Electronics Guide

The Heisenberg Uncertainty Principle in Electromagnetic Systems

The uncertainty principle, often expressed as delta-x times delta-p greater than or equal to h-bar/2, has direct implications for electromagnetic measurements. For electromagnetic fields, the conjugate variables are the electric and magnetic field quadratures, analogous to position and momentum for particles.

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

Energy fluctuations: The ground state (vacuum state) has energy E = (1/2)hf, representing irreducible zero-point energy. These vacuum fluctuations contribute noise to any measurement of field amplitude.

Photon number uncertainty: The minimum uncertainty in photon number for a coherent state is sqrt(n), where n is the mean photon number. This shot noise represents a fundamental limit on the precision of amplitude measurements.

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 to thermal noise (kT >> hf at room temperature for RF). However, at microwave frequencies approaching 10 GHz and above, or at cryogenic temperatures, quantum noise becomes significant and eventually dominant.

Standard Quantum Limit

The standard quantum limit (SQL) represents the measurement precision achievable using classical states of light (coherent states) and conventional measurement techniques. For amplitude measurements, the SQL corresponds to shot noise: the uncertainty in photon number scales as sqrt(n), giving a signal-to-noise ratio proportional to sqrt(n).

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

The signal-to-noise ratio at the SQL scales as sqrt(PT/hf), where h is Planck's constant and f is the measurement frequency. This relationship shows that improving measurement precision requires increasing power, increasing integration time, or decreasing frequency.

The SQL applies to:

• Spectrum analyzers: The minimum detectable signal for a given resolution bandwidth approaches the SQL at high frequencies or low temperatures.
• Network analyzers: Phase and amplitude measurement precision is SQL-limited at low signal levels.
• EMI receivers: Ultimate sensitivity for detecting weak emissions approaches quantum limits.

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 (T approximately 300 K), thermal energy kT equals photon energy hf at approximately 6 THz. Below this frequency, thermal noise dominates; above it, quantum noise dominates.

At cryogenic temperatures, the crossover frequency decreases proportionally. At 4 K (liquid helium temperature), the crossover is around 80 GHz. At 20 mK (dilution refrigerator temperatures used in quantum computing), the crossover is below 1 GHz, meaning quantum noise dominates at microwave frequencies.

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 Planck's constant (6.626 x 10^-34 J*s) and f is frequency. This yields extremely small energies at radio frequencies:

• At 1 GHz: E = 6.6 x 10^-25 J (approximately 0.004 meV)
• At 10 GHz: E = 6.6 x 10^-24 J (approximately 0.04 meV)
• At 100 GHz: E = 6.6 x 10^-23 J (approximately 0.4 meV)

For comparison, thermal energy at room temperature is kT approximately 25 meV, vastly exceeding microwave photon energies. Only at temperatures below about 1 K do microwave photons become individually distinguishable from thermal background.

The challenge of single microwave photon detection is further illustrated by power levels: at 10 GHz, a power of 1 femtowatt corresponds to approximately 1.5 million photons per second. Detecting individual photons requires sensitivity many orders of magnitude beyond conventional receivers.

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.

Current single photon detection at microwave frequencies requires cryogenic temperatures (typically below 1 K) and sophisticated readout electronics, limiting practical applications. However, 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. Sensitivity approaching femtotesla per root-Hertz has been demonstrated.

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 better than femtotesla per root-Hertz is achievable.

SERF magnetometers: Spin-exchange relaxation-free (SERF) magnetometers operate at high atomic density where spin-exchange collisions average out, eliminating a major noise source. SERF magnetometers achieve attotesla sensitivity, among the most sensitive magnetic field detectors ever created.

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 only on atomic properties (which are identical for all atoms of the same species), Rydberg sensors are inherently self-calibrating. This eliminates calibration drift and enables traceable measurements.

Frequency coverage: By selecting different Rydberg states, the sensitive frequency can be tuned from DC to hundreds of GHz. This wide tunability enables a single technology to cover diverse frequency ranges.

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 quantum state such that measuring one particle instantaneously affects the measurement outcomes for the other, regardless of the distance between them. For electromagnetic applications, entangled photon pairs are most relevant.

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: Entangled states with N photons in a superposition of all in one mode and all in another mode enable phase measurements with precision scaling as 1/N rather than 1/sqrt(N). This Heisenberg scaling provides quadratic improvement over classical approaches.

Quantum illumination: In quantum illumination, entangled signal and idler photons are generated together. The signal probes an object while the idler is retained. Correlating the reflected signal with the stored idler can reveal the presence of the object even when thermal noise would overwhelm classical detection.

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 4-8 GHz range, making them sensitive to microwave fields near these 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 induce transitions from excited to ground state, dissipating quantum information. The T1 time characterizes how long a qubit maintains its energy state.

Dephasing (T2): Low-frequency field fluctuations cause random variations in qubit transition frequency, accumulating phase errors in superposition states. The T2 time characterizes how long quantum coherence persists.

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: Multiple layers of high-permeability magnetic shielding (mu-metal, cryoperm) attenuate external magnetic fields. Shielding factors exceeding 10^6 are common. Active cancellation systems further reduce residual fields.

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, while not strictly electromagnetic interference, represents a noise source for cryogenic qubits. IR filters block this radiation while passing microwave control signals.

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 Hc, Type I materials are in the complete Meissner state with total field exclusion. Above Hc, superconductivity is destroyed. Lead is a common Type I superconductor used for magnetic shielding.

Type II superconductors: These materials have two critical fields. Below Hc1, they exhibit complete Meissner effect. Between Hc1 and Hc2, flux vortices penetrate the material while bulk superconductivity persists. Above Hc2, superconductivity is destroyed. Niobium and high-temperature superconductors are Type II.

Shielding factor: A properly prepared superconducting shield provides essentially perfect shielding of static magnetic fields, limited only by openings, seams, and any trapped flux. Shielding factors exceeding 10^9 are achievable.

Trapped flux: Magnetic field present when the superconductor cools through its transition temperature can be trapped inside, degrading shield performance. Proper cool-down in low ambient field is essential.

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.

Skin effect: Even though superconductors have zero DC resistance, AC fields penetrate into the surface through the London penetration depth, typically tens of nanometers. This extremely small penetration depth provides excellent high-frequency shielding in thin films.

Flux flow: In Type II superconductors at fields between Hc1 and Hc2, moving flux vortices dissipate energy, introducing an effective resistance that affects AC shielding. Pinning of vortices reduces flux flow.

RF shielding: At radio frequencies, superconducting shields can provide surface resistances orders of magnitude lower than normal metals, enabling high-Q cavities and extremely effective RF shields.

Practical Superconducting Shields

Implementing superconducting shields involves several considerations:

Material selection: Lead (Tc = 7.2 K) is easy to work with and provides Type I behavior. Niobium (Tc = 9.2 K) is harder but has higher critical field. High-temperature superconductors (Tc approximately 90 K for YBCO) allow higher operating temperatures but are harder to fabricate into seamless shields.

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 magnetic flux through the loop, with period equal to the flux quantum (Phi_0 = h/2e approximately 2.07 x 10^-15 Wb). Measuring this periodic response enables detection of flux changes as small as 10^-6 Phi_0.

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: SQUIDs convert magnetic flux to voltage with transfer functions of hundreds of microvolts per flux quantum. This steep response enables sensitive detection of small flux changes.

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, typically measured in units of Phi_0 per root-Hertz:

White noise: At frequencies above a corner frequency (typically 1-100 Hz), SQUID noise is approximately white (frequency-independent). For good DC SQUIDs, white noise levels below 10^-6 Phi_0/sqrt(Hz) are achievable, corresponding to energy sensitivities approaching the quantum limit.

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.

When coupled to appropriate input circuits, SQUIDs can detect magnetic fields with sensitivities below 1 femtotesla per root-Hertz, currents below 1 femtoampere per root-Hertz, and voltages below 1 picovolt per root-Hertz.

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 provides a fundamental relationship between frequency and voltage: V = nf * h/(2e), where n is an integer, f is frequency, h is Planck's constant, and e is the electron charge. When a Josephson junction is irradiated with microwaves at frequency f, it develops quantized voltage steps at integer multiples of approximately 2.07 microvolts times the frequency in gigahertz.

Practical Josephson standards: Modern Josephson voltage standards use arrays of thousands of junctions to generate voltages at practical levels (typically 1-10 V). These standards achieve uncertainties below 10^-10, far better than 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 standard ultimately defines the volt through fundamental constants. EMC voltage measurements are traceable to this quantum standard through calibration chains. The extreme stability of Josephson standards provides confidence in measurement system calibration.

Quantum Hall Resistance Standard

The quantum Hall effect in two-dimensional electron systems provides a resistance standard based on fundamental constants: R_H = h/e^2 approximately 25.8 kilohms. When a 2D electron system in a strong magnetic field exhibits the quantum Hall effect, its Hall resistance is quantized to exactly this value divided by an integer.

Practical implementation: Quantum Hall standards use high-mobility semiconductor heterostructures at cryogenic temperatures (typically below 1.5 K) in magnetic fields of several tesla. Resistance values are stable and reproducible at the parts per billion level.

Resistance scale: Through scaling and comparison techniques, the quantum Hall standard defines the practical resistance scale. All calibrated resistances are ultimately traceable to this quantum reference.

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 2019, the SI system defines electrical units through fixed values of fundamental constants:

• The ampere is defined through the elementary charge e
• The volt is derived from the ampere and the Josephson constant K_J = 2e/h
• The ohm is derived through the von Klitzing constant R_K = h/e^2

This redefinition means that Josephson and quantum Hall standards now define the electrical units exactly, rather than being measurements of those units. For EMC metrology, this provides improved long-term stability and eliminates uncertainties associated with realizing units from mechanical standards.

Energy-Time Uncertainty

The energy-time uncertainty relation delta-E times delta-t greater than or equal to h-bar/2 limits the precision of measurements made in finite time:

Frequency measurement: Measuring a frequency to precision delta-f requires observation time at least approximately 1/(2*pi*delta-f). This relationship is not strictly quantum mechanical but reflects the wave nature of electromagnetic signals.

Spectral analysis: The resolution-bandwidth-time relationship in spectrum analysis reflects this fundamental trade-off. Improved frequency resolution requires longer measurement time, while fast measurements have limited frequency resolution.

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 most sensitive SQUID and atomic magnetometers approach quantum noise limits. Further improvement requires quantum techniques such as squeezing or entanglement.

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.