---
title: "A Quantum Mirage at an Empty Focus Interactive Lab | Sandbox Physics"
description: "One scatterer. A signature where no atom was added. Place a nonmagnetic impurity at one focus of an elliptical corral. Compare the same surface with and without it, and find its remote spectral signature at the empty focus."
canonical_url: "https://sandboxphysics.com/labs/elliptical-quantum-mirage/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/elliptical-quantum-mirage/"
last_modified: "2026-10-20"
---

PLACE / COMPARE / REVEAL

# A Quantum Mirage at an Empty Focus

Place a nonmagnetic impurity at one focus of an elliptical corral. Compare the same surface with and without it, and find its remote spectral signature at the empty focus.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

A single-particle potential-scattering mirage, not a many-body Kondo calculation. The two foci are snapped to the model lattice. Depending on energy, the remote response can be a dip, a peak or almost absent. No atom, spin or information is teleported.

AN EMPTY FOCUS. A REAL RESPONSE.

CHANGE AT THE EMPTY FOCUS

$0$

SELECTED ENERGY

$E/J=-3.5$

Flat specimen. Color is a spectral diagnostic on the surface, not physical height. Gold: wall scatterers. White ring: probe. Coral: added impurity. The opposite focus has no added atom.

MOVE / COMPARE / ACQUIRE

## Read the pattern. Test the signature.

### The predicted surface

$(x,y)/a=(0,0)$

Drag the white cross to move the probe. Arrow keys move one site; Shift moves five. Display scale is fixed across presets.

### The spectrum under the probe

DRAG ENERGY

Cyan: with impurity. Dashed gray: same corral without it. Gold: acquired counts divided by exposure.

LOCAL DENSITY $0$

REFERENCE $0$

LOCAL CHANGE $0$

Density units: states per site per hopping energy. A local spectrum is not normalized as a position probability.

SIMULATED ACQUISITION

### Make your own noisy scan

Each surface pixel and spectral bin is an independent Poisson count. The scan reveals recorded data row by row; the prediction above stays separate.

Exposure per pixel / spectral bin $\mu=\mathcal E\,\rho J$

Counts use a fixed-height, constant tip-DOS, low-temperature narrow-bin idealization. Exposure is a dimensionless count scale, not calibrated seconds. Changing the model or probe clears records; pin first to preserve them.

What is solved · what is schematic

The retarded resolvent is evaluated on a periodic 256 by 256 square lattice. Nearest-neighbor hopping is negative; energy is in hopping units and distance in lattice spacings. The displayed window is 45 by 45 sites. Finite real on-site potentials form the wall. The model keeps multiple scattering to all orders.

$$
H=-J\sum_{\langle r,s\rangle}(|r\rangle\langle s|+|s\rangle\langle r|)+\sum_j U_j|j\rangle\langle j|
$$

$$
G(z)=(z-H)^{-1},\qquad \rho_r(E)=-\operatorname{Im}G_{rr}(E+i\eta)/\pi
$$

All maps and spectra use the same Green function. Positive spectral half-width smooths the spectrum with a Lorentzian; it is not a separately solved bath or a direct coherence-time measurement. The periodic host is numerical, not a claim that the physical surface is a torus. The published control range is checked against a doubled host.

The tip is schematic. Its height, atom size and colors are enlarged for legibility. No tunneling barrier, height feedback, finite-temperature convolution, realistic crystal band structure, spin, Coulomb interaction or Kondo screening is solved. Difference maps subtract two predictions at identical settings; they are not reconstructed from the acquired counts.

- [Fiete & Heller · Theory of Quantum Corrals and Quantum Mirages](https://arxiv.org/html/cond-mat/0211607)
- [Manoharan, Lutz & Eigler · Quantum mirages (2000)](https://www.nature.com/articles/35000508)
- [Schmid & Kampf · Non-magnetic impurity mirages](https://arxiv.org/html/cond-mat/0308115)

Physics tutorial

## A Quantum Mirage at an Empty Focus

**Background** Place a nonmagnetic impurity at one focus of an elliptical corral. Compare the same surface with and without it, and find its remote spectral signature at the empty focus.

**Why it matters** Geometry changes the states available to an electron.

Start with the essentials

- **Focus question**: One scatterer. A signature where no atom was added.
- **One-sentence intuition**: A local perturbation changes a remote spectrum through coherent propagation and multiple scattering.

### Core mathematical model

#### The spectrum at one point

$$
\rho(\mathbf r,E)=-\frac{1}{\pi}\operatorname{Im}G_{\mathbf r\mathbf r}(E+i\eta)
$$

Positive broadening produces Lorentzian spectral resolution. This is density per site per hopping-energy unit.

#### One impurity, a remote change

$$
\delta G_{rr}=\frac{U_{\rm imp}G^{\rm c}_{ri}G^{\rm c}_{ir}}{1-U_{\rm imp}G^{\rm c}_{ii}}
$$

The empty-corral propagator carries the perturbation to the probe; its complex phase can turn a dip into a peak.

### Common difficulties

#### A spectrum is not topography

**Typical misconception** Bright ridges are raised metal or electron trajectories.

**Better mental model** The surface is flat. Color encodes local spectral density at one energy. The tip is schematic and has no feedback loop here.

#### What the model omits

**Typical misconception** The image quantitatively reproduces the original Kondo mirage experiment.

**Better mental model** A single-particle potential-scattering mirage, not a many-body Kondo calculation. The two foci are snapped to the model lattice. Depending on energy, the remote response can be a dip, a peak or almost absent. No atom, spin or information is teleported.

### Run the experiment

1. 01

   #### Read the surface

   Drag the probe across the map; then drag the energy cursor across the spectrum.

   **What to observe:** Nodes and bright regions have different local spectra.

2. 02

   #### Make the comparison

   Switch between total density and impurity difference. Remove the impurity, then the wall.

   **What to observe:** A remote spectral signature needs no added atom at that position. Its sign and strength depend on energy.

3. 03

   #### Acquire a scan

   Scan the surface and spectrum, increase dwell, pin the result and export raw counts.

   **What to observe:** The acquired image contains Poisson shot noise. Its estimate divides the counts by the declared exposure, never by the image maximum.

Physical conclusion

A single-particle potential-scattering mirage, not a many-body Kondo calculation. The two foci are snapped to the model lattice. Depending on energy, the remote response can be a dip, a peak or almost absent. No atom, spin or information is teleported.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)