Quantum Optics III Two-level system
fengxiaot Lv4

In a calculation on a digital computer, information is processed in the form of so-called binary digits or bits, representing its smallest unit. Each bit can have the value 0 or 1, corresponding to two discrete states. Moving the processing of information itself into the quantum mechanical domain, one can identify a two-level system now called quantum bit, or qubit that is based on two distinguishable quantum states. Therefore, it is necessary to look on two-level system specifically.

Time-independent interaction in two-level system

Static aspect

The Hamiltonian is given by

H=H0+WH = H_0 + W

Taking the eigenstates {φ1,φ2}\{\ket{\varphi_1},\ket{\varphi_2}\} of H0H_0 with eigenenergy E1,E2E_1,E_2 as the basis of state space. Suppose WW possesses purely non-diagonal matrix element, which means that WW only describes the coupling between two levels, W11=W22=0W_{11}=W_{22}=0. The matrix representing HH is written

H=[E1W12W21E2]H = \begin{bmatrix} E_1 & W_{12} \\ W_{21} & E_2 \end{bmatrix}

in which W12=W21W_{12} = W_{21}^*. The diagonalization of matrix HH presents no problems. We find the eigenvalues

E+=12(E1+E2)+12(E1E2)2+4W122E_+ = \frac{1}{2} (E_1 + E_2) + \frac{1}{2} \sqrt{(E_1 - E_2 )^2 + 4 |W_{12}|^2}

E=12(E1+E2)12(E1E2)2+4W122E_- = \frac{1}{2} (E_1 + E_2) - \frac{1}{2} \sqrt{(E_1 - E_2 )^2 + 4 |W_{12}|^2}

Introduce two parameters

Em=12(E1+E2)Δ=12(E1E2)E_m = \frac{1}{2} (E_1 + E_2) \qquad \Delta = \frac{1}{2}(E_1 - E_2)

The eigenvalues now are written

E+=Em+Δ2+W122E=EmΔ2+W122E_+ = E_m + \sqrt{\Delta^2 + |W_{12}|^2} \qquad E_- = E_m -\sqrt{\Delta^2 + |W_{12}|^2}

with eigenvectors (not normalized)

+=ΔΔ2+W122W12φ0+φ1=Δ+Δ2+W122W12φ0+φ1\begin{gathered} \ket{\mathrm{+}} = \frac{\Delta-\sqrt{\Delta^2 + |W_{12}|^2}}{W_{12}} \ket{\varphi_0} + \ket{\varphi_1} \\ \ket{\mathrm{-}} = \frac{\Delta+\sqrt{\Delta^2 + |W_{12}|^2}}{W_{12}} \ket{\varphi_0} + \ket{\varphi_1} \end{gathered}

Dynamic aspect

Assume that the system at time t=0t=0 in the state ψ(0)=φ1\ket{\psi(0)} = \ket{\varphi_1} . The probability amplitude of finding the system at time tt in the state φ2\ket{\varphi_2} is then written

P12(t)=φ2ψ(t)=W122W122+Δ2sin2(W122+Δ2t)=W122W122+Δ2sin2(E+E2t)\mathcal{P}_{12}(t) = \braket{\varphi_2|\psi(t)} =\frac{|W_{12}|^2}{|W_{12}|^2+\Delta^2} \sin^2 \left(\sqrt{|W_{12}|^2+\Delta^2} \frac{t}{\hbar}\right) = \frac{|W_{12}|^2}{|W_{12}|^2+\Delta^2} \sin^2 \left(\frac{E_+ - E_-}{2 \hbar} t\right)

This relation shows that the probability P12(t)\mathcal{P}_{12}(t) oscillates over time with a frequency of Ωrabi=W122+Δ2/=(E+E)/2\Omega_\text{rabi} = \sqrt{|W_{12}|^2+\Delta^2} / \hbar = (E_+ - E_-) / 2 \hbar and an amplitude of W122W122+Δ2\frac{|W_{12}|^2}{|W_{12}|^2+\Delta^2} . The generalized Rabi frequency Ωrabi\Omega_\text{rabi} is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field.

/* Note: If E1=E2E_1 = E_2, which means Δ=0\Delta = 0, the amplitute of oscilation will be 11. There exists some certain times when all particles transmit from φ1\ket{\varphi_1} to φ2\ket{\varphi_2}. */

Rabi Oscillation

In this section we shall calculate exactly the probability of a transition between two atomic states g\ket{g} and e\ket{e} driven by a quasi-resonant wave of angular frequency ω\omega similar to the Bohr frequency of the transition ω0=(EeEg)/\omega_0 = (E_e - E_g) / \hbar .

Schrodinger Equation

We shall set to zero the energy EgE_g of the lower state and denote by ω0\omega_0 the atomic Bohr frequency

H0=[000ω0]H_0 = \begin{bmatrix} 0 & 0 \\ 0 & \hbar \omega_0 \end{bmatrix}

We employ the electric dipole Hamiltonian of WDE=DEW_{DE} = - \boldsymbol{D} \cdot \boldsymbol{E} , where E=E0cos(ωt+φ)\boldsymbol{E} = \boldsymbol{E_0} \cos (\omega t + \varphi). We denote

Weg=eDE0g=Ω1W_{eg} = -\braket{e|\boldsymbol{D} \cdot \boldsymbol{E}_0 | g} = \hbar \Omega_1

in which Ω1\Omega_1 is called Rabi frequency. The complete Hamiltonian reads

H=[0Ω1cos(ωt+φ)Ω1cos(ωt+φ)ω0]H = \begin{bmatrix} 0 & \hbar \Omega_1 \cos (\omega t + \varphi) \\ \hbar \Omega_1 \cos (\omega t + \varphi) & \hbar \omega_0 \end{bmatrix}

Suppose ψ(t)=cg(t)g+ce(t)eiω0te\ket{\psi(t)} = c_g(t) \ket{g} + c_e(t) \mathrm{e}^{-\mathrm{i}\omega_0 t} \ket{e} . Schrodinger equation reads

iddtψ(t)=Hψ(t)\mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d} t} \ket{\psi(t)} = H \ket{\psi(t)}

idcgdt=[Ω1eiφ2ei(ωω0)t+Ω1eiφ2ei(ω+ω0)t]ce\mathrm{i}\frac{\mathrm{d} c_g}{\mathrm{d} t} = \left[ \frac{\Omega_1 \mathrm{e}^{\mathrm{i}\varphi}}{2} \mathrm{e}^{\mathrm{i}(\omega-\omega_0)t} + \frac{\Omega_1 \mathrm{e}^{-\mathrm{i}\varphi}}{2} \mathrm{e}^{-\mathrm{i}(\omega+\omega_0)t} \right] c_e

idcedt=[Ω1eiφ2ei(ωω0)t+Ω1eiφ2ei(ω+ω0)t]cg\mathrm{i}\frac{\mathrm{d} c_e}{\mathrm{d} t} = \left[ \frac{\Omega_1 \mathrm{e}^{-\mathrm{i}\varphi}}{2} \mathrm{e}^{-\mathrm{i}(\omega-\omega_0)t} + \frac{\Omega_1 \mathrm{e}^{\mathrm{i}\varphi}}{2} \mathrm{e}^{\mathrm{i}(\omega+\omega_0)t} \right] c_g

in which we decompose cos(ωt+φ)\cos (\omega t + \varphi) into 12[ei(ωt+φ)+ei(ωt+φ)]\frac{1}{2} [\mathrm{e}^{\mathrm{i}(\omega t + \varphi)} + \mathrm{e}^{-\mathrm{i}(\omega t + \varphi)}] . Applying rotating wave approximation, ω+ω0|\omega + \omega_0| terms are negligible, we obtain

idcgdt=Ω1eiφ2ei(ωω0)tce\mathrm{i}\frac{\mathrm{d} c_g}{\mathrm{d} t} = \frac{\Omega_1 \mathrm{e}^{\mathrm{i}\varphi}}{2} \mathrm{e}^{\mathrm{i}(\omega-\omega_0)t}c_e

idcedt=Ω1eiφ2ei(ωω0)tcg\mathrm{i}\frac{\mathrm{d} c_e}{\mathrm{d} t} = \frac{\Omega_1 \mathrm{e}^{-\mathrm{i}\varphi}}{2} \mathrm{e}^{-\mathrm{i}(\omega-\omega_0)t} c_g

Corollary : Under rotating wave approximation, the Hamiltonian could be seen as

H=[0Ω12ei(ωt+φ)Ω12ei(ωt+φ)ω0]H = \hbar \begin{bmatrix} 0 & \frac{\Omega_1}{2} \mathrm{e}^{\mathrm{i} (\omega t + \varphi)} \\ \frac{\Omega_1}{2} \mathrm{e}^{-\mathrm{i} (\omega t + \varphi) } & \omega_0 \end{bmatrix}

Hence we conventionally write the Hamiltonian as

H=ω0ee+Ω12ei(ωt+φ)ge+Ω12ei(ωt+φ)eg=ω0ee+Ω12[σei(ωt+φ)+σei(ωt+φ)]H = \hbar \omega_0 |e\rangle\langle e| + \frac{\hbar \Omega_1}{2} \mathrm{e}^{\mathrm{i} (\omega t + \varphi)}|g\rangle\langle e| + \frac{\hbar \Omega_1}{2} \mathrm{e}^{-\mathrm{i} (\omega t + \varphi)}|e\rangle\langle g| = \hbar \omega_0 |e\rangle\langle e| + \frac{\hbar \Omega_1}{2} \left[\sigma \mathrm{e}^{\mathrm{i} (\omega t + \varphi)} + \sigma^\dagger\mathrm{e}^{-\mathrm{i} (\omega t + \varphi)} \right]

in which σ:=ge\sigma:= |g\rangle\langle e| is the atomic lowering operator.

Define the detuning frequency from resonance δ:=ωω0\delta := \omega - \omega_0 and perform a transformation

c~g=cgeiδt/2c~e=ceeiδt/2\tilde{c}_g = c_g \mathrm{e}^{-\mathrm{i} \delta t/2} \qquad \tilde{c}_e = c_e \mathrm{e}^{\mathrm{i} \delta t/2}

The Schrodinger equations will become first order linear ODE with constant coefficients

idc~gdt=δ2c~g+Ω1eiφ2c~e\mathrm{i}\frac{\mathrm{d} \tilde{c}_g}{\mathrm{d} t} = \frac{\delta}{2} \tilde{c}_g + \frac{\Omega_1 \mathrm{e}^{\mathrm{i}\varphi}}{2} \tilde{c}_e

idc~edt=Ω1eiφ2c~gδ2c~e\mathrm{i}\frac{\mathrm{d} \tilde{c}_e}{\mathrm{d} t} = \frac{\Omega_1 \mathrm{e}^{-\mathrm{i}\varphi}}{2} \tilde{c}_g-\frac{\delta}{2} \tilde{c}_e

whose solution is the superposition of the eigenvector of coefficient matrix, multiplied by the exponential of the corresponding eigenvalues of the matrix. /* Hint: idxdt=Mx    x=eiλ1tv1+eiλ2tv2\mathrm{i}\frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d} t} = M \boldsymbol{x} \implies \boldsymbol{x}=\mathrm{e}^{-\mathrm{i}\lambda_1 t} \boldsymbol{v}_1+\mathrm{e}^{-\mathrm{i}\lambda_2 t} \boldsymbol{v}_2 */ Therefore, it is easy to find the eigenvalues

λ±=±12Ωrabi=±12Ω12+δ2\lambda_\pm = \pm\frac{1}{2} \Omega_\text{rabi} = \pm \frac{1}{2} \sqrt{\Omega_1^2 + \delta^2}

in which Ωrabi:=Ω12+δ2\Omega_\text{rabi} := \sqrt{\Omega_1^2 + \delta^2} are defined as the generalized Rabi frequency. The eigenstates of them are called dressed states.

Assume that the system at time t=0t=0 in the state ψ(0)=g\ket{\psi(0)} = \ket{g} . We will find

Pge(t,t0)=Ω12Ω12+δ2sin2Ωrabi2(tt0)\mathcal{P}_{g \to e} (t,t_0) = \frac{\Omega_1^2}{\Omega_1^2 + \delta^2} \sin^2 \frac{\Omega_\text{rabi}}{2} (t-t_0)

Rotating Frame Transformation

Back to Rabi Oscillation. The Hamiltonian is

H=[0Ω12ei(ωt+φ)Ω12ei(ωt+φ)ω0]H = \hbar \begin{bmatrix} 0 & \frac{\Omega_1}{2} \mathrm{e}^{\mathrm{i} (\omega t + \varphi)} \\ \frac{\Omega_1}{2} \mathrm{e}^{-\mathrm{i} (\omega t + \varphi) } & \omega_0 \end{bmatrix}

Rotate it by

U=[100eiωt]U = \begin{bmatrix} 1 & 0 \\ 0 & \mathrm{e}^{\mathrm{i}\omega t} \end{bmatrix}

The transformed Hamiltonian H~(t)=UHU+iU˙U\tilde{H}(t) = UHU^\dagger+\mathrm{i}\hbar \dot{U} U^\dagger will be

H~=[0Ω12Ω12ω0ω]=[0Ω12Ω12δ]\tilde{H} = \hbar \begin{bmatrix} 0 & \frac{\Omega_1}{2} \\ \frac{\Omega_1}{2} & \omega_0 - \omega \end{bmatrix} = \hbar \begin{bmatrix} 0 & \frac{\Omega_1}{2} \\ \frac{\Omega_1}{2} & -\delta \end{bmatrix}

We can then immediately obtain the Rabi frequency Ωrabi=12Ω12+δ2\Omega_\text{rabi} = \frac{1}{2} \sqrt{\Omega_1^2 + \delta^2} using the result of time-independent two-level system.

Corollary The effect of the AC electric field of light on a two-level system whose energy interval is ω0\hbar \omega_0, is equivalent to the effect of the DC electric field on a two-energy system whose energy interval is δ=(ω0ω)- \hbar \delta = \hbar (\omega_0-\omega).

/* Note: UU can be written in non-matrix form U=eiωteeU = \mathrm{e}^{\mathrm{i}\omega t |e\rangle\langle e|} */

AC Stark shift

In spectroscopy, AC Stark effect, is a dynamical Stark effect corresponding to the case when an oscillating electric field (e.g. laser) is tuned in resonance (or close) to the transition frequency of a given spectral line, and resulting in a change of the shape of the absorption/emission spectra of that spectral line. The AC Stark effect was discovered in 1955 by American physicists Stanley Autler and Charles Townes.

The old energy levels are given by

Eg=δ/2,Ee=δ/2E_g = -\delta/2, \quad E_e = \delta/2

The new energy levels are given by

E+=12Ω12+δ2,E=12Ω12+δ2E_+ = \frac{1}{2} \sqrt{\Omega_1^2 + \delta^2}, \quad E_- = -\frac{1}{2} \sqrt{\Omega_1^2 + \delta^2}

For large detuning δΩ1\delta \gg \Omega_1, we have

E+=Ω124δ,E=δΩ124δE_+ = \frac{\Omega_1^2}{4\delta}, \quad E_- = -\delta - \frac{\Omega_1^2}{4\delta}

+=Ω12δg+e1+Ω124δ2,=2δΩ1g+e1+Ω124δ2\ket{\mathrm{+}} = \frac{-\dfrac{\Omega_1}{2\delta} \ket{g} + \ket{e}}{\sqrt{1+\dfrac{\Omega_1^2}{4\delta^2}}},\quad \ket{\mathrm{-}} = \frac{\dfrac{2\delta}{\Omega_1} \ket{g} + \ket{e}}{\sqrt{1+\dfrac{\Omega_1^2}{4\delta^2}}}

The effective Hamiltonian is given by

H=E++++EH = E_+ \ket{+}\bra{+} + E_- \ket{-}\bra{-}