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.

ATOMIC LANDSCAPES

There is no atom over there.

A single impurity changes the spectrum at an empty focus. Switch to the difference map, then remove the impurity to test what you see.

SHAPE THE SURFACE

Probe energy

E/J

-3.075

Major radius

a_{\rm c}/a

16

Minor / major radius

b_{\rm c}/a_{\rm c}

0.75

Wall scattering strength

U_{\rm wall}/J

8

Opening toward the right

\theta_{\rm gap}/{}^\circ

0

Spectral half-width

\eta/J

0.04

Impurity potential

U_{\rm imp}/J

6

Positions snap to a square lattice. Changing the radius also changes the number of wall sites. Spectral broadening averages resonances; it is not a measured decoherence rate.

MOVE THE PROBE

x/a

y/a

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

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.

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 misconceptionBright ridges are raised metal or electron trajectories.

Better mental modelThe 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 misconceptionThe image quantitatively reproduces the original Kondo mirage experiment.

Better mental modelA 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

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.
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.
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.