Interaction of a two-level atom with single-mode optical field beyond the rotating wave approximation

Ju Liu; Zhi-Yuan Li

doi:10.1364/OE.22.028671

1. Introduction

The atom-field interaction is a basic problem in physics. The model of the coupling of a single two-level atom with a single-mode optical field has always been the simplest model to reveal the essential features of the interaction [1–5], and this model is widely adopted in quantum optics [6–10] and condensed matter physics [11]. Under the well-known rotating wave approximation, it is possible to reduce this problem to a form that can be solved exactly, which is named the Jaynes-Cummings model (JCM) [12,13]. The standard semiclassical version of the JCM [14,15], in which the field is treated as a purely classical electric field but using the quantum-mechanical methodology to deal with the atom has proven to be very successful in studying optical resonant phenomena [16, 17]. In this approach, the probabilities of the atom being in the upper or lower state rely on time and the two-level atom undergoes optical Rabi oscillations at the Rabi frequency between two levels under the action of the driving electromagnetic field [18, 19].

The rotating wave approximation [20,21] is a basic and important approximation used in atom optics and magnetic resonance and has been well discussed in many books [18, 19] and articles [20–23]. In this approximation, the counter-rotating terms which oscillate rapidly are neglected. The approximation is valid only when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. So when the system is far away from the resonance, in other words, in the highly detuned system, the rotating wave approximation might be out of work and a more complete theory must be adopted to solve these systems. The aim of this work is to present an analytical solution without the rotating wave approximation that can also apply to highly detuned systems in addition to the usual weakly detuned systems.

Meanwhile, phase is an important property of optical fields and waves and the initial phase of optical field may influence the atom-field interaction to some extent. But in the rotating wave approximation, since the counter-rotating terms are neglected, while only keeping the co-rotating terms which oscillate slowly, the initial phase of optical field has nothing to do with the Rabi oscillations [24, 25] and Rabi splitting [26–28]. As a result, much important information of the atom-field interaction is lost under the rotating wave approximation.

In this paper, we present an analytical solution without the rotating wave approximation that can also apply to systems far away from resonance. We first verify the effectiveness of the analytical solution by checking it against the rigorous numerical solution. The result shows that they are consistent well with each other. Then we discuss the applicability of the analytical solution and find that it is a very general theory. Finally we explore the influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, a topic which cannot be explored in the framework of the rotating wave approximation.

2. The analytical solution without rotating wave approximation

Consider the interaction of a single two-level atom with a single-mode optical field of frequency $ω$ , as illustrated in Fig. 1. Let $| a 〉$ and $| b 〉$ represent the lower and upper level states of the atom, they are eigenstates of the unperturbed part of the Hamiltonian $H_{0}$ with the eigenvalues $E_{a} = ℏ ω_{a}$ and $E_{b} = ℏ ω_{b}$ , respectively, and $ω_{b a}$ is the transition frequency of the two levels. The incident optical field with initial phase $φ$ and amplitude $E_{0}$ to interact with the atom can be expressed as

Fig. 1 Schematic of the interaction of a single two-level atom with a single-mode optical field.

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E (t) = E_{0} \cos (ω t + φ) .

According to Ref [16], the wave function of a two-level atom can be written in the form

| ψ (t) 〉 = C_{a} (t) | a 〉 + C_{b} (t) | b 〉,

where

C_{a}

and

C_{b}

are the probability amplitudes of the two levels, respectively. The corresponding Schrödinger equation is

| \dot{ψ} (t) 〉 = - i H | ψ (t) 〉 / ℏ

with the Hamiltonian given by

H = H_{0} + H_{1},

where

H_{0}

and

H_{1}

represent the unperturbed and interaction Hamiltonian, respectively. We can write

H_{0}

and

H_{1}

as

H_{0} = ℏ ω_{a} | a 〉 〈 a | + ℏ ω_{b} | b 〉 〈 b |,

and

H_{1} = - (D_{a b} | a 〉 〈 b | + D_{b a} | b 〉 〈 a |) E (t),

respectively, and

D_{b a} = D_{a b}^{*}

is the dipole matrix element.

In order to solve for $C_{a}$ and $C_{b}$ , we first write $C_{a} = c_{a} \exp (- i ω_{a} t)$ and $C_{b} = c_{b} \exp (- i ω_{b} t)$ , where $c_{a}$ and $c_{b}$ are the slowly varying probability amplitude. In the dipole approximation, the equations of motion for $c_{a}$ and $c_{b}$ may be written as

\dot{c_{a}} (t) = i Ω_{a b} c_{b} (e^{- i (ω_{b a} - ω) t + i φ} + e^{- i (ω_{b a} + ω) t - i φ}) /2,

\dot{c_{b}} (t) = i Ω_{a b} c_{a} (e^{i (ω_{b a} - ω) t - i φ} + e^{i (ω_{b a} + ω) t + i φ}) /2,

where the Rabi frequency

Ω_{a b}

is defined as

Ω_{a b} = | D_{b a} | E_{0} / ℏ,

and

ω_{b a} = ω_{b} - ω_{a} = - ω_{a b} > 0

is the atomic transition frequency. To solve the equations, we divide them into two terms which are approximate co-rotating terms proportional to

exp[\pm i (ω_{b a} - ω)]

and counter-rotating terms proportional to

exp[\pm i (ω_{b a} + ω)]

, respectively. For the first part, it corresponds to the familiar method named rotating wave approximation (RWA), which is generally a very good approximation for resonant or weakly detuned situations. Then the equations turn into

\dot{c_{a}} (t) = i Ω_{a b} c_{b} e^{- i (ω_{b a} - ω) t + i φ} / 2,

\dot{c_{b}} (t) = i Ω_{a b} c_{a} e^{i (ω_{b a} - ω) t - i φ} /2 .

If we assume that the atom is initially in the state

| a 〉

, then

c_{a} (0) = 1

,

c_{b} (0) = 0

. Using the abbreviation

Ω_{1} = \sqrt{{(ω_{ba} - ω)}^{2} + Ω_{a b}^{2}},

the solution of amplitude of the upper level state can be written as

c_{b} (t)= i (Ω_{a b} / Ω_{1}) e^{i (ω_{b a} - ω) t / 2 - i φ} \sin (Ω_{1} t / 2) .

The probabilities of the atom being in states

| a 〉

and

| b 〉

at time

t

are then given by

{| c_{a} (t) |}^{2}

and

{| c_{b} (t) |}^{2}

, and the probability of finding the atom in the upper level state is

{| c_{b} (t) |}^{2} = c_{b} c_{b}^{*} =(Ω_{ab} / Ω_{1})^{2} \sin^{2} (Ω_{1} t / 2),

Equation (12-1) indicates the probability is a periodic function of time. Because

{| c_{a} (t) |}^{2} = 1 - {| c_{b} (t) |}^{2} =1 - {(Ω_{ab} / Ω_{1})}^{2} \sin^{2} (Ω_{1} t / 2),

it is seen that the system oscillates with the frequency of

Ω_{1}

between the two atomic levels.

Ω_{a b}

relies on the detuning

Δ = (ω_{b a} - ω)

, the amplitude of field

E_{0}

and the dipole matrix element

D_{a b}

_. In the special case when the atom is in resonance with the incident optical field

(Δ = ω_{b a} - ω = 0)

, we get

Ω_{1} = Ω_{a b}

, and Eqs. (12-1) and (12-2) transform into

{| c_{a} (t) |}^{2} = \cos^{2} (Ω_{1} t / 2),

{| c_{b} (t) |}^{2} = \sin^{2} (Ω_{1} t / 2) .

After

T = π / Ω_{ab}

, the probabilities of the atom being in state

| b 〉

is 1. That means that the initial status

{| c_{a} (T) |}^{2} = 1

and

{| c_{b} (T) |}^{2} = 0

has transformed into

{| c_{a} (T) |}^{2} = 0

and

{| c_{b} (T) |}^{2} = 1.

This is just the well-known Rabi oscillation as plotted in Fig. 2. From the discussion above, one can deduce the following messages: Firstly, the rotating wave approximation is satisfied just for near-resonance situation, because only in this case, the counter-rotating terms can be ignored. When the detuning is

| Δ | > > 0

, the approximation is no longer applicable. Secondly, the probabilities of the atom being in any state do not change with the initial phase of the optical field.

Fig. 2 The probability of the atom being in state $| b 〉$ oscillates with the frequency of $Ω_{1}$ .

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To solve the highly detuned situation and discuss the influence of initial phase of the optical field on the evolution of probabilities, the counter-rotating terms must be considered. The equations only having counter-rotating terms are written as

\dot{c_{a}} (t) = i Ω_{a b} c_{b} e^{- i (ω_{b a} + ω) t - i φ} / 2,

\dot{c_{b}} (t) = i Ω_{a b} c_{a} e^{i (ω_{b a} + ω) t + i φ} / 2,

The way to solve the equations is similar to the way above for the rotating wave approximation. The solution of the amplitude of the upper level state can be written as

c_{b} (t) = i (Ω_{ab} / Ω_{2} {)e}^{i (ω_{b a} + ω) t / 2 + i φ} \sin (Ω_{2} t / 2),

Then the probabilities of the atom being in states

| a 〉

and

| b 〉

at time

t

are

{| c_{b} (t) |}^{2} =(Ω_{ab} / Ω_{2})^{2} \sin^{2} (Ω_{2} t / 2),

{| c_{a} (t) |}^{2} = 1 - {| c_{b} (t) |}^{2} =1 - {(Ω_{ab} / Ω_{2})}^{2} \sin^{2} (Ω_{2} t / 2),

respectively, where

Ω_{2} = \sqrt{{(ω_{b a} + ω)}^{2} + Ω_{a b}^{2}} .

It shows the system oscillates with the frequency of

Ω_{2}

between the two atomic levels and the probabilities of the atom being in any states still have nothing to do with the initial phase of the optical field.

Up to now, we have solved two equations systems, approximate co-rotating terms and counter-rotating terms, respectively. Next we proceed to explore the solution for general light-atom interaction systems without the rotating wave approximation, such as a highly detuned system. Because the Eq. (7.1) and (7.2) are non-integrable, an approximate solution as considering both of the approximate co-rotating terms and the counter-rotating terms is necessary. Firstly, we regard $c_{b} (t) = i (Ω_{ab} / Ω_{1}) e^{i (ω_{b a} - ω) t / 2 - i φ} \sin (Ω_{1} t / 2)$ as $c_{b}_{1} (t)$ and $c_{b} (t) = i (Ω_{ab} / Ω_{2}) e^{i (ω_{b a} + ω) t / 2 + i φ} \sin (Ω_{2} t / 2)$ as $c_{b}_{2} (t)$ , then assume the solution for Eqs. (7-1) and (7-2) can be written as

c_{b} (t) {=c}_{b}_{1} (t) {+c}_{b}_{2} (t) = i (\frac{Ω_{ab}}{Ω_{1}}) e^{i (ω_{b a} - ω) t / 2 - i φ} \sin (\frac{Ω_{1} t}{2}) + i (\frac{Ω_{ab}}{Ω_{2}}) e^{i (ω_{b a} + ω) t / 2 + i φ} \sin (\frac{Ω_{2} t}{2}) .

To verify that the assumption is effective, we check the analytical solution against the rigorous numerical solution, which is obtained by directly solving the ordinary differential Eqs. (7-1) and (7-2) with the initial status $c_{a} (0) = 1$ and $c_{b} (0) = 0$ by using MATLAB software. Assume the frequency of the field ω is the dominant frequency of Ti:Sapphire pulse laser, corresponding to a wavelength of 800 nm, namely $ω = 2.33 \times 10^{15} Hz$ . To describe the parameters for simplicity, $ω$ is set as a non-dimensional value of 10. All other frequency and time parameters are normalized similarly as non-dimensional quantities. Therefore, if the Rabi frequency $Ω_{a b}$ is set as 1, from Eq. (8), we find that $Ω_{a b}$ stands for the laser intensity of about $I =1 \times 10^{13} {W/cm}^{2}$ equivalently. For a highly detuned system, such as $ω = 10,$ $Ω_{a b} = 1,$ and $ω_{b a} = 1,$ which correspond to the transition wavelength of the two levels as 8 μm, the rotating wave approximation is no longer applicable. Figure 3 illustrates the comparison between the rigorous numerical solution, the rotating wave approximation solution and the analytical solution of the probability of the atom being in the upper state $| b 〉$ with respect to time $t$ . The blue curve stands for the result of the rotating wave approximation, which clearly shows that the system oscillates with the frequency of $Ω_{1} = 9.06$ between the two atomic levels, that is, after time $T = π / Ω_{1} = 0.35$ , the probability reaches the maximum. But our numerical solution (the black curve) shows a different oscillation profile from the rotating wave approximation solution, either in the aspect of the oscillation frequency or the maximum probability. So for a highly detuned system, the influence of the counter-rotating terms is large enough that it cannot be ignored any more, and the analytical solution of the probabilities should be used. The red curve is the analytical solution according to Eq. (18) taking into account the counter-rotating terms. It is consistent well with the rigorous numerical solution. Therefore, the assumption made to derive the approximate analytical solution as Eq. (18) is reasonable.

Fig. 3 The comparison of the numerical solution (black curve), the rotating wave approximation (RWA) (blue curve) and the analytical solution (red curve) of the probability of the atom being in upper state $| b 〉$ at time $t$ .

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After verifying its effectiveness, we further discuss the applicability of the analytical solution. For different detuned systems, from $Δ = ω_{b a} - ω = - 10$ to $Δ = ω_{b a} - ω = 10$ , we change the Rabi frequency, which represents the field intensity, from $Ω_{a b} = 0.1$ to $Ω_{a b} = 1$ , and compare the analytical solution against the rigorous numerical solution. The results are plotted in Fig. 4. As assumed above, $ω = 10$ represents the wavelength of the optical field as 800 nm, then $ω_{b a} = 1$ means that the transition wavelength of the two levels is 8 μm in Figs. 4(a), 4(b) and 4(c), $ω_{b a} = 10$ means that the transition wavelength is 800 nm in Figs. 4(d), 4(e) and 4(f), and $ω_{b a} = 20$ means a transition wavelength of 400 nm in Figs. 4(g), 4(h) and 4(i). In Figs. 4(a), 4(d) and 4(g), $Ω_{a b} = 1$ means that the laser intensity is about $I = 1 \times 10^{13} {W/cm}^{2}$ , and $Ω_{a b} = 0.5$ means $I = 5 \times 10^{12} {W/cm}^{2}$ in Figs. 4(b), 4(e) and 4(h), while $Ω_{a b} = 0.1$ means $I = 1 \times 10^{12} {W/cm}^{2}$ in Figs. 4(c), 4(f) and 4(i). From Fig. 4, it can be seen clearly that the analytical solution agrees well with the rigorous numerical solution, and the model is appropriate for both highly detuned systems and resonant cases. In addition, the smaller the optical field intensity is, the better the model is. As $Ω_{a b} = 1$ stands for a laser intensity of $I =1 \times 10^{13} {W/cm}^{2}$ , it already corresponds to the regime of strong field. Therefore, the analytical solution has a broad range of applicability and is a very general theory.

Fig. 4 Comparison of the analytical solution (red curve) against the numerical solution (black curve) for different detuned systems with varying Rabi frequency. $ω_{b a} = 1$ in panels (a), (b) and (c), $ω_{b a} = 10$ in panels (d), (e) and (f), $ω_{b a} = 20$ in panels (g), (h) and (i), $Ω_{a b} = 1$ in panels (a), (d) and (g), $Ω_{a b} = 0.5$ in panels (b), (e) and (h), while $Ω_{a b} = 0.1$ in panels (c), (f) and (i).

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For the rotating wave approximation, the probability of finding the atom in the upper level state is ${| c_{b} (t) |}^{2} {=c}_{b} c_{b}^{*} = {(Ω_{ab} / Ω_{1})}^{2} \sin^{2} (Ω_{1} t / 2)$ , which does not change with the initial phase of optical field. But the numerical solution demonstrates that the initial phase indeed affects the probability evolution. For example, for a system with parameters $ω_{b a} = 1,$ $ω = 10,$ and $Ω_{a b} = 1,$ the numerical solution result plotted in Fig. 5 shows great difference between $φ = 0$ (black curve) and $φ = 0.5 π$ (red curve). This result cannot be explained by the rotating wave approximation, but can be illuminated well with the analytical solution according to Eq. (18).

Fig. 5 The influence of the initial phase $φ = 0$ (black curve) and $φ = 0.5 π$ (red curve) on the probability of finding the atom in the upper level state.

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In the analytical solution, the probability amplitude of the atom being in state $| b 〉$ at time $t$ is

\begin{array}{l} c_{b} (t) {=c}_{b}_{1} (t) {+c}_{b}_{2} (t) \\ = i (Ω_{ab} / Ω_{1}) e^{i (ω_{b a} - ω) t / 2 - i φ} \sin (Ω_{1} t / 2) + i (Ω_{ab} / Ω_{2}) e^{i (ω_{b a} + ω) t / 2 + i φ} \sin (Ω_{2} t / 2), \end{array}

so the probability is

\begin{array}{l} {| c_{b} (t) |}^{2} = {(\frac{Ω_{ab}}{Ω_{1}})}^{2} \sin^{2} (\frac{Ω_{1} t}{2}) + {(\frac{Ω_{ab}}{Ω_{2}})}^{2} \sin^{2} (\frac{Ω_{2} t}{2}) \\ +2 \frac{Ω_{ab}}{Ω_{1}} \frac{Ω_{ab}}{Ω_{2}} \sin (\frac{Ω_{1} t}{2}) \sin (\frac{Ω_{2} t}{2}) \cos (ω t +2 φ) . \end{array}

The introduction of the counter-rotating terms generates the cross terms written as

2 (Ω_{ab} / Ω_{1}) (Ω_{ab} / Ω_{2}) \sin (Ω_{1} t / 2) \sin (Ω_{2} t / 2) \cos (ω t +2 φ) .

The cross terms play a modulatory role in the term of cosine to the probability evolution. Since the initial phase has great influence on the probability of finding the atom in the two level states, it is easy to control the evolution of the two levels by changing these parameters. Therefore, this provides a way to manipulate the interaction of a single two-level atom with single-mode optical field.

The initial phase of optical field not only affects the probability but also influences the Rabi splitting. In the rotating wave approximation, the probability amplitude of the atom being in the upper state is

c_{b} (t) = i (Ω_{ab} / Ω_{1}) e^{i (ω_{b a} - ω) t / 2 - i φ} \sin (Ω_{1} t / 2),

and its Fourier transformation is

c_{b} (ω) = i \frac{Ω_{a b}}{2 Ω_{1}} e^{- i φ} {δ [ω - (\frac{ω_{ba} - ω}{2} + \frac{Ω_{1}}{2})] - δ [ω - (\frac{ω_{ba} - ω}{2} - \frac{Ω_{1}}{2})]} .

The Fourier transformation spectrum has two peaks located at

ω_{1} =(ω_{ba} - ω) / 2 + Ω_{1} / 2

and

ω_{2} =(ω_{ba} - ω) / 2 - Ω_{1} / 2

, and the initial phase does not influence the peak values, as illustrated in Fig. 6. For a system with parameters

ω_{b a} = 20,

ω = 10,

Ω_{a b} = 1,

two peaks are located at

ω_{1} = 10

and

ω_{2} = 0

, respectively, and the peak values are both

Ω_{a b} / 2 Ω_{1} = 0.05

. When we change the phase, the values do not change. But the numerical solution shows a different result. For the same system, the Fourier transformation spectrum in Fig. 7 has three peaks and the peak values change with the initial phase. When

φ = 0

, the Fourier transformation spectrum has three peaks located at

ω_{1} = 10,

ω_{2} = 0,

ω_{3} = 30.

In addition to the two peaks

ω_{1} = 10

and

ω_{2} = 0

that occur in the rotating wave approximation, there appears a higher-order harmonic at

ω_{3} = 30.

The peak values are 0.05, 0.067, and 0.017, respectively. When

φ = 0.5 π

, the peak values are 0.05, 0.03, and 0.017, respectively. Two peaks located at

ω_{1} = 10

and

ω_{3} = 30

remain unchanged, but one peak located at

ω_{2} = 0

changes a lot.

Fig. 6 Rabi splitting in the spectrum for the probability of finding the atom in the upper level state under the rotating wave approximation for a system with $ω_{b a} = 20,$ $ω = 10,$ $Ω_{a b} = 1.$ (a) $φ = 0$ , (b) $φ = 0.5 π$ .

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Fig. 7 Rabi splitting in the spectrum for the probability of finding the atom in the upper level state without the rotating wave approximation for a system with $ω_{b a} = 20,$ $ω = 10,$ $Ω_{a b} = 1.$ (a) $φ = 0$ , (b) $φ = 0.5 π$ .

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All of these interesting features can be explained by the analytical solution. In this theory, the amplitude of the probability is

\begin{array}{l} c_{b} (t) {=c}_{b}_{1} (t) {+c}_{b}_{2} (t) = i (Ω_{ab} / Ω_{1}) e^{i (ω_{b a} - ω) t / 2 - i φ} \sin (Ω_{1} t / 2) \\ + i (Ω_{ab} / Ω_{2}) e^{i (ω_{b a} + ω) t / 2 + i φ} \sin (Ω_{2} t / 2), \end{array}

and its Fourier transformation is

\begin{array}{l} c_{b} (ω) = i \frac{Ω_{a b}}{2 Ω_{1}} e^{- i φ} {δ [ω - (\frac{ω_{ba} - ω}{2} + \frac{Ω_{1}}{2})] - δ [ω - (\frac{ω_{ba} - ω}{2} - \frac{Ω_{1}}{2})]} \\ + i \frac{Ω_{a b}}{2 Ω_{2}} e^{i φ} {δ [ω - (\frac{ω_{ba} + ω}{2} + \frac{Ω_{2}}{2})] - δ [ω - (\frac{ω_{ba} + ω}{2} - \frac{Ω_{2}}{2})]}, \end{array}

which has four peaks located at

ω_{1} = \frac{ω_{ba} - ω}{2} + \frac{Ω_{1}}{2}, ω_{2} = \frac{ω_{ba} - ω}{2} - \frac{Ω_{1}}{2}, ω_{3} = \frac{ω_{ba} + ω}{2} - \frac{Ω_{2}}{2}, ω_{4} = \frac{ω_{ba} + ω}{2} + \frac{Ω_{2}}{2} .

As for highly detuned system like

ω_{b a} = 20

and

ω = 10,

Ω_{1} = \sqrt{{(ω_{ba} - ω)}^{2} + Ω_{a b}^{2}} \approx ω_{ba} - ω = 10

,

Ω_{2} = \sqrt{{(ω_{ba} + ω)}^{2} + Ω_{a b}^{2}} \approx ω_{ba} + ω = 30

, so the two peaks

ω_{2} = \frac{ω_{ba} - ω}{2} - \frac{Ω_{1}}{2} =0

and

ω_{3} = \frac{ω_{ba} + ω}{2} - \frac{Ω_{2}}{2} = 0

become degenerate. The corresponding peak values are

v (ω_{1}) = Ω_{a b} / 2 Ω_{1},

v (ω_{2}) =[{(Ω_{a b} / 2 Ω_{1})}^{2} + {(Ω_{a b} / 2 Ω_{2})}^{2} + 2 (Ω_{a b} / 2 Ω_{1}) (Ω_{a b} / 2 Ω_{2}) \cos (2 φ)]^{1 / 2}

, and

v (ω_{4}) = Ω_{a b} / 2 Ω_{2}

. Therefore, the analytical model explains why the Fourier transformation spectrum has three peaks located at

ω_{1} = 10,

ω_{2} = 0,

and

ω_{3} = 30

, and the corresponding peak values are

v (ω_{1} = 10) = 0.05,

v (ω_{2} = 0) = 0.067,

and

v (ω_{3} = 30) = 0.017,

respectively. When

φ = 0.5 π

,

v (ω_{1} = 10) = 0.05

and

v (ω_{3} = 30) = 0.017

still remain the same, but the value of

v (ω_{2} = 0)

becomes 0.03, changing a great deal. From the theory,

v (ω_{2} = 0) =[{(Ω_{a b} / 2 Ω_{1})}^{2} + {(Ω_{a b} / 2 Ω_{2})}^{2} + 2 (Ω_{a b} / 2 Ω_{1}) (Ω_{a b} / 2 Ω_{2}) \cos (2 φ)]^{1 / 2} = 0.03

, which agrees well with the result of the numerical solution. All these messages strongly support the effectiveness of the analytical model in handling general light-atom interaction systems.

In addition to the highly off-resonance system, the effect of the radiation field phase on the evolution of the atomic system is also significant for systems which are not so highly detuned. We have investigated the system with parameters as $ω_{b a} = 13,$ $ω = 10,$ $Ω_{a b} = 1$ and the result of atomic evolution is plotted in Fig. 8. Figure 8(a) shows the difference between the initial phase $φ = 0$ (black curve) and $φ = 0.5 π$ (red curve) on the probability of finding the atom in the upper level state. The influences on the Rabi splitting are plotted in Figs. 8(b) and 8(c), respectively. The value of peak located at $ω = 0$ changes when $φ = 0$ as shown in Fig. 8(b) and $φ = 0.5 π$ as plotted in Fig. 8(c). It is clear that the effect of the phase is evident on kinds of systems that we cannot ignore it when deal with the atom-field interaction, and our analytical model provides an effective tool.

Fig. 8 (a) The influence of the initial phase $φ = 0$ (black curve) and $φ = 0.5 π$ (red curve) on the probability of finding the atom in the upper level state. Rabi splitting in the spectrum for the probability of finding the atom in the upper level state without the rotating wave approximation for a system with $ω_{b a} = 13,$ $ω = 10,$ $Ω_{a b} = 1.$ (b) $φ = 0$ , (c) $φ = 0.5 π$ .

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3. Conclusion

In conclusion, we have presented an analytical model that can deal with the coupling of a single two-level atom with single-mode optical field without the rotating wave approximation. The model is not only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation, but also can apply to systems that are far away from the resonance, which is inapplicable under the usual rotating wave approximation. We have verified the effectiveness of the analytical solution by checking it against the rigorous numerical solution, where good agreement has been achieved in general situations. The model is appropriate for highly detuned systems as well as resonant cases, and is applicable to both strong field and weak field. We have discussed the influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, which cannot be explored by the rotating wave approximation, and found that the initial phase can greatly affect the oscillation period and the maximum probabilities of the atom being in the upper level. Besides, due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting and the peak value is affected by the initial phase. The analytical model has suggested new ways to regulate and control the quantum dynamics of a two-level atom by phase-locked optical field and it allows for exploring more essential features of the atom-field interaction.

Acknowledgments

This work was supported by the 973 Program of China at No. 2011CB922002.

References and links

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Interaction of a two-level atom with single-mode optical field beyond the rotating wave approximation

Abstract

1. Introduction

2. The analytical solution without rotating wave approximation

3. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (31)

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