Observing single quantum trajectories of a superconducting quantum bit

Lie Algebraic Quantum Phase Reduction

We introduce a general framework of phase reduction theory for quantum nonlinear oscillators. By employing the quantum trajectory theory, we define the limit-cycle trajectory and the phase according to a stochastic Schrödinger equation. Because a perturbation is represented by unitary transformation in quantum dynamics, we calculate phase response curves with respect to generators of a Lie algebra. Our method shows that the continuous measurement yields phase clusters and alters the phase response curves. The observable clusters capture the phase dynamics of individual quantum oscillators, unlike indirect indicators obtained from density operators. Furthermore, our method can be applied to finite-level systems that lack classical counterparts.

Deep quantum neural networks on a superconducting processor

Deep learning and quantum computing have achieved dramatic progresses in recent years. The interplay between these two fast-growing fields gives rise to a new research frontier of quantum machine learning. In this work, we report an experimental demonstration of training deep quantum neural networks via the backpropagation algorithm with a six-qubit programmable superconducting processor. We experimentally perform the forward process of the backpropagation algorithm and classically simulate the backward process. In particular, we show that three-layer deep quantum neural networks can be trained efficiently to learn two-qubit quantum channels with a mean fidelity up to 96.0% and the ground state energy of molecular hydrogen with an accuracy up to 93.3% compared to the theoretical value. In addition, six-layer deep quantum neural networks can be trained in a similar fashion to achieve a mean fidelity up to 94.8% for learning single-qubit quantum channels. Our experimental results indicate that the number of coherent qubits required to maintain does not scale with the depth of the deep quantum neural network, thus providing a valuable guide for quantum machine learning applications with both near-term and future quantum devices.

Experimentally probing anomalous time evolution of a single photon

In quantum mechanics, a quantum system is irreversibly collapsed by a projective measurement. Hence, delicately probing the time evolution of a quantum system holds the key to understanding curious phenomena. Here, we experimentally explore an anomalous time evolution, where, illustratively, a particle disappears from a box and emerges in a different box, with a certain moment in which it can be found in neither of them. In this experiment, we directly probe this curious time evolution of a single photon by measuring up to triple-operator sequential weak values (SWVs) using a novel probeless scheme. The naive interpretation provided by single-operator weak values (WVs) seems to imply the "disappearance" and "re-appearance" of a photon as theoretically predicted. However, double- and triple-operator SWVs, representing temporal correlations between the aforementioned values, show that spatial information about the photon does not entirely vanish in the intermediate time. These results show that local values (in space and time) alone, such as single-operator WVs, cannot fully explain all types of quantum evolution in time-higher order correlations are necessary in general, shedding new light on time evolution in quantum mechanics. The probeless measurement technique proposed here for measuring multiple-operator WVs can be straightforwardly extended to study various other cases of curious quantum evolution in time.

Unravelling quantum chaos using persistent homology

Topological data analysis is a powerful framework for extracting useful topological information from complex data sets. Recent work has shown its application for the dynamical analysis of classical dissipative systems through a topology-preserving embedding method that allows reconstructing dynamical attractors, the topologies of which can be used to identify chaotic behavior. Open quantum systems can similarly exhibit nontrivial dynamics, but the existing toolkit for classification and quantification are still limited, particularly for experimental applications. In this paper, we present a topological pipeline for characterizing quantum dynamics, which draws inspiration from the classical approach by using single quantum trajectory unravelings of the master equation to construct analog quantum attractors and extract their topology using persistent homology. We apply the method to a periodically modulated Kerr-nonlinear cavity to discriminate parameter regimes of regular and chaotic phases using limited measurements of the system.

Quantum reservoir computing in finite dimensions

Most existing results in the analysis of quantum reservoir computing (QRC) systems with classical inputs have been obtained using the density matrix formalism. This paper shows that alternative representations can provide better insight when dealing with design and assessment questions. More explicitly, system isomorphisms are established that unify the density matrix approach to QRC with the representation in the space of observables using Bloch vectors associated with Gell-Mann bases. It is shown that these vector representations yield state-affine systems previously introduced in the classical reservoir computing literature and for which numerous theoretical results have been established. This connection is used to show that various statements in relation to the fading memory property (FMP) and the echo state property (ESP) are independent of the representation and also to shed some light on fundamental questions in QRC theory in finite dimensions. In particular, a necessary and sufficient condition for the ESP and FMP to hold is formulated using standard hypotheses, and contractive quantum channels that have exclusively trivial semi-infinite solutions are characterized in terms of the existence of input-independent fixed points.

Energy dynamics, heat production and heat-work conversion with qubits: toward the development of quantum machines

We present an overview of recent advances in the study of energy dynamics and mechanisms for energy conversion in qubit systems with special focus on realizations in superconducting quantum circuits. We briefly introduce the relevant theoretical framework to analyze heat generation, energy transport and energy conversion in these systems with and without time-dependent driving considering the effect of equilibrium and non-equilibrium environments. We analyze specific problems and mechanisms under current investigation in the context of qubit systems. These include the problem of energy dissipation and possible routes for its control, energy pumping between driving sources and heat pumping between reservoirs, implementation of thermal machines and mechanisms for energy storage. We highlight the underlying fundamental phenomena related to geometrical and topological properties, as well as many-body correlations. We also present an overview of recent experimental activity in this field.

Measurement-Based Quantum Thermal Machines with Feedback Control

We investigated coupled-qubit-based thermal machines powered by quantum measurements and feedback. We considered two different versions of the machine: (1) a quantum Maxwell's demon, where the coupled-qubit system is connected to a detachable single shared bath, and (2) a measurement-assisted refrigerator, where the coupled-qubit system is in contact with a hot and cold bath. In the quantum Maxwell's demon case, we discuss both discrete and continuous measurements. We found that the power output from a single qubit-based device can be improved by coupling it to the second qubit. We further found that the simultaneous measurement of both qubits can produce higher net heat extraction compared to two setups operated in parallel where only single-qubit measurements are performed. In the refrigerator case, we used continuous measurement and unitary operations to power the coupled-qubit-based refrigerator. We found that the cooling power of a refrigerator operated with swap operations can be enhanced by performing suitable measurements.

Coherent interaction-free detection of microwave pulses with a superconducting circuit

The interaction-free measurement is a fundamental quantum effect whereby the presence of a photosensitive object is determined without irreversible photon absorption. Here we propose the concept of coherent interaction-free detection and demonstrate it experimentally using a three-level superconducting transmon circuit. In contrast to standard interaction-free measurement setups, where the dynamics involves a series of projection operations, our protocol employs a fully coherent evolution that results, surprisingly, in a higher probability of success. We show that it is possible to ascertain the presence of a microwave pulse resonant with the second transition of the transmon, while at the same time avoid exciting the device onto the third level. Experimentally, this is done by using a series of Ramsey microwave pulses coupled into the first transition and monitoring the ground-state population.

Semi-Markov processes in open quantum systems: Connections and applications in counting statistics

Using the age-structure formalism, we definitely establish connections between semi-Markov processes and the dynamics of open quantum systems that satisfy the Markov quantum master equations. A generalized Feynman-Kac formula of the semi-Markov processes is also proposed. In addition to inheriting all statistical properties possessed by the piecewise deterministic processes of wave functions, the semi-Markov processes show their unique advantages in quantum counting statistics. Compared with the conventional method of the tilted quantum master equation, they can be applied to more general counting quantities. In particular, the terms involved in the method have precise probability meanings. We use a driven two-level quantum system to exemplify these results.

Energetic Cost of Measurements Using Quantum, Coherent, and Thermal Light

Quantum measurements are basic operations that play a critical role in the study and application of quantum information. We study how the use of quantum, coherent, and classical thermal states of light in a circuit quantum electrodynamics setup impacts the performance of quantum measurements, by comparing their respective measurement backaction and measurement signal to noise ratio per photon. In the strong dispersive limit, we find that thermal light is capable of performing quantum measurements with comparable efficiency to coherent light, both being outperformed by single-photon light. We then analyze the thermodynamic cost of each measurement scheme. We show that single-photon light shows an advantage in terms of energy cost per information gain, reaching the fundamental thermodynamic cost.

Quantum Fluctuation Theorem under Quantum Jumps with Continuous Measurement and Feedback

While the fluctuation theorem in classical systems has been thoroughly generalized under various feedback control setups, an intriguing situation in quantum systems, namely under continuous feedback, remains to be investigated. In this work, we derive the generalized fluctuation theorem under quantum jumps with continuous measurement and feedback. The essence for the derivation is to newly introduce the operationally meaningful information, which we call quantum-classical-transfer (QC-transfer) entropy. QC-transfer entropy can be naturally interpreted as the quantum counterpart of transfer entropy that is commonly used in classical time series analysis. We also verify our theoretical results by numerical simulation and propose an experiment-numerics hybrid verification method. Our work reveals a fundamental connection between quantum thermodynamics and quantum information, which can be experimentally tested with artificial quantum systems such as circuit quantum electrodynamics.

Limits to Perception by Quantum Monitoring with Finite Efficiency

We formulate limits to perception under continuous quantum measurements by comparing the quantum states assigned by agents that have partial access to measurement outcomes. To this end, we provide bounds on the trace distance and the relative entropy between the assigned state and the actual state of the system. These bounds are expressed solely in terms of the purity and von Neumann entropy of the state assigned by the agent, and are shown to characterize how an agent's perception of the system is altered by access to additional information. We apply our results to Gaussian states and to the dynamics of a system embedded in an environment illustrated on a quantum Ising chain.

Quantum state tomography of molecules by ultrafast diffraction

Ultrafast electron diffraction and time-resolved serial crystallography are the basis of the ongoing revolution in capturing at the atomic level of detail the structural dynamics of molecules. However, most experiments capture only the probability density of the nuclear wavepackets to determine the time-dependent molecular structures, while the full quantum state has not been accessed. Here, we introduce a framework for the preparation and ultrafast coherent diffraction from rotational wave packets of molecules, and we establish a new variant of quantum state tomography for ultrafast electron diffraction to characterize the molecular quantum states. The ability to reconstruct the density matrix, which encodes the amplitude and phase of the wavepacket, for molecules of arbitrary degrees of freedom, will enable the reconstruction of a quantum molecular movie from experimental x-ray or electron diffraction data.

Ultrafast diffraction is fundamental in capturing the structural dynamics of molecules. Here, the authors establish a variant of quantum state tomography for arbitrary degrees of freedom to characterize the molecular quantum states, which will enable the reconstruction of a quantum molecular movie from diffraction data.

High-Efficiency Arbitrary Quantum Operation on a High-Dimensional Quantum System

The ability to manipulate quantum systems lies at the heart of the development of quantum technology. The ultimate goal of quantum control is to realize arbitrary quantum operations (AQUOs) for all possible open quantum system dynamics. However, the demanding extra physical resources impose great obstacles. Here, we experimentally demonstrate a universal approach of AQUO on a photonic qudit with the minimum physical resource of a two-level ancilla and a log_{2}d-scale circuit depth for a d-dimensional system. The AQUO is then applied in a quantum trajectory simulation for quantum subspace stabilization and quantum Zeno dynamics, as well as incoherent manipulation and generalized measurements of the qudit. Therefore, the demonstrated AQUO for complete quantum control would play an indispensable role in quantum information science.

Impact of Spectators on a Two-Qubit Gate in a Tunable Coupling Superconducting Circuit

Cross-resonance (CR) gates have emerged as a promising scheme for fault-tolerant quantum computation with fixed-frequency qubits. We experimentally implement an entangling CR gate by using a microwave-only control in a tunable coupling superconducting circuit, where the tunable coupler provides extra degrees of freedom to verify optimal conditions for constructing a CR gate. By developing a three-qubit Hamiltonian tomography protocol, we systematically investigate the dependency of gate fidelities on spurious qubit interactions and present the first experimental approach to the evaluation of the perturbation impact arising from spectator qubits. Our results reveal that the spectator qubits lead to reductions in CR gate fidelity dependent on ZZ interactions and particular frequency detunings between spectator and gate qubits. The target spectator demonstrates a more serious impact than the control spectator under a standard echo pulse scheme, whereas the degradation of gate fidelity is observed up to 22.5% as both the spectators are present with a modest ZZ coupling to the computational qubits. Our experiments uncover an optimal CR operation regime, and the method we develop here can readily be applied to improving other kinds of two-qubit gates in large-scale quantum circuits.

Tunneling Quantum Dynamics in Ammonia

Ammonia is a well-known example of a two-state system and must be described in quantum-mechanical terms. In this article, we will explain the tunneling phenomenon that occurs in ammonia molecules from the perspective of trajectory-based quantum dynamics, rather than the usual quantum probability perspective. The tunneling of the nitrogen atom through the potential barrier in ammonia is not merely a probability problem; there are underlying reasons and mechanisms explaining why and how the tunneling in ammonia can happen. Under the framework of quantum Hamilton mechanics, the tunneling motion of the nitrogen atom in ammonia can be described deterministically in terms of the quantum trajectories of the nitrogen atom and the quantum forces applied. The vibrations of the nitrogen atom about its two equilibrium positions are analyzed in terms of its quantum trajectories, which are solved from the Hamilton equations of motion. The vibration periods are then computed by the quantum trajectories and compared with the experimental measurements.

Thermodynamic Uncertainty Relation in Slowly Driven Quantum Heat Engines

Thermodynamic uncertainty relations express a trade-off between precision, defined as the noise-to-signal ratio of a generic current, and the amount of associated entropy production. These results have deep consequences for autonomous heat engines operating at steady state, imposing an upper bound for their efficiency in terms of the power yield and its fluctuations. In the present Letter we analyze a different class of heat engines, namely, those which are operating in the periodic slow-driving regime. We show that an alternative TUR is satisfied, which is less restrictive than that of steady-state engines: it allows for engines that produce finite power, with small power fluctuations, to operate close to reversibility. The bound further incorporates the effect of quantum fluctuations, which reduces engine efficiency relative to the average power and reliability. We finally illustrate our findings in the experimentally relevant model of a single-ion heat engine.

Entanglement Entropy Scaling Transition under Competing Monitoring Protocols

Dissipation generally leads to the decoherence of a quantum state. In contrast, numerous recent proposals have illustrated that dissipation can also be tailored to stabilize many-body entangled quantum states. While the focus of these works has been primarily on engineering the nonequilibrium steady state, we investigate the buildup of entanglement in the quantum trajectories. Specifically, we analyze the competition between two different dissipation channels arising from two incompatible continuous monitoring protocols. The first protocol locks the phase of neighboring sites upon registering a quantum jump, thereby generating a long-range entanglement through the system, while the second destroys the coherence via a dephasing mechanism. By studying the unraveling of stochastic quantum trajectories associated with the continuous monitoring protocols, we present a transition for the scaling of the averaged trajectory entanglement entropies, from critical scaling to area-law behavior. Our work provides an alternative perspective on the measurement-induced phase transition: the measurement can be viewed as monitoring and registering quantum jumps, offering an intriguing extension of these phase transitions through the long-established realm of quantum optics.

Weak Measurement of a Superconducting Qubit Reconciles Incompatible Operators

Traditional uncertainty relations dictate a minimal amount of noise in incompatible projective quantum measurements. However, not all measurements are projective. Weak measurements are minimally invasive methods for obtaining partial state information without projection. Recently, weak measurements were shown to obey an uncertainty relation cast in terms of entropies. We experimentally test this entropic uncertainty relation with strong and weak measurements of a superconducting transmon qubit. A weak measurement, we find, can reconcile two strong measurements' incompatibility, via backaction on the state. Mathematically, a weak value-a preselected and postselected expectation value-lowers the uncertainty bound. Hence we provide experimental support for the physical interpretation of the weak value as a determinant of a weak measurement's ability to reconcile incompatible operations.

Thermodynamics of Gambling Demons

We introduce and realize demons that follow a customary gambling strategy to stop a nonequilibrium process at stochastic times. We derive second-law-like inequalities for the average work done in the presence of gambling, and universal stopping-time fluctuation relations for classical and quantum nonstationary stochastic processes. We test experimentally our results in a single-electron box, where an electrostatic potential drives the dynamics of individual electrons tunneling into a metallic island. We also discuss the role of coherence in gambling demons measuring quantum jump trajectories.

Extending Quantum Probability from Real Axis to Complex Plane

Probability is an important question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the probability domain extends to the complex space, including the generation of complex trajectory, the definition of the complex probability, and the relation of the complex probability to the quantum probability. The complex treatment proposed in this article applies the optimal quantum guidance law to derive the stochastic differential equation governing a particle’s random motion in the complex plane. The probability distribution ρc(t,x,y) of the particle’s position over the complex plane z=x+iy is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, the probability distribution ρc(t,x,y) is verified by the solution of the complex Fokker–Planck equation. It is shown that quantum probability |Ψ|2 and classical probability can be integrated under the framework of complex probability ρc(t,x,y), such that they can both be derived from ρc(t,x,y) by different statistical ways of collecting spatial points.

Universality of Entanglement Transitions from Stroboscopic to Continuous Measurements

Measurement-driven transitions between extensive and subextensive scaling of the entanglement entropy receive interest as they illuminate the intricate physics of thermalization and control in open interacting quantum systems. While this transition is well established for stroboscopic measurements in random quantum circuits, a crucial link to physical settings is its extension to continuous observations, where for an integrable model it has been shown that the transition changes its nature and becomes immediate. Here, we demonstrate that the entanglement transition at finite coupling persists if the continuously measured system is randomly nonintegrable, and show that it is smoothly connected to the transition in the stroboscopic models. This provides a bridge between a wide range of experimental settings and the wealth of knowledge accumulated for the latter systems.

Noisy Quantum Metrology Enhanced by Continuous Nondemolition Measurement

We show that continuous quantum nondemolition (QND) measurement of an atomic ensemble is able to improve the precision of frequency estimation even in the presence of independent dephasing acting on each atom. We numerically simulate the dynamics of an ensemble with up to N=150 atoms initially prepared in a (classical) spin coherent state, and we show that, thanks to the spin squeezing dynamically generated by the measurement, the information obtainable from the continuous photocurrent scales superclassically with respect to the number of atoms N. We provide evidence that such superclassical scaling holds for different values of dephasing and monitoring efficiency. We moreover calculate the extra information obtainable via a final strong measurement on the conditional states generated during the dynamics and show that the corresponding ultimate limit is nearly achieved via a projective measurement of the spin-squeezed collective spin operator. We also briefly discuss the difference between our protocol and standard estimation schemes, where the state preparation time is neglected.

Experimental Assessment of Entropy Production in a Continuously Measured Mechanical Resonator

The information on a quantum process acquired through measurements plays a crucial role in the determination of its nonequilibrium thermodynamic properties. We report on the experimental inference of the stochastic entropy production rate for a continuously monitored mesoscopic quantum system. We consider an optomechanical system subjected to continuous displacement Gaussian measurements and characterize the entropy production rate of the individual trajectories followed by the system in its stochastic dynamics, employing a phase-space description in terms of the Wigner entropy. Owing to the specific regime of our experiment, we are able to single out the informational contribution to the entropy production arising from conditioning the state on the measurement outcomes. Our experiment embodies a significant step towards the demonstration of full-scale control of fundamental thermodynamic processes at the mesoscopic quantum scale.

Experimental Characterization of Unsharp Qubit Observables and Sequential Measurement Incompatibility via Quantum Random Access Codes

Unsharp measurements are increasingly important for foundational insights in quantum theory and quantum information applications. Here, we report an experimental implementation of unsharp qubit measurements in a sequential communication protocol, based on a quantum random access code. The protocol involves three parties; the first party prepares a qubit system, the second party performs operations that return both a classical and quantum outcome, and the latter is measured by the third party. We demonstrate a nearly optimal sequential quantum random access code that outperforms both the best possible classical protocol and any quantum protocol that utilizes only projective measurements. Furthermore, while only assuming that the involved devices operate on qubits and that detected events constitute a fair sample, we demonstrate the noise-robust characterization of unsharp measurements based on the sequential quantum random access code. We apply this characterization towards quantifying the degree of incompatibility of two sequential pairs of quantum measurements.

Sequential Dispersive Measurement of a Superconducting Qubit

We present a superconducting device that realizes the sequential measurement of a transmon qubit. The device disables common limitations of dispersive readout such as Purcell effect or transients in the cavity mode by turning on and off the coupling to the measurement channel on demand. The qubit measurement begins by loading a readout resonator that is coupled to the qubit. After an optimal interaction time with negligible loss, a microwave pump releases the content of the readout mode by upconversion into a measurement line in a characteristic time as low as 10 ns, which is 400 times shorter than the lifetime of the readout resonator. A direct measurement of the released field quadratures demonstrates a readout fidelity of 97.5% in a total measurement time of 220 ns. The Wigner tomography of the readout mode allows us to characterize the non-Gaussian nature of the readout mode and its dynamics.

Quantum Trajectory Analysis of Single Microwave Photon Detection by Nanocalorimetry

We apply quantum trajectory techniques to analyze a realistic setup of a superconducting qubit coupled to a heat bath formed by a resistor, a system that yields explicit expressions of the relevant transition rates to be used in the analysis. We discuss the main characteristics of the jump trajectories and relate them to the expected outcomes ("clicks") of a fluorescence measurement using the resistor as a nanocalorimeter. As the main practical outcome, we present a model that predicts the time-domain response of a realistic calorimeter subject to single microwave photons, incorporating the intrinsic noise due to the fundamental thermal fluctuations of the absorber and finite bandwidth of a thermometer.

Heat and Work Along Individual Trajectories of a Quantum Bit

We use a near quantum limited detector to experimentally track individual quantum state trajectories of a driven qubit formed by the hybridization of a waveguide cavity and a transmon circuit. For each measured quantum coherent trajectory, we separately identify energy changes of the qubit as heat and work, and verify the first law of thermodynamics for an open quantum system. We further establish the consistency of these results by comparison with the master equation approach and the two-projective-measurement scheme, both for open and closed dynamics, with the help of a quantum feedback loop that compensates for the exchanged heat and effectively isolates the qubit.

Topological transition in measurement-induced geometric phases

The state of a quantum system, adiabatically driven in a cycle, may acquire a measurable phase depending only on the closed trajectory in parameter space. Such geometric phases are ubiquitous and also underline the physics of robust topological phenomena such as the quantum Hall effect. Equivalently, a geometric phase may be induced through a cyclic sequence of quantum measurements. We show that the application of a sequence of weak measurements renders the closed trajectories, hence the geometric phase, stochastic. We study the concomitant probability distribution and show that, when varying the measurement strength, the mapping between the measurement sequence and the geometric phase undergoes a topological transition. Our finding may impact measurement-induced control and manipulation of quantum states-a promising approach to quantum information processing. It also has repercussions on understanding the foundations of quantum measurement.

Witnessing Quantum Resource Conversion within Deterministic Quantum Computation Using One Pure Superconducting Qubit

Deterministic quantum computation with one qubit (DQC1) is iconic in highlighting that exponential quantum speedup may be achieved with negligible entanglement. Its discovery catalyzed a heated study of general quantum resources, and various conjectures regarding their role in DQC1's performance advantage. Coherence and discord are prominent candidates, respectively, characterizing nonclassicality within localized and correlated systems. Here we realize DQC1 within a superconducting system, engineered such that the dynamics of coherence and discord can be tracked throughout its execution. We experimentally confirm that DQC1 acts as a resource converter, consuming coherence to generate discord during its operation. Our results highlight superconducting circuits as a promising platform for both realizing DQC1 and related algorithms, and experimentally characterizing resource dynamics within quantum protocols.

Observing and Verifying the Quantum Trajectory of a Mechanical Resonator

Continuous weak measurement allows localizing open quantum systems in state space and tracing out their quantum trajectory as they evolve in time. Efficient quantum measurement schemes have previously enabled recording quantum trajectories of microwave photon and qubit states. We apply these concepts to a macroscopic mechanical resonator, and we follow the quantum trajectory of its motional state conditioned on a continuous optical measurement record. Starting with a thermal mixture, we eventually obtain coherent states of 78% purity-comparable to a displaced thermal state of occupation 0.14. We introduce a retrodictive measurement protocol to directly verify state purity along the trajectory, and we furthermore observe state collapse and decoherence. This opens the door to measurement-based creation of advanced quantum states, as well as potential tests of gravitational decoherence models.

Conditional Dynamics of Optomechanical Two-Tone Backaction-Evading Measurements

Backaction-evading measurements of mechanical motion can achieve precision below the zero-point uncertainty and quantum squeezing, which makes them a resource for quantum metrology and quantum information processing. We provide an exact expression for the conditional state of an optomechanical system in a two-tone backaction-evading measurement beyond the standard adiabatic approximation and perform extensive numerical simulations to go beyond the usual rotating-wave approximation. We predict the simultaneous presence of conditional mechanical squeezing, intracavity squeezing, and optomechanical entanglement. We further apply an analogous analysis to the multimode optomechanical system of two mechanical and one cavity mode and find conditional mechanical Einstein-Podolski-Rosen entanglement and genuinely tripartite optomechanical entanglement. Our analysis is of direct relevance for ultrasensitive measurements and measurement-based control in high-cooperativity optomechanical sensors operating beyond the adiabatic limit.

Spontaneous Symmetry Breaking Induced by Quantum Monitoring

Spontaneous symmetry breaking (SSB) is responsible for structure formation in scenarios ranging from condensed matter to cosmology. SSB is broadly understood in terms of perturbations to the Hamiltonian governing the dynamics or to the state of the system. We study SSB due to quantum monitoring of a system via continuous quantum measurements. The acquisition of information during the measurement process induces a measurement backaction that seeds SSB. In this setting, by monitoring different observables, an observer can tailor the topology of the vacuum manifold, the pattern of symmetry breaking, and the nature of the resulting domains and topological defects.

Active protection of a superconducting qubit with an interferometric Josephson isolator

Nonreciprocal microwave devices play critical roles in high-fidelity, quantum-nondemolition (QND) measurement schemes. They impose unidirectional routing of readout signals and protect the quantum systems from unwanted noise originated by the output chain. However, cryogenic circulators and isolators are disadvantageous in scalable superconducting architectures because they use magnetic materials and strong magnetic fields. Here, we realize an active isolator formed by coupling two nondegenerate Josephson mixers in an interferometric scheme and driving them with phase-shifted, same-frequency pumps. By incorporating our Josephson-based isolator into a superconducting qubit setup, we demonstrate fast, high-fidelity, QND measurements of the qubit while providing 20 dB of protection within a bandwidth of 10 MHz against amplified noise reflected off the Josephson amplifier in the output chain. A moderate reduction of 35% is observed in T2E when the Josephson-based isolator is turned on. Such a moderate degradation can be mitigated by minimizing heat dissipation in the pump lines.

Characterizing a Statistical Arrow of Time in Quantum Measurement Dynamics

In both thermodynamics and quantum mechanics, the arrow of time is characterized by the statistical likelihood of physical processes. We characterize this arrow of time for the continuous quantum measurement dynamics of a superconducting qubit. By experimentally tracking individual weak measurement trajectories, we compare the path probabilities of forward and backward-in-time evolution to develop an arrow of time statistic associated with measurement dynamics. We compare the statistics of individual trajectories to ensemble properties showing that the measurement dynamics obeys both detailed and integral fluctuation theorems, thus establishing the consistency between microscopic and macroscopic measurement dynamics.

Correlators Exceeding One in Continuous Measurements of Superconducting Qubits

We consider the effect of phase backaction on the correlator ⟨I(t)I(t+τ)⟩ for the output signal I(t) from continuous measurement of a qubit. We demonstrate that the interplay between informational and phase backactions in the presence of Rabi oscillations can lead to the correlator becoming larger than 1, even though |⟨I⟩|≤1. The correlators can be calculated using the generalized "collapse recipe," which we validate using the quantum Bayesian formalism. The recipe can be further generalized to the case of multitime correlators and arbitrary number of detectors, measuring non-commuting qubit observables. The theory agrees well with experimental results for continuous measurement of a transmon qubit. The experimental correlator exceeds the bound of 1 for a sufficiently large angle between the amplified and informational quadratures, causing the phase backaction. The demonstrated effect can be used to calibrate the quadrature misalignment.

Ion-Based Quantum Sensor for Optical Cavity Photon Numbers

We dispersively couple a single trapped ion to an optical cavity to extract information about the cavity photon-number distribution in a nondestructive way. The photon-number-dependent ac Stark shift experienced by the ion is measured via Ramsey spectroscopy. We use these measurements first to obtain the ion-cavity interaction strength. Next, we reconstruct the cavity photon-number distribution for coherent states and for a state with mixed thermal-coherent statistics, finding overlaps above 99% with the calibrated states.

High-resolution spectroscopy of single nuclear spins via sequential weak measurements

Nuclear magnetic resonance (NMR) of single spins have recently been detected by quantum sensors. However, the spectral resolution has been limited by the sensor’s relaxation to a few kHz at room temperature. This can be improved by using quantum memories, at the expense of sensitivity. In contrast, classical signals can be measured with exceptional spectral resolution by using continuous measurement techniques, without compromising sensitivity. When applied to single-spin NMR, it is critical to overcome the impact of back action inherent of quantum measurement. Here we report sequential weak measurements on a single ¹³C nuclear spin. The back-action causes the spin to undergo a quantum dynamics phase transition from coherent trapping to coherent oscillation. Single-spin NMR at room-temperature with a spectral resolution of 3.8 Hz is achieved. These results enable the use of measurement-correlation schemes for the detection of very weakly coupled single spins.

Quantum sensors can have exceptional properties but the limits on their performance involve nonclassical effects such as quantum backaction. Here the authors show how to mitigate the effects of backaction on the spectral resolution of an NV centre nuclear spin sensor by controlling the measurement strength.

Event-Based Quantum Mechanics: A Context for the Emergence of Classical Information

This paper explores an event-based version of quantum mechanics which differs from the commonly accepted one, even though the usual elements of quantum formalism, e.g., the Hilbert space, are maintained. This version introduces as primary element the occurrence of micro-events induced by usual physical (mechanical, electromagnetic and so on) interactions. These micro-events correspond to state reductions and are identified with quantum jumps, already introduced by Bohr in his atomic model and experimentally well established today. Macroscopic bodies are defined as clusters of jumps; the emergence of classicality thus becomes understandable and time honoured paradoxes can be solved. In particular, we discuss the cat paradox in this context. Quantum jumps are described as temporal localizations of physical quantities; if the information associated with these localizations has to be finite, two time scales spontaneously appear: an upper cosmological scale and a lower scale of elementary “particles”. This allows the interpretation of the Bekenstein limit like a particular informational constraint on the manifestation of a micro-event in the cosmos it belongs. The topic appears relevant in relation to recent discussions on possible spatiotemporal constraints on quantum computing.

Symmetry Constrained Decoherence of Conditional Expectation Values

Conditional expectation values of quantum mechanical observables reflect unique non-classical correlations, and are generally sensitive to decoherence. We consider the circumstances under which such sensitivity to decoherence is removed, namely, when the measurement process is subjected to conservation laws. Specifically, we address systems with additive conserved quantities and identify sufficient conditions for the system state such that its coherence plays no role in the conditional expectation values of observables that commute with the conserved quantity. We discuss our findings for a specific model where the system-detector coupling is given by the Jaynes-Cummings interaction, which is relevant to experiments tracking trajectories of qubits in cavities. Our results clarify, among others, the role of coherence in thermal measurements in current architectures for quantum thermodynamics experiments.

Experimental repetitive quantum channel simulation

Universal control of quantum systems is a major goal to be achieved for quantum information processing, which demands thorough understanding of fundamental quantum mechanics and promises applications of quantum technologies. So far, most studies concentrate on ideally isolated quantum systems governed by unitary evolutions, while practical quantum systems are open and described by quantum channels due to their inevitable coupling to environment. Here, we experimentally simulate arbitrary quantum channels for an open quantum system, i.e. a single photonic qubit in a superconducting quantum circuit. The arbitrary channel simulation is achieved with minimum resource of only one ancilla qubit and measurement-based adaptive control. By repetitively implementing the quantum channel simulation, we realize an arbitrary Liouvillian for a continuous evolution of an open quantum system for the first time. Our experiment provides not only a testbed for understanding quantum noise and decoherence, but also a powerful tool for full control of practical open quantum systems.

Dynamics of probing a quantum-dot spin qubit with superconducting resonator photons

The hybrid system of electron spins and resonator photons is an attractive architecture for quantum computing owing to the long coherence times of spins and the promise of long-distance coupling between arbitrary pairs of qubits via photons. For the device to serve as a building block for a quantum processer, it is also necessary to readout the spin qubit state. Here we analyze in detail the measurement process of an electron spin singlet-triplet qubit in quantum dots using a coupled superconducting resonator. We show that the states of the spin singlet-triplet qubit lead to readily observable features in the spectrum of a microwave field through the resonator. These features provide useful information on the hybrid system. Moreover, we discuss the working points which can be implemented with high performance in the current state-of-the-art devices. These results can be used to construct the high fidelity measurement toolbox in the spin-circuit QED system.

Single-Loop Realization of Arbitrary Nonadiabatic Holonomic Single-Qubit Quantum Gates in a Superconducting Circuit

Geometric phases are noise resilient, and thus provide a robust way towards high-fidelity quantum manipulation. Here we experimentally demonstrate arbitrary nonadiabatic holonomic single-qubit quantum gates for both a superconducting transmon qubit and a microwave cavity in a single-loop way. In both cases, an auxiliary state is utilized, and two resonant microwave drives are simultaneously applied with well-controlled but varying amplitudes and phases for the arbitrariness of the gate. The resulting gates on the transmon qubit achieve a fidelity of 0.996 characterized by randomized benchmarking and the ones on the cavity show an averaged fidelity of 0.978 based on a full quantum process tomography. In principle, a nontrivial two-qubit holonomic gate between the qubit and the cavity can also be realized based on our presented experimental scheme. Our experiment thus paves the way towards practical nonadiabatic holonomic quantum manipulation with both qubits and cavities in a superconducting circuit.

Information Gain and Loss for a Quantum Maxwell's Demon

We use continuous weak measurements of a driven superconducting qubit to experimentally study the information dynamics of a quantum Maxwell's demon. We show how information gained by a demon who can track single quantum trajectories of the qubit can be converted into work using quantum coherent feedback. We verify the validity of a quantum fluctuation theorem with feedback by utilizing information obtained along single trajectories. We demonstrate, in particular, that quantum backaction can lead to a loss of information in imperfect measurements. We furthermore probe the transition between information gain and loss by varying the initial purity of the qubit.

Dynamics of a qubit while simultaneously monitoring its relaxation and dephasing

Decoherence originates from the leakage of quantum information into external degrees of freedom. For a qubit, the two main decoherence channels are relaxation and dephasing. Here, we report an experiment on a superconducting qubit where we retrieve part of the lost information in both of these channels. We demonstrate that raw averaging the corresponding measurement records provides a full quantum tomography of the qubit state where all three components of the effective spin-1/2 are simultaneously measured. From single realizations of the experiment, it is possible to infer the quantum trajectories followed by the qubit state conditioned on relaxation and/or dephasing channels. The incompatibility between these quantum measurements of the qubit leads to observable consequences in the statistics of quantum states. The high level of controllability of superconducting circuits enables us to explore many regimes from the Zeno effect to underdamped Rabi oscillations depending on the relative strengths of driving, dephasing, and relaxation.

Information leaked by a quantum system into its environment causes decoherence but if it is recorded then it can be used to infer the quantum state. Ficheux et al. monitor the relaxation and dephasing of a qubit and show that this allows all three components of the qubit to be probed simultaneously.

Topological Maxwell Metal Bands in a Superconducting Qutrit

We experimentally explore the topological Maxwell metal bands by mapping the momentum space of condensed-matter models to the tunable parameter space of superconducting quantum circuits. An exotic band structure that is effectively described by the spin-1 Maxwell equations is imaged. Threefold degenerate points dubbed Maxwell points are observed in the Maxwell metal bands. Moreover, we engineer and observe the topological phase transition from the topological Maxwell metal to a trivial insulator, and report the first experiment to measure the Chern numbers that are higher than one.

Direct measurement of the quantum state of photons in a cavity

We propose a scheme to measure the quantum state of photons in a cavity. The proposal is based on the concept of quantum weak values and applies equally well to both the solid-state circuit and atomic cavity quantum electrodynamics (QED) systems. The proposed scheme allows us to access directly the superposition components in Fock state basis, rather than the Wigner function as usual in phase space. Moreover, the separate access feature held in the direct scheme does not require a global reconstruction for the quantum state, which provides a particular advantage beyond the conventional method of quantum state tomography.

Simulation of molecular spectroscopy with circuit quantum electrodynamics

Spectroscopy is a crucial laboratory technique for understanding quantum systems through their interactions with the electromagnetic radiation. Particularly, spectroscopy is capable of revealing the physical structure of molecules, leading to the development of the maser-the forerunner of the laser. However, real-world applications of molecular spectroscopy are mostly confined to equilibrium states, due to computational and technological constraints; a potential breakthrough can be achieved by utilizing the emerging technology of quantum simulation. Here we experimentally demonstrate through a toy model, a superconducting quantum simulator capable of generating molecular spectra for both equilibrium and non-equilibrium states, reliably producing the vibronic structure of diatomic molecules. Furthermore, our quantum simulator is applicable not only to molecules with a wide range of electronic-vibronic coupling strength, characterized by the Huang-Rhys parameter, but also to molecular spectra not readily accessible under normal laboratory conditions. These results point to a new direction for predicting and understanding molecular spectroscopy, exploiting the power of quantum simulation.