Quantum noise in time

The noise calculation linearizes about the complete recorded trajectory, pump, signals, depletion and intermodulation products together, and propagates the fluctuations of the physical baths, the port terminations and the resistors, through that time dependent linear system: the Gaussian approximation of harmonic balance's noise calculation carried into time, small fluctuations about a classical mean rather than the full quantum state. It runs on the tangent and the adjoint of the recorded steps, so it is exact for the discrete trajectory it is given and needs no interpolation, replay or quadrature grid of its own.

Temporal modes

transientquantumplan defines photon normalized measurements of the outgoing power waves in sqrt(W). Where transientiq returns classical peak phasors, this plan returns real quadratures in the row order X1, P1, X2, P2, ..., with [X, P] = im, vacuum covariance I/2, and coherent photon number (X^2 + P^2)/2 for a normalized mode.

A record of N uniform samples covers the half open interval [t0, t0 + N*dt); its positive frequency bins are f = k/(N*dt) for k = 1:fld(N - 1, 2), without DC and without the self conjugate Nyquist bin. With T = N*dt the physical wave of one canonical bin is

A mode is a column of unit norm coefficients over those bins, and the measured annihilator is A_j = sum(conj(c[k, j])*a_k), with the 1/sqrt(h*f) weighting applied per bin before the bins are combined, which matters for a pulse spread over gigahertz. The frequency form of the constructor makes bin aligned monochromatic modes, or projects sampled envelopes onto the positive bins and normalizes them. Modes on the same port need not be orthogonal: plan.gram is their overlap and plan.vacuum and plan.commutator the corresponding cross mode quadrature matrices. transientquantumvjp! is the exact transpose of the measurement.

Baths and the initial fluctuations

transientnoisebaths(problem) builds one independent equilibrium bath per matched port termination and per finite internal resistor, from the compiled port ownership, the bound values and the component temperatures; a typed Circuit expresses which resistor a port owns. For positive frequency f and quadrature weight df in Hz, a bath of resistance R injects cosine and sine Norton currents of peak amplitude

whose independent quadratures have variance thermaloccupation(2pi*f, T)/2 = nbar + 1/2, so the bilateral symmetrized current spectral density is h*f/R*coth(h*f/(2kT)), the classical 2kT/R at high temperature. A port's bath is injected as a port source is, into the port, so its current enters the port's wave as a source's does; an internal resistor's bath is injected between the resistor's terminals. As targets of transienttangent and transientadjoint the baths are what the noise drives.

The record must start at a classical equilibrium with a constant passive prehistory, and the check reads the whole of the state the solve started from: the node fluxes and rates, the waves on the lines, which carry a direct current, and the states of the rational blocks, which must be at rest under the incident waves there. For each bath and frequency the stationary response [-w^2 C + i w G + L + J'(x_0)] x = b of the circuit at the initial state starts the cosine and sine responses, so the noise already stored in the capacitors and inductors, and its correlation with the forcing to come, are kept; zero classical initial voltage does not mean zero quantum noise.

The calculation

using JosephsonCircuits

# a solve recording the junction phases, which is all the noise reads
problem = transientproblem(circuit; sources = [TransientSource(1, pump)])
n, T = 20000, 100e-9
solution = transientsolve(problem, (0.0, T*(n - 1)/n); dt = T/n, record = :phases)
measurement = transientquantumplan(solution, solution.times, [5e9]; ports = [2])
inputs = transientquantumplan(solution, solution.times, [5e9]; ports = [1])
noise = transientnoise(solution, measurement; inputs)

noise.diagnostics
noise.covariance          # the symmetrized output covariance
noise.commutator          # the propagated bath commutator
noise.expectedcommutator  # the measurement's own mode algebra
noise.gain                # the incremental quadrature response to the inputs

The bath is periodic over the record by default: every positive Fourier bin k/T of its duration T with the weight 1/T, up to the record's Nyquist frequency, which is the complete bath of the recorded steps. Its cost grows with the frequency count: the adjoint method contracts a sum per bath, frequency and time, and the stationary prehistory of the baths is one factorization of the stationary operator per frequency and distinct initial state, so the complete bath of a long record is not cheap. A cutoff in Hz bounds it, and frequencies with weights replace it, which is what a circuit with loss spread along a line wants: a few bands around the tones and their idlers at the resolution of the window; the forward method, two directions per bath and frequency, wants a cutoff always.

method = :adjoint, the default, propagates the measured quadratures backward through transientadjoint: its derivatives with respect to the bath currents at every recorded time are the bath kernels, and each column of them is contracted against the cosine and sine of every frequency at the grid or stage time the adjoint hands it over at, a rank one update of a (bath, objective) by (frequency, quadrature) accumulator on the backend, so nothing is stored per recorded time and the frequency count costs sums rather than integrations; the columns are buffered over a block of times and contracted by one product. That accumulator holds 16 nb m nf N bytes for nb baths, m measured quadratures, nf frequencies and N conditions, which for distributed loss and a dense band is the memory of the method: within a quarter of the backend's free memory it is one, beyond it the conditions are tiled, each tile one adjoint over its conditions, and then the frequencies, each tile one more adjoint over the same conditions, a trade of passes for memory rather than a bound; the frequency count itself is set by where the measured modes respond, a few bands around the tones and their idlers at the resolution of the window, not by the whole record. The covariance, the commutator and the gain accumulate from each tile on the backend, and the stationary operator is factorized once per frequency for all conditions sharing an initial state. The stationary initial term enters through the adjoint's initial flux and rate with one stationary solve per frequency and objective, so no bath's initial state is ever built; the memory of the adjoint method is the record of the solve itself plus that accumulator. method = :forward propagates the bath quadratures forward, two tangent directions per bath and frequency through transienttangent, all directions of every condition of a batch on each step's factorization; it is the reference the adjoint is checked against, and the two agree to roundoff since they contract the same responses.

Two gains are on offer. transientgain(solution, measurement, inputs) drives each input mode's unit quadratures as incident waves inside the input window only, evaluated at the grid and the stage times of the steps inside it and zero elsewhere, through the tangent, and reads them in the measurement's modes: the causal, pulsed gain, with the transients of the probe's own edges, independent of any bath grid, and nothing of it reaches a window that ends before the probe starts. The gain of transientnoise with an input plan is the response to the periodic Fourier mode of the window extended over the record and started stationary, which is what a stationary amplifier's harmonic balance gain is; it needs the bath on the window's Fourier bins with weights 1/T. The two agree as the window grows past the circuit's memory.

Every dissipative element is a bath: the port terminations, the internal resistors, and the lossy scattering blocks, whose emitted noise wave has the covariance I - S S' of Bosma's relation, the same the linearized solver uses, factored into independent channels that enter the block's port current rows as the source of the hybrid equation; a lossless block, a circulator or a through, emits nothing. A RationalScattering block's covariance depends on frequency, so its ports are channels correlated by I - S S' at each bath frequency, contracted directly with that covariance so that a block lossless to a part in a million is never the difference of two large terms, and its states enter the stationary response and its initial term; a lossy rational block leaves the vacuum the vacuum when cold and has the excess hblinsolve gives it when warm. Nothing is inferred about a rational block's loss: it keeps its channels however small its loss, and a declared Lossless() is validated by the norms of the block and of its inverse over all frequencies, which can only refuse it. A block's temperature is its ThermalEquilibrium(T), a resistor's its temperature, and the rest the analysis default, in both solvers alike, so a warm attenuator in time has the covariance hblinsolve gives it, and equals its own resistive network at the same temperature. A block which states its noise with a NoiseCovariance, an amplifier given by its scattering parameters, has its ports as correlated channels of the same kind, contracted with the covariance it states and with the commutator I - S S' of its gain, which differ; it adds the noise the linearized solver gives it and its output obeys the commutation relations, at whatever temperature the analysis runs. A pumped block which states its noise is contracted over pairs of bath frequencies on one ladder of its pump, those differing by a multiple of the pump frequency and those summing to one, with the covariance and the commutator of its noise over all the bath frequencies at once, its harmonic covariances and its multi-mode scattering matrix assembled over them with every input which feeds them and, for a fitted block, completed there, so the correlations of an amplifier's noise between its signal and its idler are kept, and those the completion adds, and referred to the start of the record as the bath quadratures are; the bath frequencies must include the partners. A completed covariance is completed over the padded ladder of the bath frequencies as the linearized solver completes it over the padded ladder of a sweep's modes, so the two read one noise model. The block couples nothing between frequencies which are not a multiple of the pump apart, so the bath frequencies are grouped into the ladders of the pump and the block's scattering matrix, its commutator and its covariance are assembled, completed and read one ladder at a time, and no matrix spans two of them. At a given number of ports and a given width of ladder that cost grows with the number of ladders, the grouping of the frequencies and the ordering of the pairs adding their sort. A ladder is as wide as the bath frequencies it holds, so a bath whose frequencies are all a multiple of the pump apart is one ladder, whose covariance and pair list grow with the square of its width and whose completion with the cube. Every pumped block is checked over the bath frequencies before the calculation, as the linearized solver checks it over the modes of a sweep, which a fitted block, its covariance completed to the commutation relations of its own filters, passes but for the finite, Hermitian entries any covariance must have; and since the pair terms correlate the bath frequencies the frequencies of a calculation with a pumped block are contracted in one tile, which the memory budget must hold.

A transmission line stores the fluctuations that entered it before the record began. The stationary response of the circuit to each bath tone is solved on the flux phasors together with the phasors of the waves leaving the line ports, nothing inverted in the lines, so it is regular at a line's half wave resonances; the forward method hands the tangent those waves over the prehistory the delays reach into, and the adjoint's initial term contracts its cotangents of that prehistory through the transposed system. A cold line between mismatched loads then leaves the vacuum the vacuum, and a pumped amplifier behind a cable has the gain and the quantum efficiency the linearized solver gives it with the same line.

The bath quadrature is any set of positive frequency nodes with positive weights in Hz; a list of driven tones and idlers is not a complete bath for a pulse, since the loaded trajectory mixes the whole band into a measured window. The optional input plan returns the incremental quadrature gain from the same responses, and requires the measurement's record, bin aligned frequencies with weights 1/T, and coverage of every input coefficient.

What convergence looks like on a pulsed, distributed loss case, a two hundred cell line of series junctions with a loss resistor in every cell, pumped by a pulse and measured in Hann windows of 2 ns on the rise, the plateau and the fall of the pulse with the output window delayed by the line's transit: the step converges at fourth order, the covariance and the pulsed gain moving sixteen times less from 2.5 ps to 1.25 ps than from 5 ps to 2.5 ps; the bath spacing converges as the window's spectrum is resolved, at a part in ten thousand by a quarter of the window's bin; and the cutoff must pass the junction plasma frequency of the line, near 29 GHz there, where the line responds most and the pump mixes into the measured band, a band stopping short of it missing a part in a thousand and one past it changing nothing further. A rectangular window converges slowly in the cutoff, since its sidelobes carry noise from far away; a smooth envelope does not. The test suite runs that case.

The diagnostics check the commutator against the measurement's algebra and the uncertainty relation covariance + im*expectedcommutator/2 >= 0, and a failure is returned as such rather than rescaled away. Passing is necessary, not sufficient: refine the bath cutoff, the frequency spacing, the record and the step independently. On a passive two port the quadrature gain agrees with hblinsolve to a part in ten thousand at 128 samples of a 3 GHz record and to three parts in a hundred thousand at 512, converging at second order with the step.

transientquantumefficiency reduces a phase preserving single mode result, a scaled rotation for the gain and an isotropic covariance, to photon gain G, added noise v/G - 1/2, QE = G/(2v) and the package's ideal G/(2G - 1); phase sensitive gain or anisotropic noise is rejected and the full matrices remain. On a pumped Josephson amplifier measured after 200 ns of settling with the stationary Floquet frequencies as the bath, the gain and the quantum efficiency converge to hbsolve at the order of the stepping rule: under the trapezoidal rule, whose frequency warping a resonator's detuning and an amplifier's bifurcation magnify, the gain is 12% high at 5 ps and half a percent at 0.16 ps; under GaussLegendre it agrees to 1.5e-3 at 5 ps and to 2e-5 at 2.5 ps, with the quantum efficiency to 6e-6, in about a second of solve and noise together. On a traveling wave amplifier of two hundred series junctions pumped to 2.6 radians of node flux, where the accumulated propagation phase and the phase matching are the constraint, the periodic gain and the quantum efficiency agree with hbsolve to 4e-4 at 5 ps and 3e-5 at 2.5 ps, again at fourth order. Harmonic balance takes milliseconds on these stationary problems and stays their reference, the transient being for pulses and drives too heavily loaded for a harmonic grid.