Monte Carlo simulations of electromagnetically induced transparency in a square lattice of Rydberg atoms*

Shang-Yu Zhai(翟尚宇) and Jin-Hui Wu(吴金辉)

1Center for Quantum Sciences,Northeast Normal University,Changchun 130117,China

2School of Physics,Northeast Normal University,Changchun 130024,China

Keywords: electromagnetically induced transparency,Rydberg atomic lattices,Monte Carlo simulations

1. Introduction

Much attention has been paid in theoretical and experimental research on Rydberg atoms considering that they have essential applications, e.g., in the flourishing fields of quantum information and simulation,due to exaggerated properties like long radiative lifetimes,large dipole moments,and strong interatomic interactions.[1-8]In particular,rich many-body behaviors displayed in Rydberg atoms have been found to make a promising prospect for efficiently implementing quantum detection,gates,entanglement,devices,etc. as indispensable elements in future quantum networks.[9-17]Most of these implementations benefit form the so-called dipole blockade or anti-blockade effect, which prohibits or enhances more than one Rydberg excitations in a mesoscopic volume when the energy shift induced by interatomic interactions is prominent or counteracted.[18-21]

Of our special interest, dynamic propagation behaviors of classical or quantized light fields have been well studied in various Rydberg media in the regime of electromagnetically induced transparency (EIT).[22-27]This is a linear optical phenomenon exploiting quantum destructive interference to eliminate (enhance) resonant absorption (dispersion)in coherently dressed multi-level atomic systems, and has been extended to reversible light storage,[28-31]enhanced optical nonlinearities,[32-37]tunable photonic band-gaps,[38,39]etc. Combined with Rydberg atoms, EIT becomes instead a nonlinear optical phenomenon facilitating the efficient generation and manipulation of single-photon sources, switchings, and transistors.[40-43]This is why much work has been done in exploring nontrivial features of the Rydberg-EIT media. In particular, Pritchardet al.used EIT technique in a highly excited Rydberg gas and predicted a third-order nonlinearity due to blockade from repulsive interactions.[44]Simonset al.studied the effect of band-limited white Gaussian noise on EIT and Autler-Townes(AT)splitting when performing radio-frequency field strength measurements in hot Rydberg atoms.[45]Xuet al.proposed an EIT-based scheme to generate stable spatiotemporal solitons in cold Rydberg atoms exhibiting a Bessel lattice potential.[46]

Note however that it is very difficult or impossible to investigate the EIT spectra of randomly distributed Rydberg atoms by solving density matrix equations(DMEs). Great effort has been made to reduce the computation complexity by developing approximation theories for recovering relevant experiments. For instance, a superatom (SA) model developed in the mean field sense is shown to be effective in explaining most spectral features of the Rydberg-EIT media.[47,48]On the other hand, Monte Carlo (MC) simulations based on rate equations(REs)can also reproduce essential Rydberg-EIT features upon the adiabatic elimination of off-diagonal density matrix elements.[8,49]Meanwhile this method is found to be effective in examining non-equilibrium phenomena like antiferromagnetic phases, bistable phases, and topological superfluids in two-dimensional lattices of periodically distributed Rydberg atoms.[50-55]

Here we investigate the steady EIT spectra of cold Rydberg atoms arranged into a square lattice via MC simulations based on both DMEs and REs. A direct comparison shows that DMEs are more accurate than REs especially when the Rydberg lattice has a large dimension and thus complicated van der Waals (vdW) interactions. We find in particular that the absorption and dispersion of EIT spectra become more and more asymmetric until reaching the saturation regime as the lattice dimension increases. More importantly, the transparency window as a main EIT sign typically suffer from a notable reduction in depth due to dephasings arising from the inhomogeneous vdW interactions. The center of this transparency window is determined however by the average value of vdW induced level shifts. These nontrivial features are evident only when the probe field is not too weak and may also be controlled by modulating the coupling field detuning to counteract the average vdW shift.

Fig.1. (a)A three-level ladder atomic system with ground state|g〉,intermediate state|e〉,and Rydberg state|r〉driven by a probe field Ωp and a coupling field Ωc (see text for more details). (b) A n×n atomic array of period a in which each atom is driven into the three-level ladder configuration and interacts with another atom via the vdW potential Vkl if both are in state|r〉(see text for more details).

2. Model and equations

We consider a ladder configuration [see Fig. 1(a)] with ground state|g〉, intermediate state|e〉, and Rydberg state|r〉as driven by a strong coupling field of Rabi frequency(detuning)Ωc(Δc)and a weak probe field of Rabi frequency(detuning)Ωp(Δp). Then a square array ofN=n×nsuch laddertype atoms trapped,e.g.,in 2D optical lattices of perioda[see Fig.1(b)]can be described by the following interaction Hamiltonian:

whereVk=∑l/=k Vkl=C6∑l/=k|rl〉〈rl|/|rk-rl|6denotes the vdW induced shift for atomkcontributed by all other atoms,andC6is the vdW coefficient. For convenience in the following discussion, we further choose to label atomkby its coordinaterk=(xk,yk)awith integersxk ∈{1,2,...,n}andyk ∈{1,2,...,n}and defineV0=C6/a6as the unitary vdW induced shift.

Atomkin states|rk〉and|ek〉will decay via spontaneous emission to states|ek〉and|gk〉at ratesΓrandΓe,respectively.Considering a Rydberg state is typically long lived, we may setΓr →0 and obtain fromHIthe following density matrix equations(DMEs):

Assuming sufficiently strong decoherence on the probe transition (Γe ≫Ωp), however, it is viable to adiabatically eliminate the off-diagonal matrix elements in Eq. (2) by setting∂tρμν=0(μ/=ν)to attain the following set of reduced rate equations(REs):

which are much easier to solve than Eq. (2) in regard of a many-body quantum problem. From the steady solutions of Eq.(3)in the case ofΔc=0,it is straightforward to attain the off-diagonal matrix element

Fig.2. Flow chart for a single realization of the Monte Carlo method used to calculate the averaged values of density matrix elements ρμν in the steady state at time tf=20 μs.

Fig. 3. Averaged Rydberg populationρrr against cut-off radius Rc with V0 =64.3 MHz (a), V0 =130.4 MHz (b), and V0 =290.5 MHz (c), respectively. Other parameters used in calculations are given in the main text.

3. Results and discussion

Based on the MC method, we now examine in Fig. 4 the dependence of absorption Im(¯ρge)and dispersion Re(¯ρge)properties on probe detuningΔpfor a few square lattices of different dimensions. Typical absorption and dispersion spectra in ordinary EIT media, i.e., a transparent window of mirror symmetry and a normal dispersion of rotation symmetry centered atΔp=0, are observed forn=1 because vdW interactions won’t occur for a single atom. Asnincreases,both absorption and dispersion spectra first suddenly deviate from their original symmetries because vdW interactions start to take place,and then slowly approach a saturation situation.To be more concrete,as lattice dimensionnincreases,a higher proportion of atoms will become far away from boundaries to interact via the vdW shiftVkwith the same number of neighboring atoms in the cut-off radius. Meanwhile, atoms at or close to boundaries will take a lower proportion and interact with (less) different numbers of neighboring atoms.In this case, a saturation regime can be reached as lattice dimensionnis large enough so that the number of atoms at or close to boundaries can be neglected as compared to that of others. This is evident by noting that the transparent window’s center is finally stabilized atΔp/2π ≃-1.63 MHz forn≾50. That means,each atom suffers from an average vdW shift ¯V/2π ≃1.63 MHz as contributed by its neighboring atoms because ¯Vworks indeed as an effective detuning of the coupling field. It is worth noting that the average vdW shift is determined by the vdW coefficient,the average Rydberg population,and the atomic number in the cut-off radius.A depth reduction of the transparency window is also evident forn=10 andn=50 due to additional dephasings arising from the inhomogeneity of vdW shiftVk. We further note that MC calculations based on DMEs are somewhat different from those based on REs, indicating the adiabatic elimination of off-diagonal matrix elementsρμνwill result in more or less coherent information loss,especially for a large atomic lattice.

Fig. 4. Absorption Im(¯ρge) (left) and dispersion Re(¯ρge) (right) properties against probe detuning Δp attained via Monte Carlo calculations based on DMEs(red-solid)and REs(blue-dashed)with n=1(a), (b), n=2(c), (d),n=10 (e), (f), and n=50 (g), (h), respectively. Other parameters are the same as in Fig.3 except V0=64.3 MHz.

It is not difficult to imagine that the spectra of absorption and dispersion will finally recover those for two-level absorbing atoms as vdW interactions are sufficiently strong. In this case,a large enough average vdW shift ¯Vworks as an infinite effective detuning of the coupling field so that it is decoupled from the upper transition,yielding thus a two-level system involving only the lower transition. This is confirmed in Fig.5,where a square lattice ofn=50 is considered for three values ofV0. We find in particular that the transparency window becomes shallower and the dispersion slope becomes smoother as the lattice periodais reduced to attain a largerV0. It is also worth noting that the centers of absorption and dispersion curves move left together so that their right parts become more important and thus look more like those for two-level absorbing atoms. This means that the lattice periodaor the atomic density 1/a2should be carefully chosen to manipulate the blockade effect for attaining a desired optical response in a square lattice of Rydberg atoms. To be more concrete, a smaller lattice period will result in a higher atomic density and thus a stronger blockade effect because larger average vdW shifts can be attained to yield weaker atom-field couplings when more atoms are found in the cut-off radius.

Fig. 5. Absorption Im(¯ρge) (left) and dispersion Re(¯ρge) (right) properties against probe detuning Δp for a square lattice of n=50 with V0 =130.4 MHz(a),(b),V0=290.5 MHz(c),(d),and V0=360.0 MHz(e),(f),respectively. Other parameters are the same as in Fig.3.

We then check in Fig. 6 how the spectra of absorption and dispersion depend on the Rabi frequency of probe field for a square lattice ofn=50. It is clear that both Im(¯ρge)and Re(¯ρge)exhibit a nonlinear dependence onΩp,manifested as a notable change of the transparency window both in depth and in position.To be more concrete,the spectra of absorption and dispersion are found to recover those for a single atom asΩpdecreases from 0.3 MHz to 0.03 MHz,but become more asymmetric with a shallower transparency window asΩpincreases from 0.3 MHz to 0.9 MHz.This is a strong evidence of the socalled cooperative nonlinearity[8,39]due to long-range vdW interactions among Rydberg atoms. Different from atomic samples of random spatial distributions, a much larger deviation of the transparency window from its original center is found for our atomic lattice of a periodic spatial distribution.

Fig. 6. Absorption Im(¯ρge) (left) and dispersion Re(¯ρge) (right) properties against probe detuning Δp for a square lattice of n=50 with Ωp=0.03 MHz(a), (b), Ωp =0.3 MHz (c), (d), and Ωp =0.9 MHz (e), (f), respectively.Other parameters are the same as in Fig.3 except V0=64.3 MHz.

Finally, we show how to control the absorption and dispersion of the EIT spectra by modulating the coupling field detuning to compensate more or less the vdW shift for a square lattice ofn=50. As can be seen from Fig. 7, the absorption and dispersion curves disturbed by the coupling field detuning do not exhibit mirror and rotation symmetries like those for a single atom even if the transparency window is centered again atΔp≃0 forΔc/2π ≃-2.0 MHz. In this case, the average vdW shift is estimated to be ¯V/2π ≃2.0 MHz because a transparency window centered atΔp=0 requires a vanishing effective detuningΔc+ ¯V=0. This average vdW shift ¯Vis slightly different from that estimated in Fig. 4 because it depends on the Rydberg population ¯ρrrand thus the coupling detuningΔc. We further find that the transparency window moves left (right) for a larger (smaller)Δcto result in more asymmetric absorption and dispersion curves,but the transparency window’s depth does not change too much asΔcis modulated to control the transparency window’s position.Such a control of the transparency window is clearly different from those shown in Figs. 4-6 by modulating other parameters.The underlying physics is that the Rydberg populationρrrdepends on but is not very sensitive to the coupling field detuning in the case of a relatively weak probe field(Ωp=0.3 MHz vs.Γe=6.0 MHz), so that dephasings arising from the inhomogeneity of vdW shiftVkdo not change evidently asΔcchanges.

Fig. 7. Absorption Im(¯ρge) (left) and dispersion Re(¯ρge) (right) properties against probe detuning Δp for a square lattice of n=50 with Δc=-3.0 MHz(a),(b);Δc=-2.0 MHz(c),(d);Δc=-0.5 MHz(e),(f);Δc=0.5 MHz(g),(h). Other parameters are the same as in Fig.3 except V0=64.3 MHz.

4. Conclusion

In summary, we have studied a square lattice of Rydberg atoms in the ladder configuration by examining its EIT spectra of absorption and dispersion in the presence of vdW interactions. Monte Carlo calculations based on density matrix equations show that the EIT spectra becomes more and more asymmetric, until the transparency window finally centered at a position determined by the average vdW shift ¯V,as the lattice dimensionnincreases. The transparency window is found in particular to suffer from a notable reduction in depth due to the additional dephasings arising from the inhomogeneity of vdW interactions. These features are evident only when the probe Rabi frequencyΩpis not too small and may turn out to be those for two-level absorbing atoms as the unitary vdW shiftV0is large enough. Moreover, it is convenient to control these features(e.g.,roughly recover the symmetric EIT spectra) by modulating the coupling detuningΔcto counteract the average vdW shift ¯V. Our Monte Carlo calculations are more accurate than calculations based on meanfield approximations[47]and may be extended to study other properties like non-equilibrium physics[50]in finite lattices of Rydberg atoms.