Two waves and one absorber

The Matter-Wave Antilaser

How does the relative phase of two matter waves decide whether a lossy site swallows them? In phase, the two waves pile up on the lossy site, and with the loss tuned just right almost all of the probability disappears there. In opposite phase they cancel on that site, so it sits on a node and absorbs nothing at any loss strength. More loss is not always better: a very strong sink acts like a wall and sends part of the waves back.

Send two coherent matter-wave packets toward one lossy lattice site. Make them vanish together, flip their relative phase to protect them, then find out why a stronger absorber can reflect more.

This is a linear, single-particle tight-binding model with local Markovian loss. Finite wave packets are evolved directly in time; perfect absorption of a single-frequency wave is shown separately as an analytic reference. The Lab does not reproduce an interacting condensate, an electron-beam apparatus or a detector calibration.

Coherence, absorption and reflection

ABSORBED BY THIS TIME

0\%

EVOLUTION TIME

\tau=0

Count every outcome

Every bit of lost probability is counted.

Watch the encounter unfold

Time runs downward and site number increases to the right. Drag the white time cursor. Brightness is the square root of the unconditional site probability, on a fixed scale from zero to 0.06.

00.06

What remains and what was absorbed

Cyan: survival. Amber: absorbed probability. The dashed level is the monochromatic reference, not a fit to these finite packets.

MONOCHROMATIC ABSORPTION

0

MAX LOSS-BALANCE ERROR

0

The probability at this instant

Independent simulated preparations

Count the survivors and keep the losses.

Each of 1,024 fresh preparations produces one outcome: lost, or detected at a site. The event grid uses amber for loss and cyan for survival. The histogram counts only those recorded surviving sites.

The readout ideally reports either a loss or a site, with no detector inefficiency. Changing the time or the display scale keeps the record; a new preparation clears it. The export includes every loss and every site, plus a separately labelled model reference.

Equations, loss and model boundaries

\tau=Jt/\hbar,\qquad E_*=J

The hopping energy sets the units. Site number increases from left to right in the data plots. The three-dimensional layout is schematic; no physical length or calibrated apparatus is assigned.

\begin{gathered}i\dot\psi_n=-(\psi_{n-1}+\psi_{n+1})\\-i(\gamma/2)\delta_{n0}\psi_n\end{gathered}

Two equally weighted Gaussian packets start forty sites either side of the sink, each with density standard deviation eight. Opposite carrier momenta direct them inward. There are 241 sites; the outer boundary remains negligibly occupied over the displayed interval. The sink removes excitation irreversibly into a vacuum state.

\begin{gathered}v=2\sin q,\quad\alpha=\gamma/2\\A_{\rm mono}=\frac{4v\alpha}{(v+\alpha)^2}\cos^2(\phi/2)\end{gathered}

This reference assumes monochromatic, equally strong coherent inputs. Perfect absorption requires matched loss and equal phase. A finite packet has a bandwidth, so its result need not equal the reference exactly. Atom interactions, nonlinear stability, electron-beam scattering and finite detector efficiency are omitted.

\begin{gathered}S=\sum_n|\psi_n|^2,\quad P_{\rm lost}=1-S\\p_n=|\psi_n|^2,\quad p_{n|\rm survive}=p_n/S\end{gathered}

The propagator uses fourth-order Runge–Kutta with a step no larger than 0.01. It never renormalizes the evolving state. A separately integrated loss rate checks the probability balance. The bright conditional display divides by survival; raw acquisition always includes lost trials. No gain, many-body statistics or continuously observed particle paths are claimed.

Müllers et al. — Coherent perfect absorption of nonlinear matter waves (2018)

These sources give the physical background. This Lab uses the linear model described here with illustrative parameters; it does not reproduce their experimental data.

Start with the essentials

Focus question: How does the relative phase of two matter waves decide whether a lossy site swallows them?
One-sentence intuition: A symmetric input can feed the lossy site; an antisymmetric input has a node there. More loss does not always mean more absorption.

Core mathematical model

Equation of motion

i\dot\psi_n=-(\psi_{n-1}+\psi_{n+1})-\frac{i\gamma}{2}\delta_{n0}\psi_n

Time is measured in units of the inverse hopping. The state is not renormalized as it evolves.

Probability lost at the absorbing site

\frac{d}{d\tau}\sum_n|\psi_n|^2=-\gamma|\psi_0|^2

All the probability that disappears is lost at that site; integrating the local loss rate over time gives the total.

Common difficulties

A bright shape does not mean high survival

Typical misconceptionThe wave gets brighter, so probability must have been created.

Better mental modelThe display scale and whether the image is conditioned on survival are both labeled. The dynamics is passive, so the total survival probability never grows.

What the model covers

Typical misconceptionThis simulates a calibrated apparatus or an interacting many-body experiment.

Better mental modelThis is a linear, single-particle tight-binding model with local Markovian loss. Finite wave packets are evolved directly in time; perfect absorption of a single-frequency wave is shown separately as an analytic reference. The Lab does not reproduce an interacting condensate, an electron-beam apparatus or a detector calibration.

Run the experiment

01
Compare two presets
Compare the first two presets at the same time.
What to observe: The loss at the center stays the same. The phase decides whether the incoming waves feed into it.
02
Check how much probability remains
Switch between all trials and surviving trials. Keep an eye on the survival meter.
What to observe: The conditional shape always adds up to one, even when very few trials survive.
03
Keep every outcome
Acquire 1,024 independent preparations, and export the lost outcomes along with the measured sites.
What to observe: A bright conditional image does not mean more events were detected. The sampled snapshot stays fixed when you move the time.