Atomic Physics #2

By using a particle to a qubit, it has to be fixed in space. Then there is an arising question. How to apply mechanical force to trap atoms? Representatively, there are three methods:

• Optical scattering force
• Photon scattering force
• Magnetic gradient force

This time, we know about the first two forces.

 

3. Optical Dipole Force

It is a similar case problem of dispersion relation.

$$ \ddot{x} + \gamma \dot{x} + w_{0}^2 x = -{e \over m}E(t). $$

w/ $x$: electron position, $\gamma$: atom's spontaneous emission rate, $w_{0}$: atom's resonance frequency.

By using Fourier Transform, then:

$$ \tilde{x}(w) = -{e \over m}{1 \over (w_{0}^{2} - w^2) - i \gamma w} \tilde{E}(w). $$

On the other hand, the electric dipole moment$P$ is given as follows:

$$ P = -ex = \alpha E. $$

In this case, $\alpha$ is the atomic polarizability w/

$$ \alpha \equiv {e^{2} \over m}{1 \over (w_{0}^{2} - w^{2}) - i \gamma w} \in \mathbb{C}. $$

 

In particular, dipole potential energy is given as follows.

$$ u = -{1 \over 2} \left\langle \vec{p} \cdot \vec{E} \right\rangle. $$

${1 \over 2}$ is from the dipole is induced electric dipole.

 

We must know that $P$ and $E$ are complex in this case. To eliminate confusion, each complex is written as $\hat{A}$, and then, the original $A$ is considered as $Re(\hat{A})$. 

$ \hat{p} = \hat{\alpha} \hat{E} = (\alpha_{r} + i \alpha_{i})(E_{r} + i E_{i}) = (\alpha_{r} E_{r} - \alpha_{i} E_{i}) + i(\alpha_{i}E_{r} + \alpha_{r} E_{i}). $

$ \Rightarrow p = \alpha_{r} E_{r} - \alpha_{i} E_{i}. $

 

$ P \cdot E = Re(\hat{P}) \cdot Re(\hat{E}) = (\alpha_{r} E_{r} - \alpha_{i} E_{i}) E_{r} = \alpha_{r} E_{r}^{2} - \alpha_{i} E_{r}E_{i}. $

$ \Rightarrow \left\langle P \cdot E \right\rangle = \left\langle \alpha_{r} E_{r}^{2}  - \alpha_{i}E_{r}E_{i} \right\rangle = \alpha_{r} \left\langle E_{r}^{2} \right\rangle - \alpha_{i} \left\langle E_{r}E_{i} \right\rangle$

$ = \alpha_{r} E_{0}^{2} \left\langle cos^{2}(w_{E}t) \right\rangle - \alpha_{i} \left\langle  cos(w_{E}t)sin(w_{E}t) \right\rangle = {1 \over 2} \alpha_{r} E_{0}^{2} = {1 \over 2} Re(\alpha) E_{0}^{2}$ 

$ = {1 \over 2} Re(\alpha) \left({2I \over \epsilon_{0}c}\right) = {1 \over \epsilon_{0}c} Re(\alpha) I$ where $I = {1 \over 2} \epsilon_{0}c E_{0}^{2}.$

$$ \therefore u = - {1 \over 2 \epsilon_{0}c} Re(\alpha) I = -{1 \over 2 \epsilon_{0}c} {w_{0}^{2}-w^{2}\over (w_{0}^{2}- w^{2})^{2} + (\gamma w)^{2} } I.$$

 

For $\vec{F}_{dip} = \nabla u$, then

$$ \vec{F}_{dip} = {1 \over \epsilon_{0}c} {e^{2} \over m} {w_{0}^{2} - w^{2} \over (w_{0}^{2}-w^{2})^{2} + (\gamma w)^{2}} \nabla I(r) \underset{\gamma \rightarrow 0}{\underset{\delta \rightarrow 0}{\approx}} -{1 \over 4 \epsilon_{0}cw_{0}} {1 \over \delta} \nabla I $$

where $\delta = w - w_{0}$.

 

Let's shoot a monochromatic Gaussian beam, whose frequency is $w$, to the atoms, then

if $\delta < 0$, there exists stable equilibrium. It is called red detuned wavelength.

if $\delta >0$, there exists unstable equilibrium.It is called blue detuned wavelength.

It is called "optical dipole trap$($ODT$)$. 

By using this approach, there exists a cooling method, Sisyphus cooling.

 

4. Scattering Force

$W$$($power$)$ = $\left\langle \dot{x} \cdot F \right\rangle = \left\langle \dot{x} \cdot (qE) \right\rangle = \left\langle (q \dot{x}) \cdot E \right\rangle = \left\langle P \cdot E \right\rangle.$

The similar complex process of $W$, we obtained as follows.

$$ P = {w \over \epsilon_{0} c} Im(\alpha) I. $$

$ \mathit{\Gamma}_{sc}$$($scattering rate$)$ $ = {P \over \hbar w} = {1 \over \hbar \epsilon_{0}c} Im(\alpha) I. $

$ \vec{F}_{sc} = (\hbar \vec{k}) \mathit{\Gamma}_{sc}. $

$$\therefore \vec{F}_{sc} = {\vec{k} \over \epsilon_{0}c} {\gamma w \over (w_{0}^{2} - w^{2})^{2} + (\gamma w)^{2}} \underset{\gamma \rightarrow 0}{\underset{\delta \rightarrow 0}{\propto}} {1 \over \delta^{2}} I. $$

 

원자가 $E=\hbar w, \vec{p}=\hbar \vec{k}$의 photon을 받으면$($이때 $w < w_{0}, \delta \approx 0$ $)$, photon의 방출을 무작위적인 방향으로 이루어진다. 따라서 원자가 받는 힘은 photon이 오는 방향이다. 따라서 서로 반대되는 방향의 빛을 조사하면 stable point를 만들 수 있고, 이를 3차원으로 확장하면 원자를 scattering force를 이용해 공간 상에 가두어 둘 수 있다. $w<w_{0}$이기 때문에, 흡수한 에너지보다 방출한 에너지가 조금씩 크게 된다. 이것이 계속해서 반복되다 보면, 원자는 냉각되게 된다. 이를 Doppler cooling이라고 한다. 

 

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수업 내용을 바탕으로, 원자물리 책으로 독학하면서 공부한 것이다 보니 틀린 내용이 있을 수 있어 감안하고 읽으면 좋을 것 같다.