Atomic corral / Quantum mirage

A Quantum Mirage at an Empty Focus

Can an atom at one focus of an elliptical corral change the electrons at the other, empty focus? Yes. An ellipse sends waves from one focus to the other, so electron waves scattered by the impurity are carried across the corral and interfere at the empty focus. The energy spectrum measured there changes even though no atom was added. Depending on the energy, the change is a dip, a peak or almost nothing, and without the fence it is much weaker. No atom or spin moves to the second focus.

Place a nonmagnetic impurity at one focus of an elliptical corral. Compare the surface with and without it, then look for its spectral signature at the other focus, where no atom sits.

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.

Set up the corral

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

The mirage here comes from single-particle potential scattering; the many-body Kondo effect is not calculated. Each focus sits on the nearest lattice site of the model. Depending on energy, the response at the far focus can be a dip, a peak or almost nothing. No atom, spin or information is teleported.

The empty focus still responds

CHANGE AT THE EMPTY FOCUS

0

SELECTED ENERGY

E/J=-3.5

Compare the prediction with your own scan

Read the pattern and test what causes it

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: Can an atom at one focus of an elliptical corral change the electrons at the other, empty focus?
One-sentence intuition: An impurity at one spot changes the spectrum far away, because electron waves travel coherently and scatter many times.

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)

A positive broadening gives each level a Lorentzian line shape, which sets the spectral resolution. The result is a density of states per lattice site per unit of hopping energy.

How one impurity changes the spectrum far away

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

The propagator of the empty corral carries the impurity's effect to the probe. Because the propagator is complex, its phase can turn a dip into a peak.

Common difficulties

A spectrum is not topography

Typical misconceptionThe bright ridges are bumps in the metal, or paths the electrons follow.

Better mental modelThe surface is flat. The color shows the local spectral density at one energy. The tip is drawn only as a sketch, and this model has no feedback loop.

What the model leaves out

Typical misconceptionThe image reproduces the original Kondo mirage experiment quantitatively.

Better mental modelThe mirage here comes from single-particle potential scattering; the many-body Kondo effect is not calculated. Each focus sits on the nearest lattice site of the model. Depending on energy, the response at the far focus can be a dip, a peak or almost nothing. 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: Dark nodes and bright regions have different local spectra.
02
Change the setup and compare
Switch between the total density and the impurity difference. Remove the impurity, then remove the wall.
What to observe: A spectral signature appears at the far focus even though no atom was added there. Whether it is a dip or a peak, and how strong it is, depends on energy.
03
Acquire a scan
Scan the surface and the spectrum, increase the dwell time, pin the result and export the raw counts.
What to observe: The scanned image has Poisson shot noise. The estimate divides the counts by the set exposure; it is never scaled to the brightest pixel.