Analysis of Dissipation Mechanisms for Cesium Rydberg Atoms in Magic-Wavelength Optical Trap

Abstract

A magic optical dipole trap (ODT) can confine atoms in the ground state and a highly excited state with the same light shifts, resulting in a long-range coherent lifetime between them, which plays an important role in high-fidelity quantum logic gates, multi-body physics and other quantum information. Here, we use a sum-over-states model to calculate the dynamic polarizabilities of the 6S_1/2 ground state and 46S_1/2 Rydberg state of Cs atoms and identify corresponding magic wavelengths and magic detunings for trapping the two states in the range of 900–1950 nm. Then, we analyze the robustness of the magic condition and the feasibility of the experimental operation. Furthermore, we estimate the trapping lifetime of Cs Rydberg atoms by considering different dissipation mechanisms, such as photon scattering and photoionization in the magic ODT. The photoexcitation and photoionization of Cs atoms under the action of three-step laser pulses are calculated by the rate equation. The presented results for magic-wavelength ODTs are of great significance for quantum information and quantum computing based on Rydberg atoms.

1. Introduction

A Rydberg atom refers to an atom in a highly excited state with a principal quantum number n > 10 [1]. Compared with low-excited atoms, Rydberg atoms have a longer radiation lifetime, a larger electric dipole moment, and more long-range controllable interactions, which make them ideal systems in the fields of multi-body physics, quantum information and precision measurements [2,3,4,5,6,7]. In the past, the capture of Rydberg atoms using magnetic traps and electrostatic traps has been both proposed and experimentally verified [8,9], but these trap architectures are inflexible. Therefore, in most experiments on cold atoms involving the confinement of ground-state atoms in a far-off-resonance optical dipole trap (ODT) and Rydberg excitation [10,11], the cold atomic sample is prepared in an ODT to hold the atoms in one position for a significant amount of time. However, traditional far-off-resonance red-detuned ODTs have an attractive potential for ground-state atoms, but they show a repulsive potential for Rydberg atoms. During their operation on Rydberg atoms, traditional ODTs make the atoms lose coherence quickly, which leads to low logic gate fidelity and a low experiment repetition rate [12,13,14,15].

In order to reduce this effect, some methods are proposed to trap Rydberg atoms [16,17,18]. In 2008, Hogan et al. demonstrated the three-dimensional electrostatic trapping of state-selected Rydberg atoms [19]. In 2011, Anderson et al. achieved the optical trapping of ⁸⁷Rb Rydberg atoms in a 1064 nm optical lattice with one-dimensional ponderomotive potential through complex experimental operations [20]. Later, Barredo et al. demonstrated the three-dimensional trapping of ⁸⁷Rb nS (n = 60–90) Rydberg atoms in a hollow bottle beam trap generated by a spatial light modulator, which extended the trapping lifetime relatively [21]. However, there are some issues in the above experiment. Firstly, the experimental system is rather complex and difficult to operate. Secondly, the simultaneous trapping of ground- and Rydberg-state atoms has not been achieved, leading to a relatively low experimental repeatability. To solve the above problems, a magic-wavelength ODT was proposed to trap ground- and low-excited-state atoms effectively.

To better understand the behavior and applications of Rydberg atoms, we need to study specific atomic species. Among various atoms, alkali metal atoms have only one electron in the outermost layer, and their structure is relatively simple, without the need to consider complex multi-electron problems. Furthermore, due to the lower melting points of rubidium and cesium atoms, which are slightly above room temperature, they are often used in laser cooling and trapping experiments for atoms. Compared to rubidium atoms, cesium atoms have a larger orbital radius. For atomic states with the same principal quantum number, cesium atoms are more sensitive to external electric fields and have stronger interactions between them. Therefore, cesium atoms are a more suitable choice.

Here, we extend the magic-wavelength ODT to trap Cs 6S_1/2 ground-state and 46S_1/2 Rydberg-state atoms simultaneously by adjusting its laser wavelength to blue detuning for auxiliary transitions, where the magic ODT has an attractive potential for ground- and Rydberg-state atoms. Firstly, we calculate the dynamic polarizabilities of the Cs 6S_1/2 ground state and 46S_1/2 Rydberg state in the range of 900–1950 nm, and then we find the magic wavelengths for trapping the two states. Secondly, we analyze the robustness of the magic trapping conditions and the feasibility of the experimental operation. Finally, we calculate and analyze the trapping lifetime of Rydberg atoms in the magic ODT by considering dissipation mechanisms such as photon scattering and photoionization during Rydberg excitation.

2. Trapping Principle

For alkali metal Rydberg atoms, the valence electron is far away from the nucleus, so the inner electrons and the nucleus can be regarded as an ‘atomic core’ with a positive charge, which can be treated as a hydrogen-like atom [22,23]. In order to maintain an atomic coherent state, it is necessary to trap the atom in an ODT. The trap potential depth, U, can be expressed as follows [24,25]:

$$U=-\alpha \left(\omega \right){\displaystyle \frac{I}{2{\epsilon }_{0}c}}$$

(1)

where $\alpha \left(\omega \right)$ is the polarizability of the atom, I is the light intensity of the incident ODT laser, ε₀ is the vacuum permittivity, and c is the speed of light in a vacuum. The interaction potential, U, is influenced by atomic polarizability and light intensity. When a certain laser intensity is selected, the different potential traps are determined by the atomic polarizability. Polarizability describes the degree to which the electron cloud distribution of atoms or molecules deviates from the normal distribution under the action of external electric fields, which can be expressed as follows [26]:

$${\alpha }_i\left(\omega \right)={\alpha }_i^S\left(\omega \right)+A\cos {\theta }_k{\displaystyle \frac{m_{j_i}}{2j_i}}{\alpha }_i^v\left(\omega \right)+\left({\displaystyle \frac{3{\cos }^2{\theta }_p-1}{2}}\right){\displaystyle \frac{3m_{j_i}^2-j_i\left(j_i+1\right)}{j_i\left(2j_i-1\right)}}{\alpha }_i^T\left(\omega \right)$$

(2)

where A represents the degree of circular polarization, and j and m_j represent the total angular momentum quantum number and magnetic quantum number, respectively. θ_k is the angle between the wave vector k and the quantized axis e_z, which satisfies the relation cosθ_k = k·e_z (for the experimental setup, the quantization magnetic field B defines e_z [27,28]). θ_p is related to the direction of the polarizability vector ε and the quantized axis e_z. θ_p and θ_k are required to satisfy the geometric relation cos²θ_k + cos²θ_p ≤ 1.

For the expressions of dynamic polarizability given in Equation (2), ${\alpha }_i^s$, ${\alpha }_i^v$, and ^{$a_i^T$} are scalar, vector, and tensor polarizabilities, respectively. When the total angular momentum is less than 1, the tensor polarizability does not exist. As stated in reference [24], the rate of change in the relative depth of the potential well caused by the ellipticity variation of linearly polarized ODT lasers is extremely small. In this paper, all calculations were performed for linear-polarization light, where A = 0, cosθ_k = 1, and cosθ_p = 0 (at this time, the selected quantized magnetic field is parallel to the wave vector; that is, B//k). So, there are no vector and tensor components. We do not need to consider the effects of changes in θ_k and θ_p on polarizability and the magic wavelength in our research. For the 6S_1/2 and 46S_1/2 states, the total angular momentum quantum number j = 1/2, so its total polarizability is expressed as

$$\alpha \left(\omega \right)={\alpha }_i^S\left(\omega \right)={\displaystyle \sum _n\frac{f_{in}}{\Delta E_{ni}^2-{\omega }^2}}+{\alpha }_{core}$$

(3)

The core polarizability of Cs atoms is 15.81 a.u. The oscillator strength is expressed as follows [29]:

$$f_{in}={\displaystyle \frac{2\Delta E_{ni}}{3\left(2j_i+1\right)}}{\left|\left.〈{\psi }_i\right|\left|r{C}^1\left(r\right)\right|\left|{\psi }_n\right.〉\right|}^2$$

(4)

where ΔE_ni = E_n − E_i and $\left.〈{\psi }_i\right|\left|r{C}^1\left(r\right)\right|\left|{\psi }_n\right.〉$ are the transition energy and the reduced matrix element from $\left|{\psi }_n\right.〉$ to $\left|{\psi }_i\right.〉$, respectively. C¹(r) is the first-order spherical tensor. The transition energies and related reduced matrix elements were calculated using the program in reference [30].

3. Magic Trapping of Ground and Rydberg States

3.1. Magic Condition for 6S_1/2 and 46S_1/2 States

The two-photon excitation scheme for the transition from the 6S_1/2 ground state to the 46S_1/2 Rydberg state is shown in Figure 1b. The traditional far-off-resonance red-detuned ODT has an attractive potential for ground-state atoms, but it shows a repulsive potential for Rydberg atoms, as shown in Figure 1a. After adjusting the ODT laser wavelength to blue detuning for the 46S_1/2–7P_3/2 auxiliary transition, ground- and Rydberg-state atoms have the same potential trap.

From Equations (3) and (4), the polarizability of the $\left|{\psi }_i\right.〉$ state is mainly the sum of all contributions of other atomic states, $\left|{\psi }_n\right.〉$, that satisfy the selection rule. Because the photon energy of the transition from the highly excited state to the ground state and the low excited state is large and the matrix element of the electric dipole transition is small, the contribution of the highly excited state to its polarizability is very small. When calculating the polarizability of the nS_1/2 state, the contributions of the nP_1/2 and nP_3/2 states should be considered simultaneously. For the 6S_1/2 and 46S_1/2 states, the ranges of principal quantum numbers during the calculation are n = 6–20 and 6–100.

Since the magic wavelength for a Rydberg atom is close to the strong transition wavelength, it is necessary to consider the contributions from more atomic states that satisfy the dipole transition selection rule to the polarizability of Rydberg states in calculations [31,32]. Meanwhile, we also consider the contribution from continuum states. In order to verify the contributions of the neighboring intermediate states to the polarizability of the 46S_1/2 state, we use the sum-over-states method [33] to calculate dynamic polarizabilities.

We give the dynamic polarizabilities of the 6S_1/2 and 46S_1/2 states in the range of 900–1950 nm and find the corresponding magic wavelength, as shown in Figure 2a. There are four auxiliary transitions corresponding to 46S_1/2 ↔ 7P_1/2, 46S_1/2 ↔ 7P_3/2, 46S_1/2 ↔ 8P_1/2, and 46S_1/2 ↔ 8P_3/2 transitions, respectively. The magic detuning and relevant wavelengths are calculated and listed in

Table 1. The relevant wavelengths and magic conditions for the $46S_{1/2}\leftrightarrow \left|a\right.〉$ auxiliary transition.

Transition	Transition Wavelength (nm)	Magic Wavelength (nm)	Polarizability (a.u.)	Magic Detuning (GHz)
46S1/2 ↔ 7P1/2	1043.980	1043.974	1125.554	1.6515
46S1/2 ↔ 7P3/2	1064.090	1064.084	1049.664	1.5897
46S1/2 ↔ 8P1/2	1774.540	1774.523	502.563	1.6196
46S1/2 ↔ 8P3/2	1800.960	1800.934	498.655	2.4049

. For the 46S_1/2 ↔ 7P_3/2 auxiliary transition, its resonance wavelength is 1064.09 nm. The dynamic polarizabilities of the 6S_1/2 and 46S_1/2 states near 1064.09 nm are shown in Figure 2b. The intersection of 6S_1/2 and 46S_1/2 near the auxiliary transition is the magic wavelength. The polarizability at the intersection is positive, so the ground state and Rydberg state form the same potential trap.

3.2. The Robustness of the Magic ODT

In order to evaluate the robustness of magic conditions, we present the relationship between the relative potential well depth (the ratio of the depth difference in potential wells for the ground state and Rydberg state to the well depth of the ODT at the magic wavelength) and the laser wavelength relative to the magic wavelength, as shown in Figure 3. When the wavelength of the ODT laser changes by ±0.1 nm, the change rate of the well depth of the magic ODT at 1064.084 nm is within ±2.0%. Although the 1800.934 nm magic ODT shows a relatively small change rate within ±1.2%, the 1064.084 nm wavelength offers distinct advantages. At around 1064 nm, corresponding lasers and high-power ytterbium-doped fiber amplifiers (YDFAs) are commercially available, which provide great convenience and cost-effectiveness for experimental setups. And this wavelength exhibits a relatively small relative error. Furthermore, the 1064.084 nm magic ODT is easier to expand into a multi-atomic array or optical lattice.

The typical well depth is about 10–100Er, where Er = h²/(2mλ²) is the single-photon recoil energy, h is Planck’s constant, m is the mass of a single atom, and λ is the wavelength of the light field scattered by the atom. The depth of the optical trap is U = k_BT, while the peak optical intensity is I = 2P/(πω²). Here, k_B is the Boltzmann constant, T is the equivalent temperature of the atoms in the ODT, and P and ω are the power of the ODT laser and the radius of the waist spot at the beam waist position, respectively.

When the well depths of the ODT are 50Er, 100Er, and 150Er, the corresponding temperature, T, is about 5, 10, and 15 μK, respectively. By substituting the above parameters into the equation U = k_BT, the relationship between the laser power and beam waist radius of different optical traps is illustrated in Figure 4. The laser power of the ODT increases with an increase in the beam waist radius. When the waist radius is the same, the potential well depth increases with an increase in power. Typically, a certain potential well is formed when T = 1 mK and the beam waist radius is 50 μm, and the required laser power is calculated to be about 16 W. For a 1064 nm laser, its output power can reach the hundred-watt level, which is sufficient for normal experimental requirements.

4. Trapping Lifetime of Cs Rydberg State in Magic ODT

Here, we analyze the dissipative mechanism that affects the decoherence time of Rydberg states. The effective population decay lifetime is given by [22,34,35]

$${\displaystyle \frac{1}{{\tau }_{eff}}}={\displaystyle \frac{1}{{\tau }_n^{\left(0\right)}}}+{\displaystyle \frac{1}{{\tau }_n^{\left(bb\right)}}}+{\displaystyle \frac{1}{{\tau }_{sc}}}+{\displaystyle \frac{1}{{\tau }_{PI}}}$$

(5)

The lifetimes of Rydberg atoms are affected by spontaneous radiation, ${\tau }_n^{\left(0\right)}$; blackbody radiation, ${\tau }_n^{\left(bb\right)}$; photon scattering, τ_sc; and photoionization, ${\tau }_{PI}$.

4.1. Spontaneous Radiation and Blackbody Radiation

Spontaneous radiation and blackbody radiation are given by [34,35]

$${\tau }_n^{\left(0\right)}={\tau }^{\left(0\right)}{\left(n^{*}\right)}^a$$

(6)

$${\tau }_{\mathrm{n}}^{\left(bb\right)}={\displaystyle \frac{3\hslash {\left(n^{*}\right)}^2}{4{\alpha }_{fs}^3{k}_{B}T}}$$

(7)

where τ⁽⁰⁾ = 1.43 and a = 2.96 for the Cs nS state, n* = n − δ_n,l,j is the effective principal quantum number, α_fs = 1/137 is the fine-structure constant, and T = 300 K. The expansion coefficients for the quantum defect of the Cs atom are given in [36]. By substituting the above specific parameters into Equations (6) and (7), we can calculate ${\tau }_n^{\left(0\right)}$ = 90.92 μs and ${\tau }_n^{\left(bb\right)}$ = 8.6411 × 10⁻⁵ s⁻¹.

4.2. Photon Scattering

As shown in Figure 1, compared with a traditional ODT, a magic ODT can only be found near auxiliary transitions. In an ODT, photon scattering can cause atoms to be heated and quickly become decoherent. In order to estimate the trapping lifetime of Cs atoms, it is necessary to calculate the photon scattering rate, R_sc, which is expressed as follows [37]:

$$R_{sc}={\displaystyle \frac{\mathsf{\Gamma }}{2}}{\displaystyle \frac{I/I_{sat}}{1+4{\left(\Delta /\mathsf{\Gamma }\right)}^2+\left(I/I_{sat}\right)}}$$

(8)

where Ι and Ι_sat are the ODT laser intensity and the saturated light intensity, respectively. Δ is the detuning of the ODT laser relative to the auxiliary transition. Γ is the spontaneous rate (Einstein A coefficient) of the transition from $\left|n^{\prime }{j}^{\prime }\right.〉$ to $\left|nj\right.〉$, which can be expressed as follows [33]:

$$A_{n^{\prime }{j}^{\prime }nj}={\displaystyle \frac{e^2{\omega }_0^2}{2\pi {\epsilon }_0{m}_e{c}^3}}{\displaystyle \frac{2j+1}{2j^{\prime }+1}}{f}_{in}$$

(9)

where e is the elementary charge, m_e is the electron mass, and ω₀ is the transition angular frequency. f_in is the oscillator strength. For the magic ODT, for the Cs 6S_1/2 and 46S_1/2 states, the 1064.084 nm ODT laser’s detuning relative to the 46S_1/2 ↔ 7P_3/2 auxiliary transition is not so far. Therefore, we calculate the photon scattering rate of the auxiliary transition, as shown in

Table 2. Photon scattering rates induced by the 1064.084 nm magic ODT laser near the 46S_1/2 ↔ 7P_3/2 auxiliary transition.

Transition	Transition Wavelength (nm)	Reduced Matrix Element (ea0)	Einstein A Coefficient (s−1)	Frequency Detuning (GHz)	Photon Scattering Rate (s−1)
46S1/2 ↔ 7P3/2	1064.09	0.039	3.89 × 103	1.5897	262.89
46S1/2 ↔ 7P1/2	1043.98	0.029	4.36 × 103	−5429.197	2.3 × 10−5

. This shows that the photon scattering from the auxiliary transition is much more important. The photon scattering rate is directly related to frequency detuning and the Einstein A coefficient of the auxiliary transition. For Rydberg states, the Einstein A coefficient is much smaller than that of the low-excited states.

4.3. Photoionization

To evaluate the photoionization of Rydberg atoms in the magic ODT induced by excitation light, rate equations are established by using a three-step excitation process [38,39,40], as shown in Figure 5. Here, $\left|0\right.〉=6S_{1/2}$ is the ground state, $\left|1\right.〉=6P_{3/2}$ and $\left|2\right.〉=46S_{1/2}$ are the first and second excited states, and $\left|i\right.〉$ represents the ionized state under the third laser. Firstly, Cs atoms are excited from the ground state to a low-excited state and then to a Rydberg state by two laser beams. Secondly, the atoms in the Rydberg state are ionized by the third laser beam. By establishing the population equations of each atomic state in different time periods, the relevant analytical expressions are derived. We used Matlab 2017b software to build the images of each state by adding different laser parameters and atomic parameters.

In Figure 5, σ_a1, σ_a2, σ_s1, and σ_s2 are the resonance absorption cross-sections and stimulated emission cross-sections of the $\left|0\right.〉\leftrightarrow \left|1\right.〉$ and $\left|1\right.〉\leftrightarrow \left|2\right.〉$ transitions, respectively, and σ_i is the ionization cross-sectional area. ${\mathsf{\Gamma }}_1$ and ${\mathsf{\Gamma }}_2$ are the spontaneous emission rates of the $\left|1\right.〉$ and $\left|2\right.〉$ states, respectively. I₁, I₂, and I_i are the intensities of the three-step lasers, respectively.

We establish rate equations for each atomic state at different stages based on the excitation process described above. It is important to consider that, with the action of laser pulses, atoms are redistributed into different energy states at different times. In other words, the total number of particles in the atomic population is always conserved throughout the entire excitation process. In the current theoretical framework, we assume that the atomic transition process adheres to the Markovian property [41,42,43], meaning that the probability of an atom transitioning from one energy state to another at a specific time depends solely on the current state and is independent of its previous transition history. Therefore, based on the excitation process shown in Figure 5, we provide the corresponding rate equation system. Since all atoms are in the ground state at the initial moment, we can set t = 0, n₀ = 100, and n₁ = n₂ = n₃ = 0.

When the statistical weights of each state are ignored, σ_a1 = σ_s1 = σ₁ and σ_a2 = σ_s2 = σ₂. Under the given excitation mode, the atoms are first excited by the first two laser steps and then ionized by the third laser step. Therefore, the first two laser steps end at time t₁, and the third laser step ends at time t₂. And the rate equations can be written as follows:

(1)

For 0 < t < t₁,

$$\begin{array}{l}{\displaystyle \frac{d{n}_0}{dt}}=-{\sigma }_1{I}_1{n}_0+\left({\mathsf{\Gamma }}_1+{\sigma }_1{I}_1\right)n_1\\ {\displaystyle \frac{d{n}_1}{dt}}={\sigma }_1{I}_1{n}_0-\left({\mathsf{\Gamma }}_1+{\sigma }_2{I}_2+{\sigma }_1{I}_1\right)n_1+\left({\mathsf{\Gamma }}_2+{\sigma }_2{I}_2\right)n_2\\ {\displaystyle \frac{d{n}_2}{dt}}={\sigma }_2{I}_2{n}_1-\left({\mathsf{\Gamma }}_2+{\sigma }_2{I}_2\right)n_2\end{array}$$

(10)

(2)

For t₁ < t < t₂,

$$\begin{array}{l}{\displaystyle \frac{d{n}_0}{dt}}={\mathsf{\Gamma }}_1{n}_1\\ {\displaystyle \frac{d{n}_1}{dt}}=-{\mathsf{\Gamma }}_1{n}_1+{\mathsf{\Gamma }}_2{n}_2\\ {\displaystyle \frac{d{n}_2}{dt}}=-\left({\mathsf{\Gamma }}_2+{\sigma }_i{I}_i\right)n_2\\ {\displaystyle \frac{d{n}_i}{dt}}={\sigma }_i{I}_i{n}_2\end{array}$$

(11)

For Equations (10) and (11), these parameters can be expressed as follows [38]:

$$\sigma ={\displaystyle \frac{{\lambda }^{2}A}{2\pi \Delta \nu }}$$

(12)

$$I=5.0\times {10}^{13}{\displaystyle \frac{\lambda P}{\delta }}$$

(13)

$$\Delta \nu =7.16\times {10}^{-7}{\left({\displaystyle \frac{T}{M}}\right)}^{1/2}{v}_0$$

(14)

where A is the Einstein spontaneous emission coefficient of the atom, $\Delta \nu$ is the Doppler linewidth at temperature T, M is the mass of the atom, λ and ν₀ are the resonance wavelength and frequency of the atomic transition, P is the peak power of the three-step laser, and $\delta$ is the focusing area of the laser beam. The equation for calculating the ionization cross-sectional area is as follow [44]:

$${\sigma }_{nl}={\displaystyle \frac{4{\pi }^2\alpha a_0^2}{3}}{\displaystyle \frac{h\nu }{2l+1}}\times \left[l{\left|R_{l,l-1}\right|}^2+\left(l+1\right){\left|R_{l,l+1}\right|}^2\right]$$

(15)

where α = 1/137 is the fine-structure constant of the atom, a₀ is the Bohr radius, hv is the ionization energy, and R_l,l±1 is the radial matrix element. According to Equation (15), the ionization cross-sectional area of the 46S_1/2 Rydberg state is σ_46S = 5.035 × 10⁻¹⁵ cm².

In order to simplify the calculation process, we assume that

$${\mathrm{c}}_1={\sigma }_1{I}_1,c_2={\mathsf{\Gamma }}_1,{\mathrm{c}}_3={\sigma }_2{I}_2,c_4={\mathsf{\Gamma }}_2,{\mathrm{c}}_5={\sigma }_i{I}_i$$

(16)

The rate equations can be simplified as follows:

(1)

For 0 < t < t₁,

$$\begin{array}{l}{\displaystyle \frac{d{n}_0}{dt}}=-c_1{n}_0+\left(c_1+c_2\right)n_1\\ {\displaystyle \frac{d{n}_1}{dt}}=c_1{n}_0-\left(c_1+c_2+c_3\right)n_1+\left(c_3+c_4\right)n_2\\ {\displaystyle \frac{d{n}_2}{dt}}=c_3{n}_1-\left(c_3+c_4\right)n_2\end{array}$$

(17)

(2)

For t₁ < t < t₂,

$$\begin{array}{l}{\displaystyle \frac{d{n}_0}{dt}}=c_2{n}_1\\ {\displaystyle \frac{d{n}_1}{dt}}=-c_2{n}_1+c_4{n}_2\\ {\displaystyle \frac{d{n}_2}{dt}}=-\left(c_4+c_5\right)n_2\\ {\displaystyle \frac{d{n}_i}{dt}}=c_5{n}_2\end{array}$$

(18)

For the three-step photoexcitation and photoionization processes of the Cs atom shown in Figure 5, we provide the relevant transition parameters for Equations (12)–(14) in

Table 3. Parameters related to three-step excitation and ionization path of Cs atom.

Cs at M = 132.9 g/mol, T = 944.15 K
Laser	Excitation Path	ν0 (Hz)	A (s−1)	λ (nm)	P (W)	δ (cm2)
First laser	6S1/2 ↔ 6P3/2	351.72571 × 1012	3.226 × 107	852	100	0.03
Second laser	6P3/2 ↔ 46S1/2	587.94712 × 1012	2.246 × 104	509	3 × 104	0.03
Third laser	46S1/2 ↔|i>			1064	1000	0.03

. First, these specific values are substituted into Equation (16). Subsequently, we use Matlab software to program the rate equation system based on Equations (17) and (18) and plot the population distribution images for each state.

The magic wavelength of the 6S_1/2 ground state and 46S_1/2 Rydberg state is 1064.084 nm near the 46S_1/2 ↔ 7P_3/2 auxiliary transition, so a 1064 nm laser is selected as the ODT laser, which will result in the photoionization of the 46S_1/2 state. According to Equations (10)–(18), when other atomic parameters and laser parameters are determined, the photoionization rate will be affected by the excitation light power and laser pulse width, as shown in Figure 6 and Figure 7. In Figure 6, the photoionization rate of the Cs Rydberg state gradually increases with the laser power. The rate gradually increases to 30% and finally becomes basically stable. Before the ionization rate reaches a stable value, its change with the second-step laser power is relatively slow compared with the third-step laser power.

In this mode, the rate equations are established based on different time periods. During the calculation process, we set t₁ = 10 ns, while t₂ is treated as a variable. Then, we use the Matlab software to plot the relationship between the population rate of each state of the Cs atom and the laser pulse width, as shown in Figure 7. In Figure 7a, under the action of a laser, Cs atoms in the ground state continue to converge to the excited state, while the population rate of the ground state atoms continues to decrease. After 10 ns, when the two-step lasers stop acting, the ground-state atomic population no longer decreases but increases. In Figure 7b,c, the population rate of the 6P_3/2 state starts to decrease after reaching an extreme value in a very short time. Meanwhile, the population rate of the 46S_1/2 state starts to decrease after reaching a maximum value at 10 ns. In Figure 7d, there are only two excitation light emissions in the first 10 ns. During this period, the ionization rate is zero. But after 10 ns, the ionization rate gradually increases under the action of the third-step laser. After 30 ns, the ionization rate changes a little with the laser pulse width. The ionization rate reaches a maximum value of 31.49% at 32.36 ns, corresponding to ${\tau }_{PI}=0.96\times {10}^9{s}^{-1}$.

A large number of excited atoms will return to the ground state or lower excited state through spontaneous and stimulated radiation, so the population rate of the Cs ground-state atoms decreases slowly in Figure 7a. In Figure 7b,c, the population rates of the 6P_3/2 and 46S_1/2 states cease to increase after reaching the extreme values. This shows that because of the lifetime of energy levels and the influence of stimulated radiation, the transitions of atoms to high energy levels and low energy levels undergo a complex interplay, and finally the system reaches a dynamic equilibrium. In Figure 7a–c, the population decay of the corresponding three atomic states tends to plateau. After 10 ns, the two-step excitation light emissions stop, and under the action of the third-step laser, the population of the 46S_1/2 state decreases rapidly, while the ionization rate rapidly increases till saturation. Meanwhile, the atomic population of the 6P_3/2 state slowly decreases and returns to the ground state, as shown in Figure 7b. This causes a turning point and slow increase in the population in the ground state after 10 ns, as shown in Figure 7a.

In summary, under this excitation mode, as long as the power of the three-step laser remains constant, the populations of the two excited states will decrease as the duration of the ionization laser’s operation increases. We can calculate the photoionization rate when the population of the ionized state reaches a stable level so as to evaluate the impact of photoionization on the lifetime of Rydberg atoms.

After considering dissipation mechanisms such as spontaneous emission, blackbody radiation, photon scattering, and photoionization, the trapping lifetime of 46S_1/2 Rydberg atoms in a magic-wavelength trap can reach several tens of microseconds. Generally, the laser pulse width for manipulating and exciting Rydberg atoms in experiments falls within the nanosecond range. This enables the experimental repetition rate to reach the order of thousands within the atomic decoherence time. This is of great significance for quantum computing and quantum simulation based on Rydberg atoms.

5. Conclusions

In conclusion, we calculated the dynamic polarizabilities of the Cs 6S_1/2 ground state and 46S_1/2 Rydberg state by using the sum-over-states method. We found the magic wavelength for trapping the two atomic states and calculated the magic detuning relative to the auxiliary transition where the ground state and Rydberg state have the same polarizability. At around a wavelength of 1064 nm, corresponding lasers and high-power ytterbium-doped fiber amplifiers (YDFAs) are commercially available, so we chose a magic wavelength of 1064.084 nm to analyze the robustness and feasibility of the experimental operation. The trapping lifetime of the Cs Rydberg state in the magic ODT was estimated by considering different dissipation mechanisms, such as photon scattering and photoionization. When atoms are trapped in an ODT, the interaction between Rydberg atoms and the ODT laser leads to photoionization. Furthermore, we discussed the change in the ionization rate with laser parameters and calculated the photoionization rate of the 46S_1/2 state. This provides an important theoretical basis for evaluating the trapping lifetime of Rydberg atoms in an ODT, which is important for precision measurements based on Rydberg atoms.