† Corresponding author. E-mail:
Project supported by the Natural Science Foundation of Hainan Province, China (Grant No. 2019RC179).
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.
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.
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]
![](cpb_29_6_060305/cpb_29_6_060305_ieqn1.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn2.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn3.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn4.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn5.gif)
It is found that if the parameters f0,1, ν1, and g satisfy certain specific parameter relations (see details in Appendix
![](cpb_29_6_060305/cpb_29_6_060305_ieqn6.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn7.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn8.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn9.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn10.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn11.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn12.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn13.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn14.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn15.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn16.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn17.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn18.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn19.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn20.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn21.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn22.gif)
Finally, we show that our exact analytical results are also applicable for the non-Hermitian situation. For example, the final transition probability
![](cpb_29_6_060305/cpb_29_6_060305_ieqn30.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn31.gif)
![](cpb_29_6_060305/cpb_29_6_060305_ieqn32.gif)
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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[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] |