--- title: "Dynamical Casimir Photon Pairs Interactive Lab | Sandbox Physics" description: "Empty input. Correlated output. Modulate a superconducting boundary and examine two matched microwave modes. Rotate the receivers, compare equally bright thermal light, and collect the correlated fluctuations." canonical_url: "https://sandboxphysics.com/labs/dynamical-casimir-photon-pairs/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/dynamical-casimir-photon-pairs/" last_modified: "2026-10-19" --- PUMP / PAIR / LISTEN # Dynamical Casimir Photon Pairs Modulate a superconducting boundary and examine two matched microwave modes. Rotate the receivers, compare equally bright thermal light, and collect the correlated fluctuations. > Interactive simulation: open this page in a web browser to change the parameters and run the model. An effective two-mode squeezing model inspired by a SQUID-terminated waveguide. Energy comes from the prescribed pump. Equal thermal input occupations and symmetric vacuum loss; no broadband circuit solution, calibrated flux-to-gain conversion or pump depletion. The scene is schematic; the field mean is zero. A MODULATED BOUNDARY. TWO MATCHED MODES. JOINT NOISE / VACUUM REFERENCE $0$ BRIGHTNESS OF EITHER OUTPUT MODE $0$ **01** PUMP / SQUID **02** FREQUENCY SORTER **03** RECEIVERS A / B Schematic chip. Envelope height scales with square root of photon occupation, above vacuum. Purple blocks mark loss. Curves are not instantaneous fields; selected frequencies sum to the pump frequency. $r=0.9$ 01 / JOINT FLUCTUATIONS ### The shape tells you what brightness cannot. Cyan density: model for the X setting. Gold dots: saved X records only. Drag horizontally to rotate both receivers; arrows change one degree. Square-root density colors, fixed scale. $X_A\ \longrightarrow\quad X_B\ \uparrow$ 02 / ROTATE THE LISTENING AXIS ### Find the quiet direction. Model joint noise versus shared receiver phase; gold marks the selected angle. The dashed reference is one. Both orthogonal measurement settings are required for the witness. $\theta\ ({}^\circ)\ \longrightarrow\quad W$ PREPARE AGAIN. COLLECT AGAIN. ## Listen together. Keep every sample. 0 / 20000 Independent matched-mode preparations, alternating the X and orthogonal P settings. Each trial jointly measures one quadrature per mode; incompatible quadratures are never measured on the same copy. Ideal homodyne reception after the declared loss. Fixed seed, all samples retained. Recorded joint noise $\text{—}$ Sampling standard error $\text{—}$ Model joint noise $\text{—}$ Uncertainty check $\text{—}$ ### Narrow together, noisy alone. Gold: recorded combinations, X difference and P sum pooled. Cyan: the model Gaussian with integrated bin probabilities. Outer bins include every tail beyond −5 and 5. Latest 20 records · export all | Trial | Setting | A | B | | --- | --- | --- | --- | Model, conventions and sources The real pump phase defines zero. Both receivers rotate through the same angle. Each vacuum quadrature has variance one half. Separate X and P preparations give the two variances; a witness below one is sufficient for entanglement in this trusted ideal measurement model. A value above one at a poor angle does not prove separability. $$ a'=a\cosh r+b^\dagger\sinh r $$ $$ X=(a+a^\dagger)/\sqrt2,\quad P=(a-a^\dagger)/(i\sqrt2) $$ $$ W=\tfrac12[\operatorname{Var}(X_A-X_B)+\operatorname{Var}(P_A+P_B)] $$ $$ W_\theta=1-\eta+\eta(2n_{\mathrm{th}}+1)[\cosh(2r)-\sinh(2r)\cos(2\theta)] $$ Effective two selected temporal-frequency modes: the exponential squeeze map is not the full broadband SQUID input–output calculation. The two input modes have equal occupation, which is not a claim of a common physical temperature at different frequencies. Symmetric attenuation mixes each output with independent vacuum. No amplifier noise, phase drift, bandwidth inversion, pump depletion or circuit calibration. - [Johansson et al. — Dynamical Casimir effect in microwave circuits, sections III, V, VI (2010)](https://arxiv.org/html/1007.1058v1) - [Wilson et al. — Observation in a superconducting circuit (2011)](https://arxiv.org/html/1105.4714v1) - [Duan et al. — Continuous-variable inseparability criterion (2000)](https://arxiv.org/abs/quant-ph/9908056) Physics tutorial ## Dynamical Casimir Photon Pairs **Background** Modulate a superconducting boundary and examine two matched microwave modes. Rotate the receivers, compare equally bright thermal light, and collect the correlated fluctuations. **Why it matters** Use statistics to distinguish quantum light from a bright classical signal. Start with the essentials - **Focus question**: Empty input. Correlated output. - **One-sentence intuition**: Each mode alone looks thermal. Their shared fluctuations reveal the pair process. ### Core mathematical model #### A quantitative signature $$ W=\frac{\operatorname{Var}(X_A-X_B)+\operatorname{Var}(P_A+P_B)}{2} $$ With vacuum variance one half per quadrature, separable states have W at least one. The two combinations are measured on different preparations. ### Common difficulties #### Respect the energy source **Typical misconception** The vacuum supplies free energy. **Better mental model** The externally driven boundary supplies energy. Squeezing describes the effective quantum transformation. #### Read the measurement contract **Typical misconception** A displayed theoretical curve is already acquired evidence. **Better mental model** An effective two-mode squeezing model inspired by a SQUID-terminated waveguide. Energy comes from the prescribed pump. Equal thermal input occupations and symmetric vacuum loss; no broadband circuit solution, calibrated flux-to-gain conversion or pump depletion. The scene is schematic; the field mean is zero. ### Run the experiment 1. 01 #### Predict the contrast Compare the squeezed output with the equally bright thermal preset. **What to observe:** Mean brightness agrees, while the joint cloud changes shape. 2. 02 #### Break the effect Rotate both receivers or add thermal input and loss. **What to observe:** A poor measurement angle can hide correlations without changing the source brightness. 3. 03 #### Collect and inspect Collect trials, pin a run, alter one control and restore it. Export every record as CSV. **What to observe:** The gold data come only from saved simulated records; cyan curves are model references. Physical conclusion An effective two-mode squeezing model inspired by a SQUID-terminated waveguide. Energy comes from the prescribed pump. Equal thermal input occupations and symmetric vacuum loss; no broadband circuit solution, calibrated flux-to-gain conversion or pump depletion. The scene is schematic; the field mean is zero. 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)