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Quantum computers promise to solve certain problems exponentially faster than possible classically but are challenging to build because of their increased susceptibility to errors. Remarkably, however, it is possible to detect and correct errors without destroying coherence by using quantum error correcting codes [1]. The simplest of these are the three-qubit codes, which map a one-qubit state to an entangled three-qubit state and can correct any single phase-flip or bit-flip error of one of the three qubits, depending on the code used [2]. Here we demonstrate both codes in a superconducting circuit by encoding a quantum state as previously shown [3,4], inducing errors on all three qubits with some probability, and decoding the error syndrome by reversing the encoding process. This syndrome is then used as the input to a three-qubit gate which corrects the primary qubit if it was flipped. As the code can recover from a single error on any qubit, the fidelity of this process should decrease only quadratically with error probability. We implement the correcting three-qubit gate, known as a conditional-conditional NOT (CCNot) or Toffoli gate, using an interaction with the third excited state of a single qubit, in 63 ns. We find 85\pm1% fidelity to the expected classical action of this gate and 78\pm1% fidelity to the ideal quantum process matrix. Using it, we perform a single pass of both quantum bit- and phase-flip error correction with 76\pm0.5% process fidelity and demonstrate the predicted first-order insensitivity to errors. Concatenating these two codes and performing them on a nine-qubit device would correct arbitrary single-qubit errors. When combined with recent advances in superconducting qubit coherence times [5,6], this may lead to scalable quantum technology., 10 pages, 7 figures

Superconducting quantum circuits based on Josephson junctions have made rapid progress in demonstrating quantum behavior and scalability. However, the future prospects ultimately depend upon the intrinsic coherence of Josephson junctions, and whether superconducting qubits can be adequately isolated from their environment. We introduce a new architecture for superconducting quantum circuits employing a three dimensional resonator that suppresses qubit decoherence while maintaining sufficient coupling to the control signal. With the new architecture, we demonstrate that Josephson junction qubits are highly coherent, with $T_2 \sim 10 \mu$s to $20 \mu$s without the use of spin echo, and highly stable, showing no evidence for $1/f$ critical current noise. These results suggest that the overall quality of Josephson junctions in these qubits will allow error rates of a few $10^{-4}$, approaching the error correction threshold., Comment: 5 pages, 3 figures, 1 table. Accepted to Phys. Rev. Lett

Quasiparticle tunneling across a Josephson junction sets a limit for the lifetime of a superconducting qubit state. We develop a general theory of the corresponding decay rate in a qubit controlled by a magnetic flux. The flux affects quasiparticles tunneling amplitudes, thus making the decay rate flux-dependent. The theory is applicable for an arbitrary quasiparticle distribution. It provides estimates for the rates in practically important quantum circuits and also offers a new way of measuring the phase-dependent admittance of a Josephson junction., 4.5 pages, 1 figure; minor changes to text & references, to appear in Phys. Rev. Lett

As low-loss non-linear elements, Josephson junctions are the building blocks of superconducting qubits. The interaction of the qubit degree of freedom with the quasiparticles tunneling through the junction represent an intrinsic relaxation mechanism. We develop a general theory for the qubit decay rate induced by quasiparticles, and we study its dependence on the magnetic flux used to tune the qubit properties in devices such as the phase and flux qubits, the split transmon, and the fluxonium. Our estimates for the decay rate apply to both thermal equilibrium and non-equilibrium quasiparticles. We propose measuring the rate in a split transmon to obtain information on the possible non-equilibrium quasiparticle distribution. We also derive expressions for the shift in qubit frequency in the presence of quasiparticles., Comment: 25 pages, 6 figures

The authors describe preliminary mixer results using resistively shunted Nb/AlO/sub x//Nb tunnel junctions in a 100-GHz waveguide mixer mount. The mixer utilizes robust, lithographically defined devices which have nonhysteretic I-V curves. A receiver temperature of 390 K (D5B) has been obtained with a conversion loss of -6.5 dB. The receiver's behavior agrees qualitatively with the behavior predicted by the resistively shunted junction model. Substantial improvements in performance are expected with the use of better-optimized shunted junctions and numerical simulations suggest that, if devices with higher I/sub C/R/sub N/ (critical-current normal-resistance) products can be obtained. Josephson effect mixers could be competitive with superconductor-insulator-superconductor (SIS) mixers at high frequencies. >

We have fabricated an aluminum single-electron transistor and characterized it at frequencies up to 10 MHz by measuring the reflected signal from a resonant tank in which the transistor is embedded. We measured the charge sensitivity of this radio-frequency single-electron transistor to be 3.2×10−6 e/Hz, which corresponds to the uncoupled energy sensitivity of 4.8 ℏ. Our measurements indicate that with further improvements, the radio-frequency single-electron transistor could reach the shot-noise limit estimated to be about 1 ℏ.

We employ simultaneous shaping of in-phase and out-of-phase resonant microwave drives to reduce single-qubit gate errors arising from the weak anharmonicity of transmon superconducting artificial atoms. To reduce the effect of higher levels present in the transmon spectrum, we apply Gaussian and derivative-of-Gaussian envelopes to the in-phase and out-of-phase quadratures, respectively, and optimize over their relative amplitude. Using randomized benchmarking, we obtain a minimum average error per gate of $0.007\ifmmode\pm\else\textpm\fi{}0.005$ using 4-ns-wide pulses, which is limited by decoherence. This simple optimization technique works for multiple transmons coupled to a single microwave resonator in a quantum bus architecture.

We demonstrate a qubit readout scheme that exploits the Jaynes-Cummings nonlinearity of a superconducting cavity coupled to transmon qubits. We find that in the strongly-driven dispersive regime of this system, there is the unexpected onset of a high-transmission "bright" state at a critical power which depends sensitively on the initial qubit state. A simple and robust measurement protocol exploiting this effect achieves a single-shot fidelity of 87% using a conventional sample design and experimental setup, and at least 61% fidelity to joint correlations of three qubits., Comment: 5 pages, 4 figures

Electron and nuclear spins have good coherence times and an ensemble of spins is a promising candidate for a quantum memory. By employing holographic techniques via field gradients a single ensemble may be used to store many bits of information. Here we present a coherent memory using a pulsed magnetic field gradient, and demonstrate the storage and retrieval of up to 100 weak 10 GHz coherent excitations in collective states of an electron spin ensemble. We further show that such collective excitations in the electron spin can then be stored in nuclear spin states, which offer coherence times in excess of seconds., Comment: Corrected reference and typos

A single-channel joint readout is used to analyze highly entangled two-qubit states in a circuit quantum electrodynamics architecture. The measurement model for the readout is fully characterized, demonstrating a large sensitivity to two-qubit correlations. We quantify the high degree of entanglement by measuring a violation of the Clauser-Horne-Shimony-Holt inequality with a value of $2.61\ifmmode\pm\else\textpm\fi{}0.04$, without optimizing the preparation of the two-qubit state. In its present form, this joint readout can resolve improvements to the fidelity of two-qubit operations and be extended to three or four qubits.

Traditionally, quantum entanglement has played a central role in foundational discussions of quantum mechanics. The measurement of correlations between entangled particles can exhibit results at odds with classical behavior. These discrepancies increase exponentially with the number of entangled particles. When entanglement is extended from just two quantum bits (qubits) to three, the incompatibilities between classical and quantum correlation properties can change from a violation of inequalities involving statistical averages to sign differences in deterministic observations. With the ample confirmation of quantum mechanical predictions by experiments, entanglement has evolved from a philosophical conundrum to a key resource for quantum-based technologies, like quantum cryptography and computation. In particular, maximal entanglement of more than two qubits is crucial to the implementation of quantum error correction protocols. While entanglement of up to 3, 5, and 8 qubits has been demonstrated among spins, photons, and ions, respectively, entanglement in engineered solid-state systems has been limited to two qubits. Here, we demonstrate three-qubit entanglement in a superconducting circuit, creating Greenberger-Horne-Zeilinger (GHZ) states with fidelity of 88%, measured with quantum state tomography. Several entanglement witnesses show violation of bi-separable bounds by 830\pm80%. Our entangling sequence realizes the first step of basic quantum error correction, namely the encoding of a logical qubit into a manifold of GHZ-like states using a repetition code. The integration of encoding, decoding and error-correcting steps in a feedback loop will be the next milestone for quantum computing with integrated circuits., 7 pages, 4 figures, and Supplementary Information (4 figures).

The last two decades have seen tremendous advances in our ability to generate and manipulate quantum coherence in mesoscopic superconducting circuits. These advances have opened up the study of quantum optics of microwave photons in superconducting circuits as well as providing important hardware for the manipulation of quantum information. Focusing primarily on charge-based qubits, we provide a brief overview of these developments and discuss the present state of the art. We also survey the remarkable progress that has been made in realizing circuit quantum electrodynamics (QED) in which superconducting artificial atoms are strongly coupled to individual microwave photons., Proceedings of Nobel Symposium 141: Qubits for Future Quantum Information

We propose to couple an on-chip high finesse superconducting cavity to the lateral-motion and spin state of a single electron trapped on the surface of superfluid helium. We estimate the motional coherence times to exceed 15 microseconds, while energy will be coherently exchanged with the cavity photons in less than 10 nanoseconds for charge states and faster than 1 microsecond for spin states, making the system attractive for quantum information processing and cavity quantum electrodynamics experiments. Strong interaction with cavity photons will provide the means for both nondestructive readout and coupling of distant electrons., 4 pages, 3 figures, supplemental materials

We propose to encode a register of quantum bits in different collective electron spin wave excitations in a solid medium. Coupling to spins is enabled by locating them in the vicinity of a superconducting transmission line cavity, and making use of their strong collective coupling to the quantized radiation field. The transformation between different spin waves is achieved by applying gradient magnetic fields across the sample, while a Cooper Pair Box, resonant with the cavity field, may be used to carry out one- and two-qubit gate operations., Comment: Several small corrections and modifications. This version is identical to the version published in Phys. Rev. Lett

We propose an approach for studying quantum information and performing high resolution spectroscopy of rotational states of trapped molecular ions using an on-chip superconducting microwave resonator. Molecular ions have several advantages over neutral molecules. Ions can be loaded into deep (1 eV) RF traps and are trapped independent of the electric dipole moment of their rotational transition. Their charge protects them from motional dephasing and prevents collisional loss, allowing 1 s coherence times when used as a quantum memory, with detection of single molecules possible in, 9 pages, 1 figure

By harnessing the superposition and entanglement of physical states, quantum computers could outperform their classical counterparts in solving problems of technological impact, such as factoring large numbers and searching databases. A quantum processor executes algorithms by applying a programmable sequence of gates to an initialized register of qubits, which coherently evolves into a final state containing the result of the computation. Simultaneously meeting the conflicting requirements of long coherence, state preparation, universal gate operations, and qubit readout makes building quantum processors challenging. Few-qubit processors have already been shown in nuclear magnetic resonance, cold ion trap and optical systems, but a solid-state realization has remained an outstanding challenge. Here we demonstrate a two-qubit superconducting processor and the implementation of the Grover search and Deutsch-Jozsa quantum algorithms. We employ a novel two-qubit interaction, tunable in strength by two orders of magnitude on nanosecond time scales, which is mediated by a cavity bus in a circuit quantum electrodynamics (cQED) architecture. This interaction allows generation of highly-entangled states with concurrence up to 94%. Although this processor constitutes an important step in quantum computing with integrated circuits, continuing efforts to increase qubit coherence times, gate performance and register size will be required to fulfill the promise of a scalable technology., Comment: 6 pages, 1 table, 4 figures, and Supplementary Information (3 pages, 3 figures); Expanded author list, updated references, and minor improvements to text and figures

The topic of quantum noise has become extremely timely due to the rise of quantum information physics and the resulting interchange of ideas between the condensed matter and AMO/quantum optics communities. This review gives a pedagogical introduction to the physics of quantum noise and its connections to quantum measurement and quantum amplification. After introducing quantum noise spectra and methods for their detection, we describe the basics of weak continuous measurements. Particular attention is given to treating the standard quantum limit on linear amplifiers and position detectors using a general linear-response framework. We show how this approach relates to the standard Haus-Caves quantum limit for a bosonic amplifier known in quantum optics, and illustrate its application for the case of electrical circuits, including mesoscopic detectors and resonant cavity detectors., Substantial improvements over initial version; include supplemental appendices.

On the level of single atoms and photons, the coupling between atoms and the electromagnetic field is typically very weak. By employing a cavity to confine the field, the strength of this interaction can be increased many orders of magnitude to a point where it dominates over any dissipative process. This strong-coupling regime of cavity quantum electrodynamics has been reached for real atoms in optical cavities, and for artificial atoms in circuit QED and quantum-dot systems. A signature of strong coupling is the splitting of the cavity transmission peak into a pair of resolvable peaks when a single resonant atom is placed inside the cavity - an effect known as vacuum Rabi splitting. The circuit QED architecture is ideally suited for going beyond this linear response effect. Here, we show that increasing the drive power results in two unique nonlinear features in the transmitted heterodyne signal: the supersplitting of each vacuum Rabi peak into a doublet, and the appearance of additional peaks with the characteristic sqrt(n) spacing of the Jaynes-Cummings ladder. These constitute direct evidence for the coupling between the quantized microwave field and the anharmonic spectrum of a superconducting qubit acting as an artificial atom., 6 pages, 4 figures. Supplementary Material and Supplementary Movies are available at http://www.eng.yale.edu/rslab/publications.html

We present an experimental realization of the transmon qubit, an improved superconducting charge qubit derived from the Cooper pair box. We experimentally verify the predicted exponential suppression of sensitivity to 1/f charge noise [J. Koch et al., Phys. Rev. A 76, 042319 (2007)]. This removes the leading source of dephasing in charge qubits, resulting in homogenously broadened transitions with relaxation and dephasing times in the microsecond range. Our systematic characterization of the qubit spectrum, anharmonicity, and charge dispersion shows excellent agreement with theory, rendering the transmon a promising qubit for future steps towards solid-state quantum information processing., Comment: 4+ pages, 4 figures

Qubits encoded in longlived rotational states of polar molecules interact strongly with single photons of a superconducting stripline cavity. We discuss potential applications of such a hybrid device for quantum information processing.

We present a detailed characterization of coherence in seven transmon qubits in a circuit QED architecture. We find that spontaneous emission rates are strongly influenced by far off-resonant modes of the cavity and can be understood within a semiclassical circuit model. A careful analysis of the spontaneous qubit decay into a microwave transmission-line cavity can accurately predict the qubit lifetimes over two orders of magnitude in time and more than an octave in frequency. Coherence times $T_1$ and $T_2^*$ of more than a microsecond are reproducibly demonstrated., Comment: 4 pages, 4 figures

We review the main theoretical and experimental results for the transmon, a superconducting charge qubit derived from the Cooper pair box. The increased ratio of the Josephson to charging energy results in an exponential suppression of the transmon's sensitivity to 1/f charge noise. This has been observed experimentally and yields homogeneous broadening, negligible pure dephasing, and long coherence times of up to 3 microseconds. Anharmonicity of the energy spectrum is required for qubit operation, and has been proven to be sufficient in transmon devices. Transmons have been implemented in a wide array of experiments, demonstrating consistent and reproducible results in very good agreement with theory., Comment: 6 pages, 4 figures. Review article, accepted for publication in Quantum Inf. Proc

We present measurements of single-qubit gate errors for a superconducting qubit. Results from quantum process tomography and randomized benchmarking are compared with gate errors obtained from a double pi pulse experiment. Randomized benchmarking reveals a minimum average gate error of 1.1+/-0.3% and a simple exponential dependence of fidelity on the number of gates. It shows that the limits on gate fidelity are primarily imposed by qubit decoherence, in agreement with theory., Comment: 4 pages, 4 figures, plus supplementary material

Superconducting circuits are promising candidates for constructing quantum bits (qubits) in a quantum computer; single-qubit operations are now routine, and several examples of two qubit interactions and gates having been demonstrated. These experiments show that two nearby qubits can be readily coupled with local interactions. Performing gates between an arbitrary pair of distant qubits is highly desirable for any quantum computer architecture, but has not yet been demonstrated. An efficient way to achieve this goal is to couple the qubits to a quantum bus, which distributes quantum information among the qubits. Here we show the implementation of such a quantum bus, using microwave photons confined in a transmission line cavity, to couple two superconducting qubits on opposite sides of a chip. The interaction is mediated by the exchange of virtual rather than real photons, avoiding cavity induced loss. Using fast control of the qubits to switch the coupling effectively on and off, we demonstrate coherent transfer of quantum states between the qubits. The cavity is also used to perform multiplexed control and measurement of the qubit states. This approach can be expanded to more than two qubits, and is an attractive architecture for quantum information processing on a chip., 6 pages, 4 figures, to be published in Nature

Short dephasing times pose one of the main challenges in realizing a quantum computer. Different approaches have been devised to cure this problem for superconducting qubits, a prime example being the operation of such devices at optimal working points, so-called "sweet spots." This latter approach led to significant improvement of $T_2$ times in Cooper pair box qubits [D. Vion et al., Science 296, 886 (2002)]. Here, we introduce a new type of superconducting qubit called the "transmon." Unlike the charge qubit, the transmon is designed to operate in a regime of significantly increased ratio of Josephson energy and charging energy $E_J/E_C$. The transmon benefits from the fact that its charge dispersion decreases exponentially with $E_J/E_C$, while its loss in anharmonicity is described by a weak power law. As a result, we predict a drastic reduction in sensitivity to charge noise relative to the Cooper pair box and an increase in the qubit-photon coupling, while maintaining sufficient anharmonicity for selective qubit control. Our detailed analysis of the full system shows that this gain is not compromised by increased noise in other known channels., 21 pages, 12 figures; title changed, small changes in the text, additional figure

We study how the spontaneous relaxation of a qubit affects a continuous quantum non-demolition measurement of the initial state of the qubit. Given some noisy measurement record $\Psi$, we seek an estimate of whether the qubit was initially in the ground or excited state. We investigate four different measurement protocols, three of which use a linear filter (with different weighting factors) and a fourth which uses a full non-linear filter that gives the theoretically optimal estimate of the initial state of the qubit. We find that relaxation of the qubit at rate $1/T_1$ strongly influences the fidelity of any measurement protocol. To avoid errors due to this decay, the measurement must be completed in a time that decrease linearly with the desired fidelity while maintaining an adequate signal to noise ratio. We find that for the non-linear filter the predicted fidelity, as expected, is always better than the linear filters and that the fidelity is a monotone increasing function of the measurement time. For example, to achieve a fidelity of 90%, the box car linear filter requires a signal to noise ratio of $\sim 30$ in a time $T_1$ whereas the non-linear filter only requires a signal to noise ratio of $\sim 18$., Comment: 12 pages, 6 figures

Dispersive readouts for superconducting qubits have the advantage of speed and minimal invasiveness. We have developed such an amplifier, the Cavity Bifurcation Amplifier (CBA) [10], and applied it to the readout of the quantronium qubit [2]. It consists of a Josephson junction embedded in a microwave on-chip resonator. In contrast with the Josephson bifurcation amplifier [17], which has an on-chip capacitor shunting a junction, the resonator is based on a simple coplanar waveguide imposing a pre-determined frequency and whose other RF characteristics like the quality factor are easily controlled and optimized. Under proper microwave irradiation conditions, the CBA has two metastable states. Which state is adopted by the CBA depends on the state of a quantronium qubit coupled to the CBA's junction. Due to the MHz repetition rate and large signal to noise ratio we can show directly that the coherence is limited by 1/f gate charge noise when biased at the sweet spot - a point insensitive to first order gate charge fluctuations. This architecture lends itself to scalable quantum computing using a multi-resonator chip with multiplexed readouts., Comment: 6 pages, 5 figures To be published in Physical Review B

A systematic study of the intermediate frequency noise bandwidth of Nb thin-film superconducting hot-electron bolometers is presented. We have measured the spectrum of the output noise as well as the conversion efficiency over a very broad intermediate frequency range (from 0.1 to 7.5 GHz) for devices varying in length from 0.08 μm to 3 μm. Local oscillator and rf signals from 8 to 40 GHz were used. For a device of a given length, the spectrum of the output noise and the conversion efficiency behave similarly for intermediate frequencies less than the gain bandwidth, in accordance with a simple thermal model for both the mixing and thermal fluctuation noise. For higher intermediate frequencies the conversion efficiency decreases; in contrast, the noise decreases but has a second contribution which dominates at higher frequency. The noise bandwidth is larger than the gain bandwidth, and the mixer noise is low, between 120 and 530 K (double side band).

We theoretically study single and two-qubit dynamics in the circuit QED architecture. We focus on the current experimental design [Wallraff et al., Nature 431, 162 (2004); Schuster et al., Nature 445, 515 (2007)] in which superconducting charge qubits are capacitively coupled to a single high-Q superconducting coplanar resonator. In this system, logical gates are realized by driving the resonator with microwave fields. Advantages of this architecture are that it allows for multi-qubit gates between non-nearest qubits and for the realization of gates in parallel, opening the possibility of fault-tolerant quantum computation with superconduting circuits. In this paper, we focus on one and two-qubit gates that do not require moving away from the charge-degeneracy `sweet spot'. This is advantageous as it helps to increase the qubit dephasing time and does not require modification of the original circuit QED. However these gates can, in some cases, be slower than those that do not use this constraint. Five types of two-qubit gates are discussed, these include gates based on virtual photons, real excitation of the resonator and a gate based on the geometric phase. We also point out the importance of selection rules when working at the charge degeneracy point., 23 pages, 8 figures

Electromagnetic signals are always composed of photons, though in the circuit domain those signals are carried as voltages and currents on wires, and the discreteness of the photon's energy is usually not evident. However, by coupling a superconducting qubit to signals on a microwave transmission line, it is possible to construct an integrated circuit where the presence or absence of even a single photon can have a dramatic effect. This system is called circuit quantum electrodynamics (QED) because it is the circuit equivalent of the atom-photon interaction in cavity QED. Previously, circuit QED devices were shown to reach the resonant strong coupling regime, where a single qubit can absorb and re-emit a single photon many times. Here, we report a circuit QED experiment which achieves the strong dispersive limit, a new regime of cavity QED in which a single photon has a large effect on the qubit or atom without ever being absorbed. The hallmark of this strong dispersive regime is that the qubit transition can be resolved into a separate spectral line for each photon number state of the microwave field. The strength of each line is a measure of the probability to find the corresponding photon number in the cavity. This effect has been used to distinguish between coherent and thermal fields and could be used to create a photon statistics analyzer. Since no photons are absorbed by this process, one should be able to generate non-classical states of light by measurement and perform qubit-photon conditional logic, the basis of a logic bus for a quantum computer., Comment: 6 pages, 4 figures, hi-res version at http://www.eng.yale.edu/rslab/papers/numbersplitting_hires.pdf

We investigate a hybrid quantum circuit where ensembles of cold polar molecules serve as long-lived quantum memories and optical interfaces for solid state quantum processors. The quantum memory realized by collective spin states (ensemble qubit) is coupled to a high-Q stripline cavity via microwave Raman processes. We show that, for convenient trap-surface distances of a few microm, strong coupling between the cavity and ensemble qubit can be achieved. We discuss basic quantum information protocols, including a swap from the cavity photon bus to the molecular quantum memory, and a deterministic two qubit gate. Finally, we investigate coherence properties of molecular ensemble quantum bits.

This project supplies support for an additional graduate student on experimental investigations on quantum coherence, entanglement, and quantum computation in a solid-state, electronic realization of quantum bits based on superconducting single-electron devices, namely the single Cooper-pair box. Since these qubits can be microfabricated in large numbers on a single chip, addressed and coherently manipulated electronically, and then entangled with each other, they represent one of the most scalable implementations for possible quantum computers. A radio-frequency single electron transistor (RF-SET) is used for readout of the charge state of the qubit. Of particular importance in this scheme is an understanding and the control of the backaction of the measurement system, which can affect the lifetime and coherence of the qubit. Therefore, this backaction is studied with the goal of optimizing the sensitivity and lifetime in order to attain single-shot readout of the qubit.

We describe the integration of polar molecules with mesoscopic solid state devices in a way that produces robust, coherent, quantum‐level control with applications for quantum information processing. The exceptional features of polar molecules, i.e. long‐lived rotational states in combination with electric dipole moments of several Debye, provide the necessary ingredients to achieve strong coupling to the quantized field of a high‐Q microwave cavity. We discuss two scenarios, where quantum information is stored either in rotational states of a single molecule or in collective spin excitation of an ensemble of molecules. In the latter case we benefit from an enhanced coupling strength, which allows a coherent transfer of quantum information between molecules and solid state qubits.

The "quantronium" is a superconducting qubit consisting of a split Cooper pair box in which a large tunnel junction is inserted. This circuit has a special bias point where the Larmor frequency is, to first order, insensitive to fluctuations in the bias parameters -- the charge of the box island and the phase of the large junction. At this optimal working point, the state of the qubit can be determined by dispersive measurements that probe the second derivative of the state energy with respect to these bias parameters. We use the quantronium phase degree of freedom to perform a non-linear, dispersive measurement of its inductive response using bifurcation amplification. This novel readout projects the state of the qubit in a few nanoseconds, and its latching property allows us to record the resulting information in a few hundred nanoseconds. We have measured, using this technique, Rabi oscillations and Ramsey fringes with an improved signal to noise ratio and contrast. The speed of this new readout scheme also opens the door for a new class of experiments which would characterize the relaxation processes associated with the measurement protocol., updated references and revised discussion of experimental results

This project supplies support for an additional graduate student, Mr. David Schuster, on experimental investigations on quantum coherence, entanglement, and quantum computation in solid-state, electronic realization of quantum bits based on superconducting single-electron devices, namely the single Cooper-pair box. Mr. Schuster has developed a process for fabrication of Al/AlOx/Al tunnel qubits with integrated transmission line resonators, and used these devices for cavity-QED manipulations and readout of qubits. This work has led to the first strong coupling of a solid-state qubit to a single photon, the first high-fidelity non-demolition readout of superconducting qubits, and the first high-visibility quantum control of superconducting qubits.

We report an experimental observation of the backaction of a single-electron transistor (SET) measuring the Coulomb staircase of a single-electron box. As current flows through the SET, the charge state of the SET island fluctuates. These fluctuations capacitively couple to the box and cause changes in the position, width, and asymmetry of the Coulomb staircase. A sequential tunneling model accurately recreates these effects, confirming this mechanism of the backaction of a SET. This is a first step toward understanding the effects of quantum measurement on solid-state qubits.

In a Rabi oscillation experiment with a superconducting qubit we show that a visibility in the qubit excited state population of more than 90 % can be attained. We perform a dispersive measurement of the qubit state by coupling the qubit non-resonantly to a transmission line resonator and probing the resonator transmission spectrum. The measurement process is well characterized and quantitatively understood. The qubit coherence time is determined to be larger than 500 ns in a measurement of Ramsey fringes., Comment: 4 pages, 5 figures, version with high resolution figures available at http://www.eng.yale.edu/rslab/Andreas/content/science/PubsPapers.html

We performed a novel phase-sensitive microwave reflection experiment which directly probes the dynamics of the Josephson plasma resonance in both the linear and the nonlinear regime. When the junction was driven below the plasma frequency into the nonlinear regime, we observed for the first time the transition between two different dynamical states predicted for nonlinear systems. In our experiment, this transition appears as an abrupt change in the reflected signal phase at a critical excitation power. This controlled dynamical switching can form the basis of a sensitive amplifier, in particular, for the readout of superconducting qubits.

Mixing experiments have been performed at frequencies from 4 to 20 GHz on Nb thin‐film superconducting hot‐electron bolometers varying in length from 0.08 to 3 μm. The intermediate frequency (IF) bandwidth is found to vary as L−2, with L the bridge length, for devices shorter than √12 Le−ph≊1 μm, with Le−ph the electron‐phonon length. The shortest device has an IF bandwidth greater than 6 GHz, the largest reported for a low‐Tc superconducting bolometric mixer. The conversion efficiencies range from −5 to −11 dB (single sideband, SSB). For short bridges, the mixer noise temperature is found to be as low as 100 K (double sideband, DSB), with little length dependence. The local oscillator power required is small, ≊10 nW. Such mixers are very promising for low‐noise THz heterodyne receivers.

Heterodyne measurements have been made at 533 GHz using a novel superconducting hot‐electron bolometer in a waveguide mixer. The bolometer is a 0.3 μm long niobium microbridge with a superconducting transition temperature of 5 K. The short length ensures that electron diffusion dominates over electron‐phonon interactions as the electron cooling mechanism, which should allow heterodyne detection with intermediate frequencies (if) of several GHz. A Y‐factor response of 1.15 dB has been obtained at an if of 1.4 GHz with 77 and 295 K loads, indicating a receiver noise temperature of 650 K DSB. The −3 dB rolloff in the if response occurs at 1.7 GHz.

We present a thermometer based on the electrical noise from a tunnel junction. In this thermometer, temperature is related to voltage across the junction by a relative microwave noise measurement with only the use of the electron charge, Boltzmann's constant, and assumption that electrons in a metal obey Fermi-Dirac statistics (Lafe Spietz, 2003)

We propose a realizable architecture using one-dimensional transmission line resonators to reach the strong coupling limit of cavity quantum electrodynamics in superconducting electrical circuits. The vacuum Rabi frequency for the coupling of cavity photons to quantized excitations of an adjacent electrical circuit (qubit) can easily exceed the damping rates of both the cavity and the qubit. This architecture is attractive both as a macroscopic analog of atomic physics experiments and for quantum computing and control, since it provides strong inhibition of spontaneous emission, potentially leading to greatly enhanced qubit lifetimes, allows high-fidelity quantum non-demolition measurements of the state of multiple qubits, and has a natural mechanism for entanglement of qubits separated by centimeter distances. In addition it would allow production of microwave photon states of fundamental importance for quantum communication., 14 pages, 9 figures

This chapter discusses the prospects for strong cavity quantum electrodynamics (cQED) with superconducting circuits. It propose that the combination of one-dimensional superconducting transmission line resonators, which confine their zero point energy to extremely small volumes, and superconducting charge qubits, which are electrically controllable qubits with large electric dipole moments, constitutes an interesting system to access the strong-coupling regime of cavity quantum electrodynamics. This combined system constitutes an advantageous architecture for the coherent control, entanglement, and readout of quantum bits for quantum computation and communication. Among the practical benefits of this approach are the ability to suppress radiative decay of the qubit while still allowing one-bit operations, a simple and minimally disruptive method for readout of single and multiple qubits, and the ability to generate tunable two-qubit entanglement over centimeter-scale distances. In the structures described in the chapter, the emission or absorption of a single photon by the qubit is tagged by a sudden large change in the resonator transmission properties making them potentially useful as single photon sources and detectors.

We have spectroscopically measured the energy level separation of a superconducting charge qubit coupled non-resonantly to a single mode of the electromagnetic field of a superconducting on-chip resonator. The strong coupling leads to large shifts in the energy levels of both the qubit and the resonator in this circuit quantum electrodynamics system. The dispersive shift of the resonator frequency is used to non-destructively determine the qubit state and to map out the dependence of its energy levels on the bias parameters. The measurement induces an ac-Stark shift of 0.6 MHz per photon in the qubit level separation. Fluctuations in the photon number (shot noise) induce level fluctuations in the qubit leading to dephasing which is the characteristic back-action of the measurement. A cross-over from lorentzian to gaussian line shape with increasing measurement power is observed and theoretically explained. For weak measurement a long intrinsic dephasing time of T_2 > 200 ns of the qubit is found., 4 pages, 5 figures, submitted to Physical Review Letters, version with high resolution figures available at http://www.eng.yale.edu/rslab/Andreas/content/science/PubsPapers.html

We present fabrication and characterization procedures of devices for circuit quantum electrodynamics (cQED). We have made 3 GHz cavities with quality factors in the range 10^4--10^6, which allow access to the strong coupling regime of cQED. The cavities are transmission line resonators made by photolithography. They are coupled to the input and output ports via gap capacitors. An Al-based Cooper pair box is made by ebeam lithography and Dolan bridge double-angle evaporation in superconducting resonators with high quality factor. An important issue is to characterize the quality factor of the resonators. We present an RF-characterization of superconducting resonators as a function of temperature and magnetic field. We have realized different versions of the system with different box-cavity couplings by using different dielectrics and by changing the box geometry. Moreover, the cQED approach can be used as a diagnostic tool of qubit internal losses., Comment: 4 pages, 6 figures, Applied Superconductivity Conference 2004

Measurements of the thermal properties of nanoscale electron systems have ignored the effect of electrical noise radiated between the electron gas and the environment, through the electrical leads. Here we calculate the effect of this photon-mediated process, and show that the low-temperature thermal conductance is equal to the quantum of thermal conductance, ${G}_{Q}={\ensuremath{\pi}}^{2}{k}_{B}^{2}T/3h$, times a coupling coefficient. We find that, at very low temperatures, the photon conductance is the dominant route for thermal equilibration, while at moderate temperatures this relaxation mode adds one quantum of thermal conductance to that due to phonon transport.

Electrical engineers and physicists are naturally very interested in noise in circuits, amplifiers and detectors. With the advent of quantum computation and other high frequency electronics operating at low temperatures, we have entered a regime where quantum noise and quantum-limited detectors are important. Here we describe the general concept of a two-level system as a quantum spectrum analyzer and apply it to a simple superconducting qubit, the Cooper-pair box. We then discuss the coupling of a Cooper-pair box to its electromagnetic environment, whose noise leads to a finite polarization and excited-state lifetime of the qubit. Finally, we describe a theoretical technique for treating a qubit coupled to a measurement system, which allows one to calculate the full quantum noise of the measurement device. We present results for such a calculation for the case of a normal SET.

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.