), but with an added longitudinal DC electric field of 0.1 V/cm. The DC field breaksthe fine-structure coupling, and the PECs correspond with position-independent adiabatic states | F, m (cid:96) , m s (cid:105) . Symbol sizeand color show PI rate on the given color scale. the spin-up and -down states are degenerate. The PI-rateratios between the PECs in Fig. 3(d), at a fixed Z , arenow governed by Eq. 6, with (cid:96) = 3 and (cid:96) (cid:48) = 2 or 4, andthe shell-averaged PI cross sections ¯ σ (cid:15) (cid:48) ,(cid:96) (cid:48) F =650 barn for (cid:96) (cid:48) = 2 and 3494 barn for (cid:96) (cid:48) = 4. Factoring in all depen-dencies in Eq. 6 and summing over (cid:96) (cid:48) , the PI rates at thelattice intensity maxima, for conditions as in Fig. 3(d),vary between 21 × s − for m (cid:96) = 3 and 13 × s − for m (cid:96) = 0. In comparison, for the rates of black-body-radiation-induced bound-bound (Bbb) transitions andblack-body photoionization (Bpi) [42, 43] we calculateΓ Bbb, F = 10 . × s − and Γ Bpi, F = 0 . × s − ,respectively, for a radiation temperature of 300 K and forall m (cid:96) . The lattice-induced PI should therefore be dom-inant over black-body-induced transitions.In potential experimental work, a lattice as in Fig. 3could be achieved, for instance, by focusing two counter-propagating 1064-nm laser beams, with a power of 1 Weach, into a confocal spot with w = 20 µ m. The PECs ofRb nF states could then be studied via three-photon laserexcitation from Rb 5 S / . A laser-spectroscopic measure-ment of PI-limited PEC widths of 50F states in latticesas in Fig. 3 would require a laser linewidth (cid:46) V. CONCLUSION
We have obtained photoionization cross sections anddecay rates of Rydberg atoms in plane-wave opticalfields and in optical lattices. We then proceeded to in-clude photoionization into calculations of potential en-ergy curves of Rydberg atoms in optical lattices. Thedegree of lattice-induced (cid:96) -mixing has been related tothe ratio between lattice depth and the intrinsic energyscale of the Rydberg atoms. We have compared the sig-nificance of lattice-induced photoionization with that ofnatural decay and black-body effects.The strong Rydberg-atom lattices discussed inSec. IV A are suitable, for instance, for all-optical quan-tum initialization of high-angular-momentum states [5]and other quantum-control applications. Weak Rydberg-atom lattices, as discussed in Sec. IV B, are attractive forapplications that include quantum computing and sim-ulation [1–3], and high-precision spectroscopy [7, 8, 22].Weak Rydberg-atom lattices at magic wavelengths [44]can minimize trap-induced shifts of certain transitions [6,7]. Further, the nF j Rydberg states we have consideredin our examples can serve as launch states for circular-state production [5, 7]. Some of these and other appli-cations of Rydberg-atom optical lattices are subject tolimitations from spectroscopic line broadening and de-coherence caused lattice-induced photoionization. Thephotoionization rates as calculated in our paper will beuseful in detailed feasibility estimates for these efforts.
ACKNOWLEDGMENTS
This work was supported by NSF Grant No. PHY-1806809 and NASA Grant No. NNH13ZTT002N.
Appendix A: Atom-Field Interaction
In the Appendices we validate the electric-dipole ap-proximation (EDA) in optical transitions and photo-ionization (PI) of µ m-sized Rydberg atoms with light.Expressions are extended to PI in an optical lattice. Inthe following, the “ e ”-subscript on the relative electroncoordinates, used in the main text, is dropped, and alllowercase coordinates are relative electron coordinates.The non-relativistic Hamiltonian for an N -electronatom with nuclear charge Z is given byˆ H = N (cid:88) i =1 (cid:32) ˆ p i m e − Ze π(cid:15) ˆ r i (cid:33) + 14 π(cid:15) N (cid:88) i>j =1 e | ˆ r i − ˆ r j | . (A1)The first sum includes the kinetic and potential energyof each electron in the Coulomb field of the nucleus, andthe second the electrostatic repulsion between pairs ofelectrons. The interaction of the atom with an electro-magnetic field can be taken into account by replacing ˆ p i with ˆ p i + e A (ˆ r i , t ), where A (ˆ r i , t ) is the vector potential.The resulting interaction added to Eq. A1 isˆ H = N (cid:88) i =1 { e m e [ˆ p i · A (ˆ r i ) + A (ˆ r i ) · ˆ p i ]+ e m e A (ˆ r i ) } . The A (ˆ r i ) term gives rise to the ponderomotive poten-tial that is responsible for the trapping of Rydberg atomsin an optical lattice [1, 30]. In a QED treatment, theFeynman diagram of the A (ˆ r i ) term is a vertex with twoinstantaneous photons [45]. The ˆ A · ˆ p -term causes a widerange of atom-field interactions, including light-induced and black-body-radiation-induced PI. In the Coulombgauge, ∇ · A = 0, the operators ˆ p i and A (ˆ r i ) commute,and the ˆ A · ˆ p interaction writesˆ H int = (cid:88) i (cid:18) em e A (ˆ r i ) · ˆ p i (cid:19) . In the present work we consider a Rydberg atom withone active electron. In this case, the sum can be dropped,and the position and momentum operators ˆ r and ˆ p arejust for the Rydberg electron. In a source-free field,the electric field and vector potential are related by E = − ( ∂ A /∂t ) [46]. We consider a linearly polarizedplane wave with electric-field amplitude E , and choosethe x -axis in propagation and the z -axis in field direction, A ( r , t ) = E iω ˆ z e i ( kx − ωt ) + cc . There, ω is the angular fre-quency and k the wavenumber.The matrix element (cid:104) f | ˆ H int | i (cid:105) is, in the rotatingframe [12, 47], (cid:104) f | ˆ H int | i (cid:105) = − e (cid:126) E m e ω (cid:90) ψ ∗ f e ikx ∂∂z ψ i d r, (A2)where | i (cid:105) and | f (cid:105) are the initial and final states withwavefunctions ψ i and ψ f . Using Fermi’s golden rule, thetransition rate isΓ = 2 π (cid:126) |(cid:104) f | ˆ H int | i (cid:105)| ρ ( (cid:15) ) , (A3)with the density of states ρ ( (cid:15) ) at the final-state energy.The rates are proportional to the intensity, regardless ofwhether the EDA, which amounts to setting e ikx = 1,can be made or not.To compute the matrix elements M A = (cid:82) ψ ∗ f e ikx ∂∂z ψ i d r , which include both the angular andradial parts, we use the usual notations ψ n,(cid:96),m (cid:96) ( r, θ, φ )= R n,(cid:96) ( r ) Y m (cid:96) (cid:96) ( θ, φ ) [47], and R n,(cid:96) ( r ) = u n,(cid:96) ( r ) /r . Thequantum numbers ( n, (cid:96), m (cid:96) ) and ( n (cid:48) , (cid:96) (cid:48) , m (cid:48) (cid:96) ) are forthe initial and final states, respectively. The radialwavefunctions are calculated according to Ref. [24],using model potentials from Ref. [25]. The Jacobi-Angerrelation [48], e ia cos φ = ∞ (cid:88) (cid:101) m = −∞ i (cid:101) m J (cid:101) m ( a ) e i (cid:101) mφ , expresses e ikx as an azimuthal Fourier series. The matrixelement M A , including both angular and radial factors,then becomes M A = i m (cid:48) (cid:96) − m (cid:96) (cid:115) (cid:96) (cid:48) + 12 (cid:96) + 1 ( (cid:96) (cid:48) − m (cid:48) (cid:96) )!( (cid:96) (cid:48) + m (cid:48) (cid:96) )! ( (cid:96) − m (cid:96) )!( (cid:96) + m (cid:96) )! × (cid:40)(cid:90) u n (cid:48) ,(cid:96) (cid:48) ( r )[ u (cid:48) n,(cid:96) ( r ) − u n,(cid:96) ( r ) r ( (cid:96) + 1)] (cid:20)(cid:90) J m (cid:48) (cid:96) − m (cid:96) ( kr sin θ ) P m (cid:48) (cid:96) (cid:96) (cid:48) (cos θ ) P m (cid:96) (cid:96) +1 (cos θ )( (cid:96) − m (cid:96) + 1) sin θ dθ (cid:21) dr + (cid:90) u n (cid:48) ,(cid:96) (cid:48) ( r )[ u (cid:48) n,(cid:96) ( r ) + u n,(cid:96) ( r ) r (cid:96) ] (cid:20)(cid:90) J m (cid:48) (cid:96) − m (cid:96) ( kr sin θ ) P m (cid:48) (cid:96) (cid:96) (cid:48) (cos θ ) P m (cid:96) (cid:96) − (cos θ )( (cid:96) + m (cid:96) ) sin θ dθ (cid:21) dr (cid:41) . (A4)For PI the transitions are from a bound to a free state.In this case, the radial wavefunction u n (cid:48) ,(cid:96) (cid:48) is replacedby a free radial wavefunction u (cid:15) (cid:48) ,(cid:96) (cid:48) . The free radial wave-functions are normalized in energy, (cid:82) u (cid:15) (cid:48) ,(cid:96) (cid:48) ( r ) u (cid:15),(cid:96) (cid:48) ( r ) dr = δ ( (cid:15) − (cid:15) (cid:48) ), and the density of states ρ ( (cid:15) ) = 1. Appendix B: General behavior of the matrixelements
The range of relevant PI channels, i.e. the rangeof the (cid:96) (cid:48) and m (cid:48) (cid:96) quantum numbers for which the ma-trix elements M A for a given initial state are large,largely depends on the magnitude of the Bessel-functionarguments. The EDA, e ikx = 1, corresponds with J m (cid:48) (cid:96) − m (cid:96) ( kr sin θ ) = δ m (cid:48) (cid:96) ,m (cid:96) . Here we assess how well theEDA applies to Rydberg-atom PI with light. At firstglance, one may suspect the EDA to be invalid because kr ∼
1. Before making the EDA
Equation A4 yields a selection rule that arises from thethree functions within the θ integrals (one Bessel functionand two associated Legendre functions), which all havewell-defined parity about π/
2. Considering the paritybehavior of the associated Legendre functions with (cid:96) and m (cid:96) , and noting that the Bessel function terms are alwayseven, we find the selection rule that (cid:96) + m (cid:96) + (cid:96) (cid:48) + m (cid:48) (cid:96) + 1must be even (meaning that about half of the transitionsout of a state with given (cid:96) and m (cid:96) are allowed).In the limit kr → J m (cid:48) (cid:96) − m (cid:96) ( kr sin θ ) = δ m (cid:48) (cid:96) ,m (cid:96) . The orthogonality of theLegendre functions then yields the usual (very restric-tive) electric-dipole selection rules m (cid:48) (cid:96) − m (cid:96) = ∆ m (cid:96) = 0(for z -polarized light) and (cid:96) (cid:48) − (cid:96) = ∆ (cid:96) = ± σ , the rate Γ and the light intensity I follow σ = (cid:126) ω Γ /I , which after insertion of Eqs. A2 and A3 yields σ (cid:15) (cid:48) ,(cid:96) (cid:48) ,m (cid:48) (cid:96) z,n,(cid:96),m (cid:96) = πe (cid:126) (cid:15) m ωc | M A | (cid:18) E H a (cid:19) . (B1)The result is in SI units, m , the matrix element M A in atomic units, according to Eq. A4, and the term in() converts | M A | from atomic into SI units. To il-lustrate the typical PI behavior of Rydberg atoms inlight fields, we calculate matrix elements and cross sec-tions following Eqs. A4 and B1 for PI of a Rb Ryd-berg atom by 532-nm light. In Fig. 4(a), we display σ (cid:15) (cid:48) ,(cid:96) (cid:48) ,m (cid:48) (cid:96) z,n,(cid:96),m (cid:96) for PI of Rb | n = 15 , (cid:96) = 3 , m = 0 (cid:105) to thecontinuum states | (cid:15) (cid:48) = 0 . , (cid:96) (cid:48) , m (cid:48) (cid:96) (cid:105) . It is seen thatthe only transitions that have a non-negligible PI crosssection are the electric-dipole-allowed transitions in theassumed z -polarized light, ∆ m (cid:96) = 0 and ∆ (cid:96) = ±
1. Theweaker of the two electric-dipole-allowed PI channels isinto | (cid:15) (cid:48) = 0 . , (cid:96) (cid:48) = 2 , m (cid:48) (cid:96) = 0 (cid:105) and has a cross sec-tion of 4483 barn. The strongest electric-dipole-violatingchannel is into | (cid:15) (cid:48) = 0 . , (cid:96) (cid:48) = 5 , m (cid:48) (cid:96) = ± (cid:105) and hasa calculated cross section of 0.18 barn, which is smallerthan that of the weaker electric-dipole-allowed channelby a factor of 4 × − . It thus appears the EDA appliesexquisitely well to Rydberg-atom PI by light.The strong validity of the EDA for Rydberg-atom PImay appear somewhat unexpected, because both initialand final states have sizes on the order of or exceedingthe optical wavelength, and the usual argument madewhen invoking the EDA, namely that e ikx = 1 withinthe atomic volume, is actually not valid. To explore con-ditions under which electric-dipole-violating transitionswould be important, we increase the wavenumber k inthe e ikx phase factor (and in the Bessel function argu-ment in Eq. A4) by an artificial factor κ , so as to arti-ficially enhance EDA-violation, while holding everythingelse fixed (including the energy of the continuum state).While this is physically not possible, the numerical exer-cise allows us to explore where the unexpected validity ofthe EDA arises from when performing the integration inEq. A4. By increasing the argument of the Bessel func-tions by κ , we artificially increase the variation of theBessel functions in the matrix-element integration. Crosssections for PI of | n = 15 , (cid:96) = 3 , m = 0 (cid:105) to the continuum0 FIG. 4. (a) Cross sections for PI of the Rb | n = 15 , (cid:96) = 3 , m = 0 (cid:105) state with z -polarized 532-nm light for transitions intothe continuum states | (cid:15) (cid:48) = 0 . , (cid:96) (cid:48) , m (cid:48) (cid:96) (cid:105) , plotted for a range of values of final (cid:96) (cid:48) and m (cid:48) (cid:96) . In the calculations the EDA isnot applied. The only transitions with matrix elements of non-negligible amplitude are the electric-dipole-allowed transitions.(b) PI cross sections for the same transitions as in (a), but with the wavelength of the field artificially reduced by a factor of κ = 1000. Transitions that violate the electric-dipole selection rules now have larger values, often exceeding those of the twoelectric-dipole-allowed transitions.FIG. 5. Cross sections in units σ = 15220 barn for PI of the Rb | n = 15 , (cid:96) = 3 , m = 0 (cid:105) state with z -polarized 532-nm light forthe electric-dipole-allowed transition into the continuum state | (cid:15) (cid:48) = 0 . , (cid:96) (cid:48) = 4 , m (cid:48) (cid:96) = 0 (cid:105) (left), and for the electric-dipole-violating transition into the continuum state | (cid:15) (cid:48) = 0 . , (cid:96) (cid:48) = 3 , m (cid:48) (cid:96) = 1 (cid:105) (right), as a function of the upper integration limitin the matrix-element calculation, r , and for the indicated parameters κ (see text). states | (cid:15) (cid:48) = 0 . , (cid:96) (cid:48) , m (cid:48) (cid:96) (cid:105) calculated with κ = 1000are shown in Fig. 4(b). The EDA is evidently not validany more, as a large number of electric-dipole-forbiddentransitions occur. The strongest electric-dipole-allowedchannel now is to | (cid:15) (cid:48) = 0 . , (cid:96) (cid:48) = 4 , m (cid:96) = 0 (cid:105) , witha cross section of 5451 barn, while the strongest EDA-violating channel is to | (cid:15) (cid:48) = 0 . , (cid:96) (cid:48) = 3 , m (cid:96) = ± (cid:105) ,with a cross section of 17850 barn. Fig. 4(b) also shows a“checker board” pattern, which reflects the selection rulethat (cid:96) + m (cid:96) + (cid:96) (cid:48) + m (cid:48) (cid:96) + 1 must be even (which still holdsfor EDA-violating transitions).For more insight, in Fig. 5 we plot the cross sections for a few cases of PI of | n = 15 , (cid:96) = 3 , m (cid:96) = 0 (cid:105) with532-nm light and the indicated values of κ as a func-tion of cut-off radius of the radial integration in Eq. A4.Considering the physical case first, for which κ = 1, wefind that the matrix element of the EDA-allowed tran-sition integrates close to its final value already within aradius of about 50 a and then oscillates around the finalvalue, with the oscillations damping away in the outerreaches of the atomic volume. The oscillations originatefrom the structure of bound- and free-state wavefunc-tions. The effective range of the atom-field interactionappears to be confined to r (cid:46) a . One may say1that the Rydberg atom tends to photoionize close to itscenter, a finding that is in accordance with calculationsperformed elsewhere [49]. Since the matrix element in-tegrates close to its final value within a volume that isindeed much smaller than the physical wavelength, thephase variation of the field in the outer regions of theatom, r (cid:38) a , becomes irrelevant, making the EDAapplicable even though the atom diameter is on the orderof the optical wavelength.The results in Fig. 5 for artificially reduced wavelength, i.e. with the wavenumber k in the e ikx phase factor mul-tiplied with a κ >
1, show that substantial changes of thecross sections from their physical values require κ -valuesapproaching 1000, corresponding to effective wavelengthsin the phase factor (and the Bessel-function argumentsin Eq. A4) as low as several tens on a . In that case,the phase of the field does vary substantially over thevolume within which the physical, κ =1-matrix elementintegrates to near its asymptotic value. For κ approach-ing 1000, the EDA breaks down, leading to substantialchanges of the cross sections of electric-dipole-allowed PI channels, as well as to the emergence of large cross sec-tions in dipole-forbidden PI channels. We conclude thatthe validity of the EDA is linked to the behavior that thephysical matrix elements integrate to near their asymp-totic values within a small volume of only several tens of a in radius around the atomic center. The oscillations inthe integrals in Fig. 5 that occur outside that volume areinconsequential, as they damp out. Hence, it is sufficientfor the field phase in e ikx to be flat over a volume of justseveral tens of a in radius, regardless of how large theatom is. This very relaxed condition reflects the some-what surprising validity of the EDA for PI of Rydbergatoms with light.
2. With the electric dipole approximation
Making the EDA by setting kr = 0 in Eq. A4, onefinds for the matrix elements relevant to the main textof this paper M A = (cid:115) ( (cid:96) > + m (cid:96) )( (cid:96) > − m (cid:96) )(2 (cid:96) > + 1)(2 (cid:96) > − × (cid:40) (cid:82) u n (cid:48) ,(cid:96) (cid:48) ( r )[ u (cid:48) n,(cid:96) ( r ) − u n,(cid:96) ( r ) r (cid:96) > ] dr if (cid:96) > = (cid:96) (cid:48) = (cid:96) + 1 (cid:82) u n (cid:48) ,(cid:96) (cid:48) ( r )[ u (cid:48) n,(cid:96) ( r ) + u n,(cid:96) ( r ) r (cid:96) > ] dr if (cid:96) > = (cid:96) = (cid:96) (cid:48) + 1 , (B2)with the usual electric-dipole selection rules for thechanges in angular-momentum quantum numbers for z -polarized light, ∆ (cid:96) = ± m (cid:96) = 0. The full interac-tion matrix element then is (cid:104) f | ˆ H int | i (cid:105) = − | e | (cid:126) E m e ω M A , with expressions for the resultant PI rates still given byEq. A3. For linearly polarized light with arbitrary polar-ization direction ˆ n , M A = ˆ n · (cid:90) ψ ∗ f (cid:126) i ∇ ψ i d r , These forms of the matrix elements are known as the“velocity” form. With the EDA valid, the matrix ele-ments can be expressed in other forms using commuta-tion relations between operators. The relation [ˆ r , ˆ H ] = i (cid:126) m ˆ p , which applies to systems with field-free Hamiltoni-ans ˆ H that have momentum-independent potentials [47],allows the matrix elements to be written in terms of theposition operator. In this form, known as “length” form,it is (cid:104) f | ˆ H int | i (cid:105) = eE n · (cid:90) ψ ∗ f r ψ i d r , (B3)with the commonly used dipole matrix element M A , r =ˆ n · (cid:82) ψ ∗ f r ψ i d r (see, for instance, [47] equation 60.7f). Finally, if the potential in ˆ H is a Coulomb poten-tial, the matrix elements may be expressed in “accelera-tion” form, in which the commutation relation [ˆ p , ˆ H ] = − i (cid:126) ∇ ˆ V with atomic potential ˆ V is used to express thematrix elements in terms of the Coulomb acceleration( Z ˆ r ) /r [47]. In the length, velocity, and accelerationforms, the matrix elements accumulate to their asymp-totic values at large, intermediate, and small values of r , respectively [47, 49, 50]. In the present work, thevelocity form, the most-generally valid form, must beused because it allows for (cid:96) -dependent model potentialswith non-Coulombic corrections [25], which is what weuse in the computation of the wavefunctions. We havechecked that length- and velocity forms yield identicalresults for high (cid:96) , where the model potential becomes (cid:96) -independent. Even at (cid:96) = 0, the worst case, the length-and velocity forms yield PI cross sections that differ byless than 15%. For completeness it is further notedthat for bound-bound microwave transitions of Rydbergatoms the length form is generally acceptable, becausein that case the matrix elements are dominated by theouter reaches of the Rydberg wavefunctions, where themodel potentials are essentially (cid:96) -independent. Appendix C: PI in an optical lattice
In a one-dimensional optical lattice formed by twocounter-propagating beams with equal field amplitude2 E , polarization along the z -axis, and with beams prop-agating (anti)parallel with the x -axis, the electric field is E = ˆ z E (cid:104) cos (cid:8) k ( x − X ) − ωt (cid:9) +cos (cid:8) − k ( x − X ) − ωt (cid:9)(cid:105) , where X denotes the center-of-mass (CM) displacementof the atom from an intensity anti-node of the lattice. Itis then found that the matrix element to be used in placeof Eq. A4, with the EDA not being made, is M A = i m (cid:48) (cid:96) − m (cid:96) (cid:115) (cid:96) (cid:48) + 12 (cid:96) + 1 ( (cid:96) (cid:48) − m (cid:48) (cid:96) )!( (cid:96) (cid:48) + m (cid:48) (cid:96) )! ( (cid:96) − m (cid:96) )!( (cid:96) + m (cid:96) )! × (cid:40)(cid:90) u n (cid:48) ,(cid:96) (cid:48) ( r )[ u (cid:48) n,(cid:96) ( r ) − u n,(cid:96) ( r ) r ( (cid:96) + 1)] (cid:20)(cid:90) J m (cid:48) (cid:96) − m (cid:96) ( kr sin θ ) P m (cid:48) (cid:96) (cid:96) (cid:48) (cos θ ) P m (cid:96) (cid:96) +1 (cos θ )( (cid:96) − m (cid:96) + 1) sin θ dθ (cid:21) dr + (cid:90) u n (cid:48) ,(cid:96) (cid:48) ( r )[ u (cid:48) n,(cid:96) ( r ) + u n,(cid:96) ( r ) r (cid:96) ] (cid:20)(cid:90) J m (cid:48) (cid:96) − m (cid:96) ( kr sin θ ) P m (cid:48) (cid:96) (cid:96) (cid:48) (cos θ ) P m (cid:96) (cid:96) − (cos θ )( (cid:96) + m (cid:96) ) sin θ dθ (cid:21) dr (cid:41) × (cid:40) kX ) , m (cid:48) (cid:96) − m (cid:96) even2 i sin( kX ) , m (cid:48) (cid:96) − m (cid:96) odd . (C1)The PI rates scale with | M A | . For even m (cid:48) (cid:96) − m (cid:96) , therates are proportional to the lattice-field intensity, whichis 4 I [cos( kX )] , with I denoting the intensity of a singlelattice beam, while for odd m (cid:48) (cid:96) − m (cid:96) the rates scale withthe derivative-square of the lattice electric field along the x -direction.In the analysis performed in this Appendix we haveassumed a light polarization pointing along z and a fieldpropagating along x , because this allows for a trans-parent evaluation of the matrix elements in the generalcase that the EDA does not apply. Now we have estab-lished that the EDA applies, for the physics presentedhere. It follows that only the electric-dipole-allowed case m (cid:48) (cid:96) − m (cid:96) = 0 in Eq. C1 is relevant. The equation greatlysimplifies and takes the form of Eq. B2, with an X -dependent term 2 cos( kX ) multiplied on it. In essencethis means that, if the EDA applies, as in our case, the PIcross section in an optical lattice is the same as in a planewave, and that the field intensity to be used for comput- ing the PI rate from this cross section is the field intensityat the CM location of the atom. If the EDA were sub-stantially violated (which is not the case), electric-dipole-forbidden transitions with odd m (cid:48) (cid:96) − m (cid:96) would, in princi-ple, become allowed, and the PI rates following from M A would not generally be proportional to intensity at theatomic CM location. In that case, the usual concept ofa PI cross section would become, fundamentally, invalid.In the main text of this paper, it is more conve-nient to assume an atomic quantization axis along z ,one-dimensional optical-lattice laser beams propagatingalong z , and laser polarization along x . This allows usto take advantage of azimuthal symmetry in the calcula-tion of the PECs of the lattice, substantially reducing thecomputational effort. The Rydberg-atom CM position,denoted X in the Appendix, turns into Z in the maintext of the manuscript. Further, the matrix elements M in the main text are radial matrix elements, with theangular ( m (cid:96) -dependent) parts factored out. [1] S. Zhang, F. Robicheaux, and M. Saffman, Phys. Rev. A , 043408 (2011).[2] T. L. Nguyen, J.-M. Raimond, C. Sayrin, R. Corti-nas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko,S. Gleyzes, S. Haroche, G. Roux, et al., Phys. Rev. X , 011032 (2018).[3] D. Barredo, V. Lienhard, P. Scholl, S. de L´es´eleuc,T. Boulier, A. Browaeys, and T. Lahaye, Phys. Rev. Lett. , 023201 (2020).[4] J. Wilson, S. Saskin, Y. Meng, S. Ma, R. Dilip, A. Burg-ers, and J. Thompson, arXiv:1912.08754v2 (2019). [5] R. Cardman and G. Raithel, Phys. Rev. A , 013434(2020).[6] K. R. Moore and G. Raithel, Phys. Rev. Lett. ,163003 (2015).[7] A. Ramos, K. Moore, and G. Raithel, Phys. Rev. A ,032513 (2017).[8] V. Malinovsky, K. Moore, and G. Raithel, Phys. Rev. A , 033414 (2020).[9] F. Nez, F. Biraben, R. Felder, and Y. Millerioux, Opt.Commun. , 432 (1993).[10] W. Zhang, J. Robinson, L. Sonderhouse, E. Oelker, C. Benko, J. Hall, T. Legero, D. Matei, F. Riehle,U. Sterr, et al., Phys. Rev. Lett. , 243601 (2017).[11] S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D.Oppong, R. McNally, L. Sonderhouse, J. Robinson,W. Zhang, B. Bloom, et al., Science , 90 (2017).[12] H. Friedrich,
Theoretical Atomic Physics (Springer,Berlin, 2004).[13] R. R. Freeman, T. J. McIlrath, P. H. Bucksbaum, andM. Bashkansky, Phys. Rev. Lett. , 3156 (1986).[14] P. H. Bucksbaum, R. R. Freeman, M. Bashkansky, andT. J. McIlrath, J. Opt. Soc. Am. B , 760 (1987).[15] L. Pan, L. Armstrong, Jr., and J. H. Eberly,J. Opt. Soc. Am. B , 1319 (1986).[16] P. H. Bucksbaum, D. W. Schumacher, and M. Bashkan-sky, Phys. Rev. Lett. , 1182 (1988).[17] D. L. Freimund, K. Aflatooni, and H. Batelaan, Nature , 142 (2001).[18] S. E. Anderson, K. C. Younge, and G. Raithel, Phys. Rev.Lett. , 263001 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.263001 .[19] J. Lampen, H. Nguyen, L. Li, P. R. Berman,and A. Kuzmich, Phys. Rev. A , 033411 (2018),URL https://link.aps.org/doi/10.1103/PhysRevA.98.033411 .[20] B. Knuffman and G. Raithel, Phys. Rev. A , 053401(2007).[21] K. C. Younge, S. E. Anderson, and G. Raithel, New J.Phys. , 023031 (2010).[22] K. R. Moore, S. E. Anderson, and G. Raithel, Nat. Com-mun. , 6090 (2015).[23] S. E. Anderson and G. Raithel, Nat. Commun. , 2967(2013).[24] A. Reinhard, T. C. Liebisch, B. Knuffman, andG. Raithel, Phys. Rev. A , 032712 (2007).[25] M. Marinescu, H. R. Sadeghpour, and A. Dalgarno, Phys.Rev. A , 982 (1994), URL https://link.aps.org/doi/10.1103/PhysRevA.49.982 .[26] M. Marinescu, H. R. Sadeghpour, and A. Dalgarno, Phys.Rev. A , 5103 (1994), URL https://link.aps.org/doi/10.1103/PhysRevA.49.5103 .[27] M. Marinescu, V. Florescu, and A. Dalgarno, Phys. Rev.A , 2714 (1994), URL https://link.aps.org/doi/10.1103/PhysRevA.49.2714 .[28] J. W. Cooper, Phys. Rev. , 681 (1962), URL https://link.aps.org/doi/10.1103/PhysRev.128.681 .[29] O. Zatsarinny and S. S. Tayal, Phys. Rev. A ,043423 (2010), URL https://link.aps.org/doi/10. 1103/PhysRevA.81.043423 .[30] S. K. Dutta, J. R. Guest, D. Feldbaum, A. Walz-Flannigan, and G. Raithel, Phys. Rev. Lett. , 5551(2000).[31] J. Tallant, D. Booth, and J. Shaffer, Phys. Rev. A ,063406 (2010).[32] X. Wang and F. Robicheaux, J. Phys. B , 164005(2016), URL https://doi.org/10.1088/0953-4075/49/16/164005 .[33] K. C. Younge, B. Knuffman, S. E. Anderson, andG. Raithel, Phys. Rev. Lett. , 173001 (2010).[34] S. E. Anderson and G. Raithel, Phys. Rev. Lett. ,023001 (2012).[35] T. F. Gallagher, Rydberg Atoms (Cambridge UniversityPress, Cambridge, 1994).[36] M. L. Zimmerman, J. C. Castro, and D. Kleppner, Phys.Rev. Lett. , 1083 (1978).[37] J. C. Gay and D. Delande, Comments At. Mol. Phys. ,275 (1983).[38] P. Cacciani, E. Luc-Koenig, J. Pinard, C. Thomas, andS. Liberman, Phys. Rev. Lett. , 1124 (1986).[39] T. Van der Veldt, W. Vassen, and W. Hogervorst,J. Phys. B: At. Mol. Opt. Phys. , 1945 (1993).[40] Y. J. Chen, S. Zigo, and G. Raithel, Phys. Rev. A ,063409 (2014).[41] A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard,Rev. of Mod. Phys. , 1051 (2009).[42] M. Traxler, R. E. Sapiro, K. Lundquist, E. P. Power, andG. Raithel, Phys. Rev. A. , 053418 (2013).[43] D. A. Anderson, A. Schwarzkopf, R. E. Sapiro, andG. Raithel, Phys. Rev. A. , 031401(R) (2013).[44] M. S. Safronova, C. J. Williams, and C. W. Clark, Phys.Rev. A , 040303 (2003).[45] J. J. Sakurai, Modern Quantum Mechanics (2nd Edition) (Pearson, Essex, England, 2010), ISBN 0805382917, URL .[46] J. D. Jackson,
Classical Electrodynamics (John Wiley &Sons, New Jersey, 1999), 3rd ed.[47] H. A. Bethe and E. E. Salpeter,
Quantum Mechanics ofOne- and Two-Electron Atoms (Dover Publications, NewYork, 2008).[48] G. B. Arfken and H. J. Weber,
Mathematical Methods forPhysicists (Elsevier, Massachusetts, 2005).[49] A. Giusti-Suzor and P. Zoller, Phys. Rev. A , 5178(1987).[50] H. G. M¨uller and H. B. van den Heuvell, Laser Phys.3