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The fact that electrical current is carried by individual charges has been known for over 100 years, yet this discreteness has not been directly observed so far. Almost all current measurements involve measuring the voltage drop across a resistor, using Ohm's law, in which the discrete nature of charge does not come into play. However, by sending a direct current through a microelectronic circuit with a chain of islands connected by small tunnel junctions, the individual electrons can be observed one by one. The quantum mechanical tunnelling of single charges in this one-dimensional array is time correlated, and consequently the detected signal has the average frequency f=I/e, where I is the current and e is the electron charge. Here we report a direct observation of these time-correlated single-electron tunnelling oscillations, and show electron counting in the range 5 fA-1 pA. This represents a fundamentally new way to measure extremely small currents, without offset or drift. Moreover, our current measurement, which is based on electron counting, is self-calibrated, as the measured frequency is related to the current only by a natural constant., Comment: 9 pages, 4 figures; v2: minor revisions, 2 refs added, words added to title, typos corrected

We have fabricated a single Cooper-pair box (SCB) in close proximity to a single electron transistor (SET) operated in the radio-frequency mode (RF-SET) with an inductor and capacitor lithographed directly on chip. The RF-SET was used to measure the charge state of the SCB revealing a 2e periodic charge quantization. We performed spectroscopy measurements to extract the charging energy (E C ) and the Josephson coupling energy (E J ). Control of the temporal evolution of the quantum charge state was achieved by applying fast dc pulses to the SCB gate. The dephasing and relaxation times were extracted from these measurements.

We have fabricated a Josephson charge qubit by capacitively coupling a single-Cooper-pair box (SCB) to an electrometer based upon a single-electron transistor configured for radio-frequency readout (RF-SET). Charge quantization of 2e is observed and microwave spectroscopy is used to extract the Josephson and charging energies of the box. We perform coherent manipulation of the SCB by using very fast DC pulses and observe quantum oscillations in time of the charge that persist to ~=10ns. The observed contrast of the oscillations is high and agrees with that expected from the finite E_J/E_C ratio and finite rise-time of the DC pulses. In addition, we are able to demonstrate nearly 100% initial charge state polarization. We also present a method to determine the relaxation time T_1 when it is shorter than the measurement time T_{meas}., accepted for publication in Phys. Rev. B

We describe fabrication and measurements of Single Electron Transistors operated in the Radio Frequency mode (RF-SET). We demonstrate very high sensitivity for RF-SETs and we evaluate the back-action from an RF-SET, when used as a read-out device for charge qubits. We conclude that single-shot read-out is possible with a signal to noise ratio greater than one, and that niobium qubits would substantially improve the signal to noise ratio. Furthermore we describe how the RF-SET could be used to read out a differential qubit in a gradiometer coupling.

We measure the average charge on the island of a single-electron box, with an accuracy of two thousandths of an electron. Thermal fluctuations alone cannot account for the dependence of the average charge on temperature, on external potential, or on the quasiparticle density of states in the metal from which the box is formed. In contrast, we find excellent agreement between these measurements and a theory that treats the quantum fluctuations of charge perturbatively.

We are developing superconducting direct detectors for submillimeter astronomy that can in principle detectindividual photons. These devices, Single Quasiparticle Photon Counter (SQPC), operate by measuring thequasiparticles generated when single Cooper-pairs are broken by absorption of a submillimeter photon. Thisphotoconductive type of device could yield high quantum efficiency, large responsivity, microsecond responsetimes, and sensitivities in the range of 10 -20 Watts per root Hertz. The use of antenna coupling to a smallabsorber also suggests the potential for novel instrument designs and scalability to imaging or spectroscopicarrays. We will describe the device concept, r ecent results on fabrication and electrical characterization of thesedetectors, issues related to saturation and optimization of the device parameters. Finally, we have developedpractical readout amplifiers for these high-impedance cryogenic detectors based on the Radio-FrequencySingle-Electron Transistor (RF-SET). We will describe results of a demonstration of a transimpedanceamplifier based on closed-loop operation of an RF-SET, and a demonstration of a wavelength-divisionmultiplexing scheme for the RF-SET. These developments will be a key ingredient in scaling to large arrays ofhigh-sensitivity detectors.Keywords: submillimeter, superconducting detector

Single electron transistors (SETs) are very sensitive electrometers and they can be used in a range of applications. In this paper we give an introduction to the SET and present a full quantum mechanical calculation of how noise is generated in the SET over the full frequency range, including a new formula for the quantum current noise. The calculation agrees well with the shot noise result in the low frequency limit, and with the Nyquist noise in the high frequency limit. We discuss how the SET and in particular the radio-frequency SET can be used to read out charge based qubits such as the single Cooper pair box. We also discuss the backaction which the SET will have on the qubit. The back action is determined by the spectral power of voltage fluctuations on the SET island. We will mainly treat the normal state SET but many of the results are also valid for superconducting SETs.

A new four‐layer resist system was developed to fabricate small capacitance niobium tunnel junctions. Nb/AlOx/Al junctions were used to make a superconducting single electron transistor composed of two dc SQUIDs in series. The current‐voltage characteristics were measured and two types of resonant tunneling of Cooper pairs were observed at different magnetic fields.

We demonstrate that a continuously measured microelectronic circuit, the Cooper-pair box measured by a radio-frequency single-electron transistor, approximates a quantum two-level system. We extract the Hamiltonian of the circuit through resonant spectroscopy and measure the excited-state lifetime. The lifetime is more than 10(5) times longer than the inverse transition frequency of the two-level system, even though the measurement is active. This lifetime is also comparable to an estimate of the known upper limit, set by spontaneous emission, for this circuit.

We study analytically and numerically a problem of self-heating in small mesa structures. Our results show that the self-heating is proportional to a characteristic in-plane size of the mesa. Experimental data for small high-$T_c$ superconductor Bi2212 mesas are in qualitative agreement with our calculations. We estimate the self-heating in Bi2212 mesas with different sizes and demonstrate that the self-heating can effectively be obviated in small mesa structures., Comment: 3 pages, 2 figures. In the 2-nd version a misprint in the expression for self-heating was correcter

We have studied niobium superconducting thin wires deposited onto a SrTiO$_{3}$ substrate. By measuring the reflection coefficient of the wires, resonances are observed in the superconducting state in the 130 MHz to 4 GHz range. They are interpreted as standing wave resonances of one-dimensional plasma modes propagating along the superconducting wire. The experimental dispersion law, $\omega$ versus $q$, presents a linear dependence over the entire wave vector range. The modes are softened as the temperature increases close the superconducting transition temperature. Very good agreement are observed between our data and the dispersion relation predicted by Kulik and Mooij and Sch\"on., Comment: Submitted to Physical review Letters

We have measured current-voltage characteristics of two-dimensional arrays of small tunnel junctions at temperatures from 1.5 K to 4.2 K. This corresponds to thermal energies larger than the charging energy. We show that 2D-arrays can be used as primary thermometers in the same way as 1D-arrays, and even have some advantages over 1D-arrays. We have carried out Monte Carlo simulations, which agree with our experimental results., Comment: 4 pages, 4 eps figures. Also available from Journal of Applied Physics (http://link.aip.org/link/?jap/86/3844)

We present intrinsic tunneling spectroscopy measurements on small Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ mesas. The tunnel conductance curves show both sharp peaks at the superconducting gap voltage and broad humps representing the $c$-axis pseudo-gap. The superconducting gap vanishes at $T_c$, while the pseudo-gap exists both above and below $T_c$. Our observation implies that the superconducting and pseudo-gaps represent different coexisting phenomena., Comment: 5 pages, 4 figures

We have measured the transport properties of capacitively coupled one-dimensional arrays of small aluminum Josephson junctions. Under suitable biasing conditions, the samples act as current mirrors; the currents in the two arrays couple to each other and become equally large. The coupling is found both for currents flowing in the same direction and in opposite directions. We have also measured the time dependent fluctuations of the currents in the two arrays, and we find a strong correlation or anticorrelation of the fluctuations depending on the relative direction of the two currents.

We measure the resistance of a normal mesoscopic sample with two Superconducting mirrors and find two regimes with qualitatively different behavior. At temperatures below 90 mK peaks in the conductance were found when the phase difference between the two superconductors is an odd multiple of rr. The peak heights increase with decreasing temperature. Above 100 mK the observed peaks give way to dips in the conductance. While the high-temperature behavior can be explained in terms of the thermal effect [Phys. Rev. Lett, 76, 823 (1996)], we propose that the low-temperature behavior is: a manifestation of resonant transmission of low-energy quasiparticles through Andreev states. [S0163-1829(99)11641-2].

Although it is over 100 years since Millikan showed that electric charge is quantized, it is only with the development of a new generation of charge pumps that this fact can be used to define the fundamental unit of electrical current, the ampere.

A single electron transistor based on Al-AlO_x-Nb tunnel junctions was fabricated by shadow evaporation and in situ barrier formation. Its output current noise was measured, using a transimpedance amplifier setup, as a function of bias voltage, gain, and temperature, in the frequency range 1...300 Hz. The spot noise at 10 Hz is dominated by a gain dependent component, indicating that the main noise contribution comes from fluctuations at the input of the transistor. Deviations from ideal input charge noise behaviour are found in the form of a bias dependence of the differential charge equivalent noise, i. e. the derivative of current noise with respect to gain. The temperature dependence of this effect could indicate that heating is activating the noise sources, and that they are located inside or in the near vicinity of the junctions., 16 pages, 9 figures (EPS)

The conductance (G) of mirror-symmetric, disordered normal (N) metal mesoscopic structures with two interfaces to superconductors (S) has been studied experimentally with applied condensate phase differences Delta phi between the N/S interfaces. At Delta phi = 2n pi(n = 0,1,2,3,...) the conductance showed reentrance to the normal state below the temperature corresponding to the Thouless energy. The current-voltage characteristics were found to be: strongly nonlinear even at distances between the N/S interfaces largely exceeding the normal-metal coherence length. An influence of superconductors almost completely disappeared at Delta phi = (2n + 1) pi where the structures showed normal behavior. Calculations based on a quasiclassical theory have been performed offering a quantitative explanation of such a phase-periodic reentrance. The value of the superconducting gap Delta(eff) at the Ag/Al interface has been obtained. We find that Delta(eff)(T,V-->0) = beta .Delta(BCS)(T) with beta = 0.2 independent of temperature in the temperature interval of 0.1 K < T < 1.6 K; Delta(BCS)(T) is the BCS gap vs T function in Al. [S0163-1829(98)01746-9].

An extensive investigation of low frequency noise in single electron transistors as a function of gain is presented. Comparing the output noise with gain for a large number of bias points, it is found that the noise is dominated by external charge noise. For low gains we find an additional noise contribution which is compared to a model including resistance fluctuations. We conclude that this excess noise is not only due to resistance fluctuations. For one sample, we find a record low minimum charge noise of qn = 9*10^-6 e/sqrt(Hz) in the superconducting state and qn = 9*10^-6 e/sqrt(Hz) in the normal state at a frequency of 4.4 kHz., Comment: 10 pages, LaTex 2.09, 4 figures (epsfig)

We describe transport properties of two-dimensional arrays of low capacitance tunnel junctions, such as the current voltage characteristic and its dependence on external magnetic field and temperature. We discuss several experiments in which the small capacitance of the junctions plays an important role. In arrays where the junctions have a relatively large charging energy, (i.e. when they have a low capacitance) and a high normal state resistance, the low bias resistance increases with decreasing temperature and eventually at very low temperature the array becomes insulating even though the electrodes in the array are superconducting. This transition to the insulating state can be described by thermal activation. In an intermediate region where the junction resistance is of the order of the quantum resistance and the charging energy is of the order of the Josephson coupling energy, the arrays can be tuned between a superconducting and an insulating state with a magnetic field. We describe measurements of this magnetic-field-tuned superconductor insulator transition, and we show that the resistance data can be scaled over several orders of magnitude. Four arrays follow the same universal function. At the transition the transverse (Hall) resistance is found to be very small in comparison with the longitudinal resistance. However, for magnetic field values larger than the critical value.we observe a substantial Hall resistance. The Hall resistance of these arrays oscillates with the applied magnetic field. features in the magnetic field dependence of the Hall resistance can qualitatively be correlated to features in the derivative of the longitudinal resistance, similar to what is found in the quantum Hall effect., Comment: 29 pages, 16 eps figures, uses aipproc.sty and epsfig.sty, contribution to Euroschool on "Superconductivity in Networks and Mesoscopic Systems", held in Siena, Italy (8-20 september 1997)

The charging energy is important for a mesoscopic body with small capacitance to the environment. Connecting the small island via low capacitance, high resistance tunnel junctions, the current through the resulting structure will depend upon the charge of the island - charge that can be applied via a gate electrode. For a normal metal island, the current varies periodically with the charge; the periodicity is the electron charge e. If the island is superconducting, the period, e or 2e, depends upon whether the superconducting energy gap is less than or larger than the charging energy. This parity effect can be used to study possible nodes in the gap function of a high-Tc superconductor.

We describe a circuit model for a flux-driven Superconducting QUantum Interference Device (SQUID). This is useful for developing insight into how these devices perform as active elements in parametric amplifiers. The key concept is that frequency mixing in a flux-pumped SQUID allows for the appearance of an effective negative resistance. In the three-wave, degenerate case treated here, a negative resistance appears only over a certain range of allowed input signal phase. This model readily lends itself to testable predictions of more complicated circuits.

Proper grounding between different ground planes in coplanar superconducting qubit circuits is important to avoid spurious resonances which increase decoherence. Here, the authors present a possible solution to suppress such undesired modes using superconducting aluminum air-bridges which have been fabricated on top of aluminum coplanar waveguide transmission lines. 3D electromagnetic simulations were done to guide the design of the air-bridges such that the input reflection (S11) of the bridges was kept at a minimum level. A fabrication method based on optical lithography techniques was developed and it resulted in air-bridges with a height of approximately 10 μm and lengths of up to 500 μm. The method can be generalized to arbitrary length air-bridge with heights even exceeding 15 μm.

An interaction layer is found at the Al/SiO2 interface in Al/AlOx/Al tunnel junctions grown on SiO2 substrates. The amorphous intermixing layer has an average thickness of about 5 nm. We present the detailed structure of this interfacial layer as determined by transmission electron microscopy. The layer contains alumina with aluminum being octahedrally coordinated according to electron energy loss spectroscopy analysis rather than tetrahedrally coordinated, where the latter coordination is the most common type in amorphous alumina. Depth profiles of the Al-O and Si-O bonding characteristics were also investigated using energy loss near edge structure.

We have measured the temperature dependence and magnetic field dependence of the zero-bias resistance ({ital R}{sub 0}) as well as the current-voltage ({ital I}-{ital V}) characteristics for several two-dimensional arrays of small aluminum Josephson junctions. {ital R}{sub 0}({ital T}) decreases with decreasing temperature, which can be described in terms of two types of vortex motion: flux, flow, and vortex tunneling. At temperatures higher than the Kosterlitz-Thouless transition temperature ({ital T}{gt}{ital T}{sub {ital c}}) or at a bias current greater than the current corresponding to the onset of the nonlinear {ital I}-{ital V} characteristics ({ital I}{gt}{ital I}{sub {ital d}}), the effective damping resistance which characterizes flux-flow motion is found to be approximately equal to the junction normal-state resistance {ital R}{sub {ital N}}. At low temperatures and at small bias current, {ital R}{sub 0} is temperature independent and remains finite down to our minimum attainable temperature. This finite resistance is found to be dependent on the array size as well as the junction parameters. {copyright} {ital 1996 The American Physical Society.}

We have fabricated an aluminum single-electron transistor (SET) and characterized it at frequencies up to 700 kHz. The relatively high frequency was achieved by reducing the capacitance at the SET output. The SET was bonded to an InP high-electron-mobility transistor (HEMT), and biased through a small (100 \ensuremath{\mu}m) on-chip resistor made of 150 tunnel junctions in series. The output-voltage swing of the SET decreases with increasing HEMT current because of heating. Thus, the gain of the HEMT was limited and therefore the noise performance of the system was limited to $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\frac{e}{\sqrt{\mathrm{Hz}}}$ at 10 kHz.

We have investigated several single and double aluminum tunnel junctions in the nonsuperconducting state. Current-voltage (I-V) characteristics were measured up to voltages corresponding to several hundred times e/C, where C is the capacitance of each junction. The individual junctions had capacitances in the order of 0.5 fF and resistances of 50--150 k\ensuremath{\Omega}. The offset voltage, defined as ${\mathit{V}}_{\mathrm{off}}$=V-I/(dI/dV), is resistance independent for given junction capacitances, but depends on the bias voltage. For a single junction the offset voltage is very small for low voltages, but it approaches the local-rule value of e/2C at high voltages. For a double junction the global rule applies at low voltages and the offset voltage equals ${\mathit{V}}_{\mathrm{off}}$=e/${\mathit{C}}_{\mathrm{\ensuremath{\Sigma}}}$, where ${\mathit{C}}_{\mathrm{\ensuremath{\Sigma}}}$=2C+${\mathit{C}}_{0}$. At large voltages the local rule applies and each junction contributes with a voltage of e/2C to the offset voltage, so that ${\mathit{V}}_{\mathrm{off}}$=e/C. This paper shows that a careful analysis of the offset voltage has to be made to determine the values of the capacitances involved.

The conductance of a normal $\left(N\right)$ mesoscopic silver or antimony conductor, to which a superconducting $\left(S\right)$ aluminum wire is attached at two points, oscillates when a control current in the superconductor is varied. The amplitude of oscillations ranges from $2\ifmmode\times\else\texttimes\fi{}{10}^{2}({e}^{2}/h)$ for Ag/Al to $5.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}({e}^{2}/h)$ for Sb/Al structures at $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.02\phantom{\rule{0ex}{0ex}}\mathrm{K}$. The oscillations are attributed to phase transfer from the superconducting condensate to normal electrons via Andreev reflections at the $N\ensuremath{-}S$ interfaces (``mirrors''). The influence of the magnetic field and of the Aharonov-Bohm effect on the oscillations is studied.

An investigation was made of conductance oscillations in normal metal mesoscopic structrures to which superconducting wires were attached. The oscillations were due to phase transfer from the superconducting condensate to normal electrons via Andreev reflections at the normal metal/superconductor interfaces. The phase difference between the interfaces was controlled in a wide range by varying a sub-critical current through the superconductor and, alternatively, by applied magnetic field. This technique may be used to measure phase gradients in superconductors directly, or to control dissipative currents by a non-dissipative supercurrent. Drastic difference was found between the spectra of oscillations in the structures with normal rings and the structures with normal cross-like conductors.

We have operated a single-electron transistor (SET) in the radio-frequency mode. In this way both the bandwidth and the sensitivity of the traditional SET can be increased several orders of magnitude. By optimizing the system we reached a best sensitivity of 3.2×10 −6 e / Hz . We use this system to study charge noise up to 1 MHz and to demonstrate pulsed, short-time measurements of charge.

We have used a radio-frequency single electron transistor to read-out a single Cooper pair box. We observe a Coulomb staircase with 2e periodicity, but with a shorter step corresponding to odd number ofcharges in the box. From the size of this short step we can extract the even–odd energy di3erence, ˜_, and study its magnetic 5eld dependence. We have also characterized the box with microwave spectroscopy and extracted the charging energy and the Josephson coupling energy to be Ec=kB = 1:65 K and EJ0=kB = 0:69 K..

We have measured the current-voltage (IV) characteristics of several two-dimensional arrays of small Josephson junctions as a function of temperature, T and magnetic field B. The junctions have relatively large charging energies ${\mathit{E}}_{\mathit{C}}$\ensuremath{\approxeq}1 K, and normal-state resistances ${\mathit{R}}_{\mathit{N}}$ in the range of 4\char21{}150 k\ensuremath{\Omega}. From the IV characteristics we can deduce the zero-bias resistance ${\mathit{R}}_{0}$ and the threshold voltage ${\mathit{V}}_{\mathit{t}}$ which reveal important information about the dynamics and statics of charge solitons in the array. ${\mathit{R}}_{0}$(T) increases with decreasing temperature and may be described by thermal activation of charge solitons, characterized by an activation energy ${\mathit{E}}_{\mathit{a}}$. When the electrodes are in the normal state, ${\mathit{E}}_{\mathit{a}}$ is close to 1/4${\mathit{E}}_{\mathit{C}}$. At low T, the thermal activation behavior breaks down, and ${\mathit{R}}_{0}$(T) levels off to a value that can be attributed to the quantum fluctuations in the array. This interpretation places limitations on the observability of the charge unbinding, Kosterlitz-Thouless-Berezinskii transition for single electrons. When the electrodes are superconducting, ${\mathit{E}}_{\mathit{a}}$ is much larger and dependent on B. In several samples, both ${\mathit{E}}_{\mathit{a}}$ and ${\mathit{V}}_{\mathit{t}}$ oscillate with B, having a period corresponding to one flux quantum per unit cell. For increasing magnetic fields, ${\mathit{V}}_{\mathit{t}}$ increases until B\ensuremath{\approxeq}250\char21{}450 G where it starts to decrease rapidly. We interpret the B dependence of ${\mathit{E}}_{\mathit{a}}$ and ${\mathit{V}}_{\mathit{t}}$ as a result of competition between Cooper-pair solitons and single-electron solitons.

Several of the characteristic and interesting single charge tunneling effects have been observed in ID arrays of tunnel junctions, i. e., in series coupled arrays of many, closely spaced, ultrasmall junctions. For example, the time correlation of tunnel events was first observed in such an array. The interpretation of the results obtained with an array may be complicated as compared to a single junction, but the additional degrees of freedom for an array, like the formation of charge “solitons”, may be of advantage in developing the single charge effects and, in particular, for applications of these effects. One of the advantages of series coupled junctions is that it is easier to fabricate high resistance tunnel junctions closely connected to the junction under study than high resistance conventional resistors.

Several one-dimensional arrays of ultrasmall aluminum tunnel junctions have been fabricated and measured under microwave irradiation at temperatures down to 35 mK. Microwaves, of frequency f, were coupled either to the edges of the array or to the center of the array via a gate electrode. In both cases steps appear in the I-V characteristics at currents I=nef. When the electrodes are superconducting the steps are weak and can only be seen in the derivative, however when most of the superconductivity is suppressed by a magnetic field the steps are clearly seen directly in the I-V curve. We observe peaks with n up to 4 in the derivative. The steps are clearly observed in the I-V curve even up to 0.5 K. The sharpness of the steps are found to be dependent on the coRC product and the type of microwave coupling used. Gate modulation gave sharper steps, and the sharpness and symmetry of the steps could be altered by applying a dc voltage to the gate electrode. The influence of the background charge is discussed.

We have fabricated and measured a two-dimensional (2-D) and a one-dimensional (1-D) array of small aluminum tunnel junctions. The 2-D array is in a previously unexplored regime where the charging energy is much larger than the Josephson coupling energy, x=EC/EJ”l. The I-V characteristics and their dependence on temperature and magnetic field have been investigated at temperatures down to 35mK. The I-V characteristics show a large Coulomb blockade. The low voltage conductance decreases rapidly as the temperature is lowered. Comparison of the 2-D and 1-D case shows a nearly identical temperature dependence in both the superconducting and normal state. Although there is no Josephson current an oscillatory behavior corresponding to one flux quantum (h/2e) per unit cell is observed.

We have measured current–voltage characteristics of a two-dimensional array of small tunnel junctions at temperatures from 1.4 to 4.2 K. This corresponds to thermal energies larger than the charging energy. We show that 2D-arrays can be used as primary thermometers in the same way as 1D-arrays, and have several advantages over 1D-arrays.

We have observed very low-energy collective modes in a thin superconducting niobium wire below Tc. The extracted dispersion relation ωp(q) follows a quasi-linear q-dependence which is very well described by the 1D plasma dispersion.

One-dimensional arrays of small capacitance SQUIDs have been studied experimentally. The effective Josephson coupling between neighboring electrodes is tunable in situ. The arrays can be tuned from a Josephson-like state, with low resistance and sharp critical current, to a Coulomb blockade state, with infinite resistance and sharp threshold voltage. A quantum phase transition occurs at the crossover between these two types of behavior, which is evident from an analysis of the temperature dependence of the zero bias resistance.

We describe the transport properties of 2d arrays of aluminium–aluminium oxide–aluminium tunnel junctions. The dependence of these properties on temperature and external magnetic field is examined. The arrays have been fabricated on oxidized silicon substrates using e-beam lithography and shadow evaporation. By altering the fabrication parameters used we are able to control the ratio between the charging energy and the Josephson coupling energy and hence the dynamics of the arrays. The various phenomena exhibited by these arrays, such as a magnetic field tuned superconductor–insulator transitions, are discussed.

We have studied 1D-arrays of Josephson junctions with a gate capacitively coupled to the middle of the array. The SQUID shape of each junction enables tuning of the Josephson coupling energy, EJ. By adding a signal to the gate with a radio-frequency f we create a step in the I–V characteristics at the value I=2ef. The step is clearly visible as a peak in the differential resistance dI/dV and can be seen in the whole frequency range studied (10–90 MHz).

We have measured the Hall resistance in two-dimensional arrays of ultrasmall aluminium Josephson junctions. We found that the Hall resistance was periodical with respect to an external magnetic fie ...

Quantum mechanics is the most ground-breaking and fascinating theoretical concept developed in physics during the past century. Much of our present understanding of the microscopic world and its extension into the macroscopic world, including modern technical applications, is based upon quantum mechanics. We have experienced a remarkable development of information and communication technology during the past two decades, to a large extent depending upon successful fabrication of smaller and smaller components and circuits. However, we are finally approaching the physical limits of component miniaturization as we enter a microscopic world ruled by quantum mechanics. Present technology is mainly based upon classical physics such as mechanics and electromagnetism. We now face a similar paradigm shift as was experienced two hundred years ago, at the time of the industrial revolution. Engineered construction of systems is currently increasingly based on quantum physics instead of classical physics, and quantum information is replacing much of classical communication. Quantum computing is one of the most exciting sub-fields of this revolution. Individual quantum systems can be used to store and process information. They are called quantum bits, or qubits for short. A quantum computer could eventually be constructed by combining a number of qubits that act coherently. Important computations can be performed much more quickly than by classical computers. However, while we control and measure a qubit, it must be sufficiently isolated from its environment to avoid noise that causes decoherence at the same time. Currently, low temperature is generally needed to obtain sufficiently long decoherence times. Single qubits of many different kinds can be built and manipulated; some research groups have managed to successfully couple qubits and perform rudimentary logic operations. However, the fundamental problems, such as decoherence, entanglement, quantum measurements and error correction, have yet to be solved. It has been predicted that quantum computers will be able to perform certain complicated computations or simulations in minutes or hours instead of years as with present computers. So far there exist very few useful quantum algorithms; however there is hope that the development of these will be stimulated once there is a breakthrough in hardware. Remarkable progress has been made in quantum engineering and quantum measurements, but a large scale quantum computer is still far off. Quantum communication and cryptography are much closer to the market than a quantum computer. The development of quantum information has meant a large push in the field of quantum physics, that previously could only be studied in the microscopic world. Artificial atoms, realized by circuit technology and mimicking the properties of 'natural' atoms, are one example of the new possibilities opened up by quantum engineering. Several different types of qubits have been suggested. Some are based upon microscopic entities, like atoms and ions in traps, or nuclear spins in molecules. They can have long coherence times (i.e. a long period allowing many operations, of the order of 10 000, to be performed before the state needs to be refreshed) but they are difficult to integrate into large systems. Other qubits are based upon solid state components that facilitate integration and coupling between qubits, but they suffer from interactions with the environment and their coherent states have a limited lifetime. Advanced experiments have been performed with superconducting Josephson junctions and many breakthroughs have been reported in the last few years. They have an advantage in the inherent coherence of superconducting Cooper pairs over macroscopic distances. We chose to focus the Nobel Symposium on Qubits for Future Quantum Information on superconducting qubits to allow for depth in discussions, but at the same time to allow comparison with other types of qubits that may prevail in the long run. The purpose of the symposium was to bring together leading researchers in adjoining fields. Often, microscopic qubits are considered at conferences within atomic, molecular and optical physics, while macroscopic ones belong to the solid state community. At the symposium, we experienced objective comparisons between different types of qubits?pros and cons as well as prospects. One example was the topic of quantum electrodynamics of superconducting circuits where qubits are coupled to a high-Q microwave resonator. This breakthrough technology was covered in several talks and was compared, in detail, with the corresponding case of light coupled to atoms in a cavity. A highlight was the presentation of how arbitrary photon states can be created in a cavity and the measurement of the corresponding Wigner functions. A Nobel Symposium provides an excellent opportunity to bring together a group of outstanding scientists for a stimulating exchange of ideas and results. The present symposium took place in Gothenburg, 25?28 May 2009. In order to allow local researchers and students to get a feeling of what is happening in the field, the first day of the symposium was held at the Chalmers campus. The remaining three days were spent at the mansion built by William Chalmers, the benefactor behind Chalmers University of Technology. Thirty-three speakers gave popular lectures open to the general public, overviews of different types of qubits, quantum phenomena, and quantum computing requirements, as well as specialized contributions in six sessions, and ten posters were displayed. The list of participants, program, abstracts and summaries of presentations is given at www.chalmers.se/mc2/EN/nobel-symposium-2009. In order to encourage constructive interactions and discussions, ample time was given to extensive critical discussions and to individual meetings in relaxing and stimulating environments. Questions and discussions followed all talks but longer, more extensive, discussions of about one hour ended each session. These discussions were initiated by a special questioner (a kind of 'devil's advocate'). Receptions were given by the President of Chalmers and by the City of Gothenburg. The participants also sailed with SS Bohusl?n in the archipelago outside the city. The symposium was sponsored by the Nobel Foundation through its Nobel Symposium Committee and was organized by Thilo Bauch, Tord Claeson, Per Delsing, Ann-Marie Frykestig, Eva Hellberg, G?ran Johansson, G?oran Wendin, and Chris Wilson. Special thanks are given to the program committee: John Clarke, Daniel Est?ve, Steve Girvin, Anne l'Huillier, Anthony Leggett, and Mikko Paalanen. The editor of the proceedings is G?ran Johansson.