Rydberg Blockade Mechanism

The Rydberg blockade is the fundamental mechanism enabling entangling gates between neutral atoms in optical tweezer arrays and optical lattices. It exploits the enormous electric dipole moments of Rydberg states ($n\gtrsim 50$) to create strong, long-range interactions that conditionally prevent double excitation of nearby atom pairs.

Physics of the Interaction

When a neutral atom is excited to a Rydberg state $|r⟩$ with principal quantum number $n$, its orbital radius scales as $n^2{a}_0$ and its polarizability scales as $n^7$. Two atoms both in Rydberg states experience a van der Waals interaction:

$V_{\text{vdW}}(R)=\frac{C_6}{R^6}$

where $R$ is the interatomic distance and the $C_6$ coefficient scales as $n^{11}$. For typical experimental parameters ($n\sim 70$, $R\sim 3-10\mu \text{m}$), this interaction strength reaches $V/\hslash \sim 2\pi \times 10-1000\text{MHz}$, vastly exceeding qubit transition linewidths and Rabi frequencies.

The Blockade Condition

Consider two atoms separated by distance $R$, each driven by a laser with Rabi frequency $\Omega$ on the transition $|g⟩\to |r⟩$. The doubly-excited state $|rr⟩$ is shifted in energy by $V(R)=C_6/R^6$. If:

$V(R)\gg \hslash \Omega$

then the laser is far off-resonance for exciting the second atom when the first is already in $|r⟩$. This defines the blockade regime: at most one atom in the pair can be in the Rydberg state.

The blockade radius is defined as the distance where the interaction equals the excitation linewidth:

$R_b={\left(\frac{C_6}{\hslash \Omega }\right)}^{1/6}$

For $C_6/\hslash \sim 2\pi \times 800\text{GHz}\mu {\text{m}}^6$ (rubidium $|70S⟩$) and $\Omega /2\pi \sim 1\text{MHz}$, $R_b\approx 10\mu \text{m}$, encompassing several lattice sites.

Entangling Gate Protocol

The Rydberg blockade enables a controlled-Z (CZ) gate via the following protocol:

1. Apply a $\pi$ pulse to atom 1: $|g{⟩}_1\to |r{⟩}_1$
2. Apply a $2\pi$ pulse to atom 2: $|g{⟩}_2\to |r{⟩}_2\to |g{⟩}_2$ (acquires a $\pi$ phase)
3. Apply a $\pi$ pulse to atom 1: $|r{⟩}_1\to |g{⟩}_1$

If atom 1 is in $|g{⟩}_1$ (the non-Rydberg qubit state), step 1 excites it, step 2 is blockaded (no phase acquired), and step 3 de-excites it. If atom 1 starts in the other qubit state (not coupled to Rydberg), step 2 proceeds normally and acquires phase $\pi$. The net truth table is:

$|00⟩\to |00⟩,|01⟩\to |01⟩,|10⟩\to |10⟩,|11⟩\to -|11⟩$

which is a CZ gate.

Blockade Fidelity

The CZ gate fidelity is limited by:

• Finite blockade strength: Residual population in $|rr⟩$ scales as $(\Omega /V{)}^2$, giving a blockade error:

${ϵ}_{\text{blockade}}\sim {\left(\frac{\Omega R^3}{C_6^{1/2}}\right)}^2$

• Rydberg state decay: Lifetime ${\tau }_r\propto n^3$ at zero temperature, reduced by blackbody radiation at finite $T$.

• Doppler and motional effects: Atomic motion during the gate changes $R$ and introduces phase noise. Resolved by using tightly trapped atoms at $\mu \text{K}$ temperatures.

State-of-the-art implementations achieve CZ fidelities of 99.5% (Evered et al. 2023) with ongoing improvements from pulse optimization and higher Rydberg states.

Historical Context

• Jaksch et al. (2000) proposed the Rydberg blockade gate mechanism.
• Lukin et al. (2001) independently proposed “dipole blockade” for quantum information.
• Urban et al. (2009) and Gaëtan et al. (2009) first observed blockade between individual atoms.
• Saffman et al. (2010) provided a comprehensive review establishing the field.
• Evered et al. (2023) demonstrated 99.5% CZ fidelity in ${}^{87}\text{Rb}$ tweezer arrays.

References

• rydberg-neutral-atom-qubit
• neutral-atom-qubit
• coherence-time-hierarchy