Max Hofheinz

Quantum ground state and single-phonon control of a mechanical resonator

Quantum mechanics provides a highly accurate description of a wide variety of physical systems. However, a demonstration that quantum mechanics applies equally to macroscopic mechanical systems has been a long-standing challenge, hindered by the difficulty of cooling a mechanical mode to its quantum ground state. The temperatures required are typically far below those attainable with standard cryogenic methods, so significant effort has been devoted to developing alternative cooling techniques. Once in the ground state, quantum-limited measurements must then be demonstrated. Here, using conventional cryogenic refrigeration, we show that we can cool a mechanical mode to its quantum ground state by using a microwave-frequency mechanical oscillator—a ‘quantum drum’—coupled to a quantum bit, which is used to measure the quantum state of the resonator. We further show that we can controllably create single quantum excitations (phonons) in the resonator, thus taking the first steps to complete quantum control of a mechanical system.

Synthesizing arbitrary quantum states in a superconducting resonator.

The superposition principle is a fundamental tenet of quantum mechanics. It allows a quantum system to be ‘in two places at the same time’, because the quantum state of a physical system can simultaneously include measurably different physical states. The preparation and use of such superposed states forms the basis of quantum computation and simulation. The creation of complex superpositions in harmonic systems (such as the motional state of trapped ions, microwave resonators or optical cavities) has presented a significant challenge because it cannot be achieved with classical control signals. Here we demonstrate the preparation and measurement of arbitrary quantum states in an electromagnetic resonator, superposing states with different numbers of photons in a completely controlled and deterministic manner. We synthesize the states using a superconducting phase qubit to phase-coherently pump photons into the resonator, making use of an algorithm that generalizes a previously demonstrated method of generating photon number (Fock) states in a resonator. We completely characterize the resonator quantum state using Wigner tomography, which is equivalent to measuring the resonator’s full density matrix.

Violation of Bell's inequality in Josephson phase qubits

The measurement process plays an awkward role in quantum mechanics, because measurement forces a system to ‘choose’ between possible outcomes in a fundamentally unpredictable manner. Therefore, hidden classical processes have been considered as possibly predetermining measurement outcomes while preserving their statistical distributions. However, a quantitative measure that can distinguish classically determined correlations from stronger quantum correlations exists in the form of the Bell inequalities, measurements of which provide strong experimental evidence that quantum mechanics provides a complete description. Here we demonstrate the violation of a Bell inequality in a solid-state system. We use a pair of Josephson phase qubits acting as spin-1/2 particles, and show that the qubits can be entangled and measured so as to violate the Clauser–Horne–Shimony–Holt (CHSH) version of the Bell inequality. We measure a Bell signal of 2.0732 ± 0.0003, exceeding the maximum amplitude of 2 for a classical system by 244 standard deviations. In the experiment, we deterministically generate the entangled state, and measure both qubits in a single-shot manner, closing the detection loophole. Because the Bell inequality was designed to test for non-classical behaviour without assuming the applicability of quantum mechanics to the system in question, this experiment provides further strong evidence that a macroscopic electrical circuit is really a quantum system.

Generation of Fock states in a superconducting quantum circuit

Spin systems and harmonic oscillators comprise two archetypes in quantum mechanics. The spin-1/2 system, with two quantum energy levels, is essentially the most nonlinear system found in nature, whereas the harmonic oscillator represents the most linear, with an infinite number of evenly spaced quantum levels. A significant difference between these systems is that a two-level spin can be prepared in an arbitrary quantum state using classical excitations, whereas classical excitations applied to an oscillator generate a coherent state, nearly indistinguishable from a classical state. Quantum behaviour in an oscillator is most obvious in Fock states, which are states with specific numbers of energy quanta, but such states are hard to create. Here we demonstrate the controlled generation of multi-photon Fock states in a solid-state system. We use a superconducting phase qubit, which is a close approximation to a two-level spin system, coupled to a microwave resonator, which acts as a harmonic oscillator, to prepare and analyse pure Fock states with up to six photons. We contrast the Fock states with coherent states generated using classical pulses applied directly to the resonator.

Microwave dielectric loss at single photon energies and millikelvin temperatures

The microwave performance of amorphous dielectric materials at very low temperatures and very low excitation strengths displays significant excess loss. Here, we present the loss tangents of some common amorphous and crystalline dielectrics, measured at low temperatures (T<100mK) with near single-photon excitation energies, E∕ℏω0∼1, using both coplanar waveguide and lumped LC resonators. The loss can be understood using a two-level state defect model. A circuit analysis of the half-wavelength resonators we used is outlined, and the energy dissipation of such a resonator on a multilayered dielectric substrate is theoretically considered.

Emulation of a Quantum Spin with a Superconducting Phase Qudit

Higher-Level Quantum Emulation At the heart of a quantum computer is the device on which information is to be encoded. This is typically done with a qubit, a two-level quantum system analogous to the two-level bit that encodes 0 and 1 in classical computers. However, there need not be just two quantum energy levels. There could be three (a qutrit), or more generally, d-levels (a qudit) in the device. Neeley et al. (p. 722; see the Perspective by Nori) demonstrate a five-level quantum device and show that their qudit can be used to emulate the processes involved in manipulating quantum spin. The use of multilevel qudits may also have potential in quantum information processing by simplifying certain computational tasks and simplifying the circuitry required to realize the quantum computer itself. A multilevel superconducting device is used to emulate the manipulation of quantum spin systems. In quantum information processing, qudits (d-level systems) are an extension of qubits that could speed up certain computing tasks. We demonstrate the operation of a superconducting phase qudit with a number of levels d up to d = 5 and show how to manipulate and measure the qudit state, including simultaneous control of multiple transitions. We used the qudit to emulate the dynamics of single spins with principal quantum number s = 1/2, 1, and 3/2, allowing a measurement of Berry’s phase and the even parity of integer spins (and odd parity of half-integer spins) under 2π-rotation. This extension of the two-level qubit to a multilevel qudit holds promise for more-complex quantum computational architectures and for richer simulations of quantum mechanical systems.

Improving the coherence time of superconducting coplanar resonators

The quality factor and energy decay time of superconducting resonators have been measured as a function of material, geometry, and magnetic field. Once the dissipation of trapped magnetic vortices is minimized, we identify surface two-level states (TLS) as an important decay mechanism. A wide gap between the center conductor and the ground plane, as well as use of the superconductor Re instead of Al, are shown to decrease loss. We also demonstrate that classical measurements of resonator quality factor at low excitation power are consistent with single-photon decay time measured using qubit-resonator swap experiments.

Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits

Quantum process tomography provides a means of benchmarking the components and algorithms of a quantum computer in a quantitative fashion, independent of the particular architecture used. Such a procedure has now been demonstrated for a universal entangling gate in a solid-state system. Quantum gates must perform reliably when operating on standard input basis states and on complex superpositions thereof. Experiments using superconducting qubits have validated truth tables for particular implementations of, for example, the controlled-NOT gate1,2, but have not fully characterized gate operation for arbitrary superpositions of input states. Here we demonstrate the use of quantum process tomography3,4 (QPT) to fully characterize the performance of a universal entangling gate between two superconducting qubits. Process tomography permits complete gate analysis, but requires precise preparation of arbitrary input states, control over the subsequent qubit interaction and ideally simultaneous single-shot measurement of output states. In recent work, it has been proposed to use QPT to probe noise properties5 and time dynamics6 of qubit systems and to apply techniques from control theory to create scalable qubit benchmarking protocols7,8. We use QPT to measure the fidelity and noise properties5 of an entangling gate. In addition to demonstrating a promising fidelity, our entangling gate has an on-to-off ratio of 300, a level of adjustable coupling that will become a requirement for future high-fidelity devices. This is the first solid-state demonstration of QPT in a two-qubit system, as QPT has previously been demonstrated only with single solid-state qubits9,10,11.

Measurement of the Decay of Fock States in a Superconducting Quantum Circuit

We demonstrate the controlled generation of Fock states with up to 15 photons in a microwave coplanar waveguide resonator coupled to a superconducting phase qubit. The subsequent decay of the Fock states, due to dissipation, is then monitored by varying the time delay between preparing the state and performing a number-state analysis. We find that the decay dynamics can be described by a master equation where the lifetime of the n-photon Fock state scales as 1/n, in agreement with theory. We have also generated a coherent state in the microwave resonator, and monitored its decay process. We demonstrate that the coherent state maintains a Poisson distribution as it decays, with an average photon number that decreases with the same characteristic decay time as the one-photon Fock state.

1/f Flux noise in Josephson phase qubits.

We present a new method to measure 1/f noise in Josephson quantum bits (qubits) that yields low-frequency spectra below 1 Hz. A comparison of the noise taken at positive and negative bias of a phase qubit shows the dominant noise source to be flux noise and not junction critical-current noise, with a magnitude similar to that measured previously in other systems. Theoretical calculations show that the level of flux noise is not compatible with the standard model of noise from two-level state defects in the surface oxides of the films.