Giant-Atom Effects on Population and Entanglement Dynamics of Rydberg Atoms

Giant atoms are attracting interest as an emerging paradigm in the quantum optics of engineered waveguides. Here we propose to realize a synthetic giant atom working in the optical regime starting from a pair of interacting Rydberg atoms driven by a coherent field and coupled to a photonic crystal waveguide. Giant-atom effects can be observed as a phase-dependent decay of the double Rydberg excitation during the initial evolution of this atomic pair while (internal) atomic entanglement is exhibited at later times. Such an intriguing entanglement onset occurs in the presence of intrinsic atomic decay toward non-guided vacuum modes and is accompanied by an anti-bunching correlation of the emitted photons. Our findings may be relevant to quantum information processing, besides broadening the giant-atom waveguide physics with optically driven natural atoms.

Nevertheless, it is important to explore new physics of giant atoms within different atomic architectures, especially those working beyond the microwave or acoustic regime and with high-lying Rydberg atoms.Owing to unique properties unlike atoms in the ground or low-lying excited states, including strong dipole-dipole interactions and long radiative lifetimes [25], Rydberg atoms turn out to be an excellent building block in quantum information processing, with potential applications for realizing quantum logic gates [26][27][28][29], single-photon sources [30][31][32][33], and various entangled states [34][35][36][37].Rydberg atoms have also been successfully employed in waveguide quantum electrodynamics by coupled to engineered structures like optical nanofibers [38], photonic crystal waveguides [39], and coplanar microwave waveguides [40].
In this Letter, we consider a pair of two-level Rydberg atoms coupled to a photonic crystal waveguide (PCW) through two lower transitions and driven by a coherent field through two upper transitions in the two-atom basis to display giant-atom effects in the optical regime.The self-interference effect appears specifically at shorter times of the pair's dynamic evolution dominated by two competing two-photon resonant transitions, thereby realizing a synthetic giant atom with two coupling points at a variable distance d about the atomic separation R. For longer times, on the other hand, we observe the onset of atomic entanglement through mutual Rydberg excitations of the two atoms as the phase φ accumulated from a coupling point to the other takes specific values, which depends on d and is tunable.This effect is characterized by a detailed examination on quantum correlations of the emitted photons and further understood in terms of dark states decoupled from both coherent field and waveguide modes.Our findings remain valid even for continuous atom-waveguide couplings, which are more appropriate for Rydberg atoms with large spatial extents.
Model.-We start by illustrating in Fig. 1(a) that a pair of Rydberg atoms with ground |g 1,2 and Rydberg |r 1,2 states separated by frequency ω e are trapped in the vicinity of a PCW [41,42] at x 1 = 0 and x 2 = d, respectively.The two atoms are illuminated by a coherent field of frequency ω c and interact through a (repulsive) van der Waals (vdW) potential V 6 = C 6 /R 6 , when both excited to the Rydberg states, with C 6 being the vdW coefficient and R the interatomic distance, which could be different from d [43].The vdW interaction will result in a largely shifted energy 2 ω e + V 6 for the double-excitation state |r 1 r 2 while the single-excitation states |r 1 g 2 and |g 1 r 2 remain to exhibit energy ω e .In the two-atom basis, for a large enough V 6 , we have the four-level configuration detuning δ k = ω k − ω e and strength g k .This scheme is supported by the following two considerations.First, ω c is far away from ω e but close to ω e + V 6 with ∆ c V 6 , so that the coherent field can only drive two upper transitions.Second, ω e and ω e + V 6 fall, respectively, within the lower propagating band and the band gap of a PCW as sketched in Fig. 1(c), so that the waveguide mode can only drive two lower transitions.
Then, for a continuum of waveguide modes interacting only with two lower transitions, the Hamiltonian in the rotating-wave approximation is ( = 1) Here, we have introduced the atomic raising operators k and a k refer, respectively, to the creation and annihilation operators of a waveguide mode ω k .We have also assumeed constant coupling strengths, i.e. g k g, in the Weisskopf-Wigner approximation.
As we address only the two-atom dynamics, the waveguide modes of density D(k) at frequency ω k can be traced out (Born-Markov approximation), yielding the master equation for density operator ρ [43,46,47] where is the Lindblad superoperator describing two-atom decay processes, with Γ(k) = 4πg 2 D(k) denoting the decay rate into relevant modes of the waveguide (guided modes) [2,3] while γ being the mean decay rate into other electromagnetic modes (non-guided modes).Under the two-photon resonance condition ∆ c + δ k 0, assumed to hold for all guided modes, the atomic Hamiltonian is with J ex = Γsinφ and Γ ex = Γcosφ denoting, respectively, coherent and dissipative parts of the exchange interaction mediated by the waveguide.Here we have defined φ(k being the group velocity of a guided mode bearing the "linearized" dispersion ω k kv g (k).It is worth stressing that Γ ex serves as a reservoir to engineer the atomic decay through selected guided modes ω k (or bandwidth of modes) and their density distribution D(k).In turn, such an engineered reservoir together with the atomic separation d represents a set of knobs to control the phase φ(k) acquired by an emitted photon between the contact points x 1 and x 2 of a giant-atom (see below).
Synthetic two-level giant atom: the short-time regime.-Maintaining∆ c + δ k 0 and further requiring |∆ c | Ω c , g, the two atoms initially in the double Rydberg state |r 1 r 2 would behave like a two-level giant atom decaying directly to the ground state |g 1 g 2 at two coupling points x 1 and x 2 .This expectation will be verified by numerically comparing the dynamics of the four-level atomic pair to that of the synthetic two-level giant atom.In the latter picture, the two atoms interact with the waveguide modes only through the two-photon resonant transition |r 1 r 2 ↔ |g 1 g 2 whereby an external photon of frequency ω c and a waveguide photon of frequency ω k are emitted (or absorbed) at the same time.Upon the adiabatic elimination of states |r 1 g 2 and |g 1 r 2 [48,49], the effective With the above assumptions, we write down the synthetic two-level giant-atom Hamiltonian as where Υ = 4πξ 2 D(1 with its real and imaginary parts being, respectively, the phase-dependent decay rate and Lamb shift. In the remaining part, we perform numerical analysis in support of the predictions anticipated above.We first consider that for large enough (driving) detunings |∆ c |, the adiabatic elimination of two single-excitation states can be actually made, providing in turn an adequate evidence of the equivalence between the (four-level) atomic pair and the (two-level) giant atom in the short-time regime.This has been examined in Fig. 2(a) by comparing time evolutions of atomic-pair population ρ r1r1,r2r2 based on Eq. ( 2) and giant-atom population rr based on Eq. ( 5) with matched parameters.Taking φ = 40.5πas an example and starting from ρ r1r1,r2r2 (0) = rr (0) = 1, we find that ρ r1r1,r2r2 (t) and rr (t) exhibit a better agreement for a larger |∆ c | so that the adiabatic elimination leading to a giant atom becomes reliable for |∆ c |/Ω c 30.It is also worth noting that ρ r1r1,r2r2 (t) and rr (t) de in a stronger coupling strength ξ and thereby a larger decay rate Re(Υ) of the synthetic giant atom.
The giant-atom self-interference effect can instead be established by plotting ρ r1r1,r2r2 (t) in Fig. 2(b) for ∆ c = 30 MHz and different values of φ.It is clear that an enhanced (reduced) decay occurs for ρ r1r1,r2r2 (t) in the case of φ = 2mπ (φ = 2mπ + π) with m ∈ Z due to a perfect constructive (destructive) interference between two coupling points, as can be seen from Im(Υ) = 0 and Re(Υ) = Γ(1 + cos φ)Ω 2 c /∆ 2 c .The atomic pair is found in particular to show an undamped double-excitation population [ρ r1r1,r2r2 (t) ≡ 1] for φ = 2mπ + π and γ = 0, which is one of the most remarkable features of giant atoms [2] due to a complete decoupling from the waveguide (Υ = 0) and a vanishing intrinsic decay (γ = 0).In the case of φ = 2mπ ± π/2, we have and Re(Υ) = ΓΩ 2 c /∆ 2 c , which accounts for the identical population dynamics with a moderate decay since opposite detunings (Lamb shifts) make no difference.
We finally detail how one can actually adjust phase φ while leaving potential V 6 unchanged.To this end, a pair of 87 Rb atoms with ground state 2π × 1009 THz are taken here as an example.In this case, we have γ 1.0 kHz for the intrinsic Rydberg lifetime τ 964 µs while V 6 20 GHz for R 3.1 µm and C 6 2π × 2.8 × 10 12 s −1 µm 6 [50,51].When the atomic pair is placed exactly along the waveguide, we have d = R and hence φ 41.6π by assuming that v g is a half of the vacuum light speed c.When the atomic pair is misaligned along the waveguide, however, it is viable to attain d 2.95 µm and hence φ 39.6π with neither R nor V 6 changed [43].
Atomic entanglement onset: the long-time regime.-Nowleaving the short-time regime where the atomic pair can be modeled as a giant atom according to Eq. ( 5), we turn to the long-time regime where the decay toward nonguided vacuum modes becomes important as less and less population can be found in the double-excitation state.A peculiar aspect of the long-time regime is internal entanglement of the atomic pair (i.e., a specific superposition of two single-excitation states) generated by the coherent interaction described by H at , which can be quantified by the Wootters concurrence [52][53][54] with λ 1 > λ 2 > λ 3 > λ 4 being the four eigenvalues of matrix X defined by of the photons emitted by two Rydberg atoms.As usual, ρ r1r1 = r 1 |Tr 2 ρ|r 1 and ρ r2r2 = r 2 |Tr 1 ρ|r 2 are obtained from reduced density matrices of different atoms, while g ph > 1 and g ph < 1 refer to the effects of photon bunching and anti-bunching, respectively.
We plot in Fig. 3(a) time evolutions of C at for different values of φ starting from the same double-excitation state and find that C at becomes suddenly nonzero at a critical time for φ = 2mπ but remains vanishing for other values of φ.Such an onset of internal atomic entanglement happens when the photon correlation function g (2) ph evolves from the regime of bunching to that of anti-bunching as shown in Fig. 3(b).This signifies a rigid correspondence between the emergence of photon anti-bunching and the generation of atomic entanglement.
It is worth noting that g ph also indicates how the two Rydberg atoms are distributed in the single-and doubleexcitation states, which would be helpful to understand the underlying physics of the entanglement sudden-onset dynamics.This inspires us to further plot ρ gg , ρ rr , ρ ++ , and ρ −− in Fig. 3(c) for φ = 2mπ and Fig. 3(d with γ ± = γ + Γ ± Γ ex .It is clear that |− is decoupled from field Ω c and will become a dark state if it is further immune to the waveguide modes in the case of φ = 2mπ.However, |+ is always a bright state in that its dynamics depends on field Ω c all the time. Note in particular that γ + = γ + 2Γ γ − = γ in the case of φ = 2mπ, which explains why ρ ++ remains to be vanishing while ρ −− does not in Fig. 3(c) so that we have g (2) ph 4ρ rr /ρ 2 −− .The atomic entanglement onset occurs for φ = 2mπ just because a dark state immune to field Ω c allows the transition from 4ρ rr > ρ 2 −− to 4ρ rr < ρ 2 −− .In the case of φ = 2mπ + π, however, we have a nonzero ρ ++ and a vanishing ρ −− in Fig. 3(d) and hence g (2) ph 4ρ rr /ρ 2 ++ due to γ − = γ + 2Γ γ + = γ.The atomic entanglement onset is absent for φ = 2mπ +π just because a bright state interacting with field Ω c always results in 4ρ rr > ρ 2 ++ .Finally, we stress that ρ ++ > 0 (ρ −− > 0) and ρ −− = 0 (ρ ++ = 0) do not mean that the decomposition of a mixed state ρ must include a pure state |+ (|− ), hence do not mean that we must have C at > 0, which further explains why C at could suddenly become nonzero only for φ = 2mπ.
Continuous couplings.-Workingwith point-like atomwaveguide couplings, as assumed so far, is just a rough approximation for highly-excited Rydberg states of size r ∝ n 2 , with n being the principal quantum number.We then extend our discrete-coupling configuration results to the continuum limit whereby the coupling region becomes a large ensemble of coupling points, each with a different strength.For an exponential continuous distribution [8] of such coupling strengths as in Fig. 4(a) spread about each contact point with a characteristic width Θ, we find that the master equations ( 2) and ( 5) can be generalized to the continuous-coupling limit by replacing Γ → Γ , Γ ex → Γ ex , J ex → J ex and Υ → Υ while simultaneously introducing an interaction term J (σ 1 + σ 1 − + σ 2 + σ 2 − ) in Eq. (3).While explicit expressions of these modified parameters are given in [43], we here use them to plot in Figs.4(b) and 4(c) the time evolutions of ρ r1r1,r2r2 and C at in the short-time and long-time regimes, respectively, for different values of φ and a fixed Θ.
It is easy to see that the giant-atom effects of phasedependent population decay and entanglement onset remain observable for a remarkable coupling broadening.Moreover, the dynamic behavior of ρ r1r1,r2r2 for Θ = 5π/2 (continuous couplings) is identical to that for Θ = 0 (point-like couplings) in the case of φ = 2mπ + π since there is no decay toward the waveguide with Re(Υ ) = 0. Note, however, that a nonzero Lamb shift with Im(Υ ) = 0 always exists for continuous couplings [43], which does not affect atomic population decay but would be relevant to other problems such as photon scattering.
Conclusions.-We have proposed a feasible scheme for constructing a synthetic giant atom with two interacting Rydberg atoms coupled to a PCW and driven by a coherent field.Giant-atom effects are manifested by a phasedependent population dynamics in the short-time regime and an entanglement sudden-onset dynamics in the longtime regime.These effects are observable even for continuous atom-waveguide couplings, indicating their robust- ness against the unavoidable coupling broadening.Compared to typical schemes utilizing superconducting quantum circuits, our Rydberg scheme provides a promising platform for studying giant-atom physics in the optical regime.Hence, our findings have potential applications in quantum network engineering and quantum information processing based on optical photons.
Supplemental Material for "Giant-Atom Effects on Population and Entanglement Dynamics of Rydberg Atoms" Yao-Tong Chen 1 , Lei Du This supplementary material gives further details on the misaligned arrangement of two atoms along a waveguide (Sec.I), the master equation of a two-atom four-level configuration (Sec.II), the master equation of a giant-atom two-level configuration (Sec.III), and the continuous couplings of Rydberg atoms and waveguide modes (Sec.IV) that are omitted in the main text.

I. MISALIGNED ARRANGEMENT OF TWO ATOMS ALONG A WAVEGUIDE
In this section, we discuss how to manipulate the van der Waals (vdW) potential V 6 and the accumulated phase φ separately.As mentioned in the main text, V 6 and φ are determined by the (straight-line) distance R between a pair of Rydberg atoms and the separation d (along the waveguide) between two coupling points, respectively.When this atomic pair is placed exactly along the waveguide, we have R ≡ d so that φ or V 6 cannot be changed alone as shown in Fig. S1(a).In this case, it is easy to attain φ ω e d/v g = ω e R/v g = 41.6π with R 3.1 µm, ω e 2π × 1009 THz, and v g 0.5c as considered in the main text.In order to change φ and V 6 separately, we can choose a misaligned arrangement of this atomic pair as shown in Fig. S1(b), where d is clearly smaller than R.In this case, keeping R 3.1 µm and hence V 6 = 20 GHz unchanged, it is viable to tune φ in the range of [41.6, 39.6]π by reducing d from 3.1 µm with a vanishing misaligned angle to 2.95 µm with a 0.09π misaligned angle.

II. MASTER EQUATION OF A TWO-ATOM FOUR-LEVEL CONFIGURATION
In this section, we provide the derivation procedures from Eq. ( 1) on Hamiltonian H to Eq. ( 2) on density operator ρ in the main text.As shown in Then the Hamiltonian describing the atom-waveguide interaction can be written as defined as in the main text.To study the population dynamics of this atomic pair, we can eliminate the waveguide field via a standard procedure and calculate the following master equation for a reduced density operator [1,2] where Tr w represents a partial tracing over the waveguide degrees of freedom and ρ w = |0 0| is the initial vacuum state of the waveguide modes.Substituting Eq. (S1) into Eq.(S2) we have as well as Γ ex = Re{ Γ} = Γcosφ and J ex = Im{ Γ} = Γsinφ.In the above derivation, we have also considered the δ function definition +∞ −∞ dke ±ikx = 2πδ(x) and the waveguide mode density D(k) = ∂k/∂ω k [3][4][5].Note that Eq. (S3) just describes the interactions between a continuum of waveguide modes and two lower atomic transitions |g 1 g 2 ↔ |g 1 r 2 and |g 1 g 2 ↔ |r 1 g 2 .Further taking into account the intrinsic atomic decay toward nonguided modes in the free space as well as the neglected interactions between a coherent field and two upper atomic transitions |g 1 r 2 ↔ |r 1 r 2 and |r 1 g 2 ↔ |r 1 r 2 in H, one can easily obtain the master equation (2) in the main text.This equation, if expanded in the two-atom four-level configuration, will turn out to be As mentioned in the main text, it is helpful to understand the long-time entanglement onset dynamics by replacing single-excitation states |g 1 r 2 and |r 1 g 2 with their superpositions |± = 1/ √ 2(|r 1 g 2 ± |g 1 r 2 ).Population evolutions in the two symmetric and anti-symmetric states can be calculated from the above equations as which are exactly Eq. ( 8) in the main text if we further introduce γ ± = γ + Γ ± Γ ex .

III. MASTER EQUATION OF A GIANT-ATOM TWO-LEVEL CONFIGURATION
In the case that the single-excitation states |r 1 g 2 and |g 1 r 2 are not populated initially, if we have |∆ c | Ω c , g and ∆ c + δ k 0, our two-atom four-level configuration can be reduced to a (synthetic) giant-atom two-level configuration by eliminating |r 1 g 2 and |g 1 r 2 in the short-time regime.In view of this, a pair of Rydberg atoms will decay from the double-excitation state |r 1 r 2 directly to the ground state |g 1 g 2 by simultaneously emitting a coherent-field photon of frequency ω c and a waveguide-mode photon of frequency ω k , through two competing two-photon resonant transitions exhibiting effective coupling strengths ξ 1 = −gΩ c /∆ c ≡ ξ and ξ 2 = ξe iφ , respectively.This can be substantiated by the following discussions starting from an effective Hamiltonian defined as [6,7] with being the total interaction Hamiltonian involving both waveguide modes and coherent field of our two-atom four-level configuration.Substituting Eq. (S8) into Eq.(S7), one has (S12)

IV. CONTINUOUS COUPLINGS OF RYDBERG ATOMS AND WAVEGUIDE MODES
In this section, we try to derive the explicit expressions of relevant constants describing various interactions between two Rydberg atoms and a continuum of waveguide modes modified in the case of continuous couplings.As shown in Fig. 4(a) in the main text, the two continuous couplings around x 1 = 0 and x 2 = d exhibit a common characteristic width Θ, with which relevant exponential distribution functions can be expressed as ν 1 (ϕ) = Here ϕ = φx/d ω e x/v g describes the phase accumulated from x 1 to x by a propagating photon along the waveguide and will become φ in the case of x = x 2 , which has been considered above for two discrete couplings.In this way, one can immediately generalize the master equation (2) in the main

FIG. 1 .
FIG. 1.(a) The two-level configuration (single-atom basis).Two Rydberg atoms placed at a distance R interact via a potential V6(R) and couple to a waveguide mode a k at x1 = 0 or x2 = d while driven by a coherent field ωc.(b) The four-level configuration (two-atom basis).Blue (red) lines represent single (double) Rydberg excitations, whereby two lower (upper) transitions are coupled by mode a k of strength g k (field ωc of strength Ωc).(c) The dispersion relation of a PCW: frequency ωe of the lower transitions falls within the propagating band while frequency ωe + V6 of the upper transitions falls within the band gap (red shaded).(d) An equivalent two-level giant atom with |g ≡ |g1g2 and |r ≡ |r1r2 , upon adiabatic elimination of the single-excitation states, bears separate couplings of strengths ξ at x1 and ξe iφ at x2, with φ being the accumulated phase between x1 and x2.
in terms of the transition operator σ + = (σ − ) † = |r g| with |r ≡ |r 1 r 2 and |g ≡ |g 1 g 2 .Again, by tracing out the waveguide modes, we arrive at the master equation for giant-atom density operator[43]

3 FIG. 4 .
FIG. 4. (a) Schematic of two Rydberg atoms close to a waveguide with identical exponential coupling distributions around x1 = 0 and x2 = d.Time evolutions of population ρr 1 r 1 ,r 2 r 2 (b) and concurrence Cat (c) with Θ = 5π/2 and different values of φ.Other parameters are the same as in Fig. 2(b).
FIG. S1.Arrangement details of two Rydberg atoms with respect to a one-dimensional waveguide shown as a blue area.The atomic pair is (a) placed along the waveguide with d = R; (b) misaligned along the waveguide with d < R.
Fig 1(b), two upper transitions |g 1 r 2 ↔ |r 1 r 2 and |r 1 g 2 ↔ |r 1 r 2 are driven by the external coherent field Ω c while two lower transitions |g 1 g 2 ↔ |g 1 r 2 and |g 1 g 2 ↔ |r 1 g 2 are coupled to the waveguide modes.Under the two-photon resonance condition (i.e., ∆ c −δ k with ∆ c = ω c − ω e − V 6 and δ k = ω k − ω e ), we neglect the interactions resulted from coherent field ω c in H for the moment and move to the interaction picture with