Estimates the residual thermal photon occupation n̄ and equivalent noise temperature at the end of a microwave input line in a dilution refrigerator, given the attenuation and physical temperature of each stage. Radiation from each stage propagates through a chain of thermalized attenuators (beam-splitter model). Edit any field and results update immediately; use "Copy link" to share your configuration.
| Stage | Temperature (K) | Attenuation (dB) |
|---|
Attenuation is the total dissipative loss thermalized at that stage (attenuators plus cable loss). A stage with 0 dB contributes no thermal radiation of its own.
Each curve is that source's contribution to the photon occupation at the device, after all downstream attenuation. The vertical marker is the chosen frequency f0. Hover or focus the chart to read values; the table below carries the same numbers at f0.
| Source | T (K) | Attenuation after source (dB) | n̄ contribution | Share |
|---|
A resistive attenuator at physical temperature T both attenuates incident noise and emits its own thermal (Johnson–Nyquist) radiation. For a matched 50 Ω line, an attenuator with power attenuation A = 10dB/10 transforms the photon occupation of the incoming field as
n̄out = n̄in / A + (1 − 1/A) · n̄BE(T, f)
where the Bose–Einstein occupation of a thermal source at temperature T is
n̄BE(T, f) = 1 / (ehf/kBT − 1).
The (1 − 1/A) emissivity follows from Kirchhoff's law (emissivity equals absorptivity), or equivalently from modeling the attenuator as a beam splitter with transmissivity 1/A that mixes the incoming field with a thermal mode at temperature T — the standard input–output treatment of a lossy element (Clerk et al.). Applying this stage by stage from the 300 K input down to the device gives the total occupation n̄, equivalent to summing each source's emission attenuated by everything downstream of it, as done for cryogenic attenuation chains in Krinner et al. The equivalent temperature is the temperature of a single blackbody source that would produce the same occupation:
Teff = (hf / kB) / ln(1 + 1/n̄).
When the input is specified as a power spectral density S in dBm/Hz, the input occupation is n̄in = S / hf (thermal emission convention, vacuum energy excluded). For reference, a matched 300 K source emits about −174 dBm/Hz at microwave frequencies. Assumptions: matched impedances (no reflections), attenuators fully thermalized at their stage temperature, and no attenuation between the last stage and the device.
References: A. A. Clerk et al., "Introduction to quantum noise, measurement, and amplification", Rev. Mod. Phys. 82, 1155 (2010) for the quantum treatment of noise and lossy elements, and S. Krinner et al., "Engineering cryogenic setups for 100-qubit scale superconducting circuit systems", EPJ Quantum Technol. 6, 2 (2019), arXiv:1806.07862 for the application to cryogenic attenuation chains.