Quantum-state engineering with Josephson-junction devices

Yuriy Makhlin; Gerd Schön; Alexander Shnirman

doi:10.1103/RevModPhys.73.357

Quantum-state engineering with Josephson-junction devices

Yuriy Makhlin, Gerd Schön, and Alexander Shnirman

Rev. Mod. Phys. 73, 357 – Published 8 May 2001

Abstract

Quantum-state engineering, i.e., active control over the coherent dynamics of suitable quantum-mechanical systems, has become a fascinating prospect of modern physics. With concepts developed in atomic and molecular physics and in the context of NMR, the field has been stimulated further by the perspectives of quantum computation and communication. Low-capacitance Josephson tunneling junctions offer a promising way to realize quantum bits (qubits) for quantum information processing. The article reviews the properties of these devices and the practical and fundamental obstacles to their use. Two kinds of device have been proposed, based on either charge or phase (flux) degrees of freedom. Single- and two-qubit quantum manipulations can be controlled by gate voltages in one case and by magnetic fields in the other case. Both kinds of device can be fabricated with present technology. In flux qubit devices, an important milestone, the observation of superpositions of different flux states in the system eigenstates, has been achieved. The Josephson charge qubit has even demonstrated coherent superpositions of states readable in the time domain. There are two major problems that must be solved before these devices can be used for quantum information processing. One must have a long phase coherence time, which requires that external sources of dephasing be minimized. The review discusses relevant parameters and provides estimates of the decoherence time. Another problem is in the readout of the final state of the system. This issue is illustrated with a possible realization by a single-electron transistor capacitively coupled to the Josephson device, but general properties of measuring devices are also discussed. Finally, the review describes how the basic physical manipulations on an ideal device can be combined to perform useful operations.

DOI:https://doi.org/10.1103/RevModPhys.73.357

Authors & Affiliations

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

^aSince 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.
^bThroughout this review we frequently use temperature units for energies.
^cIn 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 $T^{*} \approx Δ {/ (k}_{B} \ln N_{eff}),$ 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.
^dWhile 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.
^eIf 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 $Φ_{x} = Φ_{0} / 2$ the effective Josephson coupling vanishes.
^fWhile expression (2.18) is valid only in leading order in an expansion in $E_{J}^{i} / ħ ω_{LC}^{N},$ higher terms also vanish when the Josephson couplings are put to zero. Hence the decoupling in the idle periods persists.
^gIn later experiments the same group reported phase coherence times as long as 5 ns (Nakamura et al., 2000).
^hSeeMooij et al. (1999) for suggestions on how to control ${\tilde{Φ}}_{x}$ independent of $Φ_{x} .$
ⁱNote 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 $σ_{z} = \cos η ρ_{z} + \sin η ρ_{x} .$ Furthermore, we neglect renormalization effects, since they are weak for $α ≪ 1.$
^jNakamura et al. (1999) reached an even smaller ratio for the qubit, but the probe circuit introduced a high stray capacitance.
^kIn the experiments ofNakamura 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 byChoi et al. (2001).
^lSee, however, recent work ofKrupenin 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.
^mIn 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.
ⁿMore precisely, the leading contributions are cotunneling processes, which are weak in high-resistance junctions.
^oThis quantity is denoted as the “energy sensitivity” byAverin (2000b), butDevoret and Schoelkopf (2000) use this term for a different quantity.
^pThis 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 ofNakamura et al. (1999) a stray capacitance in the probe circuit, larger than $C_{g},$ renders the dephasing time shorter.
^q $\sqrt{NOT}$ is not uniquely defined: the matrices $i^{- n / 2} U_{x} (α = n π / 2)$ with $n = 1, 3, 5, 7$ produce NOT when squared.