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# Two-Level Atom

# Rabi oscillation

\quad The two-level system is an elementary but basic tool in quantum optics. In the previous note, we have discussed the interaction of a two-level atom with a classical field. Here, let's review this topic again with more concise symbols and new understandings.
\quad Let ∣g⟩\ket{g} and ∣e⟩\ket{e} denote the ground and excited states of an atom, with respective energies Eg=ℏωgE_g=\hbar\omega_g and Ee=ℏωeE_e=\hbar\omega_e. The corresponding frequency of the atomic transition is ωeg=ωe−ωg\omega_{eg}=\omega_e-\omega_g. Placed in an electromagnetic field, the Hamiltonian of the atom is

H^=H^A+V^AF,(1.1)\hat{\mathcal{H}} = \hat{\mathcal{H}}_A+\hat{\mathcal{V}}_{AF}\ , \tag{1.1}

where H^A\hat{\mathcal{H}}_A represent the intrinsic Hamiltonian of the atom, while V^AF\hat{\mathcal{V}}_{AF} denotes the interaction Hamiltonian describing the coupling between the atom and the field. They are respectively given by

H^A=ℏωg∣g⟩⟨g∣+ℏωe∣e⟩⟨e∣,(1.2a)\hat{\mathcal{H}}_A = \hbar\omega_g\ket{g}\bra{g} +\hbar\omega_e\ket{e}\bra{e}\ , \tag{1.2a}

V^AF=−p^⋅E⃗(t),(1.2b)\hat{\mathcal{V}}_{AF} = -\hat{\mathbf{p}}\cdot\vec{E}(t)\ , \tag{1.2b}

where p^\hat{\mathbf{p}} is the electric dipole operator, and E⃗\vec{E}is the electric field given by

E⃗=E⃗cos⁡(ωt−φ)=E⃗2(ei(ωt−φ)+e−i(ωt−φ))(1.3)\vec{E} = \vec{\mathcal{E}}\cos(\omega t-\varphi) = \frac{\vec{\mathcal{E}}}{2}\left(e^{i(\omega t-\varphi)}+e^{-i(\omega t-\varphi)}\right) \tag{1.3}

with ω\omega being its frequency, φ\varphi the phase, and E\mathcal{E} the amplitude. Here we are assuming the polarization of E⃗(t)\vec{E}(t) is linear. If you need, it can be extended to circular simply. Additionally, we also can simply replace the interaction between p⃗\vec{p} and E⃗(t)\vec{E}(t) by the interaction between the magnetic moment μ⃗\vec{\mu} and a harmonically varying magnetic field B⃗(t)\vec{B}(t).
\quad Since ∣g⟩\ket{g} and ∣e⟩\ket{e} constitute a complete set of orthonormal basis vector for the two-level atom system, The state of the atom at any time can be expressed as

∣Ψ(t)⟩=cg(t)∣g⟩+ce(t)∣e⟩,(1.4)\ket{\Psi(t)} = c_g(t)\ket{g}+c_e(t)\ket{e}\ , \tag{1.4}

where cgc_g and cec_e are the time-dependent complex amplitude of the atomic states. The time-evolution of ∣Ψ(t)⟩\ket{\Psi(t)} is governed by the Schro ̈dinger equation, i.e.,

∂∂t∣Ψ(t)⟩=−iℏH^∣Ψ(t)⟩.(1.5)\frac{\partial}{\partial t}\ket{\Psi(t)} = -\frac{i}{\hbar}\hat{\mathcal{H}}\ket{\Psi(t)}\ . \tag{1.5}

Substituting Eq. (1.1)~Eq. (1.4) into Eq. (1.5), taking the inner product with ∣g⟩\ket{g} and ∣e⟩\ket{e}, and using the orthonormality of ∣g⟩\ket{g} and ∣e⟩\ket{e}, we can obtain

∂∂tcg=−iωgcg+ice⟨g∣p^∣e⟩2ℏ⋅E⃗(ei(ωt−φ)+e−i(ωt−φ)),(1.6a)\frac{\partial}{\partial t}c_g = -i\omega_gc_g + ic_e\frac{\bra{g}\hat{\mathbf{p}}\ket{e}}{2\hbar}\cdot\vec{\mathcal{E}}\left(e^{i(\omega t-\varphi)}+e^{-i(\omega t-\varphi)}\right)\ , \tag{1.6a}

∂∂tce=−iωece+icg⟨e∣p^∣g⟩2ℏ⋅E⃗(ei(ωt−φ)+e−i(ωt−φ)).(1.6b)\frac{\partial}{\partial t}c_e = -i\omega_ec_e + ic_g\frac{\bra{e}\hat{\mathbf{p}}\ket{g}}{2\hbar}\cdot\vec{\mathcal{E}}\left(e^{i(\omega t-\varphi)}+e^{-i(\omega t-\varphi)}\right)\ . \tag{1.6b}

Here we have assumed that diagonal matrix elements of p^\hat{\mathbf{p}} are zero, ⟨g∣p^∣g⟩=⟨e∣p^∣e⟩=0\bra{g}\hat{\mathbf{p}}\ket{g}=\bra{e}\hat{\mathbf{p}}\ket{e}=0. As is customary, the off-diagonal elements can be chosen to be real, leading to the relation ⟨g∣p^∣e⟩=⟨e∣p^∣g⟩\bra{g}\hat{\mathbf{p}}\ket{e}=\bra{e}\hat{\mathbf{p}}\ket{g}.
\quad We define

ΩR=⟨g∣p^∣e⟩⋅E⃗ℏ(1.7)\Omega_R = \frac{\bra{g}\hat{\mathbf{p}}\ket{e}\cdot\vec{\mathcal{E}}}{\hbar} \tag{1.7}

as the Rabi frequency of the transition between two-levels under the field E⃗\vec{\mathcal{E}}. Using Eq. (1.7), we can write Eq. (1.6) as

∂∂tcg=−iωgcg+iceΩR2(ei(ωt−φ)+e−i(ωt−φ)),(1.8a)\frac{\partial}{\partial t}c_g = -i\omega_gc_g + ic_e\frac{\Omega_R}{2}\left(e^{i(\omega t-\varphi)}+e^{-i(\omega t-\varphi)}\right)\ , \tag{1.8a}

∂∂tce=−iωece+icgΩR2(ei(ωt−φ)+e−i(ωt−φ)).(1.8b)\frac{\partial}{\partial t}c_e = -i\omega_ec_e + ic_g\frac{\Omega_R}{2}\left(e^{i(\omega t-\varphi)}+e^{-i(\omega t-\varphi)}\right)\ . \tag{1.8b}

Making the transformations cμ(t)=c~μ(t)e−iωμt(μ=g,e)c_\mu(t)=\tilde{c}_\mu(t)e^{-i\omega_\mu t}\ (\mu=g,e), which is equivalent to adopting the interaction picture, as was done in the previous note. We obtain the differential equations for the slowly varying amplitudes cμ(t)c_\mu(t) as

∂∂tc~g=ic~eΩR2(ei(ω−ωeg)te−iφ+e−i(ω+ωeg)teiφ)≃ic~eΩR2eiΔte−iφ,(1.9a)\frac{\partial}{\partial t}\tilde{c}_g = i\tilde{c}_e\frac{\Omega_R}{2}\left(e^{i(\omega-\omega_{eg})t}e^{-i\varphi}+e^{-i(\omega+\omega_{eg})t}e^{i\varphi}\right) \simeq i\tilde{c}_e\frac{\Omega_R}{2} e^{i\Delta t}e^{-i\varphi}\ , \tag{1.9a}

∂∂tc~e=ic~gΩR2(e−i(ω−ωeg)teiφ+ei(ω+ωeg)te−iφ)≃ic~gΩR2e−iΔteiφ,(1.9b)\frac{\partial}{\partial t}\tilde{c}_e = i\tilde{c}_g\frac{\Omega_R}{2}\left(e^{-i(\omega-\omega_{eg})t}e^{i\varphi}+e^{i(\omega+\omega_{eg})t}e^{-i\varphi}\right) \simeq i\tilde{c}_g\frac{\Omega_R}{2} e^{-i\Delta t}e^{i\varphi}\ , \tag{1.9b}

where we have defined the detuning Δ≡ω−ωeg\Delta\equiv\omega-\omega_{eg} of the radiation frequency from the resonance frequency ωeg\omega_{eg} of the atomic transition. We shall focus on near-resonant transitions. In this regime, where Δ≪ωeg≈ω\Delta\ll\omega_{eg}\approx\omega, we can employ rotating wave approximation (RWA), which amounts to neglecting the rapidly oscillating terms proportional to e±i(ω+ωeg)e^{\pm i(\omega+\omega_{eg})}. Laplace transforms provide an effective method for solving the above differential equations, and revisiting this approach is quite insightful. With the initial conditions c~g(0)=1\tilde{c}_g(0)=1 and c~e(0)=0\tilde{c}_e(0)=0, we present the detailed solution and the result of Eq. (1.9) as

The detailed solution of Eq. (1.9) using Laplace transforms

\quad Let us denote

Lj(s)=L[c~j(t)]=∫0∞dte−stc~j(t)(1)L_j(s) = \mathcal{L}\left[\tilde{c}_j(t)\right] = \int^\infin_0dt\ e^{-st}\tilde{c}_j(t) \tag{1}

as the Laplace transform of c~j(t)\tilde{c}_j(t), where the symbol L\mathcal{L} represents the Laplace transform. Utilizing the properties of the Laplace transform as follow:

L[f(t)]=L(s)⟹L[f′(t)]=sL(s)−f(0),(2.1)\mathcal{L}\left[f(t)\right] = L(s) \Longrightarrow \mathcal{L}\left[f'(t)\right] = sL(s)-f(0)\ , \tag{2.1}

L[f(t)]=L(s)⟹L[es0tf(t)]=L(s−s0),(2.2)\mathcal{L}\left[f(t)\right] = L(s) \Longrightarrow \mathcal{L}\left[e^{s_0t}f(t)\right] = L(s-s_0)\ , \tag{2.2}

we can simply apply the Laplace transform to Eq. (1.9) and derive the following equation:

sLg(s)−1=iΩR2e−iφLe(s−iΔ),(3a)sL_g(s)-1 = i\frac{\Omega_R}{2}e^{-i\varphi}L_e(s-i\Delta)\ , \tag{3a}

sLe(s)=iΩR2eiφLg(s+iΔ),(3b)sL_e(s) = i\frac{\Omega_R}{2}e^{i\varphi}L_g(s+i\Delta)\ , \tag{3b}

with initial conditions c~g(0)=1\tilde{c}_g(0)=1 and c~e(0)=0\tilde{c}_e(0)=0.
\quad From Eq. (3.2) we have

(s−iΔ)Le(s−iΔ)=iΩR2eiφLg(s).(4)(s-i\Delta)L_e(s-i\Delta) = i\frac{\Omega_R}{2}e^{i\varphi}L_g(s)\ . \tag{4}

Utilizing the above equation and substituting it into Eq. (3a), we obtain

Lg(s)=s−iΔs2−iΔs+Ω2.(5)L_g(s) = \frac{s-i\Delta}{s^2-i\Delta s+\Omega^2}\ . \tag{5}

The roots of the denominator in Eq. (5) are

s±=i[Δ2±Ω2+(Δ2)2].(6)s_\pm = i\left[\frac{\Delta}{2}\pm\sqrt{\Omega^2+\left(\frac{\Delta}{2}\right)^2}\right]\ . \tag{6}

Thus we have

Lg(s)=s−iΔ(s−s+)(s−s−)=1s+−s−[s+−iΔs−s+−s−−iΔs−s−].(7)L_g(s) = \frac{s-i\Delta}{(s-s_+)(s-s_-)} = \frac{1}{s_+-s_-}\left[\frac{s_+-i\Delta}{s-s_+}-\frac{s_--i\Delta}{s-s_-}\right]\ . \tag{7}

In view of the form of the above equation, utilizing

L[ekt]=1s−k,(8)\mathcal{L}\left[e^{kt}\right]=\frac{1}{s-k}\ , \tag{8}

which is a common Laplace transform, we can simply apply the inverse Laplace transform to Lg(s)L_g(s), thereby obtaining c~g(t)\tilde{c}_g(t) as following

c~g(t)=s+−iΔs+−s−es+t+s−−iΔs+−s−es−t.(9)\tilde{c}_g(t) = \frac{s_+-i\Delta}{s_+-s_-}e^{s_+t} + \frac{s_--i\Delta}{s_+-s_-}e^{s_-t}\ . \tag{9}

Then, by substituting Eq. (7) into Eq. (9) and performing a few straightforward steps of trigonometric algebra, we obtain the expression for c~g(t)\tilde{c}_g(t). Following the same procedure, we also derive the expression for c~e(t)\tilde{c}_e(t).

c~g(t)=eiΔ2t[cos⁡(Ωˉt)−iΔ2Ωˉsin⁡(Ωˉt)],(1.10a)\tilde{c}_g(t) = e^{i\frac{\Delta}{2}t}\left[\cos(\bar{\Omega}t)-i\frac{\Delta}{2\bar{\Omega}}\sin(\bar{\Omega}t)\right]\ , \tag{1.10a}

c~e(t)=ie−iΔ2t−iφΩΩˉsin⁡(Ωˉt),(1.10b)\tilde{c}_e(t) = ie^{-i\frac{\Delta}{2}t-i\varphi}\frac{\Omega}{\bar{\Omega}}\sin(\bar{\Omega}t)\ , \tag{1.10b}

where Ωˉ≡Ω2+(Δ/2)2\bar{\Omega}\equiv\sqrt{\Omega^2+(\Delta/2)^2} is an effective Rabi frequency for non-zero detuning Δ\Delta, and which satisfy the normalization condition ∣c~g(t)∣2+∣c~e(t)∣2=1|\tilde{c}_g(t)|^2+|\tilde{c}_e(t)|^2=1 for all t≥0t\ge0.
\quad Clearly, when Δ=0\Delta=0, we have

c~e(t)=cos⁡(Ωt),c~e(t)=ie−iφsin⁡(Ωt)(1.11)\tilde{c}_e(t) = \cos(\Omega t),\quad\tilde{c}_e(t)=ie^{-i\varphi}\sin(\Omega t) \tag{1.11}

which explain why Ω\Omega is the Rabi frequency, as it represents the frequency of the oscillation of the two-level system between its two states under the driving by the external field. This oscillation is essentially caused by the combined effects of absorption transitions and stimulated emission of the atom under the external field.
\quad Fig 1.1 illustrates the population variation during the Rabi oscillation.

Fig 1.1

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