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 $\backslash Omega\_0$ and $\backslash Omega\_1$, detuned from some excited state (of energy $\backslash omega$) by an amount $\backslash Delta$. 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 $a\_0\backslash rightarrow\; e^\{-it(\backslash omega-\backslash Delta)\}a\_0$ and $a\_1\backslash rightarrow\; e^\{-it(\backslash omega-\backslash Delta)\}a\_1$, 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 $\backslash Delta\backslash gg\backslash Omega\_i$, 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 $a\_e$, 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 $\backslash lambda/2$, where $\backslash lambda$ 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 $E\_R=\backslash hbar^2\; k^2/(2m)$ is the recoil energy, with
$k=2\backslash pi/\backslash lambda$, m is the mass of the atoms and $V\_0$ 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 $a\_s$ is the s-wave scattering length of the colliding atoms. The sum is taken over all lattice sites (i) and atomic states ($\backslash sigma\backslash in\backslash \{0,1\backslash \}$).
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 $J^\backslash sigma/U\_\{\backslash sigma\; \backslash sigma\text{'}\}$. 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{{J^0}^2+{J^1}^2}{2 U_{01}} - \frac{{J^0}^2}{ U_{00}}- \frac{{J^1}^2}{U_{11}} \,\,,\,\,\,
\lambda^{(2)}=- \frac{J^0 J^1}{U_{01}} .
The local field $\backslash vec\{B\}$ can be
arbitrarily tuned by applying appropriately detuned laser fields after
canceling single particle phase rotations of the form $B\_z\; \backslash sum\_i\backslash sigma^z\_i$ with
B_z = -\frac{2{J^0}^2}{U_{00}}+\frac{2{J^1}^2}{U_{11}}.
The effective couplings $\backslash lambda^\{(i)\}$ 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 $J^0$, we can obtain the
diagonal interaction $Z\_i\; Z\_\{i+1\}$ 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 $X\_i\; X\_\{i+1\}\; +Y\_i\; Y\_\{i+1\}$. 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 $\backslash pi$ 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 $U\_\{01\}$ is induced. So, if we wait for a time, t, a phase $U\_\{01\}t$ 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.
For more information, see quant-ph/0403152
== External links ==
*[http://www.optical-lattice.com/ Introduction to optical lattices]
*[http://en.wiki.org/wiki/Bose-Einstein_condensate Bose-Einstein condensate] in Wikipedia
*[http://en.wikipedia.org/wiki/Raman_spectroscopy Raman spectroscopy] in Wikipedia
Category:Introductory Tutorials
Category:Physical Realisations
Category:Handbook of Quantum Information

## Last modified:

Monday, October 26, 2015 - 17:56