Making a Quantum Oscillator Behave Like a Kuramoto Oscillator

quantum
Published

2026-07-14

Modified

2026-07-14

This post is a draft of experimental work that is undergoing revision. You have been warned.

The question

The Kuramoto model is one of the simplest descriptions of synchronization. Each oscillator is reduced to a phase on a circle, and the oscillators pull on one another through the sine of their phase difference. This raises a deceptively difficult quantum question:

What physical constraints make an open quantum system acquire Kuramoto phase dynamics?

Our current answer uses two quantum van der Pol oscillators. Each oscillator is a bosonic mode with gain and nonlinear loss. A shared dissipative channel removes their antisymmetric motion and attracts their phases. In a regime where the oscillation amplitudes are reasonably stable, the phase equation reduces to the two-node Kuramoto equation.

The simulations support that mechanism. They also show where the correspondence remains incomplete. At the tested occupation, the two modes become strongly frequency-entrained while their relative phase remains fairly broad. The dynamics is quantum and open, although the canonical steady state has no detected entanglement.

1. From a physical oscillator to a phase oscillator

A self-sustained classical oscillator follows a stable closed orbit in phase space. Examples include a pendulum maintained by a feedback drive, a laser above threshold, and an electronic oscillator with nonlinear saturation. Motion toward the orbit removes amplitude disturbances. Motion along the orbit is neutral and can be described by an angle theta.

Once amplitude disturbances relax faster than the phase evolves, the detailed oscillator can be replaced by a phase equation:

\[ \frac{d\theta_i}{dt} = \omega_i + \text{interaction} + \text{noise}. \]

Kuramoto chose the leading periodic interaction permitted by rotational symmetry:

\[ \frac{d\theta_i}{dt} = \omega_i + \sum_j K_{ij}\sin(\theta_j-\theta_i). \]

For two equally coupled oscillators, define the relative phase

\[ \phi = \theta_2-\theta_1, \qquad \delta = \omega_2-\omega_1. \]

The two phase equations combine into

\[ \frac{d\phi}{dt} = \delta-\Gamma\sin(\phi). \]

Here Gamma is the total restoring strength in the relative coordinate. A fixed point exists when

\[ |\delta| \leq \Gamma. \]

Below that condition, the relative phase continually slips. Above it, the two oscillators share one long-time frequency and settle near a preferred phase difference. Noise broadens this sharp deterministic picture into a crossover with occasional phase slips.

This reduction contains an important physical assumption: a Kuramoto oscillator is already a self-oscillator. Coupling coordinates that have no stable oscillation amplitude does not automatically produce the Kuramoto model.

2. Why the quantum version must be open

A closed quantum state evolves unitarily,

\[ \frac{d\rho}{dt} = -i[H,\rho]. \]

Unitary evolution preserves information and has no generic attracting steady state. Kuramoto locking is dissipative: amplitude errors decay and relative phases are drawn toward a stable relation. An environment, reservoir, feedback channel, or equivalent open-system construction therefore has to participate.

We describe the open dynamics with a Lindblad master equation:

\[ \begin{aligned} \frac{d\rho}{dt} &= -i[H,\rho]+\sum_\mu \mathcal{D}[L_\mu]\rho,\\ \mathcal{D}[L]\rho &= L\rho L^\dagger -\frac{1}{2}\left(L^\dagger L\rho+\rho L^\dagger L\right). \end{aligned} \]

The operators L_mu are commonly called collapse operators or jump operators. The word collapse can cause confusion. It does not mean that repeated projective measurements are responsible for synchronization. In the unconditional description, rho evolves smoothly under the master equation. If the environment is monitored, the same equation can be unravelled into stochastic quantum trajectories containing individual jumps.

The phase attraction in this project comes from an engineered dissipative channel. Measurement is optional. Strong monitoring would add backaction and could even suppress motion through a quantum-Zeno regime.

3. The quantum van der Pol construction

We model two truncated bosonic modes with annihilation operators a_1 and a_2. Their free Hamiltonian is

\[ H = \omega_1 a_1^\dagger a_1 + \omega_2 a_2^\dagger a_2. \]

The full master equation is

\[ \begin{aligned} \frac{d\rho}{dt} ={}&-i[H,\rho] +\gamma_{\mathrm{up}}\sum_i\mathcal{D}[a_i^\dagger]\rho\\ &+\gamma_2\sum_i\mathcal{D}[a_i^2]\rho +\Gamma\mathcal{D}[a_1-a_2]\rho. \end{aligned} \]

Each dissipative term has a separate job.

Channel Physical role What happened when we removed it
sqrt(gamma_up) a_i^dag Linear gain, or negative damping The state relaxed to the vacuum
sqrt(gamma_2) a_i^2 Nonlinear two-quantum loss that limits the amplitude Probability accumulated at the artificial Fock cutoff
sqrt(Gamma) (a_1-a_2) Loss of the antisymmetric mode and relative-phase attraction The relative-phase distribution became uniform

Linear gain alone would drive the occupation upward. Nonlinear loss grows faster with amplitude and eventually balances the gain. Mean-field balance gives the nominal limit-cycle occupation

\[ n_0=r_0^2 =\frac{\gamma_{\mathrm{up}}}{2\gamma_2}. \]

The collective loss channel damps the difference a_1-a_2. It allows the symmetric motion of the two modes to survive more easily than their antisymmetric motion. This is the mechanism that aligns their phases.

This model follows the quantum van der Pol synchronization construction studied by Walter, Nunnenkamp, and Bruder. Related proposals use trapped-ion motion Lee and Sadeghpour, and a quantum van der Pol oscillator has recently been realized experimentally Li et al..

4. Where the Kuramoto sine law comes from

Let

\[ \alpha_i =\langle a_i\rangle =r_i e^{-i\theta_i}. \]

The minus sign is a convention chosen so that theta_i advances at the positive bare frequency omega_i. Applying the adjoint Lindblad equation and using coherent-state factorization gives

\[ \begin{aligned} \frac{d\alpha_i}{dt} ={}&\left( -i\omega_i+\frac{\gamma_{\mathrm{up}}}{2} -\gamma_2|\alpha_i|^2 \right)\alpha_i\\ &+\frac{\Gamma}{2}(\alpha_j-\alpha_i). \end{aligned} \]

Separating amplitude and phase yields

\[ \frac{d\theta_i}{dt} =\omega_i +\frac{\Gamma}{2}\frac{r_j}{r_i} \sin(\theta_j-\theta_i). \]

When radial relaxation is fast and r_1 is close to r_2, the radius ratio can be replaced by one:

\[ \frac{d\theta_i}{dt} =\omega_i +\frac{\Gamma}{2}\sin(\theta_j-\theta_i). \]

The relative phase then obeys

\[ \frac{d\phi}{dt} =\delta-\Gamma\sin(\phi). \]

which is exactly the deterministic two-oscillator Kuramoto equation. Systematic semiclassical phase reduction of quantum limit-cycle oscillators also produces phase diffusion and higher-order corrections; see Kato, Yamamoto, and Nakao.

The reduction is controlled best when:

  • gain and nonlinear loss establish a developed limit cycle;
  • the mean occupation is large enough to support a reasonably narrow phase coordinate;
  • radial relaxation is faster than detuning and coupling;
  • the Fock-space truncation leaves its highest states effectively empty;
  • every Lindblad rate is nonnegative, preserving complete positivity.

Our canonical ratios, abs(delta)/gamma_up = 0.4 and Gamma/gamma_up = 0.6, are moderate. The calculation probes a small, noisy quantum oscillator rather than an asymptotic classical limit.

5. How phase is observed in the quantum model

Quantum phase requires some care. A bosonic number operator has a lower boundary, so a globally well-behaved Hermitian phase operator is unavailable in the same way as position. We use two complementary diagnostics.

Relative-phase distribution

The finite Fock basis defines a Pegg-Barnett phase POVM. Applying it to both modes and integrating over their common phase gives a normalized distribution P(phi) for the relative phase. Its first circular moment is

\[ R_\phi =\left| \int P(\phi)e^{i\phi}\,d\phi \right|. \]

R_phi = 0 describes a uniform relative phase. Values approaching one describe a sharply concentrated phase relation.

Emission-frequency entrainment

Two-time correlations such as

\[ \left\langle a_i^\dagger(t)a_i(0) \right\rangle. \]

are calculated using quantum regression. Their spectra locate each oscillator's emission peak. Convergence of the two peaks tests frequency entrainment directly.

The steady state retains global phase-rotation symmetry, so <a_i> vanishes even when the relative phase is structured. Relative observables and spectra are therefore more informative than an absolute mean field.

6. What we simulated

The production calculation used QuTiP to solve the finite-dimensional Lindblad equation on a classical computer. The canonical parameters were:

Parameter Value
Fock dimension per oscillator 12
omega_1, omega_2 +0.2, -0.2
Bare detuning abs(delta) 0.4
Linear gain gamma_up 1.0
Nonlinear loss gamma_2 0.25
Canonical collective loss Gamma 0.6
Calibrated free-oscillator occupation 2.543
Calibrated phase diffusion 0.182274

The evidence suite performed:

  1. an instantaneous coherent-state drift test over 25 relative phases;
  2. a coupling sweep from Gamma = 0 through 1.2;
  3. emission spectra from two-time quantum correlations;
  4. a stationary noisy classical Kuramoto comparison using the calibrated phase diffusion;
  5. gain, nonlinear-loss, and collective-coupling ablations;
  6. an occupation trend;
  7. Fock cutoffs from dimensions 6 through 14;
  8. mutual-information and entanglement-negativity diagnostics.

7. Evidence and interpretation

Construction check: the collective channel generates the sine drift

At Gamma = 0.6, the instantaneous drift fit returned:

fitted detuning          -0.4000000000
expected detuning        -0.4000000000
fitted sine coefficient  -0.5999329823
expected coefficient     -0.6000000000
residual RMS              4.05e-16

Instantaneous phase drift and the Kuramoto sine-law prediction

Figure 1. The coherent-state drift follows the expected sine law.

This is a construction check. It confirms the algebra and implementation of the collective channel. The independent evidence comes from solving the full mixed-state steady state, measuring its phase distribution and spectrum, and running the controls below.

Coupling concentrates phase and entrains frequency

Gamma Quantum phase concentration Matched classical concentration Normalized quantum peak detuning
0.0 0.000000 0.000000 1.000001
0.4 0.344334 0.360101 0.486807
0.6 0.453803 0.517722 0.178109
0.8 0.528799 0.644774 0.078576
1.2 0.617810 0.798904 0.024300

Phase concentration and frequency entrainment across the coupling sweep

Figure 2. Increasing collective loss concentrates relative phase and reduces emission-frequency detuning.

The uncoupled emission peaks lie at +0.2 and -0.2. At Gamma = 0.6 they move to approximately +0.03562 and -0.03562. At Gamma = 1.2, only 2.43 percent of the bare peak separation remains.

The deterministic Kuramoto threshold is Gamma = 0.4. With diffusion, the transition becomes a crossover. Using R_phi = 0.5 as an operational marker gives:

Model R_phi = 0.5 crossover
Noiseless Kuramoto equation 0.400000
Diffusion-matched classical phase equation 0.576571
Full quantum model 0.715200

The quantum oscillator therefore requires stronger coupling to reach the same phase concentration. Frequency entrainment develops earlier and more strongly than phase localization. That distinction is one of the clearest physical results of the experiment.

Quantum and matched classical relative-phase distributions

Figure 3. Collective loss changes the uniform uncoupled phase distribution into a peak near the stable relative phase.

Agreement improves as occupation rises

At fixed Gamma = 0.6, the difference between quantum and matched classical phase concentrations decreases over the tested range:

Free occupation Phase diffusion Absolute concentration difference
1.000000 0.500748 0.087070
1.535902 0.324544 0.081744
2.542996 0.182274 0.063918

This trend is consistent with semiclassical phase reduction. Three occupation points below three quanta are insufficient for an asymptotic scaling law, so the evidence remains preliminary on that question.

Ablations identify necessary mechanisms

Case Mean occupation Phase concentration Highest-level population
Full model 2.062469 0.453803 0.000120
No collective coupling 2.542996 0.000000 0.000276
No gain 1.29e-14 2.69e-15 1.78e-17
No nonlinear loss 7.416728 0.357919 0.201295

The ablations make the mechanism falsifiable. Collective loss is required for a preferred relative phase. Gain is required for self-oscillation. Nonlinear loss is required to keep the oscillator away from the artificial truncation boundary.

The numerical cutoff is controlled

Dimension 14 was used as the convergence reference. At dimension 12, phase concentration differs from the reference by 0.002057, or about 0.45 percent, and the highest-level population is 1.20e-4. The dimension-6 result is far from converged, which confirms that an aggressively small encoding changes the physics.

Occupation trend, cutoff convergence, and channel ablations

Figure 4. Classical-reduction trend, Fock-cutoff control, and channel ablations.

Correlation does not imply entanglement

At the canonical point, the mutual information is 0.309879. Bipartite negativity is zero within a numerical tolerance of about 2.1e-12. The modes share classical and quantum correlations in a mixed open-system state, while this particular entanglement witness finds no entanglement.

The evidence supports Kuramoto-like dynamics in a quantum system. It does not establish a quantum computational advantage or a uniquely nonclassical form of synchronization. Such a claim would require an additional witness, for example entanglement in another regime, Wigner negativity, discord, or a demonstrated failure of every appropriate classical stochastic phase model.

8. What makes the system behave like Kuramoto?

The simulation gives a concrete answer to the motivating question.

  1. Self-oscillation: gamma_up > 0 prevents decay to the vacuum.
  2. Amplitude saturation: gamma_2 > 0 creates a stable radial scale.
  3. Attractive phase coupling: Gamma > 0 in D[a_1-a_2] damps relative motion.
  4. Timescale separation: radial disturbances should relax faster than phase drift.
  5. Sufficient occupation: a larger limit cycle makes phase reduction more accurate and reduces intrinsic phase diffusion.
  6. Physical open dynamics: nonnegative Lindblad rates preserve complete positivity.
  7. Numerical convergence: the Fock cutoff must be high enough that its boundary has negligible population.

The quantum system does not undergo a single projective collapse into a locked phase. Its stationary relative-phase probability gradually becomes concentrated as coupling grows. A monitored trajectory would show noisy phase motion and occasional slips, while averaging many trajectories reproduces the smooth density-matrix evolution.

9. Relation to compact quantum rotors

The project's original axioms proposed a compact rotor with Hilbert space L2(S1), integer angular momentum, and a phase unitary U = exp(i theta). That remains a rigorous route to a quantum Kuramoto model. Delmonte et al. study interacting open quantum rotors and recover classical Kuramoto behavior in an overdamped high-temperature limit.

The active experiment uses a bosonic limit-cycle oscillator instead. Its phase is an emergent coordinate described by a phase POVM, and the finite Fock space maps naturally to qubits. This choice gives physically interpretable gain and saturation channels and a direct phase reduction. It only partially satisfies the strict compact-rotor axiom because its fundamental Hilbert space is a truncated bosonic ladder rather than L2(S1).

These are two legitimate research routes:

  • the rotor route starts from phase as a fundamental compact coordinate;
  • the limit-cycle route starts from a quantum amplitude whose slowly varying coordinate is phase.

The current evidence concerns the second route.

10. What quantum-computer simulation would require

The present calculations run on a classical computer. A gate-based quantum simulation would encode each truncated oscillator and implement both Hamiltonian and dissipative evolution.

For the canonical dimension:

Resource Estimate
Fock levels per mode 12
Qubits per mode under binary encoding 4
System qubits for two modes 8
Unused binary basis states per mode 4
Lindblad channels 5
Reusable environment qubit for a first-order single-channel dilation 1

The truncated ladder operators can be decomposed into Pauli strings. A time step can alternate Hamiltonian evolution with channel dilations for gain, nonlinear loss, and collective loss, resetting an environment ancilla between channels. Efficient algorithms for Lindblad evolution are described by Cleve and Wang, and Di Bartolomeo et al. discuss simulation using a single environmental qubit.

Several practical questions remain unanswered: circuit depth, Trotter error, leakage into the four unused computational states, ancilla reset errors, shot cost, and the influence of hardware noise. The eight-qubit figure is therefore an encoding estimate rather than a completed hardware design.

11. What we learned

The main lessons from this phase of the project are:

  1. An open quantum limit-cycle oscillator can acquire the Kuramoto sine interaction through collective dissipation.
  2. The word collapse refers to Lindblad channels in the master equation; projective measurement is not required.
  3. Gain, nonlinear saturation, and relative-mode damping are independently necessary in the tested construction.
  4. Finite quantum fluctuations turn deterministic locking into a broad crossover.
  5. Frequency entrainment can be strong while the relative phase remains diffuse.
  6. The current occupation trend points toward the classical noisy Kuramoto equation, although the tested range is too small for a controlled convergence claim.
  7. The canonical state is correlated and has no detected entanglement.

12. Next experiment

The most useful next study is an occupation scaling experiment at approximately n = 1, 2, 4, 8. The Fock cutoff should adapt at every point until both the highest-level population and the primary observables are converged. Detuning and collective loss should remain below about 0.2 gamma_up to improve radial-timescale separation.

At each occupation we should compare the full Lindblad model with its independently calibrated noisy phase equation using:

  • the L1 distance between complete relative-phase distributions;
  • circular phase concentration;
  • emission-peak detuning;
  • quantum-trajectory phase-slip rate.

That experiment would answer whether the discrepancy decreases systematically with occupation and whether individual quantum phase slips approach the statistics of the reduced noisy Kuramoto model. A compiled dissipative circuit becomes scientifically worthwhile after this scaling test establishes a controlled target.

Reproducing the evidence

python3 -m venv .venv
.venv/bin/python -m pip install -r requirements.txt
.venv/bin/python main.py
.venv/bin/python main.py --evidence
.venv/bin/python -m unittest -v

The evidence sweep writes strict JSON with a unique experiment ID. The stable record used for this journal entry is:

experiments/quantum-vdp-kuramoto-evidence-20260714.json

The full verification command rebuilds the JSON, figures, tests, and shareable technical-note PDF:

.venv/bin/python scripts/verify.py

References

Project files

  • main.py - active quantum and classical simulations
  • technical-note.md - derivation, full numerical results, and limitations
  • experiments/quantum-vdp-kuramoto-evidence-20260714.json - machine-readable evidence
  • output/pdf/quantum-kuramoto-technical-note.pdf - shareable technical note
  • three-axioms.md, informal-phase.md, and background.md - original theoretical design notes