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Xie Qiong-Tao, Liu Xiao-Liang. Exact analytical results for a two-level quantum system under a Lorentzian-shaped pulse field. Chinese Physics B, 2020, 29(6): 060305  
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Exact analytical results for a two-level quantum system under a Lorentzian-shaped pulse field
Xie Qiong-Tao1, †, Liu Xiao-Liang2
College of Physics and Electronic Engineering, Hainan Normal University, Haikou 571158, China
School of Physics and Electronics, Central South University, Changsha 410083, China

 

† Corresponding author. E-mail: qiongtaoxie@yahoo.com

Project supported by the Natural Science Foundation of Hainan Province, China (Grant No. 2019RC179).

Abstract

We investigate a two-level quantum system driven by a Lorentzian-shaped pulse field. An analytical solution is presented in terms of the confluent Heun functions. It is shown that for specially chosen parameter conditions, there are a number of the exact analytical solutions in an explicit form. The dependence of the final transition probabilities in the two levels on the system parameters is derived analytically and confirmed numerically.

PACS: ;03.65.Aa;;02.30.Gp;;42.50.-p;;42.50.Ct;
Keyword:;exact solution;;Lorentzian-shaped pulse;;two-level system;
1. Introduction

During the past few decades, two-level quantum systems driven by various laser pulses have attracted extensive interest of research.[1–6] It has been shown that these systems not only exhibit many interesting physical phenomena, but also have important applications in quantum information and quantum simulation. Naturally, an exact analytical solution is of fundamental importance for understanding these phenomena and applications. In most of the previous works, three typical pulse shapes have been studied theoretically and experimentally, i.e., the hyperbolic secant, the Lorentzian, and the Gaussian shapes.[1–6] For the hyperbolic secant pulse, the exact analytical solution has been found.[7] The exact analytical solutions for the latter two typical pulses are still lacking. It is thus an important problem how to find the exact analytical solutions for the two typical pulses.

In recent years, Heun equation and its confluent forms have been used to construct the analytical solutions for a number of specific pulse shapes.[8–13] The analytical solutions to these pulses are given by the Heun-related special functions defined by infinite series. These equations contain the well-known hypergeometric and confluent hypergeometric equations as particular cases which have widely been applied to construct the exact analytical solutions for various types of pulse shapes.[14–21] Different from the hypergeometric and confluent hypergeometric functions, the coefficients in these Heun-related special functions obey a three-term recurrence relation, and their asymptotic behavior in different parameter regions is not well-known.[22,23] The final transition probabilities in the two levels cannot be obtained in an explicit form.[8–13] For a general parameter condition, these analytical solutions cannot provide a good understanding of the properties of the systems.

In the present work, our main aim is to give certain exact analytical results for the well-known model where a two-level quantum system is driven by a Lorentzian-shaped pulse field. Two main findings are presented. Firstly, it is shown that this model is related to the confluent Heun equation (CHE), and its analytical solution is given in terms of the confluent Heun functions (CHFs). Secondly, it is found that for some specially chosen system parameters, the CHFs become finite series. This allows us to find an infinite number of exact analytical solutions. The dependence of the final transition probability on the system parameters is derived analytically. In addition, we show that the exact analytical results can be valid in some non-Hermitian situations.

2. Model and exact analytical solutions

We begin with a physical model which describes the interaction between an atom and a pulse electric field. In the dipole approximation and the radiation gauge, this model is described by the following Hamiltonian:[24]

where Hatom is the free-atom Hamiltonian, d is the atomic dipole moment operator, and the electric field E(t) takes the form

with peak magnitude E0, polarization , and envelope function Fe(t). The atomic Hamiltonian has a set of eigenstates |n〉 associated with the eigenvalues ħωn, Ham |n⟩ = ħ ωn (n = 1,2,3,…). The pulse electric field is chosen suitably to drive the transition between an initial state |i〉 and a final state |f〉. The single-photon transition between the two states is not dipole allowed. They are coupled to the intermediate states |k〉 which are far detuned from the single-photon resonance. In experiments, the two states are chosen, so that they have a longer lifetime.[25] Here for brevity, we take i = 1 and j = 2. The energy separation between the two energy levels is Δ0 = ħ (ω2 − ω1). In the following, we shall show that under certain conditions, this model can be simplified into a two-level model. After expanding the state vector as

we obtain

where ωkn = ωk – ωn is the frequency separation between the two states |k〉 and |n〉. We consider the case where there is no diploe coupling between the intermediate sates, and at once obtain

where Ωkn = −dknE0/ħ is the Rabi frequencies, and dkn is the electric diploe matric elements, . Since all the intermediate states are far detuned from the single-photon resonance, they can be eliminated adiabatically, and thus the system can be approximated as a two-level system.[25] In addition, under the condition of Δ0 ≪ ω, we apply the rotating-wave approximation by dropping all the oscillating terms with the frequencies 2ω ± Δ0. This results in an effective two-level quantum system

For a detailed derivation, the reader is referred to Appendix A in Ref. [25]. Here is the profile of the pulse, Δp is the ac Stark shift of state p,

and Ω is the two-photon Rabi frequency

If one further takes the following transforms:

with

one has

where f(t) and ν(t) have the form

with

Here for simplicity, we have assumed that the Rabi frequency Ω is real. For the Lorentzian-shaped pulse field, the intensity profile G(t) is given as[25,26]

where the parameter g is related to the width scale τ, g = π/τ. By eliminating a2(t) and a1(t), we obtain two second-order differential equations for a1(t) and a2(t)

where the dot denotes the derivative with respect to t, and . It follows from the above two equations that if equation (20) has a solution of ϕ(t), equation (20) has a solution of ϕ*(t). This property will simplify our later discussions. Once the analytical solution to a1(t) is obtained, a2(t) is obtained from Eq. (15). With an appropriate variable transform, equation (15) is reduced to the CHE, and the analytical solution to a1(t) is expressed in terms of the CHFs (see details in Appendix A). Although the asymptotic behavior of the CHFs in different parameter ranges is not clear, the CHFs can become finite series under some special conditions.[22,23] This gives an infinite number of the exact analytical solutions to a1,2 in terms of elementary functions. In the present work, we mainly focus on these exact analyticalq solutions.

It is found that if the parameters f0,1, ν1, and g satisfy certain specific parameter relations (see details in Appendix A)

the exact analytical solutions of a1,2(t) can be given in an explicit form. Here N ≥ 0 is an integer, and FN(f0,f1,g) is a function of the parameters f0,1 and g. Since the expression of FN(f0,f1,g) is very cumbersome for N > 2, here we only list the three cases of N = 0,1,2 as examples

With the exact analytical solutions in hand, we can obtain the final transition probability at once, . For simplicity, it is assumed that the system is initially in the first level, a1(t0) = 1 and a2(t0) = 0. In the case of N = 0, the final transition probability is given as

In this situation, for given parameters of g and f1/g, we have f0/g = 4g/f1, and from the parameter relations (22) and (23). Here without loss of generality, we may take f1 > 0 and ν1 > 0. For a real coupling strength ν1 > 0, we require f1/g < 4. Now we discuss some properties of the final transition probability . Firstly, it is found that for arbitrarily chosen values of f1 and g, we have for t0 = –∞. This indicates that if the system is initially in the first state, it remains in this state after the duration of the pulse. Secondly, from the condition , it is found that at , reaches its minimal value

which depends on the choice of the initial dimensionless time gt0. It is evident that we have for t0 = 0. This result indicates that a complete population inversion occurs at t0 = 0, and the choice of t0 ≠ 0 leads to a smaller population inversion. The above analytical results are demonstrated in Fig. 1 where we plot P1(t) and for four different initial time t0. Since the parameters f0, f1, ν1, and g have the same units of Hz, the ratios between these parameters are thus essential. Here we fix g = 1 and . So we have and . In the case of N = 1, we focus on the situation with f1 = 0, and obtain the final transition probability

In this case, for a given parameter of g, we have and ν1/g = 6 from the parameter relations (22) and (24). The final transition probability only depends on the dimensionless time gt0. Similarly, it is found that we have for t0 = –∞. For t0 = 0, the minimal transition probability is achieved, . These analytical results are shown in Fig. 2.

Fig. 1. Transition probability P1(t) = |a1(t)|2 for (a) gt0 = –5, (b) gt0 = –1, (c) gt0 = –0.5, and (d) gt0 = 0. The parameters are given as g = 1, , , and . The initial conditions are a1(t0) = 1 and a2(t0) = 0. The dashed lines are the exact results of the final transition probability .
Fig. 2. Transition probability P1(t) = |a1(t)|2 for (a) gt0 = –5, (b)gt0 = –1, (c) gt0 = –0.5, and (d) gt0 = 0 with f1/g = 0. The parameters are given as g = 1, , and ν1/g = 6. The initial conditions are a1(t0) = 1 and a2(t0) = 0. The dashed lines are the exact results of the final transition probability .

Finally, we show that our exact analytical results are also applicable for the non-Hermitian situation. For example, the final transition probability in Eq. (26) is valid in the case of f1/g > 4 where the coupling strength ν1 becomes imaginary. Now the imaginary coupling strength has been realized in different systems.[27–33] In the resulting non-Hermitian system, for the initial conditions of a1(t0) = 1 and a2(t0) = 0, the conserved quantity becomes

instead of |a1(t)|2 + |a2(t)|2 = 1 in the Hermitian situation. In this situation, we introduce the normalized transition probability defined by P1,n(t) = P1(t)/(P1(t) + P2(t)), so that the normalized final transition probability is given by . It is found that for t0 = −∞, and it takes a minimal value at t0 = 0. In Fig. 3, we plot the normalized transition probability in the case where we take g = 1 and f1/g = 5. This leads to f0/g = –4g/f1 = –4/5, and . It is observed that our exact analytical results are also applicable for the non-Hermitian situation.

Fig. 3. Normalized transition probability P1,n(t) = |a1(t)|2/(|a1(t)|2 + |a2(t)|42) for (a) gt0 = –10, (b) gt0 = –1, (c) gt0 = –0.5, and (d) gt0 = 0. The parameters are given as g = 1, f1/g = 5, f0/g = −4g/f1 = −4/5, and . The initial conditions are a1(t0) = 1 and a2(t0) = 0. The dashed lines are the exact results of the normalized final transition probability .
3. Conclusion

We have investigated the two-level quantum system interacting with the Lorentzian-shaped pulse. The analytical solution is presented in terms of the CHFs. It is shown that under certain special parameter conditions, the CHFs become finite series. This allows us to obtain certain exact analytical solutions in an explicit form. We also derived the parametric dependence of the final transition probability analytically. It is shown that the choice of the initial time affects the final transition probability strongly. In addition, it is found that our exact analytical results are also applicable to the non-Hermitian situation where the coupling strength becomes imaginary.

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