When researchers first cooled superconducting loops to near absolute zero and threaded them with Josephson junctions, they discovered something remarkable. These devices could detect magnetic fields so faint that a hummingbird's heartbeat measured from across a room would register as a thunderclap by comparison. Yet this extraordinary sensitivity came with a cruel irony: the instruments themselves whispered with enough noise to obscure the very signals they pursued. The challenge of silencing these quantum whispers has consumed five decades of research, driving innovations that extend from the physics of individual electrons to the architecture of entire measurement facilities.
Superconducting quantum interference devices operate at the boundary between classical and quantum realms, where magnetic flux becomes quantized and screening currents flow without resistance. A typical SQUID consists of a superconducting loop interrupted by one or two Josephson junctions, which are thin insulating barriers that allow Cooper pairs to tunnel between superconducting electrodes. When magnetic flux threads the loop, the device produces voltage oscillations with a period of exactly one flux quantum, Φ₀ = h/2e ≈ 2.07 × 10⁻¹⁵ Wb, where h represents Planck's constant and e denotes the elementary charge.
The sensitivity of modern SQUIDs approaches fundamental quantum limits. Low-temperature devices operating at liquid helium temperatures around 4.2 K achieve magnetic field sensitivities approaching 3 fT/√Hz in the white noise region, while high-temperature SQUIDs cooled with liquid nitrogen reach 10 to 30 fT/√Hz. For context, Earth's magnetic field measures approximately 50 μT, meaning these sensors can detect variations smaller than one part in 10¹⁶ of the background field. Such performance enables applications ranging from mapping neural activity in the brain through magnetoencephalography to searching for magnetic monopoles left over from the Big Bang.
Fundamental Noise Mechanisms in SQUID Physics
Understanding noise in SQUID receivers requires grasping how thermal fluctuations, quantum effects, and material imperfections conspire to generate unwanted signals. The intrinsic noise of a SQUID can be characterized by its spectral density, which describes how noise power distributes across frequency. This spectral density typically exhibits two distinct regions: white noise dominating at higher frequencies and one-over-f noise increasing toward lower frequencies.
White noise in dc SQUIDs arises primarily from thermal Johnson-Nyquist noise in the resistive shunts that stabilize the Josephson junctions against hysteresis. The theoretical expression for the flux noise spectral density in the white noise region follows Sᶲ(f) ≈ 16kᵦTL²/R, where kᵦ denotes Boltzmann's constant, T represents temperature, L indicates loop inductance, and R signifies the junction shunt resistance. This relationship reveals something profound: reducing noise requires either decreasing temperature, minimizing loop inductance, or increasing shunt resistance. Yet each adjustment carries trade-offs. Smaller inductance reduces flux-to-voltage coupling, while higher resistance increases junction heating and may push the device into hysteretic regimes.
The energy resolution provides another way to quantify SQUID performance, defined as ε = Sᶲ/(2L). For optimally designed low-temperature SQUIDs with βL = 2πLIc/Φ₀ ≈ 1, where Ic represents the critical current, energy resolution approaches ε ≈ ħ, the reduced Planck constant. This quantum limit represents the point where the SQUID's noise energy per unit bandwidth equals the minimum allowed by Heisenberg's uncertainty principle. Actual devices typically achieve energy resolutions between 50ħ and 500ħ, leaving room for continued improvement.
Low-frequency noise presents a more insidious challenge. The flux noise spectral density at frequencies below several kilohertz typically follows Sᶲ(f) = A²/(f/1 Hz)^α, where α ranges from 0.6 to 1.2 and A measures approximately 1 μΦ₀/√Hz for typical thin-film devices. This one-over-f character means noise grows dramatically as frequency decreases, eventually overwhelming signals in applications requiring long integration times or slow measurements. Two primary mechanisms contribute to this behavior: critical current fluctuations in the Josephson junctions and magnetic flux noise from spins in surface materials.
Critical current noise originates from charge trapping and release events in the tunnel barrier of Josephson junctions. Electrons become trapped at defect sites in the aluminum oxide barrier, creating random telegraph signals that switch between discrete states. When many such traps with different characteristic time constants superpose, they generate the observed one-over-f spectrum. The magnitude of this noise scales inversely with junction area, motivating the use of large junctions when possible.
Magnetic flux noise arises from fluctuating magnetic moments near the SQUID surface. Recent research attributes much of this noise to paramagnetic spins in surface oxides, defects in the superconducting films, or adsorbed molecules. Each spin flips stochastically between up and down states according to thermal activation over energy barriers distributed across a broad range. The resulting noise couples into the SQUID through the magnetic fields these spins produce. Measurements show flux noise amplitude roughly independent of SQUID geometry, suggesting a surface rather than volume origin. The typical areal density of noise-producing spins estimated from measurements approaches 5 × 10¹⁷ m⁻², though the physical identity of these spins remains debated.
Gradiometric Configurations and Spatial Filtering
One powerful approach to noise reduction exploits the spatial distribution of interference sources. Environmental magnetic noise from power lines, motors, and distant machinery typically produces relatively uniform fields across the dimensions of a SQUID sensor, while biological or geological signals of interest often originate from sources nearby. Gradiometer configurations capitalize on this distinction by measuring spatial derivatives of the magnetic field rather than the field itself.
A first-order axial gradiometer consists of two pickup loops with equal areas wound in opposite senses and connected in series, separated by a baseline distance b. When a uniform external field B passes through both loops, the induced fluxes cancel. However, a nearby magnetic dipole source creates field gradients that produce different flux values in each loop, generating a net signal proportional to ∂B/∂z. The common-mode rejection ratio, which quantifies how effectively the gradiometer suppresses uniform fields relative to gradients, can exceed 10⁴ for carefully balanced devices.
The response function of a gradiometer to a point dipole source varies as 1/z⁴, where z represents distance from the sensor, compared to 1/z³ for a simple magnetometer. This steeper falloff provides excellent spatial localization, making gradiometers ideal for applications like biomagnetic imaging where sources lie close to the sensor surface while interference originates farther away. The baseline length b determines the gradiometer's spatial resolution: shorter baselines enhance sensitivity to nearby sources but reduce common-mode rejection, while longer baselines improve interference suppression at the cost of sensitivity.
Second-order gradiometers measure the second spatial derivative ∂²B/∂z² by combining three pickup loops in a configuration where the center loop has twice the area of the outer loops. The output signal proportional to 1/R⁵ provides even stronger discrimination against distant sources. Practical implementations achieve common-mode rejection better than 10⁶, enabling SQUID measurements in moderately shielded or even completely unshielded environments. The construction demands exceptional precision: imbalances of just 0.01% between loops can degrade rejection to unusable levels.
Electronic gradiometry offers an alternative to hardware gradiometers by digitally subtracting signals from spatially separated magnetometers. This approach provides greater flexibility, as gradiometer baselines and orientations can be adjusted in software. Adaptive algorithms determine optimal subtraction coefficients that minimize output noise across frequency, compensating for small differences in sensor responses. Recent implementations using high-temperature SQUIDs with electronic balancing demonstrate magnetocardiography in unshielded environments with signal-to-noise ratios exceeding 20 dB at 1 Hz.
Magnetic Shielding Strategies
While gradiometry rejects many environmental interference sources, some applications demand more complete isolation. Magnetic shielding employs passive or active techniques to create low-field environments where SQUIDs can operate with minimal external disturbance. Passive shields consist of high-permeability materials like mu-metal that redirect magnetic field lines around a shielded volume, or superconducting materials that exclude fields through the Meissner effect.
Multi-layer mu-metal rooms provide shielding factors exceeding 10⁵ at frequencies above 0.1 Hz. A typical three-layer room might achieve internal fields below 1 nT, compared to the 50 μT ambient field outside. The innermost layer often uses aluminum to attenuate radio frequency interference through eddy current induction. These magnetically shielded rooms, while effective, carry substantial costs and impose constraints on experimental flexibility. A room large enough to accommodate a human subject for magnetoencephalography can cost several million dollars and requires careful environmental control to prevent degradation of the shielding materials.
Superconducting shields offer superior performance in limited volumes. Lead or niobium cylinders cooled below their critical temperatures expel magnetic fields completely, creating internal field environments measuring a few picoteslas or less. The shields must be cooled in near-zero field to avoid trapping flux, after which field exclusion factors exceeding 10⁶ can be maintained. Practical limitations include the need for cryogenic infrastructure and the requirement that all signal inputs and outputs pass through carefully designed superconducting flux transformers to avoid compromising the shield.
Active shielding systems sense the ambient magnetic field using fluxgate or SQUID magnetometers and generate cancellation fields through feedback-controlled coils. Three-axis cancellation systems employing orthogonally arranged coil sets can reduce environmental fields by factors of 100 to 1000 across frequency ranges from near-dc to several hundred hertz. The technique proves particularly effective for compensating power line interference at 50 or 60 Hz and its harmonics, where passive shields often perform poorly due to eddy current losses.
Combined active and passive shielding leverages the complementary strengths of each approach. Active systems handle low-frequency fluctuations that penetrate passive shields, while passive materials attenuate high-frequency transients that exceed the bandwidth of active compensation. Multi-stage configurations using fluxgates for coarse control and SQUIDs in the final stage achieve remarkable performance, enabling magnetocardiography and magnetoencephalography in ordinary laboratory environments without dedicated shielded rooms.
Readout Electronics and Flux-Locked Loops
The periodic, nonlinear relationship between magnetic flux and SQUID output voltage creates challenges for signal extraction. Without linearization, the SQUID provides useful output only across a flux range of approximately Φ₀/2 before the signal wraps around. Flux-locked loop electronics solve this problem by sensing the SQUID output, amplifying it, and feeding back an exactly compensating flux through a modulation coil inductively coupled to the SQUID loop. This negative feedback linearizes the transfer characteristic and extends the dynamic range from a few flux quanta to effectively unlimited values.
The FLL operates by biasing the SQUID at the point of maximum flux-to-voltage transfer coefficient VΦ = ∂V/∂Φ, typically 50 to 500 μV/Φ₀ for low-temperature devices. Small flux variations δΦ produce voltage changes δV = VΦδΦ. The preamplifier, integrator, and feedback resistor in the FLL circuit generate a current through the modulation coil that creates flux exactly canceling the applied signal. The integrator output voltage then becomes directly proportional to the input flux, providing a linear transfer function with bandwidths extending from dc to several megahertz.
Noise in the readout electronics can degrade overall system performance if not managed carefully. The voltage noise of the preamplifier, when referred to the SQUID input, adds in quadrature with the intrinsic SQUID noise. The equivalent flux noise contribution from preamplifier voltage noise en equals en/(VΦ√2), where the factor of √2 accounts for optimization of the bias point. For a SQUID with VΦ = 200 μV/Φ₀ and a preamplifier with en = 0.5 nV/√Hz, the contribution becomes approximately 1.8 μΦ₀/√Hz, potentially dominating over the intrinsic SQUID noise if not suppressed.
Flux modulation schemes reduce preamplifier noise contribution by modulating the SQUID at frequencies far above the one-over-f corner of commercial amplifiers, typically 100 kHz to several megahertz. A modulation flux at frequency fm applied through the feedback coil amplitude-modulates the SQUID output. Synchronous demodulation at fm shifts the signal above low-frequency noise sources before amplification, then mixes it back down to baseband. This technique, analogous to lock-in detection, can reduce effective preamplifier noise by factors of 10 to 100 while simultaneously suppressing SQUID critical current noise at low frequencies.
Additional positive feedback represents another approach to preamplifier noise suppression. By feeding a fraction of the preamplifier output back to the SQUID input with positive polarity, the effective transfer coefficient VΦ increases without amplifying noise already present in the SQUID. Gains of 3 to 10 are practical before the system becomes unstable. This technique works particularly well with directly coupled readout schemes that avoid the complexity of flux modulation. Modern implementations combine direct coupling with APF to achieve white noise performance within a factor of two of the intrinsic SQUID limit while maintaining simple, compact electronics suitable for multi-channel systems.
Optimal SQUID Design Parameters
The interplay between various design parameters determines ultimate SQUID performance, creating a multi-dimensional optimization landscape. The dimensionless parameter βL = 2πLIc/Φ₀ profoundly influences both noise and dynamic characteristics. Theory predicts minimum noise energy near βL ≈ 1, where flux and voltage responsivity are optimally balanced. Devices with βL significantly different from unity exhibit higher noise: low βL reduces coupling to the pickup circuit, while high βL creates hysteretic behavior and thermal fluctuations in the junction switching.
The junction critical current density JC affects both noise and fabrication tolerance. Higher JC enables smaller junctions for a given critical current, reducing junction capacitance and critical current noise. However, very high current densities increase heating and can cause junction degradation. Typical values range from 100 A/cm² for low-temperature SQUIDs to 1000 A/cm² or higher for high-temperature devices. The optimal choice depends on operating temperature, desired critical current, and available fabrication technology.
Washer geometry significantly impacts flux noise and field sensitivity. Larger washer areas increase coupling to pickup coils and improve field sensitivity but also collect more flux noise from surface spins. The relationship appears roughly linear: doubling the washer area doubles both signal coupling and noise pickup, leaving the signal-to-noise ratio approximately constant. However, larger structures prove more susceptible to trapped flux from field cooling and require more careful magnetic shielding. Modern designs balance these factors, with typical washer dimensions ranging from 10 μm for nanoSQUIDs to 10 mm for devices in shielded environments.
Slit widths in washer SQUIDs determine coupling to the input coil while influencing screening parameter and inductance. Narrower slits reduce inductance and improve voltage responsivity but make the device more sensitive to flux noise from the slit edges. Widths from 2 to 20 μm represent common choices, with the optimal value depending on input coil design and noise requirements. Recent investigations of edge roughness show that smoother slit edges significantly reduce flux noise, likely by eliminating pinning sites for flux vortices.
Advanced Noise Reduction Techniques
Beyond conventional approaches, several advanced techniques push SQUID noise performance toward fundamental limits. Two-stage SQUID configurations employ a first-stage series array as a low-noise preamplifier for a second-stage sensor SQUID. The array, typically containing 100 to 1000 junctions, provides voltage gain of 10 to 100 with extremely low back-action noise. This arrangement effectively eliminates preamplifier contributions, allowing the system to approach the intrinsic noise of the sensor SQUID. The technique proves particularly valuable for readout of arrays of transition-edge sensors in X-ray astronomy and for nanoSQUID applications where small sensor size limits voltage responsivity.
Digital SQUID readout replaces analog FLL components with fast analog-to-digital converters and digital signal processing. The SQUID output is directly digitized at sampling rates of 10 to 100 MHz, with flux feedback calculated digitally and applied through high-speed digital-to-analog converters. This approach offers unprecedented flexibility: arbitrary transfer functions, adaptive filtering, flux counting for unlimited dynamic range, and perfect matching of multiple channels in array systems. Recent implementations achieve noise performance identical to optimized analog systems while adding capabilities like automatic reset, dynamic field compensation, and built-in diagnostics.
Resonant readout techniques exploit the impedance transformation that occurs when a SQUID couples to a resonant LC tank circuit. By matching the tank resonance to the SQUID's characteristic frequency, voltage responsivity can be enhanced by the quality factor Q of the resonator, potentially reaching values above 10 mV/Φ₀. This dramatic increase in signal amplitude overwhelms preamplifier noise even with conventional room-temperature electronics. The approach works particularly well for rf SQUIDs, where the tank circuit forms an integral part of the operating principle, but has also been applied to dc SQUIDs in specialized applications requiring extreme noise performance at specific frequencies.
Material engineering addresses flux noise at its source by modifying surface chemistry and structure. Treatments including annealing in oxygen-free environments, deposition of protective overlayers, and ion milling to remove damaged material have demonstrated flux noise reductions of 50% or more. Recent work suggests that much surface spin noise originates from amorphous oxide layers formed during device fabrication. Careful control of oxidation conditions or complete removal of oxides from critical surfaces shows promise for reducing flux noise below current levels. Chemical passivation using organic self-assembled monolayers represents another frontier, potentially blocking adsorption of paramagnetic molecules from the environment.
Cryogenic and Thermal Considerations
Temperature stability critically affects SQUID noise performance in subtle ways beyond the obvious thermal noise contribution. The critical current of Josephson junctions varies with temperature according to Ic(T) ∝ 1 - (T/TC)², where TC represents the superconducting critical temperature. Fluctuations in temperature thus create apparent critical current noise that can be mistaken for junction barrier effects. Careful measurements reveal that temperature instabilities of just 1 mK at 4.2 K can generate flux noise equivalent to 0.5 μΦ₀/√Hz at 0.1 Hz.
Thermal anchoring of SQUID packages demands meticulous attention. Even small heat loads from readout wiring or radiation from warmer stages can create temperature gradients across the chip, inducing thermoelectric voltages and position-dependent critical currents. Multi-stage thermal filters using copper heat sinks at intermediate temperatures between the SQUID and room-temperature environment help stabilize the local thermal environment. Advanced systems employ active temperature control using resistive heaters and proportional-integral-differential feedback loops to maintain SQUID temperature constant within microkelvin.
High-temperature SQUID operation at 77 K using liquid nitrogen coolant offers practical advantages over low-temperature systems but introduces additional noise challenges. Thermal noise increases by a factor of 4.3 compared to 4.2 K operation, while the superconducting gap reduces, increasing quasiparticle contributions. Junction critical current densities must be substantially higher to maintain phase coherence against thermal fluctuations. Practical high-temperature devices achieve flux noise levels of 10 to 30 μΦ₀/√Hz in the white noise region, approximately an order of magnitude above low-temperature SQUIDs. Nevertheless, for many applications, this performance suffices, and the greatly simplified cryogenic infrastructure makes high-temperature SQUIDs attractive.
Thermal cycling stability presents long-term challenges for both low and high-temperature devices. Repeated cooling and warming induces mechanical stress from differential thermal contraction between materials with different expansion coefficients. Josephson junctions prove particularly vulnerable: tunnel barriers can develop microcracks, and electrode layers may delaminate. High-temperature SQUIDs face additional degradation mechanisms as oxygen migrates within the superconducting films, altering critical temperature and current density. Robust devices require careful materials selection, optimized thin-film deposition conditions, and sometimes protective encapsulation against environmental exposure during storage and operation.
Integration Approaches for Practical Systems
Translating laboratory SQUID performance into functional measurement systems requires integrating sensors, shielding, cryogenics, and electronics into coherent architectures. Multi-channel systems for biomagnetic imaging employ arrays containing dozens to hundreds of SQUIDs arranged in helmet shapes or planar grids. Each sensor demands individual readout electronics, creating challenges of channel density, crosstalk, and thermal management. Modern systems achieve channel spacing below 20 mm while maintaining mutual electromagnetic isolation better than 60 dB through careful layout and shielding.
Flexible coupling schemes balance noise performance against operational convenience. Directly coupled planar gradiometers integrate pickup loops and SQUIDs on a single chip, minimizing parasitic inductance and achieving noise near the intrinsic SQUID limit. This approach works well for applications where the sensor can be positioned very close to the source. Wire-wound pickup coils connected to SQUIDs through superconducting flux transformers provide greater flexibility in coil geometry and positioning but introduce additional inductance that slightly degrades noise. The transformer design critically affects performance: poorly optimized transformers can increase system noise by factors of two or more compared to the bare SQUID.
Multiplexing techniques enable readout of large SQUID arrays with reduced channel counts. Time-division multiplexing sequentially reads multiple SQUIDs using a shared FLL, appropriate for applications where bandwidth can be traded against channel count. Frequency-division multiplexing assigns each SQUID a unique bias frequency, allowing simultaneous readout through a common wideband amplifier with subsequent digital signal separation. Code-division approaches modulate different SQUIDs with orthogonal sequences, extracting individual channels through correlation. Each method involves trade-offs: time multiplexing sacrifices bandwidth, frequency schemes introduce crosstalk between adjacent channels, and code division requires complex signal processing. Selection depends on specific application requirements and acceptable compromises.
Future Trajectories in Noise Reduction
The quest to quiet SQUID noise continues pushing against fundamental limits while opening new application frontiers. Recent theoretical work suggests that with optimal design and materials engineering, flux noise levels approaching 0.1 μΦ₀/√Hz might be achievable, representing an order of magnitude improvement over current best performance. Such devices would enable new classes of experiments: detecting individual nuclear spins, measuring gravitational gradients for fundamental physics tests, and probing quantum fluctuations in macroscopic systems.
Quantum non-demolition measurements using SQUIDs represent an emerging direction that inverts the traditional noise perspective. Rather than treating SQUID noise as an unwanted artifact, these techniques exploit the quantum mechanical back-action of measurement to realize ideal quantum-limited amplification. By carefully engineering the coupling between signal and SQUID, it becomes possible to amplify one signal quadrature while adding noise only to the conjugate quadrature, saturating the quantum noise limits imposed by the uncertainty principle. Applications range from gravitational wave detection to quantum information processing with superconducting circuits.
Topological protection methods inspired by condensed matter physics may revolutionize SQUID design. Topologically protected qubits and SQUIDs fabricated from exotic superconducting materials with unconventional pairing symmetries could suppress certain noise channels entirely through fundamental symmetries. While implementations remain largely theoretical, early experiments with nanowire SQUIDs and Majorana-based devices hint at possibilities. Such devices might achieve coherence times and noise levels currently unimaginable, enabling a new generation of quantum sensors.
Integration with quantum information processors creates synergies where progress in one field accelerates the other. Superconducting qubits essentially comprise ultra-sensitive SQUIDs, and advances in qubit coherence directly translate to improved SQUID performance. Conversely, techniques developed for SQUID noise reduction, including materials engineering and electromagnetic design, benefit qubit fabrication. This cross-fertilization promises rapid progress as the quantum computing field invests billions in superconducting circuit technology.
Reflections on Quantum Silence
The journey toward perfectly quiet SQUID receivers reveals something profound about the relationship between technology and fundamental physics. Every advance in noise reduction uncovers another layer of complexity: suppressing thermal noise exposes critical current fluctuations, eliminating those reveals surface spin noise, and conquering surface effects finally confronts us with quantum mechanical limits. The quest resembles peeling an onion where each layer removed reveals yet another beneath.
This progression mirrors humanity's broader quest to understand and control nature at ever-finer scales. The same precision fabrication enabling modern SQUIDs makes possible the semiconductor industry, while the cryogenic techniques supporting SQUID operation enable quantum computing and particle physics. Magnetic shielding developed for biomagnetic measurements finds application in atomic physics experiments probing fundamental constants. The web of connections extends in surprising directions.
What makes SQUID noise reduction particularly compelling transcends the technical achievements. These devices literally translate quantum mechanical phenomena into signals we can measure and understand. Every flux quantum threading a SQUID loop represents a direct manifestation of wave function phase, ordinarily hidden from observation. The noise we fight so hard to suppress arises from the universe's fundamental graininess: thermal energy parceled in quanta, magnetic moments flipping stochastically, electrons tunneling probabilistically. Success means hearing whispers from the quantum realm itself.
The practical impact already extends into daily life through medical diagnostics, though most patients remain unaware. Magnetoencephalography mapping brain activity before epilepsy surgery, magnetocardiography detecting cardiac ischemia earlier than electrocardiography, and fetal magnetocardiography monitoring unborn babies all rely on the noise reduction techniques described above. Geophysical exploration using SQUID gradiometers identifies ore deposits and characterizes underground aquifers. Non-destructive testing detects hidden cracks in aircraft turbine blades. Each application exists only because researchers learned to quiet the quantum noise.
Looking forward, the potential applications stretch imagination's limits. Arrays of ultra-sensitive SQUIDs might map brain activity with unprecedented temporal and spatial resolution, revealing how consciousness emerges from neural computation. Orbiting SQUID gradiometers could characterize asteroid compositions from safe distances, enabling resource assessment for space mining. Fundamental physics experiments searching for axions, magnetic monopoles, or violations of charge conservation depend critically on continued noise reduction. The sensors enabling these measurements do not yet exist, but the path toward creating them grows clearer.
The field demonstrates how patient, incremental progress in engineering can accomplish what initially appears impossible. Five decades ago, SQUIDs detected magnetic fields at the nanotesla level with noise measuring microteslas per root hertz. Today's devices achieve femtotesla field sensitivities with noise approaching the quantum limit. That million-fold improvement accumulated through countless small advances: better materials, cleverer designs, more sophisticated electronics, and deeper understanding of noise physics. No single breakthrough enabled it; rather, sustained effort by hundreds of researchers each contributing incremental progress compounded into transformative capability.
The story continues. Noise sources once thought fundamental have yielded to human ingenuity. New techniques constantly emerge as our understanding deepens and fabrication capabilities advance. Perhaps one day, SQUID receivers will achieve such perfect silence that they detect only the quantum fluctuations required by Heisenberg's uncertainty principle. At that point, the quantum and classical realms will truly merge in our measurement apparatus, and we will have learned to listen to the universe's quietest voice.
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