Two- Level-Like Behavior of a Three-Level Atom in V Configuration Trapped in an Optical Cavity with Multiphoton Transition

Two- Level-Like Behavior of a Three-Level Atom in V Configuration Trapped in an Optical Cavity with Multiphoton

Transition

Babak Parvin

Physics Department, Faculty of Basic Sciences, University of Maragheh, P.O. Box 55181-83111, Maragheh, Iran, parvin@maragheh.ac.ir

Abstract- This paper reviews the theoretical investigation of a three-level atom in the V configuration enclosed in a single-mode cavity with multiphoton transition under several applied conditions in the time dependent regime. The transition between the first and second atomic levels occurs through the number of q photons. A closed set of equations describes the temporal behavior of the atom-cavity system. The numerical solution of these equations is implemented by the fourth-order Runge-Kutta method. In a weak driving limit along with multiple other circumstances, the three-level atom illustrates a similar manner to an effective two-level one for each separate transition. The results of the three-level atomic simulations confirm the utilized constrains in the two-level atomic model according to the plotted curves in an appropriate way.

Keywords: atom-cavity interaction, master equation, multiphoton transition, fourth-order Runge-Kutta method, time dependent regime.

1. Introduction

The discussion of the multiphoton absorption and emission in the systems with and without decay processes have been widely investigated [1-5]. In the previous work, the treatment of a three-level atom in Λ configuration with multiphoton transition has been studied in the steady-state regime [6] but here we want to consider the temporal behavior of a three-level atom in another configuration with multiphoton transition under several specific conditions. The transformation stages of the three-level atom in the V configuration trapped in an optical cavity into an effective two-level one is discussed for various transitions. By applying multiple conditions, this transformation takes place and the plotted results in diagrams adequately verify these conversions and also in this case it can be said that this three-level atom reveals a similar behavior to the two-level one. Because of lack of space, only the general conclusions have been written and for more complete discussion about an open form of equations and other parameters [6] can be referred.

1- The Three-Level Atom in the 𝐕 Configuration

The three-level atom in the V configuration with multiphoton transition as indicated in Fig. 1 is enclosed in a single-mode Fabry-Perot optical cavity. The atomic energy levels are 1, 2 and 3. The 1-3 atomic transition is driven by a classical field with the Rabi Frequency Ω at the resonance. For convenience, the Rabi frequency of the classical field is to be considered as a real quantity. It is assumed that the transition between level 1 and level 2 of the atom occurs through the number of q photons and also the frequency of the 1-2 atomic transition is at the resonance with the q–photon transition one. The coupling constant between the 1-2 atomic transition with the cavity field mode is 𝑔. The parameter 𝛾 (Γ) is related to the spontaneous emission rate from level 3 to level 2 (level 2 to level 1). The master equation which describes the behavior of the atom-cavity system is written as follows:

𝜌̇ = −i [𝑔(𝑎^𝑞𝐴̂₂₁+ 𝑎^†𝑞𝐴̂₁₂) +Ω

2(𝐴̂₁₃+ 𝐴̂₃₁), 𝜌] + 𝛾

2(2𝐴̂23𝜌𝐴̂32− 𝐴̂33𝜌 − 𝜌𝐴̂33) + Γ

2(2𝐴̂12𝜌𝐴̂21− 𝐴̂22𝜌 − 𝜌𝐴̂22) + ^𝜅

2(2𝑎𝜌𝑎^†− 𝑎^†𝑎𝜌 − 𝜌𝑎^†𝑎), (1)

Figure 1. This figure illustrates the three-level atom in the 𝑉 configuration with multiphoton transition confined in a single-mode optical cavity.

where 𝑎 (𝑎^†) is associated with the destruction (creation) operator of the cavity mode.

The number of photons that involves in the transition between levels 1 and 2 is q. The

raising and lowering operators between atomic levels |𝑖⟩ and |𝑗⟩ are 𝐴̂_𝑖𝑗’s and the cavity decay rate is 𝜅. Instead of direct solving the master equation (1), we use the following variables for q-photon transition in general:

𝐶𝑛= 〈𝑎^†𝑛𝑎^𝑛〉, 𝑛 ≥ 0 (2)

𝐻_𝑛= 〈𝐴̂₂₂𝑎^†𝑛𝑎^𝑛〉, 𝑛 ≥ 0 (3)

𝐽𝑛= 〈𝐴̂11𝑎^†𝑛𝑎^𝑛〉, 𝑛 ≥ 0 (4)

𝐿_𝑛= i〈𝐴̂₃₁𝑎^†𝑛𝑎^𝑛− 𝐴̂₁₃𝑎^†𝑛𝑎^𝑛〉, 𝑛 ≥ 0 (5)

𝐹𝑛= 〈𝐴̂32𝑎^{†𝑛+𝑞−1}𝑎^𝑛−1+ 𝐴̂23𝑎^†𝑛−1𝑎^{𝑛+𝑞−1}〉, 𝑛 ≥ 1 (6)

𝐷𝑛= i〈𝐴̂21𝑎^†𝑛−1𝑎^{𝑛+𝑞−1}− 𝐴̂12𝑎^{†𝑛+𝑞−1}𝑎^𝑛−1〉, 𝑛 ≥ 1 (7)

the time evolution of equations (2) to (7) is obtained from the underlying relations: 𝐶̇_𝑛= 𝑔 ∑ (𝑞 𝑚)_{(𝑛−𝑚)!}^𝑛! 𝐷_{𝑛−𝑚+1} 𝑞 𝑚=1 − 𝜅𝑛𝐶_𝑛, (8)

𝐻̇𝑛= −𝑔𝐷𝑛+1+ 𝛾𝐶𝑛− 𝛾𝐽𝑛− (𝛾 + Γ + 𝜅𝑛)𝐻𝑛, (9)

𝐽̇𝑛= 𝑔 ∑ (𝑞 𝑚)_{(𝑛−𝑚)!}^𝑛! 𝐷𝑛−𝑚+1 𝑞 𝑚=0 +^Ω₂𝐿𝑛+ Γ𝐻𝑛− 𝜅𝑛𝐽𝑛, (10) 𝐿̇_𝑛= 𝑔 ∑ (𝑞 𝑚)_{(𝑛−𝑚)!}^𝑛! 𝐹_{𝑛−𝑚+1} 𝑞 𝑚=0 + Ω𝐶_𝑛− Ω𝐻_𝑛− 2Ω𝐽_𝑛− (^𝛾 2+ 𝜅𝑛) 𝐿_𝑛, (11) 𝐹̇𝑛= −𝑔𝐿𝑛+𝑞−1−^Ω 2𝐷𝑛−¹ 2(𝛾 + Γ + 𝜅(2𝑛 + 𝑞 − 2))𝐹𝑛, (12) 𝐷̇_𝑛= −2𝑔𝐽_{𝑛+𝑞−1}+ 2𝑔 ∑ (𝑞 𝑚)_{(𝑛+𝑞−𝑚−1)!}^{(𝑛+𝑞−1)!} 𝐻_{𝑛+𝑞−𝑚−1} 𝑞 𝑚=0 + ^Ω 2𝐹_𝑛−¹ 2(Γ + 𝜅(2𝑛 + 𝑞 − 2))𝐷_𝑛. (13)

In numerical solution of equations (8) to (13) along with the initial conditions, the classical fourth-order Runge-Kutta method has been utilized here [7,8]. The matrix continued fractions approach can also be applied to solve these time dependent equations [9]. To do so, we consider the column vector 𝜙_𝑛(𝑡) for 𝑛 ≥ 0 as in the following: 𝜙𝑛(𝑡) = (𝐶_𝑛 𝐻_𝑛 𝐽_𝑛 𝐿_𝑛 𝐹_𝑛 𝐷_𝑛)^Τ, (14)

it is supposed that the atom is in the ground state |1⟩ and the cavity field is in the coherent state |𝛼⟩ at the initial time, in this situation 𝜙_𝑛(𝑡) takes the following form for 𝑛 ≥ 0: 𝜙_𝑛(0) = |𝛼|^2𝑛(1 0 1 0 0 0)^Τ. (15)

Thus, the numerical solution related to the desired physical quantities is found. In the next part, the numerical solution of the second atomic level population obtained from the fourth-order Runge-Kutta method in this section is measured with the approximate solution of the second level population derived in that one. 2- How to Convert the 3-Level Atom to the 2-Level One Here, just in the three-photon transition case, how to transform the three-level atom enclosed in the cavity into the two-level one under several special circumstances is clarified in detail in the time dependent regime. In the case where 𝜅 ≫ 2𝑔, the transition from level 2 to level 1 happens at the following rate: Γ_𝐸=^8𝑔2 𝜅 , (16)

now assuming that Γ ≪ Γ_𝐸, then Γ can be neglected against Γ_𝐸 and the atom trapped in the cavity of Fig. 1 is reduced to the three-level one drawing in Fig. 2(a). Finally, with the application of a weak driving limit in which Ω ≪ 𝛾, the three-level atom created in

Fig. 2(a) becomes even simpler. In the weak driving limit, the population of the third atomic level is given by:

𝑃₃= (^Ω

𝛾)²(1 − exp (−^𝛾

2𝑡))², (17)

the weak driving limit necessitates that 𝑃₃≪ 1, so with good approximation one can ignore the third level population. In this situation the transition from level 1 to level 2 takes place at the below rate:

𝛾𝐸=^Ω_𝛾², (18)

in this way the third level is eliminated from computations and the three-level atom of Fig. 2(a) is turned into the two-level one in Fig. 2(b). By using the two-level atomic model and exerting the initial condition, the population of the second level is obtained by the following equation:

𝑃2=_𝛾^𝛾𝐸

𝐸+Γ_𝐸(1 − exp(−(𝛾_𝐸+ Γ𝐸)𝑡)). (19)

The curves of the second atomic level population 𝑃₂ in the one-photon through three- photon transitions for various numerical values have been plotted in Figs. 3-5 versus the scaled time 𝑔𝑡, respectively. In each figure, the dashed curve is calculated on the basis of the numerical calculations performed by the fourth-order Runge-Kutta technique presented in section 2. The solid graph in any figure is drawn based on the obtained outcomes in this section, the curve of equation (19) is depicted in Fig. 5 in the three- photon transition case. Comparison of the calculation associated with the three-level atomic simulations with the drawn diagrams based on the two-level atomic pattern reflects that the difference between the plotted results is of the order of thousands. Therefore, it can be said that under certain applied circumstances, the given three-level atom can show the behaviors of the two-level one.

Figure 2. Transformation of the three-level atom to the two-level one in the three- photon transition case: (a) changing the three-level atom enclosed in the cavity of

Fig. 1 into the new three-level one in the situation of 𝜅 ≫ 2𝑔, (b) converting the three-level atom of model (a) to the two-level one in the case of 𝛺 ≪ 𝛾.

Figure 3. The curves of the second level population for 𝜅/𝑔 = 30, 𝛺/𝑔 = 0.15, 𝛾/𝑔 = 2.3, 𝛤/𝑔 = 0.001 and |𝛼|²= 1 in terms of 𝑔𝑡 in the one-photon transition case.

Figure 4. The diagrams of the second level population for 𝜅/𝑔 = 30, 𝛺/𝑔 = 0.17, 𝛾/𝑔 = 2.6, 𝛤/𝑔 = 0.001 and |𝛼|²= 1 against 𝑔𝑡 in the two-photon transition case.

Figure 5. The graphs of the second level population for 𝜅/𝑔 = 30, 𝛺/𝑔 = 0.5, 𝛾/𝑔 = 7.5, 𝛤/𝑔 = 0.001 and |𝛼|²= 1 versus 𝑔𝑡 in the three-photon transition case.

3- Conclusion

The presented results in this work disclose that under specific conditions, the three- level atom in the V configuration with multiphoton transition confined in the single-

mode optical cavity can be turned into the effective two-level atom in the time dependent regime. In part 2 the closed set of equations delineates the action of the atom-cavity system for each arbitrary transition which the numerical solution of these equations along with the initial conditions is implemented by the fourth-order Runge-Kutta approach. The approximate equation describing the population of the second level is found in the three-photon transition case in the two-level atomic model in section 3.

In this same part, the numerical outcomes of the three-level atomic simulations with the derived curves from plotting the equations related to the two-level atom are shown in Figs. 3-5. The analogy of the curves points that for any separate transition, the three- level atomic simulations are very close to the two-level atomic results and one can say that the three-level atom trapped in the cavity depicts the same behavior as the two- level one.

ACKNOWLEDGMENT

The author would like to thank Dr. Howard J. Carmichael for his comment.

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