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 - 1 -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: (,)(,) (,)irtrt ArteθΨ=GGG (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 is12γθθ=−, the relation between the tunneling current I flowing through the junction and the voltage Vacross the junction is given by Josephson Relations[2]: 0sinIIγ= (1.2) 02dVdtγπΦ= (1.3) Where 15 202.07 102hTme−Φ= = × ⋅ is the flux quantum**, and 0I is the critical current of the junction. * 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: 0n⋅Φ,where 0,1,2...n =Term Project Yi Yang - 2 - 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; bI is the total current flowing through, which is called Bias Current. Thus, the equation for the total current is: 000sin( )22bjVIICV I CRRγγγππΦΦ=+ += + +  (1.4) in which the Josephson Relations are plugged. 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 (γ)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.Term Project Yi Yang - 3 - 2. Dynamic analogy for a Josephson Junction Consider a particle with mass 202mCπΦ⎛⎞=⎜⎟⎝⎠ moving in a potential: ()()00() cos2bUIIγγγπΦ=− + (2.1) subjected to a damping force202 RγπΦ⎛⎞⎜⎟⎝⎠. The equation of motion is: 20()2dUmdRγγγγπΦ⎛⎞=− −⎜⎟⎝⎠  (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 γ in potential ()Uγ, subjected to a damping force. We study this potential quantum mechanically. Note that the potential depends on the bias current bI, The sketch of the potential for 00.1bII=, 00.3bII= and 00.9bII= is shown in Fig.3.Term Project Yi Yang - 4 - Fig.3 Through adjusting the bias current bI, 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 bI 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: 22()2dHUmdγγ=− += (3.1) where ()Uγ depends on the bias current bI. With the typical parameter: 010 ; 1 ; 0.15IA C pF L nHµ===, my calculation showed that when 00.97bII=, 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 (0)Ur=as the zero potential): -210=-5.0314563659704 10E ×; -211=-5.02280861941517 10E ×; -212-5.01468472553116 10E =×;Term Project Yi Yang - 5 --213-5.00738407077634 10E =× Let us label the energy space in frequency between i-th state and j-th state as: ijω, the energy spaces are: 1082GHzω=; 2177GHzω=; 3269GHzω= 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