Electronics Guide

The Heisenberg Uncertainty Principle in Electromagnetic Systems

The uncertainty principle, often written Δx Δp ≥ ℏ/2, where ℏ is the reduced Planck constant h/2π, has direct implications for electromagnetic measurements. For an electromagnetic field the conjugate variables are the two quadratures of a mode, the sine and cosine components of its oscillation, which play the roles that position and momentum play for a particle.

For a harmonic oscillator mode of the electromagnetic field at frequency f:

• Energy fluctuations: The ground state, or vacuum state, of each mode carries energy E = hf/2, an irreducible zero-point energy that no amount of cooling removes. These vacuum fluctuations contribute noise to any measurement of field amplitude.
• Photon number uncertainty: A coherent state, which is the quantum description that comes closest to the output of a stable classical source, has a Poissonian photon-number distribution, so its photon-number uncertainty is

Δn = √n,

where n is the mean photon number. This shot noise sets the precision of amplitude measurements made with ordinary sources. It is not an absolute floor: an amplitude-squeezed state has a narrower photon-number distribution, bought at the price of a broader phase distribution.

Phase-amplitude trade-off: Reducing amplitude uncertainty necessarily increases phase uncertainty, and vice versa. The product of these uncertainties cannot be reduced below the quantum limit without special preparation of the field state.

At radio frequencies, quantum noise is typically negligible compared with thermal noise, because at room temperature the thermal energy kT greatly exceeds the photon energy hf. Quantum noise becomes significant, and eventually dominant, only where the thermal contribution is suppressed: at very high frequencies (approaching the terahertz range at room temperature) or at cryogenic temperatures, where noise at microwave frequencies can already be quantum-limited.

Standard Quantum Limit

The standard quantum limit (SQL) represents the measurement precision achievable using classical states of light, that is coherent states, and conventional measurement techniques. For amplitude measurements the SQL is simply shot noise: the photon-number uncertainty grows as the square root of the mean photon number, so the signal-to-noise ratio n/Δn grows as the square root of the mean photon number as well.

For a measurement of electromagnetic field amplitude with power P and integration time T:

The mean number of photons collected is n = PT/hf, so the signal-to-noise ratio at the SQL follows as

SNR ∝ √(PT/hf),

where h is the Planck constant and f is the measurement frequency. Improving precision therefore requires more power, a longer integration time, or a lower frequency. The square root is the expensive part of the relation: each factor of ten in signal-to-noise ratio costs a factor of one hundred in power or in time.

Ordinary EMC instruments operate nowhere near this limit, and it is worth being precise about where the SQL actually binds:

• Room-temperature spectrum analyzers and EMI receivers: The noise floor is set by the noise figure of the input stage and by the thermal noise of the source impedance, many orders of magnitude above the quantum limit. A quantum limit quoted for a conducted-emissions measurement at 30 MHz carries no practical meaning.
• Cryogenic microwave amplifiers: Here the limit is real and nearly reached. A linear phase-preserving amplifier must add at least half a quantum of noise referred to its input, which corresponds to a noise temperature of approximately 0.048 kelvin for every gigahertz of signal frequency, or about 0.3 kelvin at 6 GHz. Josephson parametric amplifiers operate within a small factor of that bound.
• Interferometric phase measurements: Phase resolution at low photon flux is SQL-limited, which is why squeezed light entered gravitational-wave interferometry before it entered any other measurement discipline.
• Millimeter-wave and terahertz radiometry: As the photon energy climbs toward kT, the quantum contribution to receiver noise stops being a footnote.

Importantly, the SQL is not a fundamental limit; it can be surpassed using squeezed states or entangled photons, which redistribute quantum uncertainty to favor measurement of one variable at the expense of another.

Thermal Noise and Quantum Noise Transition

The competition between thermal and quantum noise is characterized by comparing thermal energy kT to photon energy hf. At room temperature, taken as 300 K, the thermal energy kT equals the photon energy hf at approximately 6.3 THz. Below that frequency thermal noise dominates; above it quantum noise dominates.

At cryogenic temperatures the crossover frequency falls in proportion to the temperature, since it is simply kT/h. At 4 K, near the boiling point of liquid helium, it lies at approximately 83 GHz. At 20 mK, a typical mixing-chamber temperature in a dilution refrigerator, it falls to approximately 420 MHz, so every microwave control and readout signal in a quantum processor sits in the quantum regime.

This has practical implications:

• Cryogenic amplifiers: Amplifiers operating at cryogenic temperatures can approach quantum-limited noise performance at microwave frequencies.
• Quantum computing EMC: At the millikelvin temperatures of quantum processors, even relatively low-frequency signals (few GHz) are in the quantum regime.
• Sensitive receivers: Space-borne or cryogenically cooled receivers must consider quantum noise for accurate sensitivity predictions.

Photon Energy at Different Frequencies

The energy of a single photon is E = hf, where h is the Planck constant, fixed exactly at 6.626 070 15 × 10⁻³⁴ J·s by the 2019 revision of the SI, and f is the frequency. This yields extremely small energies at radio frequencies:

• At 1 GHz: E = 6.6 × 10⁻²⁵ J, approximately 0.004 meV
• At 10 GHz: E = 6.6 × 10⁻²⁴ J, approximately 0.041 meV
• At 100 GHz: E = 6.6 × 10⁻²³ J, approximately 0.41 meV
• At 1 THz: E = 6.6 × 10⁻²² J, approximately 4.1 meV

For comparison, the thermal energy at room temperature is kT approximately 26 meV, vastly exceeding any microwave photon energy. The more useful way to state the cooling requirement is to express the photon energy as a temperature: hf/k is 0.48 K at 10 GHz. The mean thermal occupancy of that mode, 1/(e^hf/kT − 1), is still 1.6 photons at 1 K and 0.62 photons at 0.5 K, and it falls below 0.01 only near 100 mK. Individual microwave photons therefore become distinguishable from the thermal background at temperatures of tens to a few hundred millikelvin, not at one kelvin.

Power levels make the same point from the other direction. At 10 GHz each photon carries 6.6 × 10⁻²⁴ J, so a power of 1 femtowatt amounts to roughly 150 million photons per second, while a flux of one photon per second is a power of only 6.6 × 10⁻²⁴ W. Counting individual microwave photons therefore means resolving energies some eight orders of magnitude below the femtowatt floor of a good conventional receiver, which is why the task demands a superconducting detector together with cryogenic suppression of the thermal background.

Superconducting Single Photon Detectors

Several superconducting device architectures enable single photon detection at microwave frequencies:

• Superconducting nanowire single photon detectors (SNSPDs): A narrow superconducting wire biased near its critical current absorbs an incoming photon, creating a local hotspot that temporarily disrupts superconductivity. This produces a measurable voltage pulse. SNSPDs achieve near-unity detection efficiency at optical wavelengths and are being extended to longer wavelengths.
• Transition edge sensors (TES): These devices operate at the sharp transition between superconducting and normal states. Photon absorption causes small temperature increases that produce measurable resistance changes. TES detectors offer energy resolution enabling photon counting and energy measurement simultaneously.
• Superconducting qubit detectors: Specially designed superconducting qubits can absorb microwave photons and change quantum state, enabling detection through qubit readout. These detectors operate at the single photon level with high efficiency for photons in resonance with the qubit transition.
• Josephson junction detectors: Josephson junctions in various configurations can detect single microwave photons through quantum processes. Photon-assisted tunneling and other effects produce measurable signals from individual photon absorption.

Applications in EMC Measurement

Single photon detection capabilities have potential applications in EMC:

• Ultimate sensitivity receivers: Single photon counters represent the ultimate in receiver sensitivity. For detecting extremely weak emissions from well-shielded equipment or at large distances, photon counting approaches could provide capabilities beyond conventional receivers.
• Quantum-limited EMI measurements: Establishing the true quantum limit of EMI measurements enables validation of measurement equipment and provides benchmarks for ultra-sensitive applications.
• Single event detection: In systems where individual electromagnetic events (such as particle impacts or quantum transitions) generate single or few photons, single photon detection enables direct observation of these events.

Single photon detection at microwave frequencies requires cryogenic temperatures, in practice the tens of millikelvin of a dilution refrigerator, along with sophisticated readout electronics, which limits practical application to laboratories that already operate such equipment. Continued development may enable more accessible implementations.

Nitrogen-Vacancy Centers in Diamond

Nitrogen-vacancy (NV) centers are point defects in diamond crystals consisting of a nitrogen atom adjacent to a vacant lattice site. These defects have remarkable properties for electromagnetic sensing:

• Magnetic field sensing: NV centers have spin states that can be optically initialized, manipulated, and read out. External magnetic fields shift the energy levels through the Zeeman effect, enabling precise magnetic field measurement. Ensemble NV magnetometers reach sub-picotesla per root hertz sensitivity in broadband laboratory measurements, and narrowband radio-frequency configurations have demonstrated tens of femtotesla per root hertz. Those are research results obtained with large sensing volumes and heavily optimized optics; compact and chip-scale NV sensors deliver nanotesla to picotesla per root hertz.
• Electric field sensing: The NV center's electronic states are also sensitive to electric fields, enabling electric field measurement with nanoscale spatial resolution. This capability is useful for mapping fields near circuits and devices.
• Temperature sensing: The zero-field splitting of NV centers has temperature dependence that enables precision thermometry. Combined with magnetic and electric field sensing, this allows multiparameter characterization.
• Advantages for EMC: NV sensors offer room-temperature operation, optical readout without electrical connections, nanoscale spatial resolution, and wide dynamic range. They can probe fields in locations inaccessible to conventional probes and with minimal perturbation.
• Applications: Near-field mapping of IC emissions, current imaging in circuits, magnetic material characterization, and precision field calibration are emerging applications of NV sensing in EMC-related fields.

Atomic Magnetometers

Atomic magnetometers use the response of atomic vapor to magnetic fields to achieve extremely sensitive field measurements:

• Optically pumped magnetometers: Polarized laser light optically pumps atoms into aligned spin states. Magnetic fields cause precession of these states, detectable through changes in the optical properties of the vapor. Sensitivity at the femtotesla per root hertz level is achievable, with commercial sensors reaching the low tens of femtotesla.
• SERF magnetometers: Spin-exchange relaxation-free (SERF) magnetometers operate at high atomic density and very low field, where rapid spin-exchange collisions no longer broaden the magnetic resonance, which removes the dominant relaxation channel. The best of them are genuinely sub-femtotesla instruments: a potassium-vapor gradiometer has reached 160 attotesla per root hertz over a measurement volume of 0.45 cubic centimeter, matching or surpassing a cryogenic SQUID. That figure belongs to a heavily shielded laboratory apparatus with a large sensing cell, and sensitivity improves with cell volume, so the commercial and wearable sensors built around cells a few millimeters across operate instead in the range of several femtotesla to tens of femtotesla per root hertz.
• Scalar vs. vector measurements: Different configurations measure either the magnitude of the field (scalar) or components along specific axes (vector). Total-field magnetometers are insensitive to orientation, while vector magnetometers provide directional information.

For EMC applications, atomic magnetometers offer non-contact field measurement with extraordinary sensitivity. However, their sensitivity to all magnetic fields (not just the signal of interest) requires careful magnetic shielding or gradiometer configurations to reject environmental noise.

Rydberg Atom Electric Field Sensors

Rydberg atoms are atoms excited to high principal quantum numbers, with electrons in orbits far from the nucleus. These atoms have extreme sensitivity to electric fields due to their large electric dipole moments:

• Operating principle: RF or microwave electric fields drive transitions between Rydberg states. These transitions are detected through electromagnetically induced transparency (EIT), where the field-induced transitions modify the transmission of a probe laser beam.
• Self-calibrating: Because the response depends on atomic properties that are identical for every atom of a species and do not age, a Rydberg measurement is traceable to atomic structure and to an optical frequency rather than to a calibrated artifact, and it cannot drift in the way a calibrated antenna or a field probe does. One qualification governs the accuracy achieved in practice: the atoms report the field inside the vapor cell, and the dielectric walls of that cell perturb the incident field, so relating the atomic reading to a free-space field strength requires the cell to be characterized or engineered out. That, rather than the atomic physics, limits present Rydberg field probes.
• Frequency coverage: By selecting different Rydberg states, the resonant frequency can be tuned across a remarkably broad span. The well-established range for resonant electrometry runs from roughly 500 MHz to 1 THz, and separate schemes, notably heterodyne detection against a local oscillator applied to the vapor, have carried demonstrations down through the kilohertz range and below one hertz. The span belongs to the technology as a whole rather than to any single sensor head, since each region calls for different Rydberg states and a different readout arrangement.
• EMC applications: Rydberg sensors show promise for antenna calibration, field strength measurements, and as transfer standards. Their self-calibrating nature and potential for high accuracy address key challenges in EMC metrology.

Comparison of Quantum Sensors

Different quantum sensors offer distinct advantages:

Selecting the appropriate sensor technology depends on the measurement requirements including frequency range, field type (electric or magnetic), required sensitivity, spatial resolution, and practical constraints such as operating temperature and setup complexity.

Entanglement Basics

Entangled particles share a single quantum state, so measurement outcomes on the two are correlated more strongly than any classical model of separate objects permits, however far apart they are. The correlation carries no signal: nothing done to one particle changes the statistics observed at the other, which is why entanglement transmits no information by itself and why every protocol built on it also needs a classical channel. For electromagnetic applications, entangled photon pairs are the most relevant case.

• Generation: Entangled photon pairs are typically generated through spontaneous parametric down-conversion in nonlinear crystals. A single high-energy photon splits into two lower-energy photons with correlated properties (polarization, timing, frequency).
• Correlations: Entangled photons exhibit correlations stronger than classically possible. Measurements on one photon are correlated with measurements on the other, enabling applications that exploit these correlations.
• Fragility: Entanglement is easily disrupted by interaction with the environment (decoherence). Maintaining entanglement requires careful isolation from noise sources, making EMC relevant to quantum systems.

Quantum-Enhanced Sensing

Entanglement can improve measurement precision beyond the standard quantum limit:

• Squeezed states: While not strictly entangled, squeezed states redistribute quantum uncertainty between conjugate variables. Squeezing the uncertainty in the measured variable below the SQL improves sensitivity at the expense of increased uncertainty in the conjugate variable.
• NOON states: An entangled state with N photons in a superposition of all in one arm and all in the other allows phase estimation whose uncertainty falls as 1/N, the Heisenberg limit, rather than as one over the square root of N, which is the best a coherent state of the same mean photon number achieves. The gain in precision at N photons is therefore a factor equal to the square root of N, a hundredfold at N of 10,000. Such states are also extremely fragile, since the loss of a single photon erases the phase information entirely, which is why the scaling has been demonstrated only for small N.
• Quantum illumination: In quantum illumination, entangled signal and idler beams are generated together. The signal probes the object while the idler is stored, and a joint measurement of the returned signal against the stored idler detects the object in a background that would defeat an equivalent classical probe. The counterintuitive part is that the loss and the thermal noise of the round trip destroy the entanglement itself: enough correlation survives to confer an advantage, but only in the regime of low reflectivity and strong background, and only if the receiver measures the two beams jointly. Measuring each beam separately discards the advantage completely, and building that joint receiver is the hard part of the idea.

These approaches have been demonstrated in laboratory settings but face challenges in practical implementation, particularly the difficulty of maintaining quantum correlations in the presence of loss and noise.

Secure Communication Implications

Quantum key distribution (QKD) uses quantum mechanical principles to enable provably secure communication, with implications for EMC:

• Physical layer security: QKD provides security based on physics rather than computational assumptions. Any eavesdropping attempt disturbs the quantum states, alerting legitimate parties to the intrusion.
• EMC for QKD systems: QKD systems, particularly those using single photons, are sensitive to electromagnetic interference that could cause errors or create side channels. EMC design of QKD equipment must address both the sensitive quantum components and the classical electronics.
• Side channel attacks: EMI from QKD equipment might leak information about the quantum states being processed, creating vulnerabilities. EMC measures to prevent unintended emissions are part of secure QKD system design.

Qubit Sensitivity to Electromagnetic Fields

Quantum bits (qubits) store information in quantum superposition states that are exquisitely sensitive to environmental perturbations:

• Superconducting qubits: The most common qubit implementation uses superconducting circuits operating at millikelvin temperatures. These qubits have transition frequencies in the range of 4 GHz to 8 GHz, which makes them sensitive to microwave fields near those frequencies. Stray fields can cause unwanted transitions, dephasing, or energy relaxation.
• Trapped ion qubits: Ions trapped in electromagnetic fields encode quantum information in their internal states. Electric field noise causes motional heating that degrades gate fidelity. Magnetic field fluctuations shift energy levels, causing dephasing.
• Spin qubits: Electron or nuclear spins in semiconductors or defects are sensitive to both electric and magnetic field fluctuations. Charge noise affects spin states through spin-orbit coupling, while magnetic noise directly couples to the spin.
• Photonic qubits: Qubits encoded in photon states are less sensitive to low-frequency electromagnetic fields but are affected by loss and scattering from electromagnetic sources in the optical path.

Decoherence Mechanisms

Electromagnetic interference contributes to qubit decoherence through several mechanisms:

Energy relaxation (T₁): Electromagnetic fields at the qubit transition frequency can drive the qubit from its excited state to its ground state, dissipating the information it holds. The T₁ time characterizes how long a qubit keeps its energy state; in current superconducting devices it runs from tens to a few hundred microseconds.

Dephasing (T₂): Low-frequency field fluctuations shift the qubit transition frequency at random, so phase errors accumulate in a superposition state. The T₂ time characterizes how long coherence survives. It cannot exceed 2T₁, and wherever pure dephasing dominates it falls well short of that ceiling, which is why low-frequency magnetic and charge noise receives as much attention as loss at the qubit frequency.

• Leakage: For multilevel systems, fields can induce transitions to states outside the computational subspace, causing leakage errors that are difficult to correct.
• Crosstalk: In multi-qubit systems, control fields intended for one qubit can affect neighboring qubits, causing crosstalk errors. This internal EMC challenge requires careful design of qubit layout and control systems.

The extreme sensitivity of qubits means that even fields many orders of magnitude below conventional EMI limits can cause significant errors. A microwave field that would be negligible for any classical circuit can completely destroy quantum coherence.

Shielding and Isolation Requirements

Quantum computers employ extraordinary EMC measures:

• Magnetic shielding: Nested layers of high-permeability alloy attenuate external magnetic fields, mu-metal at room temperature and cryoperm inside the cold volume, the latter because ordinary mu-metal loses much of its permeability at liquid-helium temperatures. A nested set of cans delivers low-frequency shielding factors of roughly 10³ to 10⁵, and purpose-built magnetically shielded rooms exceed 10⁶. Active cancellation coils driven from fluxgate or atomic sensors suppress the residual field and its slow drift further.
• RF shielding: Multiple nested Faraday cages provide radio frequency isolation. The cryogenic environment is enclosed in continuous metal shields, with all penetrations carefully filtered.
• Filtering: Every wire entering the shielded volume passes through multiple stages of filtering. Cryogenic filters achieve attenuation exceeding 100 dB at microwave frequencies while passing DC and low-frequency control signals.
• Infrared filtering: Thermal radiation is electromagnetic, and for a superconducting qubit it is interference of the most damaging kind. An infrared photon carries far more energy than the superconducting gap, so absorbing one breaks Cooper pairs and leaves behind quasiparticles that relax the qubit. Infrared-absorbing filters and light-tight cold enclosures block that radiation while the microwave control signals pass.
• Vibrational isolation: While not electromagnetic, vibration can modulate electromagnetic fields and couple to qubits through various mechanisms. Vibration isolation is part of the overall isolation strategy.

Internal EMC Challenges

Beyond external interference, quantum computers face internal EMC challenges:

• Control electronics: The classical electronics that control and read out qubits generate noise that can couple to the quantum processor. Careful layout, shielding, and signal conditioning minimize this coupling.
• Signal routing: Control signals must reach qubits without excessive crosstalk or loss. Coaxial cables, waveguides, and on-chip transmission lines must be carefully designed for EMC.
• Ground loops: Multiple grounds at different potentials can create current loops that generate interfering magnetic fields. Careful grounding strategy avoids problematic loops.
• Thermal noise: Components at higher temperatures radiate thermal noise that can propagate to cryogenic stages. Attenuators and filters at intermediate temperature stages absorb this noise.

Meissner Effect Shielding

Superconductors in the Meissner state expel magnetic fields from their interior, exhibiting perfect diamagnetism. A superconducting shell creates a field-free region inside, regardless of external field strength up to the critical field.

• Type I superconductors: Below the critical field H_c, Type I materials hold the complete Meissner state with total field exclusion. Above H_c, superconductivity is destroyed. Lead is the Type I superconductor most often used for magnetic shielding.
• Type II superconductors: These materials have two critical fields. Below H_c1 they show the complete Meissner effect. Between H_c1 and H_c2, flux vortices penetrate the material while bulk superconductivity persists. Above H_c2, superconductivity is destroyed. Niobium and the high-temperature superconductors are Type II, and their tolerance of much larger fields is why they serve where a Type I material would already have been driven normal.
• Shielding factor: A closed superconducting shield screens changes in the ambient field almost perfectly, and it does so by geometry rather than by a permeability that can saturate, which is the decisive advantage over a ferromagnetic shield. What the shield holds constant, however, is the flux that already threaded it when it became superconducting, not zero field. Reported shielding factors for closed shields reach 10⁶ and beyond, with the practical limit set by seams, openings, and trapped flux rather than by the superconductor itself.
• Trapped flux: This is why the cool-down procedure, rather than the material, usually decides the performance achieved. Whatever field threads the shield as it passes through its transition temperature is frozen in, and no subsequent screening removes it. Cooling inside a conventional high-permeability shield or an active cancellation coil is the standard remedy, and techniques that expand a superconducting shell after it has cooled dilute the trapped flux further, which is how the lowest magnetic fields ever measured have been produced.

Dynamic Shielding Behavior

For time-varying fields, superconducting shields exhibit frequency-dependent behavior:

• Low frequency: AC magnetic fields induce supercurrents that oppose the field, providing effective shielding. The shielding is determined by the shield geometry and the induced current patterns.
• Field penetration: Although a superconductor has no DC resistance, a field still enters its surface over the London penetration depth, typically tens of nanometers. Unlike the skin depth of a normal metal, the London depth does not shrink as frequency rises; it is a material constant rather than a frequency-dependent one. A film only a few hundred nanometers thick therefore screens well, and making it thicker buys little.
• Pair-breaking limit: The lossless behavior ends when the photon energy reaches twice the superconducting gap, because each absorbed photon then breaks a Cooper pair. Taking the weak-coupling estimate 2Δ = 3.5kT_c, the threshold falls near 700 GHz for niobium and near 500 GHz for lead. Above it a superconducting surface performs no better than a good normal metal, which places a firm ceiling on both superconducting shielding and high-Q superconducting cavities.
• Flux flow: In Type II superconductors at fields between H_c1 and H_c2, moving flux vortices dissipate energy, which introduces an effective resistance and degrades AC shielding. Pinning the vortices, on defects deliberately introduced for the purpose, holds them in place and restores much of the performance.
• RF shielding: Below that threshold, superconducting shields present surface resistances orders of magnitude lower than any normal metal, which is what enables high-Q cavities and extremely effective RF shields.

Practical Superconducting Shields

Implementing superconducting shields involves several considerations:

• Material selection: Lead, with T_c = 7.2 K, is soft, easy to form, and gives Type I behavior. Niobium, with T_c = 9.2 K, is harder to work but tolerates a much larger field. The high-temperature superconductors, with T_c near 90 K for YBCO, allow liquid-nitrogen operation but resist fabrication into the seamless closed shapes that shielding wants.
• Thermal design: The shield must be maintained below its critical temperature, requiring cryogenic infrastructure. Temperature uniformity ensures no normal regions that would compromise shielding.
• Seam and joint design: Any seam or joint in the shield must maintain superconducting continuity. Welded or seamless construction is preferred. If joints are necessary, they must be designed to exclude current flow across normal-state regions.
• Penetrations: Any opening or penetration compromises shielding. Necessary penetrations use waveguide-below-cutoff principles, where the opening dimensions are small enough that electromagnetic fields cannot propagate through.

Superconducting shields find application in quantum computing, SQUID magnetometers, and ultra-sensitive physics experiments where the highest possible magnetic shielding is required.

SQUID Operating Principle

A SQUID consists of a superconducting loop interrupted by one or two Josephson junctions, weak links where supercurrent can tunnel between superconductors:

DC SQUID: A superconducting loop with two Josephson junctions. The maximum supercurrent through the device oscillates with the magnetic flux threading the loop, with a period equal to the magnetic flux quantum

Φ₀ = h/(2e) = 2.067 833 8 × 10⁻¹⁵ Wb,

a relation the SI treats as exact, since both h and e are defining constants with fixed values. Measuring this periodic response resolves flux changes of order 10⁻⁶ Φ₀ in a one-hertz bandwidth.

• RF SQUID: A superconducting loop with a single Josephson junction, inductively coupled to an RF tank circuit. The junction acts as a flux-dependent nonlinear inductance that modulates the tank circuit response. RF SQUIDs are simpler to fabricate but typically less sensitive than DC SQUIDs.
• Flux-to-voltage transfer: A SQUID converts magnetic flux to voltage with a transfer coefficient of order 100 microvolts per flux quantum. This steep response is what allows a room-temperature amplifier of ordinary noise performance to read out a flux change far below one flux quantum.
• Feedback operation: For practical measurements, SQUIDs operate in a flux-locked loop where feedback current maintains constant flux through the SQUID. The feedback current provides a linear measure of the input flux, extending dynamic range beyond the periodic SQUID response.

SQUID Sensitivity and Noise

SQUID noise is characterized by the flux noise spectral density, expressed in units of Φ₀ per root hertz:

White noise: Above a corner frequency, typically between 1 Hz and 100 Hz, SQUID noise is approximately white. Good DC SQUIDs reach a flux noise of order 10⁻⁶ Φ₀ per root hertz, which is about 2 × 10⁻²¹ Wb per root hertz. The corresponding energy sensitivity is

ε = Φ_n²/(2L),

so such a device with a loop inductance of 100 pH reaches approximately 2 × 10⁻³² J/Hz, some two hundred times the reduced Planck constant. The best low-temperature devices close much of the remaining gap, but the frequent claim that a practical SQUID already sits at the quantum limit overstates the case by one to two orders of magnitude.

• 1/f noise: At low frequencies, excess noise with approximately 1/f spectral dependence dominates. This noise arises from fluctuating defects in the junction barriers and limits low-frequency sensitivity.
• Environmental contributions: In practical installations, environmental magnetic noise often exceeds intrinsic SQUID noise. Gradiometer configurations, where multiple SQUID loops are connected in opposition, cancel uniform background fields while detecting local field gradients.

Every other sensitivity figure follows from that flux noise and from how the input circuit couples to the loop. A field measurement gives B_n = Φ_n/A_eff, so a flux noise of 10⁻⁶ Φ₀ per root hertz with the effective area of a few square millimeters that a multiturn superconducting flux transformer provides yields about 1 femtotesla per root hertz. A current measurement gives I_n = Φ_n/M_in, and since the mutual inductance of a practical input coil reaches at most tens of nanohenries, the current noise lands between tens of femtoamperes and a few picoamperes per root hertz; reaching below one femtoampere per root hertz would demand a mutual inductance of order a microhenry, which no real input coil offers. SQUID voltmeters resolve voltages at the picovolt level, but only for source impedances low enough to couple efficiently into that same input coil.

SQUID Applications in EMC

SQUIDs have several applications relevant to EMC:

• Magnetic field mapping: SQUIDs can map extremely weak magnetic fields from circuits and devices, revealing current distributions and identifying emission sources that are invisible to conventional probes.
• Low-frequency measurements: At frequencies below a few hertz, where conventional antennas are ineffective, SQUIDs provide sensitive detection of electromagnetic signals. This is relevant for measuring emissions from slowly varying sources.
• Shielding effectiveness testing: The extreme sensitivity of SQUIDs enables measurement of shielding effectiveness at levels far beyond conventional methods. This is valuable for characterizing shields for quantum computing and other ultra-sensitive applications.
• Current sensing: SQUID-based current sensors can detect currents in conductors without direct connection, with sensitivity far exceeding conventional current probes. This enables non-invasive measurement of currents in sensitive circuits.
• Noise source identification: The combination of high sensitivity and spatial resolution enables identification of weak noise sources that contribute to system performance limitations.

SQUID System EMC Considerations

SQUID systems themselves require careful EMC design:

• Magnetic shielding: SQUIDs must be operated inside magnetic shields to reduce environmental noise to levels below the SQUID intrinsic noise. Layered mu-metal and superconducting shields provide the required attenuation.
• RF shielding: High-frequency interference can cause noise or dysfunction through rectification at Josephson junctions. RF shields and filters protect the SQUID from radio frequency pickup.
• Vibration isolation: In magnetic field gradients, vibration modulates the flux through the SQUID pickup loop, creating noise. Low-vibration mounting and gradiometer configurations address this.
• Grounding and filtering: All connections to the SQUID system must be carefully filtered to prevent interference from electronics. Ground loops are particularly problematic due to the extreme sensitivity.

Josephson Voltage Standards

The AC Josephson effect relates voltage to frequency and to nothing else. A junction irradiated with microwaves at frequency f develops constant-voltage steps at

V = n f h/(2e) = n f Φ₀,

where n is an integer. The step spacing is therefore 2.0678 microvolts for every gigahertz of drive frequency, the reciprocal of the Josephson constant K_J = 2e/h = 483 597.848 4… gigahertz per volt. No property of the material, the geometry, or the temperature of the junction enters, which is exactly what makes the effect usable as a standard.

• Practical Josephson standards: A single junction produces only microvolts, so modern standards series-connect thousands of junctions to reach useful levels, typically 1 V to 10 V. International comparisons of such standards agree to a few parts in 10¹⁰, far beyond the reach of any other voltage reference.
• Programmable standards: Josephson arrays with individually addressable junction segments can generate arbitrary voltages within their range, enabling programmable voltage sources with quantum-level accuracy.
• Implications for EMC: The Josephson effect realizes the volt; it does not define it. The definition rests on the fixed values of the defining constants, a distinction the section on the redefined SI below sets out. EMC voltage measurements reach that realization through a chain of calibrations, in practice by way of a Zener reference that a national laboratory has compared against a Josephson array. Because the quantum standard itself neither drifts nor needs recalibration, the uncertainty of an EMC voltage measurement is set entirely by the transfer chain beneath it.

Quantum Hall Resistance Standard

The quantum Hall effect in two-dimensional electron systems provides a resistance standard built from the defining constants. When such a system in a strong magnetic field enters the quantum Hall regime, its Hall resistance takes the quantized values

R_H = R_K/i,

where i is the integer filling factor, the number of filled Landau levels, and R_K is the von Klitzing constant, R_K = h/e² = 25 812.807 45… ohms. The distinction between the two symbols matters. R_K is the fixed constant of the standard; R_H is what a particular device on a particular plateau actually shows, and it equals R_K only on the i = 1 plateau. Metrology normally works on the i = 2 plateau, at 12 906.403 7 ohms, which is the most robust plateau in the gallium arsenide heterostructures traditionally used.

• Practical implementation: Quantum Hall standards use high-mobility semiconductor heterostructures at cryogenic temperatures, typically below 1.5 K, in magnetic fields of several tesla, and they reproduce at the level of parts in 10⁹. Graphene has relaxed those conditions considerably: its quantum Hall plateaus are wide enough to survive at higher temperature and lower magnetic field than gallium arsenide allows, which brings a primary resistance standard within reach of laboratories outside the largest national institutes.
• Resistance scale: Through scaling and comparison techniques, usually a cryogenic current comparator bridging the quantum Hall device to 100 ohm and 1 kilohm working standards, the quantum Hall standard realizes and disseminates the practical resistance scale. Every calibrated resistance traces back to it.
• EMC implications: Precision EMC measurements involving impedances depend on accurate resistance standards. The quantum Hall standard provides the foundation for this traceability.

Redefined SI Units

Since 20 May 2019, the SI has defined all of its units by fixing the numerical values of seven defining constants. The consequences for electrical metrology are specific, and they are often stated loosely:

• The ampere is the base electrical unit. It is defined by fixing the elementary charge at e = 1.602 176 634 × 10⁻¹⁹ C, together with the definition of the second.
• The volt and the ohm are derived units: the volt is the watt per ampere, and the ohm is the volt per ampere. No quantum effect appears anywhere in their definitions.
• What the quantum effects do is realize those derived units. Because h and e are now exact, K_J = 2e/h and R_K = h/e² are exact numbers, so a Josephson array realizes the volt and a quantum Hall device realizes the ohm with no uncertainty contributed by the constants themselves.
• In practice the ampere is most often realized from those two, through Ohm's law, rather than the other way around. A single-electron-transport device realizing the ampere directly from e remains the less accurate route.

The revision also retired the conventional values that had served since 1990, K_J-90 = 483 597.9 gigahertz per volt and R_K-90 = 25 812.807 ohms. Each of those defined a unit slightly larger than its present SI counterpart: the 2019 value of K_J is smaller than K_J-90 by 106.665 parts in 10⁹, and the 2019 value of R_K is larger than R_K-90 by 17.793 parts in 10⁹. National laboratories therefore stepped their voltage and resistance scales up by those fractional amounts in 2019, roughly 0.11 microvolt per volt and 0.018 microhm per ohm. Both steps lie far below anything an EMC measurement can resolve, but they matter to the laboratories that calibrate the references EMC measurements depend on.

For EMC metrology the practical gain is long-term stability. A quantum realization does not drift, does not need recalibration against something higher up, and cannot be damaged or lost the way an artifact standard can, so the uncertainty of a traceable measurement is dominated entirely by the transfer chain between the quantum standard and the instrument on the bench.

Energy-Time Uncertainty

The energy-time relation ΔE Δt ≥ ℏ/2 limits the precision of measurements made in a finite time. It repays care, because the version usually quoted in instrumentation contexts is not the quantum relation at all:

Frequency measurement: Resolving two frequencies separated by Δf requires an observation time T of order

T ≈ 1/Δf.

This is the Fourier limit, and it is not quantum mechanical in the least: it follows from the wave nature of the signal and applies to a classical waveform recorded by a classical instrument. The exact coefficient depends on the window function and on how resolution is defined, but it is of order one, so writing the bound as 1/(2πΔf) understates the observation time actually needed by a factor of 2π. The strict uncertainty-relation form is a different statement about different quantities: for the standard deviations of the time and frequency distributions of a pulse it reads σ_tσ_f ≥ 1/(4π), which is what ΔE Δt ≥ ℏ/2 becomes under E = hf. A spectrum analyzer's sweep time obeys the Fourier limit, not that bound.

• Spectral analysis: The resolution-bandwidth-time relationship in a spectrum analyzer follows directly from the Fourier limit. A sweep must dwell at least 1/RBW in each resolution cell and must cover Span/RBW cells, so the sweep time scales as the span divided by the square of the resolution bandwidth. Narrowing the resolution bandwidth by a factor of ten therefore lengthens the sweep a hundredfold, which is why EMI scans over a wide span with a 9 kHz or 120 kHz CISPR bandwidth take the time they do.
• Pulse measurements: Ultra-short pulses have broad frequency content due to this uncertainty. Characterizing such pulses requires measurement systems with corresponding bandwidth.

Number-Phase Uncertainty

For electromagnetic fields, the number of photons and the phase of the field are conjugate variables with minimum uncertainty product:

• Amplitude measurements: Determining field amplitude (related to photon number) with high precision causes increased phase uncertainty. The standard quantum limit for amplitude measurement arises from this uncertainty.
• Phase measurements: Determining field phase with high precision causes increased amplitude uncertainty. Interferometric measurements are ultimately limited by this trade-off.
• Squeezing: Squeezed states can reduce uncertainty in one variable below the SQL at the expense of increased uncertainty in the conjugate variable. This enables measurements that beat the SQL for specific variables.

Practical Implications for EMC

While quantum limits are often far below practical sensitivity, they become relevant in specific situations:

• Cryogenic systems: At millikelvin temperatures, thermal noise drops below quantum noise for microwave frequencies, making quantum limits directly relevant.
• Ultra-sensitive measurements: The best SQUID and atomic magnetometers come within a few orders of magnitude of their quantum noise limits, and in the atomic case spin-projection noise is already the binding constraint. Further improvement there calls for spin squeezing or entanglement rather than better engineering of the readout.
• High-frequency measurements: At terahertz frequencies and above, single photon energies become significant, and quantum effects become more important even at room temperature.
• Standards and calibration: Quantum metrology standards approach quantum limits and define the accuracy achievable for traceable measurements.