A **Josephson Junction (JJ)** is a specialized electronic construct, composed of a pair of superconducting materials divided by a delicate insulating barrier. This unique design is underscored by its inherent property known as the Josephson effect. In essence, even a minuscule voltage application can induce an oscillatory electric current across the junction, remarkably without any resistive losses. This phenomenon gives rise to a synchronized quantum state within the junction, commonly recognized as the Josephson supercurrent. The intriguing attributes of the Josephson effect have found applications in diverse domains. For instance, it has paved the way for the creation of ultra-sensitive magnetic field detectors. Furthermore, it has been instrumental in the conceptualization and realization of Superconducting Quantum Interference Devices (SQUIDs), which have proven invaluable in advanced sectors like medical imaging and material sciences. On the forefront of technological evolution, Josephson junctions have also become a pivotal element in the architecture of superconducting quantum computers. These revolutionary machines harbor the promise of vastly surpassing traditional computers in performing specific computational tasks.

The wondrous behavior of a Josephson junction is deeply rooted in the realm of quantum mechanics. The core players in this phenomenon are Cooper pairs—paired electrons that exist in superconductors at temperatures close to absolute zero. When these pairs encounter the insulating barrier of the junction, they undergo a process called quantum tunneling. Instead of being blocked by the barrier, they seamlessly “pass through” it, a counterintuitive action that defies classical physics. This movement of Cooper pairs, without the scattering and energy losses typically associated with electron movement in conventional conductors, is what gives rise to the supercurrent.

Beyond the immediate realm of fundamental physics and materials science, the implications of the Josephson junction’s properties are vast. One significant application is in voltage standards. By exploiting the relationship between the frequency of the oscillating current and the applied voltage in a Josephson junction, precise voltage standards can be established, aiding in metrology. Additionally, because of their sensitivity to magnetic fields, these junctions are pivotal in the design of ultrasensitive magnetometers. With the burgeoning interest in quantum computing, Josephson junctions are also being eyed as potential building blocks for future quantum circuits, holding promise for the next revolution in computational power.

The year 1962 was when British scientist Brian Josephson came up with a theory about the tunneling of electrons across thin insulator layers that was sandwiched in between superconductors (such the structure is now referred to as the Josephson junction). The theory was confirmed experimentally fairly quickly in the same way that Josephson himself was recognized with the Nobel Prize in Physics for it in 1973. Since that time it has been a major factor in the development of science. Josephson junction has been an important technological structure that is that is used, for instance, for superconducting quantum interference devices (SQUIDs) that are those devices that are the most sensitive used for monitoring magnetic field induction that, based on their design, employ either one and/or 2 Josephson junctions.

In the year 2006, Hans Mooji and Yuli Nazarov from Delft University in the Netherlands published a paper that was a theoretical study of the phenomenon of quantum tunneling that occurs when magnetic energy through the superconductor layer, which is sandwiched between two different materials. The phenomenon, known as the coherent quantum phase slip was claimed to be the result of quantum phase slip by Mooji as well as Nazarov to be comparable in magnitude to that of that of the Josephson effect. But, in over six years after the study was published nobody has been able to prove experimentally the existence of this phenomenon in superconductors.

## Physics of josephson junctions

Superconducting electronic devices are built on Josephson junctions which is the mathematical study of which we’ll be presenting. Equations describing the tunneling process of Cooper couples through the barriers of potential a junction created by a thin insulating component produce changes of the Ψα wavefunction (α = 1 or 2) across both covers of the junction, according to the formula:

where the subscripts α = 1 when β = 2 and α = 2 when β = 1 refer to the left and right covers of the junction shown in Figure 4, while μ is the chemical potential. The parameter c determines the mutual coupling of the wave functions in the two covers.

Let’s write the wave function in composite form:

Where nα is, respectively, the concentration of current carriers and thus Cooper pairs in both parts of the junction, while Φ α is the phase of the wave function on the left and right covers of the junction:

Let’s write the solution of equation 1 in the form:

Where ΔΦ = Φ 2 – Φ 1 is the time-dependent phase difference on the two connector covers:

h is the reduced Planck constant. Since the change in the concentration of current carriers over time implies a flow of charge, so ultimately the expression for Josephson current flow I takes the form:

Where I0 is the maximum Josephson current of the junction, while the change in chemical potentials associated with the voltage V applied to the junction, determined by the relation:

It is based on the equation. 4 that with no applied voltage, a direct current is flowing through the junction. However, at 0V, an alternating current flows across the joint, with e representing the charge of electron. There is also an alternation Josephson result that occurs with the application of voltage to connector covers as shown in Eq. 4. This now takes the form of:

Where V is the DC voltage, while υ is the perturbing microwave field. The solution of equation (7) describing the phase change of the wave function when passing through the potential barrier takes the form:

The current flowing through the tunnel junction is described in terms of a series:

Where J n are nth-order Bessel functions, while θ and φ1 are fixed parameters.

The most important conclusion from equation (9) is the occurrence of superconducting DC at voltage:

The various current spikes that are associated with the variation in the index n are known as Shapiro steps. We can derive the conclusion that as long as you apply an equal voltage V to the Josephson superconducting junction junction emits electromagnetic radiation with an amplitude of 77.03 millivolts, for an n of 1. This phenomenon is utilized in a variety of instruments for measuring as well as the standard volt created from its base. Let’s apply the relation (5) to the system consisting of 2 parallel Josephson junctions, such as those found for superconducting quantum interferometers (SQUIDs) as well as the equation for the maximum Josephson current is dependent on the magnetic field that is applied as per the equation:

Where Imax is the current amplitude, which is a function of the magnetic induction flux φ passing through the surface of the interferometer:

Interference dependance (12) on the Josephson current of a SQUID based on the magnetic field is illustrated in Figure 1. The magnetic characteristics of the current in the characteristics of a SQUID composed of superconductors with high temperatures with the d-type wave function symmetry are additionally highlighted in this document.

In reality, Josephson junctions are not point-like, but they do have some size. Therefore, let’s take a look at the second extreme scenario of the long Josephson junction that is placed in a magnetic field that is parallel. The long Josephson junction, in contrast to one-to-10 Nm junction that was previously discussed it is described using the sine-Gordon equation

where

The wave attenuation ratio C is the capacity of the unit cross-sectional area at the junction. g is it is the electrical conductivity, and J is The Josephson the current density and the current I0 is described in equation 5.

The answer to Equation 13 is the presence of a singularity, or a soliton carrying a quantum flux, which is an appearance similar to a vortex thread. Photolithography also creates Josephson connectors that have different designs that meet the requirements of the instrument as well as affecting the parameters of its electromagnetic field, like having an expotentially narrowed width and is commonly known as Eiffel tower connectors. The changes in connector shape are a result of different propagation conditions for the solitons (vortex threads) which is the solution to Eq. 13. The boundary condition shown in this figure. For an annular joint, as an instance, there exists an underlying boundary condition. The propagation of vortices is emitting electromagnetic radiation. Based on their shape, the connectors release electromagnetic radiation with frequencies of between 100 and 1000 GHz. That’s the frequency used in radio astronomy, high-speed electronics as well as satellite communication. Eiffel-type connectors are also utilized in DC/AC converters.

## Critical current Josephson junction

An Josephson junction can be described as a common tunnel junction wherein layers of superconducting materials are separated by an extremely small (about 10A) dielectric layer. When the dielectric layer is properly polarized at the junction the tunneling of Cooper pairings through the conductive layers may occur. In the real world even without an polarization voltage, an unspecified amount of current – the Josephson current is flowing through the junction, connected to the phase change in the parameter for order of wave effects of superconductors. When the voltage of polarization limit Ug exceeds, there will be an increase in the amount of the current passing into the junction. To polarize the junction by an external voltage the tunneling current can be calculated by following the following equation:

in which I – the current flowing through the junction, I0 – the critical current (junction constant), V – the junction voltage, e/h – the ratio of the electron charge to Planck’s constant.

## Josephson junction array

Josephson junction arrays stand as remarkable feats of quantum engineering, representing a lattice-like arrangement of multiple Josephson junctions in two-dimensional spaces. As their fundamental constituents, these arrays utilize superconducting materials, with each junction separated by insulating layers to ensure efficient functioning. When current traverses through these junctions, they synchronize to produce a unified quantum state. This collective phenomenon is emblematic of the Josephson junction and is aptly termed the “Josephson junction supercurrent.” Such arrays, due to their intricate construction, serve as a testament to the advancements in quantum electronics.

One of the most pivotal attributes of these arrays is the precise control they offer over the quantum states of their constitutive qubits. By adeptly modulating the voltage applied across each individual junction, it becomes feasible to dictate and modify the quantum state of every qubit housed within the array. This capability not only underlines the potential of Josephson junction arrays in quantum computations but also signifies their role in the broader spectrum of quantum technologies. With each qubit acting as a quantum bit of information, the ability to control and alter its state with precision is paramount in the progression of quantum computing.

### Why are Josephson Junction Arrays Promising for Quantum Computing?

Josephson junction arrays’ main advantage is their capacity to scale. Since each junction is an individual qubit and increases the amount of junctions can increase the number of qubits they can have, allowing bigger quantum processors with more qubits to be built. Additionally, their design can be tuned to allow precise control of interactions between qubits and, consequently, reduce the risk of mistakes.

Josephson junction arrays are characterized by very low rates of decoherence. Decoherence is the term used to describe an absence of coherence in quantum systems caused by interactions with their surroundings. Josephson junctions are constructed of superconducting materials that have low dissipation levels, which maintain quantum coherence for long durations and are therefore ideal to be used to process quantum information.

### Applications of Josephson Junction Arrays

Josephson junction arrays can be used for a variety of uses in quantum computation. One of the most promising applications for Josephson junction arrays could be making quantum annealing devices, that use quantum mechanics to tackle optimization issues. Additionally, Josephson junction arrays may be utilized to create two-dimensional Ising models that could aid in solving optimization issues too.

Josephson junction arrays can also be utilized to serve as quantum simulators. Quantum simulators employ quantum mechanics to model complex systems that would be difficult to analyze on conventional computers. Josephson junction arrays could be used as quantum simulators, by creating Hubbard models that explain the interactions between quantum particles.

## Graphene Josephson junction

Graphene, a two-dimensional material made up of carbon atoms arranged hexagonally, has become one of the most promising materials for future electronic devices. One of its unique properties is its ability to form Josephson junctions – key components in superconducting electronic devices with great potential applications in quantum computing and spintronics.

### What Is a Graphene Josephson Junction?

A graphene Josephson junction is a type of Josephson junction formed using graphene sheet sandwiched between two superconducting electrodes and created through creating narrow constrictions in its surface, acting as weak links between electrodes allowing electrons to tunnel through, creating Josephson Junction supercurrents.

### How Do Graphene Josephson Junctions Work?

Graphene Josephson junctions work according to the same principles as traditional Josephson junctions: when voltage is applied across them, current can flow freely without resistance, creating what’s known as Josephson supercurrent – with graphene serving as an effective weak link connecting two superconducting electrodes.

Josephson junctions made of graphene can exhibit spin-polarized transport. Because graphene itself exhibits spin polarity, supercurrent flowing through graphene Josephson junctions can display spin-dependent behavior. This makes graphene Josephson junctions an appealing platform for spintronics research: exploiting electron spin to develop new electronic devices.

### Applications of Graphene Josephson Junctions

Graphene Josephson junctions offer immense promise for use in quantum computing and spintronics applications. Their spin-dependent transport characteristics make them a prime platform for creating spin-based qubits – an alternative type of quantum bit used in quantum computing that may be less sensitive to environmental noise than other qubit types, making them potentially more robust solutions.

Graphene Josephson junctions can serve as building blocks for other types of spintronic devices, including spin valves and magnetic tunnel junctions. Such devices could be utilized for applications ranging from data storage to magnetic sensors.

### Attributes of Josephson Junctions

One of the primary challenges in creating graphene Josephson junctions lies in controlling their size and shape of constriction in graphene sheet, as this must allow electrons to tunnel through while not becoming so narrow as to become unstable for the junction. Another difficulty lies in making sure graphene stays in contact with its superconducting electrodes since even small gaps between sheets could disrupt supercurrent flow.

## Josephson Junction Qubit

Quantum computing is an exciting field with the potential to completely change our world of computing, and one promising avenue is Josephson junction qubits.

### What is a Josephson Junction Qubit?

A Josephson junction qubit is a quantum bit that uses the Josephson effect, a quantum mechanical phenomenon that occurs in superconducting materials, to store and manipulate information. The qubit is made up of two superconducting electrodes separated by a thin insulating layer, which acts as the Josephson junction.

### How Do Josephson Junction Qubits Work?

In a Josephson junction qubit, information is stored in the form of the phase difference between the superconducting electrodes. When a small voltage is applied across the Josephson junction, a supercurrent can flow without any resistance. By manipulating the voltage, the phase difference between the two electrodes can be controlled, allowing for the storage and manipulation of quantum information.

One of the key advantages of Josephson junction qubits is their potential for scalability. Because they can be fabricated using standard microfabrication techniques, they can be easily integrated into larger quantum computing systems. Additionally, they have been shown to have long coherence times, meaning that they can retain their quantum state for relatively long periods of time.

References:

https://home.agh.edu.pl/~km2007/misc/papers/161.pdf