Spin-Flip Blockade

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A spin-flip blockade is a phenomenon seen in the dressed basis of atoms experiencing a Rydberg blockade. The dressed basis consists of the dressed states, which are the eigenstates of a light shifted two level Hamiltonian . The dressed states experience a single atom AC Stark effect (also known as the light shift) from the bare states and the $|\widetilde{11}⟩$ dressed state gets an additional shift from the Rydberg interaction if the atoms are Rydberg blockaded. This allows for flopping from the $|00⟩$ state to the $\frac{|0\tilde{1}⟩+|\tilde{1}0⟩}{\sqrt{2}}$. This is relevant to research in Neutral atom quantum computing and has been used to prepare Bell states with a fidelity $>81(2)\mathrm{\%}$..

Physical Description

The spin-flip blockade is a mapping of the Rydberg blockade from a ground state $|1⟩$ and a Rydberg state $|r⟩$ to a different ground state $|0⟩$ and a dressed ground state $|\tilde{1}⟩$.

Rydberg Dipole-Dipole Interaction

Rydberg atoms are atoms with a Hydrogen-like structure that are excited to a high principle quantum number $n$. As $n$ increases the Van der Waals interaction between Rydberg atoms also increases. The Van der Waals interaction energy between two Rydberg atoms is $V_{rr}=\frac{C_6}{R^6}$ where $C_6$ is determined by the structure of the Rydberg states chosen.

Classic Rydberg Blockade

In a single atom that we assume to have a Hydrogen-like energy structure, we can choose two energy levels with some energy splitting and drive the transition between them with a laser according to Rabi flopping. We can refer to the lower energy state as the ground state $|g⟩$ and the higher energy state as the Rydberg state $|r⟩$, where the Rydberg state is at some high principle quantum number. By driving the system with a laser that has a small detuning ${\mathrm{\Delta }}_L$, the system will Rabi oscillate between $|g⟩$ and a state with some non-negligible component of $|r⟩$.

These diagrams are commonly shown to display how far "detuned" a laser is from resonance. The level diagram on the left shows a far (red) detuned laser, which will never pump much of the ground state into the Rydberg state. The level diagram on the right shows a closer to resonant system that will oscillate with more population in the Rydberg state. If the detuning is large enough the amount of population pumped into the Rydberg state is negligible.

If there are two atoms being driven by the same laser and there is no interaction between the atoms the dynamics look like two separate block spheres and there cannot be any entanglement. However when the two atoms in $|r⟩$ have a strong enough interaction potential energy $V_{rr}$ the state where both atoms are in the Rydberg state $|rr⟩$ has an energy shift. If this energy shift is sufficiently large, the $|rr⟩$ state will now be far enough detuned that it will not participate in the Rabi oscillation. The condition for blockade is ${\mathrm{\Omega }}_L\ll V_{rr}$. If two atoms that are close enough to experience a Rydberg blockade are symmetrically driven by a laser that would result in Rabi flopping in single atoms from $|g⟩$ to $|r⟩$ with Rabi frequency ${\mathrm{\Omega }}_L$, then $|gg⟩$ will Rabi flop to the bright state $\frac{|gr⟩+|rg⟩}{\sqrt{2}}=|B⟩$ with Rabi frequency $\sqrt{2}{\mathrm{\Omega }}_L$. We do not move population into the $|rr⟩$ state if we are in a perfect blockade. This mechamism has been proposed to allow for neutral atom entanglement for years.

Rydberg atoms have a space dependent Van der Waals potential. When the Rydberg atoms are close enough that the interaction potential is large compared to the Rabi frequency ${\mathrm{\Omega }}_L$ the $|rr⟩$ energy level is shifted such that it is too far detuned to see meaningful population transferred from $|B⟩$ into $|rr⟩$. This reduces the $|11⟩$ and $|B⟩$ state evolutions into two level dynamics, Rabi flopping at $\sqrt{2}{\mathrm{\Omega }}_L$.

Dressed States

If the laser detuning ${\mathrm{\Delta }}_L$ starts off very detuned and slowly brought closer to on resonant, the adiabatic theorem tells us that the state will always stay in instantaneous eigenstates of the Hamiltonian. These states $|\tilde{g}⟩$ and $|\tilde{r}⟩$ (known as "dressed states") will be a superposition state of $|g⟩$ and $|r⟩$.

$|\tilde{g}⟩=\cos \frac{\theta }{2}|g⟩+\sin \frac{\theta }{2}|r⟩$

$|\tilde{r}⟩=-\sin \frac{\theta }{2}|g⟩+\cos \frac{\theta }{2}|r⟩$

where $\theta =\arctan \frac{{\mathrm{\Omega }}_L}{-{\mathrm{\Delta }}_L}$.

The dressed states each have an energy shift coming from the single atom light shift so that they will have a larger energy difference as the detuning approaches 0.

Dressed states are the eigenstates of the dressed Hamiltonian. The energies of the dressed states experience the single atom light shift.

Mapping Onto Dressed Spin States

The ground state that has been defined can be recast as $|g⟩\to |1⟩$ and another ground state $|0⟩$ is introduced such that it is off resonant with the Rydberg state and therefore does not participate in the Rabi flopping. Generally the $|0⟩$ and $|1⟩$ states have a natural frequency in the microwave regime, while the natural frequency between $|1⟩$ and $|r⟩$ is in the optical regime.

If there is a Rydberg laser dressing the Hamiltonian while the atoms are in the $|00⟩$ state and are not within the Rydberg blockade regime, then a laser near to the resonance of $|0⟩$ and $|\tilde{1}⟩$ will result in flopping for each atom between the two states. However if they are blockaded, flopping will only happen from $|00⟩$ to $\frac{|0\tilde{1}⟩+|\tilde{1}0⟩}{\sqrt{2}}$. The same result seen between $|1⟩$ and $|r⟩$ has been mapped onto $|0⟩$ and $|\tilde{1}⟩$.

Mathematical Description

When an atom with states $|0⟩$, $|1⟩$, and $|r⟩$ is in an oscillating field that is near resonance for the transition between $|1⟩$ and $|r⟩$, we can ignore the coupling between $|0⟩$ and $|r⟩$.

$$\begin{array}{rl}{\hat{H}}_{\text{single atom}}& ={\mathrm{\Delta }}_L|r⟩⟨r|+\mathrm{\Omega }(|r⟩⟨1|+|1⟩⟨r|)\\ \hat{H}& ={\hat{H}}_{\text{single atom}}^{(1)}+{\hat{H}}_{\text{single atom}}^{(2)}\end{array}$$ By performing adiabatic dressing, the states can be brought from the computational basis $\{|00⟩,|01⟩,|10⟩,|11⟩\}$ into the dressed states $\{|00⟩,|0\tilde{1}⟩,|\tilde{1}0⟩,|\tilde{1}\tilde{1}⟩\}$. $$\begin{array}{rl}|\tilde{1}⟩& =\cos \frac{\theta }{2}|1⟩+\sin \frac{\theta }{2}|r⟩\\ E_{\tilde{1}}& =\frac{1}{2}({\mathrm{\Delta }}_L+\sqrt{{\mathrm{\Omega }}_L^2+{\mathrm{\Delta }}_L^2})\\ |\tilde{r}⟩& =-\sin \frac{\theta }{2}|1⟩+\cos \frac{\theta }{2}|r⟩\\ E_{\tilde{r}}& =\frac{1}{2}({\mathrm{\Delta }}_L-\sqrt{{\mathrm{\Omega }}_L^2+{\mathrm{\Delta }}_L^2})\end{array}$$

In the complete absence of the dipole–dipole interaction we get $E_{\tilde{1}\tilde{1}}=2E_{\tilde{1}}$. This means that the difference between $E_{(01+10)/\sqrt{2}}-E_{00}=E_{11}-E_{(01+10)/\sqrt{2}}$, which implies that flopping from $|00⟩$ to $\frac{|0\tilde{1}⟩+|\tilde{1}0⟩}{\sqrt{2}}$ will also flop between $\frac{|0\tilde{1}⟩+|\tilde{1}0⟩}{\sqrt{2}}$ and $|\tilde{1}\tilde{1}⟩$, producing no entanglement.

If the dipole–dipole interaction is introduced (physically bringing atoms closer), the Hamiltonian gains:

$$\begin{array}{rl}\hat{V}& =V_{rr}|rr⟩⟨rr|\\ \Rightarrow \hat{H}& ={\hat{H}}_{\text{single atom}}^{(1)}+{\hat{H}}_{\text{single atom}}^{(2)}+\hat{V}\end{array}$$

Since the $|0⟩$ states still do not participate, the computational states with a single $|\tilde{1}⟩$ are unchanged, but the dressed $|11⟩$ state now acquires an additional shift $\kappa$.

With the addition of $\hat{V}$, $|\widetilde{11}⟩\ne |\tilde{1}⟩|\tilde{1}⟩$ and $E_{\widetilde{11}}\ne 2E_{\tilde{1}}$. To find the new energy, note the Rabi frequency change:

$$\begin{array}{rl}\mathrm{\Omega }& \mapsto \sqrt{2}\mathrm{\Omega }\\ E_{\tilde{1}}& =\frac{\mathrm{\Delta }+\sqrt{{\mathrm{\Omega }}^2+{\mathrm{\Delta }}^2}}{2}\mapsto E_{\widetilde{11}}=\frac{\mathrm{\Delta }+\sqrt{(\sqrt{2}\mathrm{\Omega }{)}^2+{\mathrm{\Delta }}^2}}{2}\\ \Rightarrow \kappa & =E_{\widetilde{11}}-2E_{\tilde{1}}\end{array}$$

Spin-Flip Blockade Level Diagram. When there is no interaction potential between two atoms, there will not be any way to entangle them. In the dressed basis of the computational states, without any interaction between the atoms, the 01 state experiences the single atom light shift and the 11 state experiences double the single atom light shift. This leaves all the energy levels evenly spaced again. This only changes when the interaction is added in and the entanlging energy kappa splits the 11 state off, resulting in a blockade and allowing for entangelment to occur.

This entangling energy $\kappa$ is a shift large enough that a laser which would normally couple $\frac{|0\tilde{1}⟩+|\tilde{1}0⟩}{\sqrt{2}}$ to $|\tilde{1}\tilde{1}⟩$ will no longer couple it to the new dressed state $|\widetilde{11}⟩$.

Thus the flopping between $|00⟩$ and $\frac{|0\tilde{1}⟩+|\tilde{1}0⟩}{\sqrt{2}}$ is the only evolution when starting from $|00⟩$, producing an entangling operation.

This yields a clear mapping from the Rydberg blockade to the dressed computational basis.

Application to Neutral Atom Quantum Computing

Some well known quantum computing gate styles like the Levine-Pichler gate and adiabatic gate use an optical laser to accumulate different amounts of phase on $|01⟩$, $|10⟩$, $|11⟩$ states such that universal quantum computation is reached. This is generally done by implementing a generalized CZ gate. The spin-flip blockade opens up additional methods to implement universal quantum computing.

Bell State Preparation

Bell state preparation has been demonstrated with fidelity $\ge 81\mathrm{\%}$. The protocol dresses the Hamiltonian while the atoms are in the $|00⟩$ state and utilizes the spin-flip blockade to move the population into $\frac{|0\tilde{1}⟩+|\tilde{1}0⟩}{\sqrt{2}}$ state. This is just a $\pi$ pulse since the spin-flip blockade results in two level dynamics between $|00⟩$ and $\frac{|0\tilde{1}⟩+|\tilde{1}0⟩}{\sqrt{2}}$. Then the state is adiabatically undressed so the single atoms in $|\tilde{1}⟩$ move back to $|1⟩$. The result is a maximally entangled Bell state .

Entangling Gate

There is a proposed CZ gate that uses a microwave laser to perform a gate similar to the time optimal LP gate via the spin-flip blockade. The protocol encodes $|0⟩$ and $|1⟩$ states into the hyper-fine states of Cesium. All population in $|0⟩$ is moved into another long lived state and the original $|0⟩$ state is repurposed as an auxiliary state $|a⟩$. After dressing ($|a⟩\mapsto |\tilde{a}⟩=\cos \frac{\theta }{2}|a⟩+\sin \frac{\theta }{2}|r⟩$) we recover the dynamics of the standard Rydberg blockade where $|01⟩$ Rabi flops into $|0\tilde{a}⟩$ with Rabi frequency ${\mathrm{\Omega }}_{\mu w}$ and $|11⟩$ flops into $\frac{|1\tilde{a}⟩+|\tilde{a}1⟩}{\sqrt{2}}$. This has the same structure as the optical Rydberg blockade with a mapping for each quantity into the microwave regime. This can be used to implement the LP gate and its time optimal equivalent using optimal control

References

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