Quantum-state engineering with Josephson-junction devices

Yuriy Makhlin
Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany
Landau Institute for Theoretical Physics, Kosygin st. 2, 117940 Moscow, Russia

Gerd Schön
Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany
Forschungszentrum Karlsruhe, Institut für Nanotechnologie, D-76021 Karlsruhe, Germany

Alexander Shnirman
Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany

Since computational applications are widely discussed, we frequently employ here and below the terminology of quantum information theory, referring to a two-state quantum system as a qubit and denoting unitary manipulations of its quantum state as quantum logic operations or gates.

Throughout this review we frequently use temperature units for energies.

In the ground state the superconducting state is totally paired, which requires an even number of electrons on the island. A state with an odd number of electrons necessarily costs an extra quasiparticle energy Δ and is exponentially suppressed at low T. This “parity effect” has been established in experiments below a crossover temperature T^*≈Δ/(k_BlnN_eff), where N_eff is the number of electrons in the system near the Fermi energy (Tuominen et al., 1992; Lafarge et al., 1993; Schön and Zaikin, 1994; Tinkham, 1996). For a small island, T^* is typically one order of magnitude lower than the superconducting transition temperature.

While this cannot be guaranteed with high precision in an experiment, we note that the effective Josephson coupling can be tuned to zero exactly by a design with three junctions.

If the SQUID inductance is not small, the fluctuations of the flux within the SQUID renormalize the energy (2.10). But still, by symmetry arguments, at Φ_x=Φ₀/2 the effective Josephson coupling vanishes.

While expression (2.18) is valid only in leading order in an expansion in E_Jⁱ/ħω_LC^N, higher terms also vanish when the Josephson couplings are put to zero. Hence the decoupling in the idle periods persists.

In later experiments the same group reported phase coherence times as long as 5 ns (Nakamura et al., 2000).

See Mooij et al. (1999) for suggestions on how to control Φ̃_x independent of Φ_x.

Note that in the literature usually the evolution of 〈σ_z(t)〉 has been studied. To establish the connection to the results (4.11) and (4.12) one has to substitute Eqs. (4.9) and (4.10) into the identity σ_z=cosηρ_z+sinηρ_x. Furthermore, we neglect renormalization effects, since they are weak for α≪1.

Nakamura et al. (1999) reached an even smaller ratio for the qubit, but the probe circuit introduced a high stray capacitance.

In the experiments of Nakamura et al. (1999) much of the dephasing can actually be attributed to the measurement device, a dissipative tunnel junction that was coupled permanently to the qubit. Its tunneling resistance was optimized to be large enough not to destroy the qubit’s quantum coherence completely, but low enough to allow for a measurable current. Single-electron tunneling processes, occurring on a time scale of the order of 10 ns, destroy the state of the qubit (escape out of the two-state Hilbert space), thus putting an upper limit on the time when coherent time evolution can be observed. For a more detailed discussion of the experiment and the measurement process we refer to the article by Choi et al. (2001).

See, however, recent work of Krupenin et al. (2000) where the 1/f noise was suppressed by fabricating a metallic island on top of an electrode instead of placing it on the substrate.

In this section τ_φ denotes the dephasing time during a measurement. It is usually much shorter than the dephasing time during the controlled manipulations discussed in the previous sections. From the context it should be clear which situation we refer to.

More precisely, the leading contributions are cotunneling processes, which are weak in high-resistance junctions.

This quantity is denoted as the “energy sensitivity” by Averin (2000b), but Devoret and Schoelkopf (2000) use this term for a different quantity.

This may be overly optimistic and indicate that other sources of dephasing need to be considered as well. For instance, at these slow time scales the background charge fluctuations may dominate. We also note that in the experiment of Nakamura et al. (1999) a stray capacitance in the probe circuit, larger than C_g, renders the dephasing time shorter.

√NOT is not uniquely defined: the matrices i^-n/2U_x(α=nπ/2) with n=1,3,5,7 produce NOT when squared.