Q032 · Separate / excite / compare

Rydberg Atom Blockade

Why are two nearby atoms rarely excited to Rydberg states at the same time? Because at close range the strong interaction between their Rydberg states shifts the doubly excited state off resonance. A pulse that would excite either atom alone now excites only one of them, so double excitation becomes rare.

Move two trapped atoms closer together, drive both with the same pulse and record the state of each atom. As the atoms get closer, both are rarely excited at once.

Model at the cursor $\text{—}$

Recorded at this time $\text{—}$

01 / Control the atoms

Move closer. Excite together.

Drag the separation handle below. Both atoms see the same drive.

Schematic apparatus with fixed atom positions. Beam brightness shows the drive and is not a measurement.

Sequence time

t/t_*=0

Try a comparison

Every shot starts from a fresh preparation. The page opens with 200 simulated shots already recorded.

Drive strength

\Omega_0 t_*

1.2

Atom separation

R/R_*

1

Pulse protocol

Detuning, decay & readout

Interaction at reference distance

V_*t_*

6

Phase hold time

T/t_*

2.6

Single-photon detuning

\Delta t_*

0

Spontaneous decay rate

\gamma t_*

0

State-readout availability

\eta

1

Record seed

s

2832

Make a prediction

Which spacing gives fewer double excitations?

Each atom is a fixed, effective two-level system. One coherent drive acts on both, and they share a repulsive interaction that falls off as the inverse sixth power of their distance. Each atom decays back to the ground state independently. The state readout is ideal but can fail to return a result; real atom loss, atomic motion and the pair potentials of a specific species are left out. Counts alone do not prove that the atoms are entangled.

02 / Change one condition

Pull the atoms apart

R/R_*=1

The distance changes the interaction; the atoms themselves stay the same size. The slider gives the same control from the keyboard.

03 / Evolve, then measure

Count each joint outcome

Model populations at the cursor

Both ground0%

Only B excited0%

Only A excited0%

Both excited0%

Both groundOne excitationBoth excitedLines: model · circles: records · bars: 95% Wilson intervals

Click the plot to move the cursor. Each point combines only shots recorded at that time and those settings.

Model pair-level shift

\text{—}

Recorded double-excitation fraction

\text{—}

04 / Test your prediction

When does double excitation return?

Run a scan to collect 200 new preparations at each of 13 settings.

Setting	Read / total	Double excitations

What is simulated?

Every record is simulated. State readout is ideal when available, with state-independent missing records. Changing preparation settings clears the active data; an already acquired scan retains its own settings until you replace or reset it. Model diagnostics are separate from measured frequencies.

The interaction remains active during all pulses, including the prepare–wait–analyze protocol. This demonstrates an interaction phase; it is not a certified controlled-phase gate. Traps define fixed sites; motion and recapture loss are omitted.

Time is in reference units; angular frequencies are in inverse reference time. No species or laboratory clock is calibrated.

Scientific sources: Gaëtan et al. · Blockade · Saffman et al. · Rydberg interactions

Start with the essentials

Focus question: Why does bringing the atoms closer suppress double excitation?
One-sentence intuition: The population curves are predictions from the model; the dots come only from measured outcomes.

Core mathematical model

Reference units

\tau=t/t_*,\quad \Omega,\Delta,\delta,\gamma,V\ \text{in}\ t_*^{-1}

Every rate is either an angular frequency or a population decay rate. No real atomic species is calibrated.

Open-system dynamics

\dot\rho=-i[H/\hbar,\rho]+\sum_j\left(L_j\rho L_j^\dagger-\tfrac12\{L_j^\dagger L_j,\rho\}\right)

The full density matrix is evolved, so coherences are kept. Fourth-order Runge–Kutta steps are short enough to resolve the fastest scale in the Hamiltonian, and the integration is split exactly at pulse edges.

The doubly excited state is shifted

\frac H\hbar=\sum_{j=A,B}\left[\frac\Omega2\sigma_x^{(j)}-\Delta n_j\right]+Vn_An_B,\quad V=V_*\left(\frac{R_*}{R}\right)^6

The basis states are ground–ground, ground–excited, excited–ground and excited–excited. The repulsive coefficient is an effective parameter chosen for the Lab, not fitted to a particular rubidium state. Each atom decays to ground through its own collapse operator, and the other atom keeps its coherence.

Atoms without interaction

P_{rr}=\sin^4(\Omega t/2)\quad(V=\Delta=\gamma=0)

Without interactions the state is a product of the two atoms' states. After a pulse that fully inverts one atom, the probability of double excitation can reach one.

Collective oscillation

P_{gr}+P_{rg}\longrightarrow\sin^2(\sqrt2\,\Omega t/2)\quad(V/\Omega\to\infty)

On resonance, a large shift of the doubly excited state suppresses double excitation, and the symmetric single-excitation state oscillates faster. A finite interaction adds corrections. Population counts alone cannot prove that this state is entangled.

Waiting builds up an interaction phase

|rr\rangle\longmapsto e^{-iVT}|rr\rangle

With the drive off, the interaction adds this phase to the doubly excited state only. The prepare–wait–analyze sequence uses two pulses that are nominally quarter turns. The interaction and decay stay on during those pulses, so the sequence is not an ideal quantum gate.

Readout and uncertainty

\hat p=\frac{k}{N_{\rm read}},\quad N_{\rm unread}=N-N_{\rm read}

The interval shown is a 95% Wilson binomial interval for the fraction of target-state or double-excitation outcomes. It counts only shots that were read out, and whether a readout fails does not depend on the state. The interval holds point by point; it is not a joint band for the scan and does not cover numerical error in the model. Every shot starts from a freshly prepared ground state.

Common difficulties

A probability is not a trajectory

Typical misconceptionThe glowing atom shows the path the atom really follows.

Better mental modelThe atom marker only shows a fixed site, and the brightness of a beam shows the drive amplitude. Only the result tiles show sampled measurements.

Sources and limits

Typical misconceptionThis Lab reproduces a real, calibrated apparatus.

Better mental modelThe STIRAP model follows Vitanov et al., Reviews of Modern Physics 89, 015006 (2017), section II. The blockade model follows Gaëtan et al., Nature Physics 5, 115–118 (2009). The off-resonant interaction assumed here follows Saffman et al., Reviews of Modern Physics 82, 2313 (2010), section II.C, instead of the resonant pair potentials used in that experiment. These papers motivate the Hamiltonians, but our dimensionless rates and schematic geometry are not fitted to experimental values.

Run the experiment

01
Predict
Before collecting data, choose a contrasting case.
What to observe: Keep in mind which drive is applied and when the measurement is made.
02
Run the sequence
Drag pulse peaks or the separation handle, and scrub the time cursor.
What to observe: Playing replays the deterministic evolution. It does not create any measurement data.
03
Acquire and compare
Measure at the cursor, then collect a parameter scan. Export its raw outcomes.
What to observe: The scan measures at the cursor time captured when it started, and it keeps the preparation settings from that moment.