Noise at the ports

A port's termination is the measurement line: the source and load that send a field into the port and absorb what leaves it. Its temperature sets the field it sends in; the circuit's losses and blocks set the noise the circuit adds. This page states port temperatures and reads two quantities of a readout chain: the measurement efficiency from the device to room temperature, and the thermal occupation the chain sends back toward the device.

Warm terminations

Port(n; termination = MatchedTermination(temperature = T)) gives a port a matched termination at the physical temperature T in kelvin. It sends in a thermal field, the symmetrized noise nbar + 1/2 in each mode, with nbar = thermaloccupation(ω, T) at the mode's frequency, so a signal and its idler see different occupations. The temperature table gives the temperature of each source of noise. For a line whose noise is not that of one temperature, put its attenuators in the circuit at their own temperatures with the port at the top, as below, or give the port the effective temperature of the field that reaches it (effectivetemperature).

A passive circuit whose ports and losses share one temperature is in equilibrium: every wave leaving it is thermal.

using JosephsonCircuits, LinearAlgebra
w = 2pi*3e9
T = 0.3
warm(n) = Port(n; termination = MatchedTermination(temperature = T))
circuit = Circuit([
    (:p1, 1, 0, warm(1)), (:c1, 1, 0, Capacitor(0.3e-12)),
    (:loss, 1, 2, Resistor(30.0; temperature = T)),
    (:c2, 2, 0, Capacitor(0.5e-12)), (:p2, 2, 0, warm(2)),
])
sol = hblinsolve([w], circuit; keyedarrays = false, returnVout = true)
n = thermaloccupation(w, T)
@assert isapprox(sol.Vout[:, :, 1], (n + 1/2)*I; atol = 1e-12)
(occupation = n, nbar = sol.nbar[:, 1])

(occupation = 1.623502914385847, nbar = [1.6235029143858477, 1.6235029143858486])

What the solvers report

Noise is in quanta, the vacuum being half a photon (see conventions).

Output	Meaning
QE	photon gain over twice the noise at the output, every input in its state, the signal's own included
QEideal	the ideal phase-preserving amplifier at the same gain, its inputs in the vacuum
nbar	the occupation of the wave leaving each port mode
Vout	the output covariance, S*Diagonal(sigma)*S' + Cnoise, with sigma the nbar + 1/2 of every input (returnVout = true)
Cnoise	the noise the circuit adds, never its ports' (returnCnoise = true)
Snoise, channeltemperatures	the transfer from each internal noise channel, and the channel's temperature (returnSnoise = true)

QE and nbar read the same noise at an output, v = Vout[i,i]: QE = G/(2v) and nbar = v - 1/2. With a warm source the QE counts the thermal photons arriving with the signal, as a measurement of the signal to noise ratio does; with the source cold it is the device's own efficiency, and QE/QEideal its distance from the phase-preserving quantum limit. Every source s puts (nbar_s + 1/2)*abs2(t_s) into v, a port mode through S and an internal channel through Snoise: that is the noise budget of an output.

The occupation reaching the device

An input line brings the room-temperature load to the device through attenuators at the fridge's stages. The occupation reaching the device is the cascade of the stages, each stage's occupation weighted by its transmission to the device.

att(dB, T) = ScatteringParameters([0.0 10^(-dB/20); 10^(-dB/20) 0.0];
    zref = 50.0, noise = ThermalEquilibrium(T))
at(T) = MatchedTermination(temperature = T)
line = Circuit([
    (:device, 1, 0, Port(1; termination = at(0.02))),
    (:a1, 1, 2, att(20, 0.02)), (:a2, 2, 3, att(20, 0.1)),
    (:a3, 3, 4, att(20, 4.0)), (:room, 4, 0, Port(2; termination = at(300.0))),
])
w = 2pi*5e9
sol = hblinsolve([w], line; returnSnoise = true)
nback = sol.nbar((0,), 1, 1)
A = 0.01
cascade = thermaloccupation(w, 300.0)*A^3 + thermaloccupation(w, 4.0)*(1 - A)*A^2 +
    thermaloccupation(w, 0.1)*(1 - A)*A + thermaloccupation(w, 0.02)*(1 - A)
@assert isapprox(nback, cascade; rtol = 1e-10)
# the budget: each source's thermal photons, weighted by its transmission;
# a channel is named for its block and port
blockof(channel) = String(first(split(channel, '/')))
budget = Dict("room" => thermaloccupation(w, 300.0)*abs2(sol.S((0,), 1, (0,), 2, 1)))
for (c, T) in zip(sol.Snoise.component, sol.channeltemperatures)
    photons = thermaloccupation(w, T)*abs2(sol.Snoise((0,), c, (0,), 1, 1))
    budget[blockof(c)] = get(budget, blockof(c), 0.0) + photons
end
@assert isapprox(sum(values(budget)), nback; rtol = 1e-10)
(nbar = nback, temperature = effectivetemperature(w, nback),
    budget = sort(collect(budget); by = last, rev = true))

(nbar = 0.0038451687142565527, temperature = 0.04312162272407069, budget = ["a3" => 0.001601255126826077, "room" => 0.0012496972140558086, "a2" => 0.0009881220442042359, "a1" => 6.094329170210723e-6])

The device sees the field of a bath at the effective temperature, above the mixing chamber's 20 mK; most of it comes from the 4 K and 0.1 K stages and from the room-temperature load.

A readout chain

A JPA read out through a circulator, an isolator at the mixing chamber, a cable at 4 K and a HEMT stated by its noise temperature, to a room temperature load. The quantum efficiency from the device's port to the room temperature port is the measurement efficiency of the chain, and its budget says what limits it: the idler of the JPA, the losses in front of the HEMT, and the HEMT.

iso = ScatteringParameters([0.0 10^(-20/20); 10^(-0.45/20) 0.0]; zref = 50.0,
    noise = ThermalEquilibrium(0.02))
circulator = ScatteringParameters([0.0 0.0 1.0; 1.0 0.0 0.0; 0.0 1.0 0.0];
    zref = 50.0, noise = Lossless())
fs, fp, ip = 4.748e9, 4.75e9, 0.0058e-6
ws = 2pi*fs
GH, TN = 1e4, 2.0
nadd = noisequanta(ws, TN)
# the HEMT's input emits backward like a matched load at 4 K
hemt = ScatteringParameters([0.0 0.0; sqrt(GH) 0.0]; zref = 50.0,
    noise = NoiseCovariance([thermaloccupation(ws, 4.0)+1/2 0.0; 0.0 GH*nadd]))
chain = Circuit([
    (:device, 1, 0, Port(1; termination = at(0.02))),
    (:circ, 1, 2, 3, circulator),
    (:cc, 2, 4, Capacitor(100e-15)), (:jj, 4, 0, JosephsonJunction(1000e-12)),
    (:cj, 4, 0, Capacitor(1000e-15)),
    (:iso, 3, 5, iso), (:cable, 5, 6, att(1, 4.0)), (:hemt, 6, 7, hemt),
    (:room, 7, 0, Port(2; termination = at(300.0))),
])
sol = hbsolve([ws], (2pi*fp,), [(mode = (1,), port = 1, current = ip)], (8,), (16,),
    chain; returnSnoise = true).linearized
QE = sol.QE((0,), 2, (0,), 1, 1)
G = abs2(sol.S((0,), 2, (0,), 1, 1))
# every source's share of the noise at the output, referred to the input:
# each port in each mode, and each block's channels in every mode
wmode(m) = ws + m[1]*2pi*fp
share(n, t) = (n + 1/2)*abs2(t)/G
budget = Dict(string("port ", p, " ", m) =>
    share(thermaloccupation(wmode(m), p == 1 ? 0.02 : 300.0), sol.S((0,), 2, m, p, 1))
    for p in (1, 2) for m in sol.modes)
for (c, T) in zip(sol.Snoise.component, sol.channeltemperatures), m in sol.modes
    photons = share(thermaloccupation(wmode(m), T), sol.Snoise(m, c, (0,), 2, 1))
    budget[blockof(c)] = get(budget, blockof(c), 0.0) + photons
end
@assert isapprox(sum(values(budget)), 1/(2QE); rtol = 1e-10)
# against the Friis cascade: the JPA's idler, then each loss and the HEMT
# referred through the gain before it
ns = thermaloccupation(ws, 0.02)
ni = thermaloccupation(abs(ws - 2*2pi*fp), 0.02)
eiso, ecable = 10^(-0.45/10), 10^(-1/10)
GJ = G/(eiso*ecable*GH)
loss(eta, T) = (1 - eta)/eta*(thermaloccupation(ws, T) + 1/2)
friis = (GJ - 1)/GJ*(ni + 1/2) +
    (loss(eiso, 0.02) + loss(ecable, 4.0)/eiso + nadd/(eiso*ecable))/GJ
@assert isapprox(1/(2QE) - (ns + 1/2), friis; rtol = 1e-3)
(jpagain = 10*log10(GJ), QE = QE, fraction = QE/sol.QEideal((0,), 2, (0,), 1, 1),
    systemtemperature = noisetemperature(ws, 1/(2QE)),
    budget = sort(filter(x -> last(x) > 1e-3, collect(budget)); by = last, rev = true))

(jpagain = 20.079387615022828, QE = 0.42894390635837737, fraction = 0.8578872246024525, systemtemperature = 0.2656152211812134, budget = ["port 1 (0,)" => 0.5000112697119807, "port 1 (-2,)" => 0.4951144076178168, "hemt" => 0.12033938207546022, "cable" => 0.049514212405177484])

The budget's entries are photons referred to the device's port. They add to 1/(2QE), the signal's own half photon included; the JPA's idler contributes about half a photon at this gain, the quantum limit of a phase preserving amplifier. The system noise temperature is the same noise in kelvin (noisetemperature).

The noise the chain sends back

nbar at the device's port is the thermal occupation the chain sends back toward the device. Here it is the HEMT's backward emission and the 4 K cable's, through the 20 dB of the isolator, and the isolator's own at the mixing chamber.

nback = sol.nbar((0,), 1, 1)
gi2 = 10^(-20/10)
back = gi2*thermaloccupation(ws, 4.0) + (1 - gi2)*thermaloccupation(ws, 0.02)
@assert isapprox(nback, back; rtol = 1e-8)
(nbar = nback, temperature = effectivetemperature(ws, nback))

(nbar = 0.170598804915713, temperature = 0.11831425707748681)

A circulator of finite isolation would add the JPA's amplified output here, and a real HEMT emits backward more than a matched load at its stage temperature; both are blocks with the noise they state.

In time

The transient's port baths take the same termination temperatures, so transientnoise measures warm ports as harmonic balance does: its covariance is the output covariance Vout in the measured modes and its addedcovariance the noise the circuit adds, Cnoise, and transientquantumefficiency returns the same QE of a phase-insensitive mode. See quantum noise in time.