As promised in the previous post in this post we will cover Josephson Junction and squid and hopefully we will take our understanding of superconducting qubit to a good level

**Josephson Junction**

Just to recall we discussed in my previous post that persistent current flows in a superconductor to cancel out the externally applied magnetic field and this current does not contain electrons instead it consists of cooper pairs . This effect is called Meissner Effect .

Now in my first blog we talked about tunneling ,means if conductors are extremely near to each other than barrier cannot control the flow of electrons or flow of current from one conductor to another

Josephson Junction is an amalgamation of all these concepts, If we will separate two superconductors (any conductor behaves like a superconductor only under a critical temperature) by a thin barrier and apply external magnetic field to it. A current consists of cooper pairs start flowing in these super conductors to oppose the external magnetic field as per Meissner Effect. Now because these superconductors are separated by a very thin barrier ,This current of copper pair tunnels through the barrier and reaches to other superconductor .

So, the net current in these superconductors is the coupling of these two currents one is the actual current which is flowing because of external magnetic field and one is flowing because of tunneling.

Source:http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/Squid.html

Copper Pairs are tunnelling through the barrier, this barrier is called Josephson junction

Now in my first blog, I talked about wavefunction of electron but in case of superconductors current is flown by copper pairs and this current is also tunneled from one superconductor to another .So in these two superconductors currents are flowing in both directions simultaneously and this dual current is changing corresponding to external magnetic field ,wavefunctions of these cooper pair looks like as below

**Equation 1**

Source: http://fy.chalmers.se/~delsing/LowTemp/Labbar/SQUIDlab-rev3.pdf

Here Ψ1 represents the wave function of copper pairs present in persistent current flowing in super conductor and Ψ2 represents the wave function of copper pairs tunnel to this super conductor and K represents the coefficient of tunneled current . µ_{1} and µ_{2 }represents the energy levels of wave functions.

Here Ψ1 and Ψ2 can be represented as below

**Equation 2**

Where n1 and n2 are copper pairs density and θ1 and θ2 are phases. Here phases basically define the direction of current or copper pair movement.

Now these wave functions can be represented in terms of current as well because current is nothing but measurement of number of electrons pass in one second or in some time unit

Current is measured in ampere and below is definition of ampere

*“The SI unit of electric current is the ampere, which is the flow of electric charge across a surface at the rate of one coulomb per second”*

And Electric charge is measured in terms of coulomb which is basically defines the charge carried by one electron. Electric charge on an electron is approximately 1.6021773310−19 coulomb. So, if this value is 2 coulombs per second it means 2 electrons passed in one second

Now intensity of this current is represented by µ_{1} and µ_{2 } in Equation 1 .In much simpler way We can understand that current is flow of electrons from negative to positive ,now if we want to increase the speed of this flow, we apply voltage to this flow. Voltage pushes electrons and electrons moves with greater speed and this change in speed changes the overall energy of wave function. Please note in case of superconductor voltage pushes copper pairs, as current consists of cooper pairs not electrons.

Full derivation on this is as below

Source: http://fy.chalmers.se/~delsing/LowTemp/Labbar/SQUIDlab-rev3.pdf

Till now, hopefully we understood that current flows in both direction in these Josephson junction based superconductors ,in presence of external magnetic field and Intensity of these current or wave function of current can be controlled by voltage or by changing the external magnetic field

Now let us understand how to measure this coupled current

Source: https://file.scirp.org/Html/2-7502120_55347.htm

As you can see above JJ2 and JJ1 are Josephson junctions, H is the external magnetic field and I_{B} is the net persistent current which is flowing in this superconductor to expel the magnetic field H.

Now in JJ1 junction I_{1} current is coming in anti-clockwise direction while I_{2 }current is coming from clockwise direction, so net current which is flowing in this junction is the coupled effect of both these currents. Now current I_{B} is the current, which is expelling the magnetic field, so this current will not change until external magnetic field will not change. But currents at junctions can be controlled by application of voltage or in simple words let us say if

I_{B}= I_{1}+ I_{2}

Let us say we apply some voltage on junction 1 and I1 is increased to I_{1}´ then I2 will decrease to I_{2}´ to maintain the same persistent current I_{B} ,because external magnetic field is not changed ,so current I_{B} also can’t be changed, but I1 and I2 can change on application of voltage.

I_{B}= I_{1}´+ I_{2}´

Now overall magnetic field of this entire system which is created by current I_{B} will not change. But because current is changing at junctions, magnetic field at junctions will change and this change in magnetic field can be measured using magnetic flux. This effect is called AC Josephson Effect .

We can change the current at junction by changing the external magnetic field as well. This effect is called DC Josephson Effect.

This magnetic flux is indirectly proportional to current and voltage on junctions and we have mentioned in equation 5 and 6 above that wave function of copper pairs can be represented in terms of current and voltage. So if we can measure this magnetic flux ,we can measure this wavefunction from outside .This magnetic flux can be measured by a device called SQUID.

Now after all this explanation you might be thinking where is the qubit in all this

To understand it let us see what a bit is

Bit is 0 when current does not flow and Bit is 1 when current does flow right?

Now in case of qubit ,current is always flowing ,its never 0. But qubit means ,there are two currents flowing instead of one and wave function of this qubit is the combination of these two currents. What is changing is the intensities of these two current , if current I_{2} is extremely high then current I_{1} will be low to maintain the net persistent current I_{B}, and when these currents changes ,wave function changes ,and because magnetic flux is proportional to these currents ,so when we measure this magnetic flux ,we are measuring different values of this wavefunction, or different states of qubit.

This magnetic flux can be measured using SQUID, so we can measure the qubit as well without collapsing the wave function.

I hope you all have better understanding of qubit Now. In next blog we will read about how to identify different states of qubits from different values of flux.

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