# Quantum Picture of the Josephson Junction

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Term Project Yi Yang Quantum Picture of the Josephson Junction A promising candidate for qubits in quantum computing In this project I introduce the idea of Josephson Junction which has drawn great attention of researchers with its potential to server as a quantum bit in a quantum computer I use the approaches we discussed in class to study the quantum picture of a current biased Josephson Junction The lowest two states of such a Junction are singled out as two states of a quantum bit I calculated the tunneling rate for each state and explain how we determine which state a qubit is in with the information about the tunneling rates of each state Because of the existence of thermal noise ultra low temperature is required usually around 10 mk if we want to observe these quantum properties experimentally There are many other issues that should be considered when we study such a system such as electromagnetic isolation and decoherence and dissipation of the Junction All these are not taken into account when I sketch the quantum picture of the Junction here Term Project Yi Yang Quantum Picture of the Josephson Junction A promising candidate for quantum bits in quantum computing 1 Physical Properties of a Josephson Junction Electrons in a superconductor form Cooper pairs for temperatures below the critical temperature of the superconductor 1 These pairs are in a collective motion corresponding to the ground state of the system The state can be described with a wave function G G G r t A r t ei r t 1 1 The phase of the wave function is coherent throughout the superconductor If two superconductors are separated by a very thin layer of insulator a Josephson Junction is formed Fig 1 When the insulator thickness is small enough the electronic wave functions from the two sides can overlap The cooper pairs in these two superconductors can tunnel to the other and the phase of the wave functions in two superconductors are correlated If the phase difference is 1 2 the relation between the tunneling current I flowing through the junction and the voltage V across the junction is given by Josephson Relations 2 I I 0 sin 1 2 0 d 2 dt 1 3 V h 2 07 10 15 T m 2 is the flux quantum and I 0 is the critical 2e current of the junction Where 0 Animations of the formation of Copper Pairs and how they contribute to superconductivity can be found at http www chemsoc org exemplarchem entries igrant bcstheory noflash html Flux quantum The magnetic flux in a superconducting loop is quantized The allowed flux is n 0 where n 0 1 2 1 Term Project Yi Yang In an actual Josephson Junction there is a capacitance between the two superconducting plates and a shunt resistance parallel to the Junction In this model which is called RCSJ Model a Josephson Junction can be indicated as Fig 2 The cross is the bare Junction I b is the total current flowing through which is called Bias Current Thus the equation for the total current is V I b I j CV I 0 sin C 0 0 R 2 2 R in which the Josephson Relations are plugged 1 4 We study Josephson Junction as a Phase Qubit which means the states of the qubit 0 and 1 in classical computer are indicated by different states of the phase of the Junction I am going to make an analogy between the problem of a Josephson Junction and a dynamic problem using equation 1 4 2 Term Project Yi Yang 2 Dynamic analogy for a Josephson Junction Consider a particle with mass m C 0 moving in a potential 2 2 U 0 I 0 cos Ib 2 2 1 2 0 subjected to a damping force The equation of motion is 2 R dU 0 m d 2 R 2 2 2 which is exactly equation 1 4 So the problem of phase state of a Josephson Junction is analogized to problem of motion of a particle with displacement in potential U subjected to a damping force We study this potential quantum mechanically Note that the potential depends on the bias current I b The sketch of the potential for I b 0 1I 0 I b 0 3I 0 and I b 0 9 I 0 is shown in Fig 3 3 Term Project Yi Yang Fig 3 Through adjusting the bias current I b we can control the deepness of the well thus the number of bound states 3 Eigen states of the tilted wash board potential When a Josephson Junction is used as a qubit one need to find the bias current I b such that there are only several eigen states existing in the well As we discussed in class we can discretize the potential and diagonalize the Hamiltonian matrix to get the eigen values and eigen functions The Hamiltonian of this system is simply H d2 U 2m d 2 3 1 where U depends on the bias current I b With the typical parameter I 0 10 A C 1 pF L 0 15nH my calculation showed that when I b 0 97 I 0 there are only 4 bound states in the well which is a good condition where we can regard the Josephson Junction as a qubit The eigen energies of the bounded states are I chose U r 0 as the zero potential E0 5 0314563659704 10 21 E1 5 02280861941517 10 21 E2 5 01468472553116 10 21 4 Term Project Yi Yang E3 5 00738407077634 10 21 Let us label the energy space in frequency between i th state and j th state as ij the energy spaces are 10 82GHz 21 77GHz 32 69GHz Unlike the Harmonic Oscillator Potential the energy spaces of the wash board potential are not equal This is a significant quality of this system that makes it a candidate for qubit as discussed later The potential in my calculation and the eigenfunctions for the 4 lowest states are shown in Fig 4 The eigenfunctions look similar with eigenfunctions of Harmonic Oscillator Potential However one can see that on the right edge of the functions for state 2 and state 3 the functions are no longer zero In fact since right barrier of the well is not infinitely high actually very low in this case there must be a transmission rate or quantum tunneling rate for each state From the sketch of the functions we can roughly tell that the tunneling rate of state 2 and 3 are much larger than that of state 0 and 1 Actually this difference in tunneling rate is another basis for our effort to design a qubit with a Josephson Junction In next section I am going to calculate the tunneling rate of each state and explain how to measure the state of such a qubit through the quantum tunneling of it 3 00E 021 3 50E 021 4 00E 021 4 50E 021 5 00E 021 5 50E 021 6 00E 021 6 50E 021 0 0 0 5 1 0 1 …