# Optical Lattices

An optical lattice is simply a set of standing wave lasers. The electric field of these lasers can interact with atoms - the atoms see a potential and therefore congregate in the potential minima. In the case of a typical one-dimensional setup, the wavelength of the opposing lasers is chosen so that the light shift is negative. This means that the potential minima occur at the intensity maxima of the standing wave. Furthermore, the natural beam width constrains the system to being one-dimensional.

For quantum computation, we would like to be able to initialise the system so that there is an atom (generally, we think about alkali atoms, such as Rubidium) in every lattice site. This is accomplished by making use of the Mott insulator phase transition. This is done by starting with a Bose-Einstein condensate (BEC) and turning on the standing wave laser. At some critical value of the intensity, the atoms change from being in the BEC phase to the Mott insulator phase, which means that there are integer numbers of atoms in each lattice site. Purification procedures and suitable choice of the density of the BEC enable us to limit this to a single atom per lattice site. Usefully, we can assume that the atoms do not interact provided the intensity of the lasers is large enough.

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One reason for choosing alkali atoms is their comparatively simple atomic structure - there is only a single outermost electron. In the case of Rubidium, this is a 5s electron. The nuclear spin is I=3/2, which gives an atomic structure as shown in the figure. We can therefore select two of these hyperfine levels to act as the basis of our qubits - these are both ground states and therefore stable.

### One-qubit Gates

For one--qubit gates, we don't need to worry about the optical lattice - that's just providing us with regular confinement of the atoms. It is typical to choose our basis states to be magnetically sensitive, such as |0>=|F=1,m_f=1> and |1>=|F=1,m_f=-1>, which means that the two states are sensitive to different (circular) polarisations of light. To convert between the |0> and |1> states, we use a two-photon Raman transition (i.e. two lasers, one of each polarisation) with Raman frequencies Ω0 and Ω1, detuned from some excited state (of energy ω) by an amount Δ. This excited state is often taken to be the 5p level, although could easily be the 6p level. In the {|0>,|1>,|e>} basis, we can write the Schrodinger equation as

i\frac{\partial |\psi>}{\partial t}=\left(\begin{array}{ccc} 0 & 0 & \Omega_0^*\cos\left((\omega-\Delta)t\right) \\ 0 & 0 & \Omega_1^*\cos\left((\omega-\Delta)t\right) \\ \Omega_0\cos\left((\omega-\Delta)t\right) & \Omega_1\cos\left((\omega-\Delta)t\right) & \omega \end{array}\right)|\psi>.

We can make progress in solving this by making the substitutions a0 → e − it(ω − Δ)a0 and a1 → e − it(ω − Δ)a1, and by neglecting the fast-rotating terms (the rotating wave approximation). This reduces the above equation to

i\frac{\partial |\psi>}{\partial t}=\left(\begin{array}{ccc} 0 & 0 & \Omega_0^*/2 \\ 0 & 0 & \Omega_1^*/2 \\ \Omega_0/2 & \Omega_1/2 & \Delta \end{array}\right)|\psi>.

By working in the regime where Δ ≫ Ωi, the adiabatic approximation is valid, which means that we can set

i\frac{\partial a_e}{\partial t}=a_0\Omega_0/2+ a_1\Omega_1/2+ a_e\Delta=0

and thus eliminate ae, giving a rotation in the computational basis.

i\frac{\partial |\psi>}{\partial t}=\frac{-1}{4\Delta}\left(\begin{array}{cc} |\Omega_0|^2 &\Omega_0^*\Omega_1 \\ \Omega_0\Omega_1^* & |\Omega_1|^2 \end{array}\right)|\psi>.

The problem in optical lattices is that each atom is only separated by λ/2, where λ is the wavelength of the trapping laser. This means that the two lasers of the Raman transition have to be focused hopelessly tightly to create a rotation on a single qubit. To overcome this a number of techniques are employed, such as increasing the separation between lattice sites, or using magnetic gradients. Global control is a further alternative.

### Two--Qubit Gates

#### Tunneling Coupling

We now need to consider how to create a two--qubit gate. There are two types of coupling that could be used to create this - tunneling or collisional couplings.

The evolution of the system, for atoms restricted in the lowest Bloch-band (i.e. the ground state of each individual potential well), is described by the Bose-Hubbard Hamiltonian that is comprised of tunneling transitions of atoms between neighbouring sites of the lattice, and collisional interactions between atoms in the same site,

H=-\sum_{i\sigma} J^\sigma_i (a_{i\sigma}^\dagger a_{i+1 \sigma}+ a_{i\sigma} a_{i+1 \sigma}^\dagger)+\frac{1}{2} \sum _{i \sigma \sigma'} U_{\sigma \sigma'} a^{\dagger}_{i\sigma}a^{\dagger}_{i\sigma'}a_{i\sigma'}a_{i\sigma}

with tunneling couplings that are given by

J =\frac{E_R}{2} \exp (- \frac{\pi^2}{4} \sqrt{ \frac{V_0}{E_R}})

 \left[\sqrt{\frac{V_0}{E_R}} + \left( \frac{V_0}{E_R}
 \right)^{3/2}\right]

where ER = ℏ2k2/(2m) is the recoil energy, with k = 2π/λ, m is the mass of the atoms and V0 is the potential barrier between two successive lattice sites, while the collisional couplings are given by

U=\frac{4 a_s}{\lambda } V_0^{3/4} E_R^{1/4}

where as is the s-wave scattering length of the colliding atoms. The sum is taken over all lattice sites (i) and atomic states (σ ∈ {0, 1}). The collisional couplings can be arranged to take, in principle, arbitrarily large values via Feshbach resonances. Tunneling couplings can be varied by changing the amplitude of the laser fields comprising the optical lattice.

It is possible to expand the resulting evolution, generated by the Bose-Hubbard Hamiltonian, in terms of the small parameters Jσ/Uσσʹ. In an interaction picture with respect to the collisional Hamiltonian, one obtains the effective evolution from the perturbation expansion up to the second order with respect to the tunneling interaction given, in terms of the Pauli matrices, by

   H = \sum_{i=1}^3 \Big[ \vec{B} \cdot \vec{\sigma}_i
   +\lambda^{(1)} Z_i Z_{i+1} + \lambda^{(2)}
   (X_i X_{i+1} +Y_i Y_{i+1})  \Big].

The couplings strengths are related to the collisional and tunneling strengths by

\lambda^{(1)} = \frac - \frac- \frac \,\,,\,\,\, \lambda^{(2)}=- \frac{J^0 J^1}{U_{01}} .

The local field B⃗ can be arbitrarily tuned by applying appropriately detuned laser fields after canceling single particle phase rotations of the form Bziσiz with

B_z = -\frac{2{J^0}^2}{U_{00}}+\frac{2{J^1}^2}{U_{11}}.

The effective couplings λ(i) can be tuned at will by manipulating the amplitudes of the lasers that generate the optical lattices. In particular, by activating only one of the two tunneling couplings, say J0, we can obtain the diagonal interaction ZiZi + 1 along all the qubits of the lattice (to the second order). This, up to local qubit rotations is equivalent to a series of control phase gates (CP). However, if we activate both of the tunneling couplings with appropriate magnitudes, it is possible to activate the exchange interaction XiXi + 1 + YiYi + 1. When applied for a sufficient time interval it results in a SWAP gate, exchanging the atoms at neighbouring lattice sites.

#### Collisional Coupling and State Selective Transport

If you rotate the phase of the trapping lattice, this moves the location of the potential minima and carries the trapped states along with it. If, instead, we only rotate the phase of one of the polarisations, that would only move one of the types of states e.g. |1>, while the |0> state remains where it is. After a π rotation, the |1> component of a given qubit will be in the same lattice site as the |0> component of the neighbouring qubit. This is referred to as a collision, and an energy shift U01 is induced. So, if we wait for a time, t, a phase U01t is created. This is an entangling operation, equivalent to a controlled-phase gate.

#### Qubit Measurement

To measure the qubit, we make use of a dissipative decay from an excited state. For example, we would promote the |1> state to the 5p level. It will decay back to the |1> state (not |0> due to selection rules if we've made sensible choices for our basis states), and so we can re-excite it and, in this way, get many photons emitted. This means that we don't need to make single photon measurements, which, to perform with high fidelity, are very difficult to achieve, even if the photons are being emitted in a specific direction, which they aren't in this situation.