Trapped-ion System Hamiltonian Derivation¶

Goal

Derive the general Hamiltonian of the trapped-ion system for:

\(N\) ions
- \(J\) specified levels each
\(M\) lasers
\(L\) phonon modes

Simple 2-level Ion System¶

We'll start by deriving the Hamiltonian for a simple, 2-level system separated by energy \(\hbar\omega_0\), where \(\omega_0\) is the transition frequency. When the ion is irradiated by a laser, its hydrogen-like Hamiltonian with electronic and motional degrees of freedom \(H_{\text{el}} + H_{\text{mot}}\), is perturbed by \(H_{ED}\), the dipole operator

\[ \begin{align} H &= H_{\text{el}} + H_{\text{mot}} + H_{ED} \\ &= \frac{\hbar\omega_0}{2}\left(|1\rangle\langle1| - |0\rangle\langle 0|\right) + \hbar\nu a^{\dagger}a + H_{ED} \end{align} \]

where \(\nu\) is the ion's center-of-mass motion mode frequency in the harmonic potential.

To first order

\[ H_{ED} = -\vec{d}\cdot\vec{E} = e \vec{r}_e \cdot E_0\hat{\epsilon}\cos({\vec{k}\cdot\vec{r} - \omega t + \phi}) \]

\(\hat{\epsilon}\) is the laser's polarization
\(\vec{r}\) and \(\vec{r}_e\) are the center of mass positions of the nucleus and electron, respectively
\(\omega\) and \(\phi\) are the laser's frequency and initial phase, respectively.

Because \(\vec{r}_e\) has odd parity, the diagonal entries in the dipole operator cancel, thus we can write \(\vec{d}\cdot\hat{\epsilon}\) in terms of the Pauli raising and lower operators

\[ \begin{align} \sigma_+ &= \frac{1}{2} (\sigma_x + i \sigma_y) = |1\rangle\langle0| = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix} \\ \sigma_- &= \frac{1}{2} (\sigma_x - i \sigma_y) = |0\rangle\langle1| = \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix} \end{align} \]

We'll follow the convention that the highest level in the number basis corresponds to the unit vector whose first entry is 1:

\[ |1\rangle = \begin{pmatrix} 1\\ 0 \end{pmatrix}, \quad |0\rangle = \begin{pmatrix} 0\\ 1 \end{pmatrix} \]

Making these substitutions, expanding the cosine, and taking the Rabi frequency to be \(\Omega = \frac{eE_0}{\hbar} \langle 1|\vec{r}_e \cdot \hat{\epsilon}_l | 0\rangle\), we arrive a

\[ H_{ED} = \frac{\hbar\Omega}{2}(\sigma_+ + \sigma_-)\left[e^{i(\vec{k}\cdot\vec{r}-\omega t +\phi)} + e^{-i(\vec{k}\cdot\vec{r}-\omega t +\phi)}\right] \]

Note

The computation of the matrix elements: \(\langle 1|\vec{r}_e \cdot \vec{\epsilon}_l | 0\rangle\), for dipole and quadropole transitions is described here.

Interaction Picture for Spin¶

In principle, this Hamiltonian fully describes the motional and internal dynamics of the 2-level system. However, for computational purposes and for gaining greater intuition about the system, we boost into the frame rotating at the transition frequency \(\omega_0\):

\[ \begin{align} H_0 &= H_{\mathrm{el}} = \frac{\hbar\omega_0}{2}\sigma_z, \quad U = \exp(-iH_0t/\hbar), \quad H_I = H_{ED} \\ H'_I &= U^{\dagger}H_I U \\ &= \frac{\hbar\Omega}{2} e^{i\frac{\omega_0t}{2}\sigma_z}(\sigma_+ + \sigma_-)e^{-i\frac{\omega_0t}{2}\sigma_z}[\dots] \\ &= \frac{\hbar\Omega}{2}(e^{i\omega_0t}\sigma_+ + e^{-i\omega_0t}\sigma_-)\left[e^{i(\vec{k}\cdot\vec{r}-\omega t +\phi)} + e^{-i(\vec{k}\cdot\vec{r}-\omega t +\phi)}\right] \end{align} \]

Note

The transform is equivalent to: \(\sigma_+ \rightarrow e^{i\omega_0 t}\sigma_+\) and \(\sigma_- \rightarrow e^{-i\omega_0 t}\sigma_-\).

Motional Coupling¶

Next, we include the effects of motional coupling by taking \(\vec{k}\cdot\vec{r} = \eta(a^{\dagger} + a )\), where \(a, a^{\dagger}\) are the harmonic oscillator ladder operators and \(\eta\) is the Lamb-Dicke parameter

\[ \begin{align} H'_I &= \frac{\hbar\Omega}{2}(e^{i\omega_0t}\sigma_+ + e^{-i\omega_0t}\sigma_-)\left\{e^{i[\eta(a^{\dagger}+a)-\omega t +\phi]} + e^{-i[\eta(a^{\dagger} + a)-\omega t +\phi}\right\}\\ &= (\dots) \left[e^{i\eta(a^{\dagger} + a)}e^{-i(\omega t - \phi)} + e^{-i\eta(a^{\dagger} + a)}e^{i(\omega t - \phi)} \right]\\ &= \frac{\hbar\Omega}{2} \biggl[e^{i\eta(a^{\dagger} + a)}e^{-i[(\omega-\omega_0)t - \phi]}\sigma_+ + e^{-i\eta(a^{\dagger} + a)}e^{i[(\omega+\omega_0)t - \phi]} \sigma_+ + e^{i\eta(a^{\dagger} + a)}e^{-i[(\omega+\omega_0)t - \phi]}\sigma_- + e^{-i\eta(a^{\dagger} + a)}e^{i[(\omega-\omega_0)t - \phi]} \sigma_- \biggr] \end{align} \]

Rotating Wave Approximation (RWA)¶

Under the rotating wave approximation (RWA), we neglect the fast oscillating terms, those with frequency \(\omega_0 + \omega\), and define the detuning from resonance \(\Delta = \omega - \omega_0\)

\[ H'_I = \frac{\hbar\Omega}{2} \biggl[e^{i\eta(a^{\dagger} + a)}e^{-i(\Delta t - \phi)}\sigma_+ + e^{-i\eta(a^{\dagger} + a)}e^{i(\Delta t - \phi)} \sigma_- \biggr] \]

Interaction Picture for Phonons¶

At this point we apply one more transformation: boosting into the frame rotating at the ion's motional frequency \(\nu\).

\[ \begin{align} H''_I &= \frac{\hbar\Omega}{2} e^{i\nu t a^{\dagger}a}\left[e^{i\eta(a^{\dagger} + a)}e^{-i(\Delta t - \phi)} + e^{-i\eta(a^{\dagger} + a)}e^{i(\Delta t - \phi)} \right]e^{-i\nu t a^{\dagger}a} \end{align} \]

Note

The transform is equivalent to \(e^{i\eta(a^{\dagger} + a)} \rightarrow D(\alpha) = \exp\left(\alpha \hat{a}^{\dagger} - \alpha^* \hat{a}\right)\) where \(\alpha = i\eta e^{i \nu t}\).

\[ \begin{align} H''_I &= \frac{\hbar\Omega}{2} \biggl[e^{-i(\Delta t - \phi)}\sigma_+ D(\alpha) + e^{i(\Delta t - \phi)}\sigma_- D(-\alpha)\biggr]\\ &= \frac{\hbar\Omega}{2} \biggl[e^{-i(\Delta t - \phi)}\sigma_+ D(\alpha)\biggr] + H.C. \end{align} \]

Additional Phonon Modes¶

We assumed motion in only one dimension, but we can easily generalize to \(L\) mode coupling

\[ \vec{k} \cdot \vec{r} = \sum_l \eta_l(\hat{a}_l^{\dagger} + \hat{a}_l) \]

Because \(a^{\dagger}_l, a_l\) commute with \(a^{\dagger}_k,a_k\) (\(l\neq k\)), and \(D(\alpha)\) commutes with all other terms in the Hamiltonian, we can simply write the motional term as the product of displacement operators:

Important

\[ H''_I = \frac{\hbar\Omega}{2}\left[e^{-i(\Delta t - \phi)}\sigma_+ \prod_l^L D(\alpha_l) \right] + H.C. \]

where the coherent state parameter picks up a mode index \(\alpha_l = i\eta_l e^{i \nu_l t}\). We know that for a single ion, the \(l\) indices correspond to motion along \(\hat{e}_x\), \(\hat{e}_y\) and \(\hat{e}_z\).

Generalization¶

Let's now fully generalize the Single-ion Hamiltonian for:

\(N\) ions
- \(J\) considered electronic states each
\(M\) lasers

Additional Lasers¶

Introduce \(M\) laser fields irradiating the ion. Moving forward, \(m\) will represent the laser index, which marks all laser-specific parameters (\(\phi_m\), \(\omega_m\), \(\vec{k}_m\), \(\hat{\epsilon}_m\), \(E_{0,m}\)). \(H_{ED}\) now becomes:

\[ H_{ED} = -\vec{d}\cdot\vec{E} = e \vec{r}_e \cdot \sum_m E_{0,m} \hat{\epsilon}_m \cos(\vec{k}_m\cdot\vec{r} - \omega_m t + \phi_m) \]

The Rabi frequency becomes \(\Omega_m = \frac{eE_{0,m}}{\hbar}\langle1|\vec{r}_e\cdot\hat{\epsilon_m}|0\rangle\) as it gains laser index dependence.

\[ H_{ED} = \sum_m\frac{\hbar\Omega_m}{2}(\sigma_+ + \sigma_-)\left[e^{i\left(\vec{k}_m\cdot\vec{r}-\omega_m t -\phi_m\right)} + e^{-i\left(\vec{k}_m\cdot\vec{r}-\omega_m t -\phi_m\right)}\right] \]

Performing the same steps as before, we obtain:

Important

\[ H''_I = \sum_m^M \frac{\hbar\Omega_m}{2}\left[e^{-i(\Delta_m t - \phi_m)}\sigma_+ \prod_l^L D(\alpha_{ml}) \right] + H.C. \]

where \(\Delta_m = \omega_m - \omega_0\) and \(\alpha_{ml} = i\eta_{ml} e^{i \nu_l t}\). The Lamb-Dicke parameter gains a laser index as it's a function of the laser's alignment with the motional mode's axis.

Additional Ions¶

We now consider \(N\) ions, each of which is still a 2-level system, irradiated by the superimposed classical field from \(M\)-lasers. Moving forward, \(n\) will be the ion index.

\[ H_{ED} = \sum_n-\vec{d}_n\cdot\vec{E}_m(\vec{r}_n, t) =\sum_n e \vec{r}_{e,n} \cdot \sum_m E_{0,m} \hat{\epsilon}_m \cos(\vec{k}_m\cdot\vec{r}_n - \omega_m t + \phi_m) \]

Taking \(\Omega_{nm} = \frac{eE_{0,m}}{\hbar}\langle1|\vec{r}_{e,n}\cdot\hat{\epsilon}_m|0\rangle\), we see the Rabi frequency, raising and lowering operators all pick up an ion index.

\[ H_{ED} = \sum_{n,m}\frac{\hbar\Omega_{nm}}{2}\left(\sigma_+^{(n)} + \sigma_-^{(n)}\right)\left[e^{i\left(\vec{k}_m\cdot\vec{r}_n-\omega_m t +\phi_m\right)} + e^{-i\left(\vec{k}_m\cdot\vec{r}_n-\omega_m t +\phi_m\right)}\right] \]

Performing the same steps as before, we obtain:

Important

\[ H''_I = \sum_n^N \sum_{m}^N \frac{\hbar\Omega_{nm}}{2}\left[e^{-i(\Delta_{nm} t - \phi_m)}\sigma_+^{(n)} \prod_l^L D(\alpha_{nml}) \right] + H.C. \]

where \(\Delta_{nm} = \omega_m - \omega_{0,n}\) and \(\alpha_{nml} = i\eta_{nml} e^{i \nu_l t}\). The Lamb-Dicke parameter depends on the ion's mass.

Additional Levels¶

Finally we consider ions where we include more than two levels in their Hilbert space. Immediately, we must modify \(H_{\mathrm{el}}\) to reflect the eigen-energies associated with our \(J_n\) internal states (in the \(n^{\mathrm{th}}\) ion)

\[ H_{\text{int}}^{(n)} = \hbar\sum_{j_n}^{J_n}\omega_{0 j_n}|j_n\rangle\langle j_n| \]

\(\hbar\omega_{0 j_n}\) is the energy separation between the ground and \(j_n^{\mathrm{th}}\) level. Thus, \(\hbar\omega_{00} = 0\). \(H_{ED}\) must also be modified to account for coupling between all level pairs; coupling is ultimately determined from their respective matrix elements.

At this point, it becomes clear that the electronic ladder operators we've been using are now ambiguous: between which level pairs does it act? Thus, we introduce some notation: let \(\sigma_{+}^{(njk)} = |k_n\rangle\langle j_n|\) be the raising operator for the \(|j_n\rangle \leftrightarrow |k_n\rangle\) transition in the \(n^{\mathrm{th}}\) ion, where we take \(k_n\) to be higher in energy than \(j_n\) (the corresponding lowering operator is simply the Hermitian conjugate). Thus, \(H_{ED}\) for any such level pairs in ion \(n\), addressed by laser \(m\), become:

\[ H_{ED}^{(nmjk)} = \frac{\hbar\Omega_{nmjk}}{2}\left(\sigma_+^{(njk)} + \sigma_-^{(njk)}\right)\left[e^{i(\vec{k}_m\cdot\vec{r}_n-\omega_m t -\phi_m)} + e^{-i(\vec{k}_m\cdot\vec{r}_n-\omega_m t -\phi_m)}\right] \]

and the full perturbative term is now:

\[ H_{ED} = \sum_{n}\sum_{m}\sum_{j_n\neq k_n}H_{ED}^{(nmjk)} \]

for unique \(j_n, k_n\) pairs.

Performing the same steps as before, we obtain:

Important

\[ H''_I = \sum_n^N \sum_{m}^N \sum_{j_n \neq k_n}^{J_n} \frac{\hbar\Omega_{nmjk}}{2}\left[e^{-i(\Delta_{nmjk} t - \phi_m)}\sigma_+^{(njk)} \prod_l^L D(\alpha_{nml}) \right] + H.C. \]

where \(\Delta_{nmjk} = \omega_m - \omega_{njk}\) and \(\omega_{njk}\) is the transition frequency for \(|j_n\rangle \leftrightarrow |k_n\rangle\) in the \(n^{\mathrm{th}}\) ion.