--- title: "Andreev Reflection Interactive Lab | Sandbox Physics" description: "Send an electron. Get a hole back. Launch an electron wavepacket into a superconducting interface. Watch the returning wave switch character, raise a barrier to stop the conversion, then cross the gap to transmit into the superconductor." canonical_url: "https://sandboxphysics.com/labs/andreev-electron-hole-reflection/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/andreev-electron-hole-reflection/" last_modified: "2026-10-18" --- SUPERCONDUCTING INTERFACE / ELECTRON–HOLE CONVERSION # Andreev Reflection Launch an electron wavepacket into a superconducting interface. Watch the returning wave switch character, raise a barrier to stop the conversion, then cross the gap to transmit into the superconductor. > Interactive simulation: open this page in a web browser to change the parameters and run the model. A one-dimensional tight-binding Bogoliubov–de Gennes model with a prescribed s-wave pair potential. Electron and hole ribbons are components on the same wire, not two paths. Finite wavepacket, zero-temperature mean field; no self-consistent condensate, collisions or calibrated conductance measurement. ONE INTERFACE. TWO KINDS OF RETURN. HOLE WEIGHT IN THE NORMAL REGION $0$ GOLD PLANE · NS INTERFACE $0$ CYAN · ELECTRON CORAL · HOLE HEIGHT · PROBABILITY PER SITE The two ribbons are Nambu components on ONE wire, not two trajectories. Fixed linear probability height. The shaded right half has superconducting pairing. $0$ 01 / THE COMPLETE COLLISION ### Follow the color that comes back. Position left to right; time 0 at the top and 75 at the bottom. Cyan is electron weight, coral is hole weight. Color uses a fixed square-root gain; the gold line is the interface. DARK 0 → BRIGHT 0.04 per site 02 / A LOCAL LOOK ### Two components. One normalized state. Cyan: electron probability per site. Coral: hole probability per site. Both components use the same position axis. Gold marks the interface at site 161. $j\ \longrightarrow\quad |u_j|^2,\ |v_j|^2$ 03 / PROBABILITY BUDGET ### Where did the excitation go? Cyan: electrons in the normal region. Coral: holes in the normal region. Purple: total weight inside the superconductor. Early left-electron weight includes the incoming packet; only separated late packets describe reflection. $\tau=Jt/\hbar\ \longrightarrow\quad P$ 04 / INDEPENDENT BULK REFERENCE ### The gap decides what can propagate. Analytic bands of the infinite uniform superconductor. The gold line is the incident packet’s central energy, which has a finite spread. $k\ \longrightarrow\quad E/J$ TURN THE MODEL INTO RECORDS ## Count what came back. 0 / 8000 Ideal component-and-position Born readouts from independent preparations at the selected time. These are simulated quasiparticle snapshots, not a calibrated detector or a count of Cooper pairs. Fixed reproducible seed. Recorded normal-hole fraction $\text{—}$ 95% Wilson interval · shot noise $\text{—}$ Model probability $\text{—}$ Norm drift $\text{—}$ ### Records in gold. Model in cyan. Three exhaustive outcomes: normal electron, normal hole, or either component inside the superconductor. The bars are built only from saved records. Latest 20 records · export every trial | Trial | Site | Component | | --- | --- | --- | Model, units and sources 320 sites with open ends. The normal region ends at site 160; pairing begins at 161. A one-site barrier sits at 160. The initial electron is centered at site 81 with position standard deviation 12 sites. Position increases to the right in lattice-spacing units. Hopping sets the energy unit; time is measured in reduced Planck constant divided by hopping. Ribbons are a diagnostic representation, with schematic spacing between components. $$ H=\begin{pmatrix}h&\Delta_s\\\Delta_s&-h\end{pmatrix},\quad h_{j,j+1}=-J,\quad h_{160,160}=V $$ Symmetric split evolution uses exact onsite and disjoint-bond rotations, with time step 0.0125. No renormalization, absorbing boundary or probability clipping. The displayed interval ends before significant boundary returns; the numerical boundary weight is monitored. The pair potential is prescribed, not solved self-consistently. $P_{\mathrm{edge}}=0$ · Probability in the outer eight sites on either end. Sources: Blonder, Tinkham and Klapwijk’s interface theory and the BdG formulation in the Kwant superconductivity tutorial. The lattice wavepacket is not the continuum BTK approximation. - [Blonder–Tinkham–Klapwijk · Superconducting interfaces](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.25.4515) - [Kwant · Superconductors and electron–hole degrees of freedom](https://kwant-project.org/doc/latest/tutorial/superconductors) Physics tutorial ## Andreev Reflection **Background** Launch an electron wavepacket into a superconducting interface. Watch the returning wave switch character, raise a barrier to stop the conversion, then cross the gap to transmit into the superconductor. **Why it matters** Connect a visible change to the spectrum of a superconducting system. Start with the essentials - **Focus question**: Send an electron. Get a hole back. - **One-sentence intuition**: Pairing mixes electron and hole amplitudes. A subgap excitation can return as a hole. ### Core mathematical model #### Electron and hole amplitudes $$ i\hbar\partial_t\begin{pmatrix}u\\v\end{pmatrix}=H_{\mathrm{BdG}}\begin{pmatrix}u\\v\end{pmatrix} $$ These are two components of a quasiparticle, not independent physical paths. #### Uniform superconducting spectrum $$ E(k)^2=4J^2\cos^2 k+\Delta_s^2 $$ An energy inside the bulk gap has no propagating quasiparticle mode deep in the superconductor. ### Common difficulties #### Respect the scope **Typical misconception** A hole is a positron emitted by the interface. **Better mental model** A hole describes a missing electronic occupation relative to the reference condensate. Andreev conversion exchanges charge with that condensate. #### A diagnostic is not a photograph **Typical misconception** The raised ribbons are paths followed by individual particles. **Better mental model** Ribbon height encodes a declared probability or spectral density. The two colors share the same physical site coordinate. ### Run the experiment 1. 01 #### Make a prediction Compare the three presets, then change one continuous control. **What to observe:** The barrier suppresses hole conversion; sufficient energy permits transmission. 2. 02 #### Inspect the cause Scrub from the incoming packet to the separated outgoing packets. **What to observe:** Early-time left-electron weight includes the incoming packet; it is not yet a reflection probability. 3. 03 #### Acquire evidence Collect repeated records, inspect the histogram and interval, then export the complete CSV. **What to observe:** The model curve is a separate reference. Only recorded events build the gold histogram. Physical conclusion A one-dimensional tight-binding Bogoliubov–de Gennes model with a prescribed s-wave pair potential. Electron and hole ribbons are components on the same wire, not two paths. Finite wavepacket, zero-temperature mean field; no self-consistent condensate, collisions or calibrated conductance measurement. 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. 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