Josephson Persistent Current Qubit and Quantum Flux Measurements Caspar H. van der Wal1,∗ , J. E. Mooij1 , A. C. Wallast1 , A. C. J. ter Haar1 , L. M. van Gorkom1, C. J. P. M. Harmans1 , H. Tanaka1,+ , Lin Tian2 , J. J. Mazo3 , T. P. Orlando3, L. Levitov2 , Seth Lloyd4 1 Department of Applied Physics and DIMES, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, the Netherlands 2 Department of Physics, M.I.T., Cambridge, MA 02139, USA 3 Department of Electrical Engineering and Computer Science, M.I.T., Cambridge, MA 02139, USA 4 Department of Mechanical Engineering, M.I.T., Cambridge, MA 02143, USA + Visiting researcher from NTT Basic Research Laboratories, Atsugi, Japan ∗ Corresponding author, e-mail:
[email protected] In a quantum computer information is stored and processed on two-state quantum systems, called qubits. Practical realization of a quantum computer requires a large number of qubits that remain coherent on a time scale much longer than the typical switching time of the qubits. We present the design of a qubit that consists of a micrometersized superconducting loop, which contains three Josephson junctions [1, 2], see Fig. 1. The qubit has persistent currents of opposite sign as its two states. Estimates of the qubit’s decoherence time are longer than 1 ms for all known sources of decoherence, while switching takes 10 ns. The qubits can be integrated and fabricated in large numbers with conventional electron beam lithography. In circuits of underdamped, small-capacitance Josephson junctions, the charge and phase variables behave quantum mechanically. Superpositions of charge states and flux states have been demonstrated, as well as microwave induced quantum dynamics (see e.g. refs [3, 4]). These effects have their origin in the conjugation of the phase and charge variables of Josephson circuits. The qubit design that we present here also contains small-capacitance Josephson junctions. However, even though charge-phase conjugation underlies the quantum behavior of the qubit, it was explicitly designed to be very insensitive to charge effects. In practice, random background charges and 1/f charge noise are always present in Josephson circuits, but these charge effects have negligible influence on the qubit. The qubit loop contains three Josephson junctions. Two junctions have Josephson coupling EJ , the third junction has a coupling of αEJ . We aim at using aluminum technology, with a value of EJ corresponding to 200 GHz. A magnetic flux Φ can be applied to the loop. The loop has a very small diameter (≈ 1µm), such that inductive effects can be neglected in an analysis of the behavior of an isolated qubit. Fluxoid quantization puts a constraint on the gauge invariant phase differences of the three junctions, which leaves two phase degrees of freedom for the qubit. The parameter ranges of interest are 0.5 < α < 1, and Φ ≈ 0.5Φ0 , with Φ0 the
α C, α EJ
C, EJ
Φ
C, EJ
Figure 1: The Josephson junction qubit contains three underdamped tunnel junctions, modeled as a pure Josephson tunnel element with coupling EJ (cross), in parallel with a capacitance C. A magnetic flux Φ can be applied to the loop. The top junction is smaller than the side junctions.
superconducting flux quantum. Useful values are α = 0.75, and Φ = 0.495Φ0. For these parameters the system has two stable classical states with persistent circulating currents of opposite sign. The states are degenerate at Φ = 0.5Φ0 , but have a different energy that increases when going away from Φ = 0.5Φ0 . When taking into account the small capacitance of the junctions, a quantum analysis is needed. The results show that, for the parameters mentioned above, the system is effectively a twostate system. The two eigen states have currents of opposite sign, and a level spacing corresponding to 10 GHz. This system is insensitive to capacitively coupled charges in the environment for two reasons. First, the ratio of the Josephson coupling EJ , and the junction’s charging energy EC = e2 /2C (with C the junction capacitance) is chosen as high as EJ /EC = 80. At given parameters, the level spacing between the qubit states is then modulated less than 1 part in 103 as a function of induced charge. Even so, the capacitance C is still small enough to have quantum tunneling between the two classical states. Second, the system has a Josephson potential that is 2π-periodic in the two phase degrees of freedom. At given parameters, tunneling from unit cell to unit cell in the two-dimensional Josephson potential is negligible when compared to tunneling between the two stable states inside a unit cell.
Tunneling from unit cell to unit cell is undesirable, since this is sensitive to the induced charge. Tunneling inside the unit cell is not. This is analogous to the fact that in a crystal, electron states localized at an atom are independent of crystal momentum. The matrix elements that couple the qubit states when a small flux perturbation is applied to the loop were analyzed numerically. The coupling between the two eigen states can be modulated at resonance by irradiating the system with microwaves. With this technique the state of the qubit can be manipulated arbitrarily, similar to AC-pulse induced Rabi oscillations in NMR. Switching of the qubit by resonant microwave pulses (typically 10 GHz) is possible at a rate of 100 MHz. At this rate population of higher excited states is still negligible. We show that qubits can be coupled inductively using flux transformers, and that the coupling strength can be controlled. All operations required for quantum computing are hereby available. While the concept of our qubit is very much related to work with RF-squids on macroscopic quantum coherence [5], the fact that the qubit contains three Josephson junctions rather than one is accompanied by two advantages. First, the dimensions of the loop can be much smaller, since there is no self-inductance required for realizing an effective two-state system (the loop area can be three orders smaller). Hereby the qubit is many orders of magnitude less sensitive to flux noise. The small size also makes the decoherence time related to spontaneous emission much longer than 1 ms. Second, the design with three Josephson junctions in a loop allows for a parameter range (EJ and C) that is compatible with current fabrication technology, while constraints on exact tuning of the flux threading the loop are less strict than for the RF-squid. For experimental work on such qubits, as well as for actual realization of quantum computation, it is necessary to have magnetometers that are not destroying quantum coherence when the meter is off. Tesche pointed out that it is possible, and necessary, to use extremely underdamped DC-squids for studying the quantum dynamics of flux in a Josephson circuit [5]. The quantum state of a qubit can be probed by an underdamped DC-squid which is inductively coupled to a qubit. At zero bias current, the presence of the DC-squid does not destroy the coherence of the qubit. Readout of the underdamped DC-squid occurs via a measurement of the squid’s critical current. At finite bias current however, any damping due to the impedance of the squid’s bias circuit destroys the coherence of the qubit, and also enhances the decay of the qubit. Therefore, the measurement of the critical current must be realized very fast. Current experimental work in our group is aimed at measuring the flux resolution and intrinsic damping of DC-squids fabricated with aluminum technology.
The allowed time scale for a critical current measurement can be much longer if we first ”freeze” the relevant probabilities of the quantum state. Shnirman et al. pointed out for measurements with a SET-electrometer, that the probabilities for the results of quantum measurements can be preserved much longer than the quantum coherence of the system [6]. This can also be applied for the qubit. The third junction, with coupling αEJ , can be designed as two junctions in parallel. This allows for changing α, while keeping the other parameters fixed. A change of α corresponds to changing the tunnel barrier between the two states. Using this the height of the tunnel barrier between the two states can be increased by a factor three. A fast but adiabatic increase of this tunnel barrier causes the decay of the diagonal elements of the qubit’s density matrix (in eigen vector basis) to slow down, without affecting them. The rapid loss of quantum coherence (decay of the off-diagonal elements of the density matrix to zero) is not relevant since it has no consequence for the outcome of the measurement. This allows for a slower read out of the DC-squid magnetometer. Recent experimental results by Han et al. show that decay times for a metastable flux state in an RF-squid with similar parameters as the qubit, can be about 1 ms in the presence of considerable damping [7]. We acknowledge financial support from the USA ARO grant DAAG55-98-1-0369 and the Dutch Foundation for Fundamental Research on Matter (FOM).
References [1] J. E. Mooij, T. P. Orlando, L. Levitov, Lin Tian, Caspar H. van der Wal, Seth Lloyd, Josephson persistent current qubit, submitted to Science (1999). [2] T. P. Orlando, J. E. Mooij, Lin Tian, Caspar H. van der Wal, L. Levitov, Seth Lloyd, J. J. Mazo, A superconducting persistent current qubit, preprint (1999). [3] Y. Nakamura, this conference; Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Nature 398, 786 (1999). [4] Caspar H. van der Wal, P. Kuiper and J. E. Mooij, Microwave induced quantum transitions of a small Josephson junction array, submitted to LT22/Physica B (1999). [5] C. D. Tesche, Phys. Rev. Lett. 64, 2358 (1990). [6] A. Shnirman and G. Sch¨ on, Phys. Rev. B 57, 15400 (1998). [7] Siyuan Han, R. Rouse and J. E. Lukens, Phys. Rev. Lett. 76, 3404 (1996).
