Q-bit based on Cooper Pair Transistor

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					                  A quantum bit based on a Cooper Pair Transistor
                     David Haviland, Nanostructure Physics, KTH

In contrast to all modern electronic circuits, which obey the laws of classical
electrodynamics, the electrodynamics of small capacitance Josephson junction circuits is
quantum electrodynamics. The time dependence of the circuit variables, the flux and
charge at the various circuit nodes, is described by a time dependant Schrödinger
Equation. This unique and interesting situation arises due to the Josephson coupling
energy of superconducting tunnel junctions, which depends on the phase difference
between the circuit nodes. Furthermore, the charging energy, which depends on the
charge difference between circuit nodes, is relatively large due to the very small
capacitance of the nano-meter scale tunnel junctions of the circuit. The phase (flux) and
number (charge) become complementary variables of the circuit Hamiltonian. Quantum
physics describes the state of the electronic circuit, and the current and voltage become
quantum variables, whose true values can only be known in a probabilistic sense, after
necessarily destroying the quantum state with a measurement.

This field of study has recently (in the last 5 years) taken a great leap forward when
experiments were performed which demonstrated the ability to perform quantum control
experiments on such circuits. Such experiments consist of a step where the initial state of
the system is “prepared” as a quantum superposition of two basis states. This
superposition state then evolves in time, and after some well defined time, the state of the
system is “measured”. There is a growing interest in this type of quantum control
experiment, with the hope that such systems could be used to code information, and
realize the quantum computer. A quantum computer which could compete with present
day classical electronic computers, is a long way off. Nevertheless, the quantum
computer is an intriguing and not totally unrealistic goal, which presses the advancement
of this technology and the advancement of our fundamental understanding of quantum
physics.

One of the basic components of these small capacitance circuits is the Cooper Pair
Transistor (CPT). This device is a small capacitance superconducting island, with a
source and drain, from which Cooper pairs (superconducting charge carriers) can tunnel
on to and off of the island. A third electrode, the gate, has a capacitive coupling to the
island.




         Fig 1: A cooper pair transistor with SQUID junctions for tunable Josephson energy
The quantum energy levels of this circuit can be calculated as a function of two external
parameters, the phase difference between the source and drain, +, and the gate charge Qg.
Fig. 2 shows the result of such a calculation, displaying the lowest two energy states for
the case of a symmetric CPT with EJ=EC. The states are periodic in Qg=2e and +=2.




Fig. 2: The lowest energy bands of a Cooper pair transistor plotted versus the two external parameters of
                    the Hamiltonian, the external phase, +, and the gate charge Qg.

The gate charge is simply controlled by applying a voltage to the gate electrode,
Qg=CgCg, and the external phase can be controlled either by sending a Supercurrent
current through the CPT, or by closing the source and drain with a superconducting
element and applying a magnetic flux to the resulting loop. The CPT thus becomes an
ideal system for the study and design of quantum control experiments, with two
orthogonal, external “knobs” with which we can control the circuit Hamiltonian.

Fig. 3 shows the current-voltage characteristic of a CPT of the type shown in fig. 1. We
have studied the ground state of this device while continuously tuning the parameter EJ.1
The switching current of the CPT is the current at which the CPT jumps from the
supercurrent branch to a finite voltage level. Actually, there is a non-zero voltage across
the CPT even on the supercurrent branch of the current-voltage characteristic. This finite
voltage is due to phase diffusion, which is a result of the finite temperature and the
overdamped classical dynamics of the external phase. The critical current could be tuned
by both gate charge and magnetic flux applied to the SQUID loops. Fig. 4 shows this
modulation, and comparison with theory.

In our project within the NanoDev Center we have the goal of going beyond simple
ground state measurements to realize a quantum control experiment with the CPT using
the lowest two energy states. This qubit (quantum two level system) will be measured by


1
 Switching current of a Cooper pair transistor with tunable Josephson Junctions, P. Ågren, J. Walter, and
D. B. Haviland, Phys. Rev. B. 66, 014510, (2002)
measuring the critical current between the source and drain electrodes. Quantum control
experiments require high frequency techniques, and at present we are developing fast




      Fig. 3. The current-voltage characteristic of the CPT. The inset shows the finite voltage on the
                               Supercurrent branch due to phase diffusion.

pulse methods to quickly read out the critical current. This method has recently been
successful in measuring the state of a CPT shunted by a large Josephson junction2. We
are going to investigate if it is possible to realize quantum measurement with much
smaller currents, and when the classical dynamics of the external phase is overdamped.
Once we have developed a successful measurement, or read-out strategy, we will attempt
to perform a quantum control experiment by rapidly manipulating the gate charge.




    Fig. 4 Gate modulation of the switching current, experimental (left) and theoretical (right) curves.




2
 Manipulating the quantum state of an electrical circuit. D. Vion, A. Aassime, A. Cottet, P. Joyez, H.
Pothier, C. Urbina, D. Esteve, and M.H. Devoret, Science 296 (2002), p. 286-9.

				
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