5.3 SQUID (Superconducting Quantum Interference Device)

SQUIDs are mind-blowing devices that can detect super tiny magnetic fields. They use quantum effects in superconducting loops with Josephson junctions to measure magnetic flux with incredible precision.

These ultra-sensitive magnetometers come in two flavors: DC and RF SQUIDs. They're used in everything from brain imaging to searching for dark matter, making them crucial tools in modern physics and medicine.

SQUID Structure and Operation

Working Principle and Components

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• A SQUID (Superconducting Quantum Interference Device) is a highly sensitive magnetometer that consists of a superconducting loop with one or two Josephson junctions
• The working principle of a SQUID is based on the quantum interference of superconducting currents in the loop, which is modulated by an external magnetic flux
• The critical current of the SQUID is a periodic function of the applied magnetic flux, with a period of one flux quantum ($\Phi_{0} = h/2e \approx 2.07 \times 10^{-15}$ Wb)
• The SQUID is typically coupled to a superconducting flux transformer, which consists of a pickup coil and an input coil, to enhance its sensitivity to external magnetic fields (magnetoencephalography, magnetocardiography)

Types of SQUIDs

• There are two main types of SQUIDs: dc SQUID and rf SQUID
  • A dc SQUID has two Josephson junctions in parallel, allowing for direct measurement of the voltage across the SQUID
  • An rf SQUID has a single Josephson junction in a superconducting loop, and the magnetic flux is detected by measuring changes in the resonant frequency of an LC tank circuit coupled to the SQUID

Flux Quantization in SQUIDs

Quantum Mechanical Origin

• Flux quantization is a fundamental property of superconducting loops, where the magnetic flux threading the loop is quantized in units of the flux quantum ($\Phi_{0} = h/2e \approx 2.07 \times 10^{-15}$ Wb)
• The quantization of magnetic flux arises from the requirement that the superconducting wavefunction must be single-valued around the loop, leading to the condition: $\oint \nabla\phi \cdot dl = 2\pi n$, where $\phi$ is the phase of the wavefunction and $n$ is an integer

Impact on SQUID Operation

• In a SQUID, the presence of Josephson junctions allows the superconducting phase to change by a fraction of $2\pi$, resulting in a periodic modulation of the critical current as a function of the applied magnetic flux
• The flux quantization in a SQUID enables the detection of extremely small changes in magnetic flux, as the SQUID's output voltage or current is highly sensitive to changes in the applied flux on the scale of a single flux quantum
• This sensitivity makes SQUIDs ideal for measuring weak magnetic fields in various applications (geophysics, fundamental physics experiments)

DC vs RF SQUID Characteristics

Voltage-Flux Characteristics

• The voltage-flux characteristic of a dc SQUID is a periodic function with a period of one flux quantum ($\Phi_{0}$)
  • The maximum supercurrent through the SQUID varies as a function of the applied magnetic flux, resulting in a modulation of the voltage across the SQUID when biased with a constant current
  • The voltage-flux characteristic of a dc SQUID can be described by the equation: $V = R_{N} I_{b} \sqrt{1 - (I_{c}(\Phi) / I_{b})^{2}}$, where $R_{N}$ is the normal-state resistance of the SQUID, $I_{b}$ is the bias current, and $I_{c}(\Phi)$ is the flux-dependent critical current
• In an rf SQUID, the voltage-flux characteristic is also periodic with a period of $\Phi_{0}$, but the modulation is typically detected by measuring the change in the resonant frequency of an LC tank circuit coupled to the SQUID
  • The rf SQUID operates as a flux-to-frequency transducer, where the resonant frequency of the tank circuit is modulated by the applied magnetic flux
  • The frequency modulation is then converted to a voltage signal using suitable readout electronics

SQUID Sensitivity and Noise

Sensitivity and Noise Sources

• SQUIDs are the most sensitive magnetometers available, with typical magnetic field sensitivities in the range of 1-10 fT/$\sqrt{\text{Hz}}$ (femtotesla per square root of hertz)
• The sensitivity of a SQUID is limited by various noise sources:
  • Thermal noise from the Josephson junctions and the SQUID's shunt resistors
  • Environmental noise coupled to the SQUID through the flux transformer
• The noise performance of a SQUID is characterized by its spectral density of equivalent flux noise, $S_{\Phi}(f)$, which is typically expressed in units of $\Phi_{0}^{2}/\text{Hz}$

Noise Reduction Techniques

• Low-frequency noise (1/f noise) is a significant contributor to the overall noise in SQUIDs, particularly at frequencies below a few kilohertz
  • Techniques such as flux modulation and bias reversal are employed to minimize the impact of low-frequency noise
• The use of superconducting shielding and proper grounding techniques is crucial for achieving optimal noise performance in SQUID-based magnetic field measurements
• Careful design of the SQUID geometry and the flux transformer can also help to reduce the noise and improve the overall sensitivity of the device