From 00c09dda606369593a8ee50cc4390c473ab94e5a Mon Sep 17 00:00:00 2001 From: Wojciech Kozlowski Date: Wed, 29 Jun 2016 14:59:37 +0100 Subject: [PATCH] Finished section deriving Hamiltonian --- Chapter2/chapter2.tex | 138 ++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 133 insertions(+), 5 deletions(-) diff --git a/Chapter2/chapter2.tex b/Chapter2/chapter2.tex index 060f824..9c92c31 100644 --- a/Chapter2/chapter2.tex +++ b/Chapter2/chapter2.tex @@ -239,12 +239,13 @@ state we get \Psi(\b{r}) = \sum_i^M b_i w(\b{r} - \b{r}_i), \end{equation} where $b_i$ ($\bd_i$) is the annihilation (creation) operator of an -atom at site $i$ with coordinate $\b{r}_i$ and $M$ is the number of -lattice sites. Substituting this expression in Eq. \eqref{eq:FullH} -with $\H_{1,a} = \H_{1,a}^\mathrm{eff}$ and +atom at site $i$ with coordinate $\b{r}_i$, $w(\b{r})$ is the lowest +band Wannier function, and $M$ is the number of lattice +sites. Substituting this expression in Eq. \eqref{eq:FullH} with +$\H_{1,a} = \H_{1,a}^\mathrm{eff}$ and $\H_{1,fa} = \H_{1,fa}^\mathrm{eff}$ given by Eq. \eqref{eq:aeff} and \eqref{eq:faeff} respectively yields the following generalised -Bose-Hubbard Hamiltonian, $\H = \H_f + \H_a + \H_{fa}$, +Bose-Hubbard Hamiltonian, $\H = \H_f + \H_a + \H_{fa}$, \begin{equation} \H = \H_f + \sum_{i,j}^M J^\mathrm{cl}_{i,j} \bd_i b_j + \sum_{i,j,k,l}^M \frac{U_{ijkl}}{2} \bd_i \bd_j b_k b_l + @@ -266,8 +267,135 @@ hopping rate given by \left( -\frac{\b{p}^2}{2 m_a} + V_\mathrm{cl}(\b{r}) \right) w(\b{r} - \b{r}_i), \end{equation} -and $U_{ijkl}$ is the interaction strength given by +and $U_{ijkl}$ is the atomic interaction term given by \begin{equation} U_{ijkl} = \frac{4 \pi a_s \hbar^2}{m_a} \int \mathrm{d}^3 \b{r} w(\b{r} - \b{r}_i) w(\b{r} - \b{r}_j) w(\b{r} - \b{r}_k) w(\b{r} - \b{r}_l). \end{equation} +Both integrals above depend on the overlap between Wannier functions +corresponding to different lattice sites. This overlap decreases +rapidly as the distance between sites increases. Therefore, in both +cases we can simply neglect all terms but the one that corresponds to +the most significant overlap. Thus, for $J_{i,j}^\mathrm{cl}$ we will +only consider $i$ and $j$ that correspond to nearest neighbours. +Furthermore, since we will only be looking at lattices that have the +same separtion between all its nearest neighbours (e.g. cubic or +square lattice) we can define $J_{i,j}^\mathrm{cl} = - J^\mathrm{cl}$ +(negative sign, because this way $J^\mathrm{cl} > 0$). For the +inter-atomic interactions this simplifies to simply considering +on-site collisions where $i=j=k=l$ and we define $U_{iiii} = +U$. Finally, we end up with the canonical form for the Bose-Hubbard +Hamiltonian +\begin{equation} + \H_a = -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j + + \frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1), +\end{equation} +where $\langle i,j \rangle$ denotes a summation over nearest +neighbours and $\hat{n}_i = \bd_i b_i$ is the atom number operator at +site $i$. + +Finally, we have the light-matter interaction part of the Hamiltonian +given by +\begin{equation} + \H_{fa} = \frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m + \left( \sum_{i,j}^K J^{l,m}_{i,j} \bd_i b_j \right), +\end{equation} +where the coupling between the matter and optical fields is determined +by the coefficients +\begin{equation} + J^{l,m}_{i,j} = \int \mathrm{d}^3 \b{r} w(\b{r} - \b{r}_i) + u_l^*(\b{r}) u_m(\b{r}) w(\b{r} - \b{r}_j). +\end{equation} +This contribution can be separated into two parts, one which couples +directly to the on-site atomic density and one that couples to the +tunnelling operators. We will define the operator +\begin{equation} + \hat{F}_{l,m} = \hat{D}_{l,m} + \hat{B}_{l,m}, +\end{equation} +where $\hat{D}_{l,m}$ is the direct coupling to atomic density +\begin{equation} + \hat{D}_{l,m} = \sum_{i}^K J^{l,m}_{i,i} \hat{n}_i, +\end{equation} +and $\hat{B}_{l,m}$ couples to the matter-field via the +nearest-neighbour tunnelling operators +\begin{equation} + \hat{B}_{l,m} = \sum_{\langle i, j \rangle}^K J^{l,m}_{i,j} \bd_i b_j, +\end{equation} +and we neglect couplings beyond nearest neighbours for the same reason +as before when deriving the matter Hamiltonian. + +It is important to note that we are considering a situation where the +contribution of quantized light is much weaker than that of the +classical trapping potential. If that was not the case, it would be +necessary to determine thw Wannier functions in a self-consistent way +which takes into account the depth of the quantum poterntial generated +by the quantized light modes. This significantly complicates the +treatment, but can lead to interesting physics. Amongst other things, +the atomic tunnelling and interaction coefficients will now depend on +the quantum state of light. +\mynote{cite Santiago's papers and Maschler/Igor EPJD} + +Therefore, combining these final simplifications we finally arrive at +our quantum light-matter Hamiltonian +\begin{equation} + \H = \H_f -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j + + \frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1) + + \frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m \hat{F}_{l,m} - + i \sum_l \kappa_l \ad_l \a_l, +\end{equation} +where we have phenomologically included the cavity decay rates +$\kappa_l$ of the modes $a_l$. A crucial observation about the +structure of this Hamiltonian is that in the last term, the light +modes $a_l$ couple to the matter in a global way. Instead of +considering individual coupling to each site, the optical field +couples to the global state of the atoms within the illuminated region +via the operator $\hat{F}_{l,m}$. This will have important +implications for the system and is one of the leading factors +responsible for many-body behaviour beyong the Bose-Hubbard +Hamiltonian paradigm. + +\subsection{Scattered light behaviour} + +Having derived the full quantum light-matter Hamiltonian we will no +look at the behaviour of the scattered light. We begin by looking at +the equations of motion in the Heisenberg picture +\begin{equation} + \dot{\a_l} = \frac{i}{\hbar} [\hat{H}, \a_l] = + -i \left(\omega_l + U_{l,l}\hat{F}_{1,1} \right) \a_l + - i \sum_{m \ne l} U_{l,m} \hat{F}_{l,m} \a_m + - \kappa_l \a_l + \eta_l, +\end{equation} +where we have defined $U_{l,m} = g_l g_m / \Delta_a$. By considering +the steady state solution in the frame rotating with the probe +frequency $\omega_p$ we can show that the stationary solution is given +by +\begin{equation} + \a_l = \frac{\sum_{m \ne l} U_{l,m} \a_m \hat{F}_{l,m} + i \eta_l} + {(\Delta_{lp} - U_{l,l} \hat{F}_{l,l}) + i \kappa_l}, +\end{equation} +where $\Delta_{lp} = \omega_p - \omega_l$ is the detuning between the +probe beam and the light mode $a_l$. This equation gives us a direct +relationship between the light operators and the atomic operators. In +the stationary limit, the quantum state of the light field +adiabatically follows the quantum state of matter. + +The above equation is quite general as it includes an arbitrary number +of light modes which can be pumped directly into the cavity or +produced via scattering from other modes. To simplify the equation +slightly we will neglect the cavity resonancy shift, +$U_{l,l} \hat{F}_{l,l}$ which is possible provided the cavity decay +rate and/or probe detuning are large enough. We will also only +consider probing with an external coherent beam and thus we neglect +any cavity pumping $\eta_l$. This leads to a linear relationship +between the light mode and the atomic operator $\hat{F}_{l,m}$ +\begin{equation} + \a_l = \frac{\sum_{m \ne l} U_{l,m} \a_m} + {\Delta_{lp} + i \kappa_l} \hat{F}_{l,m} = C \hat{F}_{l,m}, +\end{equation} +where we have defined +$C = \sum_{m \ne l} U_{l,m} \a_m / (\Delta_{lp} + i \kappa_l)$ which +is essentially the Rayleigh scattering coefficient into the +cavity. Whilst the light amplitude itself is only linear in atomic +operators, we can easily have access to higher moments by simply +simply considering higher moments of the $\a_l$ such as the photon +number $\ad_l \a_l$.