Purpose The purpose of this paper is to employ an alternative numerical approach to analyze the characteristics of superconducting charge qubit based on a single Cooper pair box (SCB), also to study the influence of the bias current. Design/methodology/approach The paper starts with the circuit model of a charge qubit system based on Josephson junction using Hamiltonian formalism. Corresponding Schrodinger eigenvalue problem with periodic boundary condition is converted to the Mathieu type eigenvalue problem. By applying finite difference technique, energy spectrum of charge qubit is obtained and the solutions in the lowest band are obtained in the form of Bloch waves whose superposition provides a wave packet to investigate the effect of bias current to the Coulomb blockade. Findings The paper identifies a periodic tridiagonal Hermitian matrix form of the eigenvalue problem that is believed to be a special eigenvalue problem. The paper emphasizes that Schrodinger formalism is very useful to model superconducting qubits systems. The investigations indicate that the bias current strongly influences the Coulomb blockade and expectation value of supercurrent (as well as number of Cooper pairs) are affected by gate voltage and energy scale. Research limitations/implications In the present study, Schrodinger eigenvalue problem is time independent, therefore, current‐voltage characteristics of the charge qubit system could not be considered. The solution technique applied here can also be used to apply other type of Josephson junction based qubits and circuits. Practical implications The paper includes theoretical findings for the development of superconducting qubit that can be valuable for experimentalist. The result obtained in this study is useful for the comparison of experimental study with the expectation value of number of Cooper pairs as function of gate voltage. Working parameters of a SCB can be determined from the findings. Originality/value This paper fulfils the contribution of the numerical study of Schrodinger equation for the investigation of superconducting qubits under the influence of bias current.
Purpose The purpose of this paper is to employ an alternative numerical approach to analyze the characteristics of superconducting charge qubit based on a single Cooper pair box (SCB), also to study the influence of the bias current. Design/methodology/approach The paper starts with the circuit model of a charge qubit system based on Josephson junction using Hamiltonian formalism. Corresponding Schrodinger eigenvalue problem with periodic boundary condition is converted to the Mathieu type eigenvalue problem. By applying finite difference technique, energy spectrum of charge qubit is obtained and the solutions in the lowest band are obtained in the form of Bloch waves whose superposition provides a wave packet to investigate the effect of bias current to the Coulomb blockade. Findings The paper identifies a periodic tridiagonal Hermitian matrix form of the eigenvalue problem that is believed to be a special eigenvalue problem. The paper emphasizes that Schrodinger formalism is very useful to model superconducting qubits systems. The investigations indicate that the bias current strongly influences the Coulomb blockade and expectation value of supercurrent (as well as number of Cooper pairs) are affected by gate voltage and energy scale. Research limitations/implications In the present study, Schrodinger eigenvalue problem is time independent, therefore, current‐voltage characteristics of the charge qubit system could not be considered. The solution technique applied here can also be used to apply other type of Josephson junction based qubits and circuits. Practical implications The paper includes theoretical findings for the development of superconducting qubit that can be valuable for experimentalist. The result obtained in this study is useful for the comparison of experimental study with the expectation value of number of Cooper pairs as function of gate voltage. Working parameters of a SCB can be determined from the findings. Originality/value This paper fulfils the contribution of the numerical study of Schrodinger equation for the investigation of superconducting qubits under the influence of bias current.
This paper numerically analyzes the influence of second harmonics on the characteristics of Josephson junction quantum bits (qubits). It is shown that the splitting of energy states of phase qubits is sensitive to energy scale and unharmonicity parameters. On the other hand, for charge qubits, the expectation value of the Cooper-pair number is not affected by the inclusion of unharmonicity parameters, whereas the expectation value of the superconducting current is strongly influenced by energy scale and unharmonicity parameters. The numerical results seem to present the importance of second harmonics in the realization of superconducting qubits.
The concept of a quantum computer was invented in the beginning of the 1980s as a quantum generalization of the reversible classical computer. After discoveries in the middle of the 1990s of quantum algorithms, which wouldsolve some problems considered intractable for classical computers, the field of quantum computing has developed rapidly. A wide range of quantum systems are investigated for their possible ability to implement quantum algorithms in practice. The most promising solid state implementations are superconducting electrical circuits based on the Josephson effect, and the Coulomb blockade effect. Superconducting circuits are macroscopic quantum systems which can be fabricated using standard lithography technologies. Superconducting qubits - basic building blocks of a quantum computer - have been developed at several research laboratories, among others the MC2 here at Chalmers. This thesis presents a theoretical investigation of controllable coupling of superconducting qubits based on the single Cooper-pair box. Two coupling schemes are investigated in detail: current-controlled coupling via large Josephson junctions, and coupling via a superconducting stripline cavity. Both coupling designs are scalable and suitable for the charge regime of the single Cooper-pair box, as well as for the charge-phase regime where the qubit is better protected from environmental noise. Quantum gate protocols which are relevant for these physical couplings are discussed. The investigations show that, for both considered designs, the simplest two-qubit gate is the conditional phase-shift gate. This gate is universal for quantum computing, and it can create maximallyentangled qubit pairs in only one run. The investigation combines methods from electrical engineering, quantum mechanics and quantum computer science.
We present a series of measurements on nine single Cooper-pair boxes (SCBs), where the charging energy, E C , and the Josephson coupling energy, E J , have been varied. We have investigated both the ground state properties of the SCBs and their quantum coherent properties. The state of the SCBs could be manipulated by an external gate voltage and the charge was measured by coupling it capacitatively to a radio-frequency single-electron-transistor (RF-SET). By ramping the gate voltage and simultaneously measuring the charge of the SCBs using the RF-SET, we could measure the Coulomb staircases of the SCBs. For sufficiently low E C the SCBs showed a fully 2 e periodic Coulomb staircase. For samples with higher E C the staircase showed a short step for odd number of charges indicating quasi-particle 'poisoning'. However, if E C was not too large, the short step could be removed by applying a parallel magnetic field. We attribute this effect to a stronger suppression of the superconducting energy gap in the reservoir than in the box. Using microwave spectroscopy we have determined E C and E J for the SCBs. These values agree well with the shape of the Coulomb staircases which we measure. For a limited range of gate voltage, the SCBs were found to behave as model two-level quantum-mechanical systems. A non-adiabatic change in the induced island charge was used to bring two charge states into resonance. The resulting time evolution showed clear charge oscillations between the ground and excited state, with an amplitude above 70% and a frequency given by the energy level separation divided by Planck's constant. These oscillations had a longest coherence time of T 2 = 9 ns, at a point where the pure charge states are degenerate. The coherence time at this point was found to be limited by the relaxation rate. Away from the charge degeneracy point, the coherence time was limited by the pure dephasing rate. The dependence of T 2 on gate charge suggested that low frequency fluctuators were the main source of dephasing away from the degeneracy point.
This thesis presents the results of an experimental study on single-charge effects in nanoscale Josephson junctions and Cooper pair transistors (CPTs). In nanoscale Josephson junctions the charging energy EC becomes significant at sub-Kelvin temperatures and single-charge effects, such as the Coulomb blockade of Cooper pair tunneling, influence the transport properties. In order to observe charging effects in a single Josephson junction, the impedance of the electromagnetic environment surrounding the junction has to be larger than the quantum resistance (RQ=h/4e2≈6.45kΩ). In this work the high impedance environment is obtained by biasing the sample under test (single Josephson junction or CPT) with four one-dimensional Josephson junction arrays having SQUID geometry. The advantage of this configuration is the possibility of tuning in situ the effective impedance of the electromagnetic environment. By applying a magnetic field perpendicular to the SQUID loops, the Josephson energy EJ of the SQUIDs is suppressed, resulting in an increase of the measured zero bias resistance of the arrays of several orders of magnitude (104< R0 (Ω) <109). This bias method enables the measurement of the same sample in environments with different impedance. As the impedance of the environment is increased, the current-voltage characteristics (IVCs) of the single Josephson junction and of the CPT show a well defined Coulomb blockade feature with a region of negative differential resistance, signature of the coherent tunneling of single Cooper pairs. The measured IVCs of a single Josephson junction with SQUID geometry in the high impedance environment show a qualitative agreement with the Bloch band theory as the EJ/EC ratio of the junction is tuned with the magnetic field. We also studied a single nontunable Josephson junction with strong coupling (EJ/EC > 1), where the exact dual of the overdamped Josephson effect is realized, resulting in a dual shape of the IVC, where the roles of current and voltage are exchanged. Here, we make for the first time a detailed quantitative comparison with a theory which includes the effect of fluctuations due to the finite temperature of the environment. The measurements on CPTs in the high impedance environment showed that the Coulomb blockade voltage is modulated periodically by the gate-induced charge. The gate-voltage dependence of the CPT changes from e-periodic to 2e-periodic as the impedance of the environment is increased. The high impedance environment reduces quasiparticle tunneling rates, thereby restoring the even parity of the CPT island. This behavior suggests that high impedance leads can be used to effectively suppress quasiparticle poisoning.
We have studied dynamic properties of a special Josephson charge qubit. The device consists of a single-Cooper-pair transistor (with capacitive gate) closed by a superconducting inductive loop. The phase bias is implemented by an external flux that threads the loop. Restricting ourselves to a two-band model, we have derived analytical expressions for the oppositely circulating ring currents in terms of gate charge and phase bias. They permit the determination of important device characteristics such as critical current, loop parameters, and Josephson inductance. For the purpose of measurements in connection with macroscopic quantum coherence, we investigate the qubit's microwave response within the frame of Bloch equations with phenomenological relaxation rates. We find the (nonlinear) qubit impedance in terms of quality, detuning, and the quantum resistance unit, which reflects the competition between irradiation and dephasing.
As a macroscopic quantum system, superconducting junction devices are considered both analytically and numerically with the help of the Schrödinger wave equation. Considering the Cooper pair wave function ψ in a time-dependent nature, the Hamiltonian of the system is constructed. The phase difference θ is assumed to be a unique independent variable of the superconducting junction in order to evaluate the characteristics of the system. The probability ρ(θ) and super current j (θ) densities are determined for different applied voltages V . It is observed that both the amplitude and the period of the wave function ψ(θ) have indicated different behaviors as functions of θ for various cases of V . Similarly, ρ and j reflect drastic changes. It is shown that numerical results are in good agreement with the analytic results. Our results indicate consistency with some earlier experimental studies for different superconducting devices.