Several quantum radar concepts have been proposed that exploit the entanglement found in two-mode squeezed vacuum states of the electromagnetic field, the most prominent being radar based on quantum illumination. Classical radars are sometimes required to distinguish between true echos of their transmitted signals and signals generated by interferors or spoofers. How vulnerable to spoofing is quantum illumination? We analyze the scenario of a radar operator trying to detect the presence of a classical spoofer employing a measure-and-prepare strategy against a quantum radar. We consider two spoofing strategies - (1) direct detection and number state preparation, and (2) heterodyne detection and coherent state preparation. In each case, the radar operator performs a hypothesis test to decide if received pulses are true returns or spoofs. Since the spoofer can not reproduce the entanglement with modes retained by the radar operator, both approaches to spoofing are to some extent detectable. We quantify the effectiveness of the spoof in terms of the fidelity between the real return and the spoof return, and the probability of error in spoof detection. We find that in the absence of noise and loss, direct detection tends to produce spoofs with greater fidelity, which are therefore harder to detect. Moreover, this advantage survives the introduction of noise and loss into the model. Our results suggest that entanglement is a novel resource available to quantum radar for detecting spoofing.
Quantum sensors based on solid-state defects, in particular nitrogen-vacancy (NV) centers in diamond, enable precise measurement of magnetic fields, temperature, rotation, and electric fields. However, the sensitivity of leading NV spin ensemble sensors remains far from the intrinsic spin-projection noise limit. Here we move towards this quantum limit of performance by introducing (i) a cavity quantum electrodynamic (cQED) hybrid system operating in the strong coupling regime, which enables high readout fidelity of an NV ensemble using microwave homodyne detection; (ii) a comprehensive nonlinear model of the cQED sensor operation, including NV ensemble inhomogeneity and optical polarization; and (iii) ``spin refrigeration'' where the optically-polarized spin ensemble sharply reduces the ambient-temperature microwave thermal noise, resulting in enhanced sensitivity. Applying these advances to magnetometry, we demonstrate a broadband sensitivity of 580 fT/$\sqrt{\mathrm{Hz}}$ around 15 kHz in ambient conditions. We then discuss the implications of this model for design of future magnetometers, including devices approaching 12 fT/$\sqrt{\mathrm{Hz}}$ sensitivity. Applications of these techniques extend to the fields of gyroscope and clock technologies.
Electromagnetic remote sensing technologies such as radar can be mislead by targets that generate spoof pulses. Typically, a would-be spoofer must make measurements to characterize a received pulse in order to design a convincing spoof pulse. The precision of such measurements are ultimately limited by quantum noise. Here we introduce a model of electromagnetic spoofing that includes effects of practical importance that were neglected in prior theoretical studies. In particular, the model includes thermal background noise and digital quantization noise, as well as loss in transmission, propagation, and reception. We derive the optimal probability of detecting a spoofer allowed by quantum physics. We show that heterodyne reception and thresholding closely approaches this optimal performance. Finally, we show that a high degree of certainty in spoof detection can be reached by Bayesian inference from a sequence of received pulses. Together these results suggest that a practically realizable receiver could plausibly detect a radar spoofer by observing errors in the spoof pulses due to quantum noise.
The achievement of sufficiently fast interactions between two optical fields at the few-photon level would provide a key enabler for a broad range of quantum technologies. One critical hurdle in this endeavor is the lack of a comprehensive quantum theory of the generation of nonlinearities by ensembles of emitters. Distinct approaches applicable to different regimes have yielded important insights: i) a semiclassical approach reveals that, for many-photon coherent fields, the contributions of independent emitters add independently allowing ensembles to produce strong optical nonlinearities via EIT; ii) a quantum analysis has shown that in the few-photon regime collective coupling effects prevent ensembles from inducing these strong nonlinearities. Rather surprisingly, experimental results with around twenty photons are in line with the semi-classical predictions. Theoretical analysis has been fragmented due to the difficulty of treating nonlinear many-body quantum systems. Here we are able to solve this problem by constructing a powerful theory of the generation of optical nonlinearities by single emitters and ensembles. The key to this construction is the application of perturbation theory to perturbations generated by subsystems. This theory reveals critical properties of ensembles that have long been obscure. The most remarkable of these is the discovery that quantum effects prevent ensembles generating single-photon nonlinearities only within the rotating-wave regime; outside this regime single-photon nonlinearities scale as the number of emitters. The theory we present here also provides an efficient way to calculate nonlinearities for arbitrary multi-level driving schemes, and we expect that it will prove a powerful foundation for further advances in this area.
Optimization methods for constrained quantum control problems power quantum technologies. Such control problems are notoriously difficult because they are non-convex and plagued with local extrema. Current optimization methods must be repeated many times to find good solutions, each time requiring many simulations of the system. Here we present Quantum Control via Polynomial Optimization (QCPOp), a method that eliminates this problem by directly finding globally optimal solutions. This remarkable ability is due to global optimization methods recently developed for polynomial functions. We demonstrate the tremendous improvement over current state-of-the-art methods using a number of non-trivial examples. Global optimization also allows QCPOp to find the simplest control solutions. Since QCPOp is able to reveal the optimum performance of quantum control with high confidence, we expect that it will not only enhance the utility of such control but provide a key tool for determining the limits of quantum technologies.
Current quantum devices suffer imperfections as a result of fabrication, as well as noise and dissipation as a result of coupling to their immediate environments. Because of this, it is often difficult to obtain accurate models of their dynamics from first principles. An alternative is to extract such models from time-series measurements of their behavior. Here, we formulate this system-identification problem as a polynomial optimization problem. Recent advances in optimization have provided globally convergent solvers for this class of problems, which using our formulation prove estimates of the Kraus map or the Lindblad equation. We include an overview of the state-of-the-art algorithms, bounds, and convergence rates, and illustrate the use of this approach to modeling open quantum systems.
A fundamental result of quantum mechanics is that the fluctuations of a bosonic field are given by its temperature $T$. An electromagnetic mode with frequency $\omega$ in the microwave band has a significant thermal photon occupation at room temperature according to the Bose-Einstein distribution $\bar{n} = k_BT / \hbar\omega$. The room temperature thermal state of a 3 GHz mode, for example, is characterized by a mean photon number $\bar{n} \sim 2000$ and variance $\Delta n^2 \approx \bar{n}^2$. This thermal variance sets the measurement noise floor in applications ranging from wireless communications to positioning, navigation, and timing to magnetic resonance imaging. We overcome this barrier in continuously cooling a ${\sim} 3$ GHz cavity mode by coupling it to an ensemble of optically spin-polarized nitrogen-vacancy (NV) centers in a room-temperature diamond. The NV spins are pumped into a low entropy state via a green laser and act as a heat sink to the microwave mode through their collective interaction with microwave photons. Using a simple detection circuit we report a peak noise reduction of $-2.3 \pm 0.1 \, \textrm{dB}$ and minimum cavity mode temperature of $150 \pm 5 \textrm{K}$. We present also a linearized model to identify the important features of the cooling, and demonstrate its validity through magnetically tuned, spectrally resolved measurements. The realization of efficient mode cooling at ambient temperature opens the door to applications in precision measurement and communication, with the potential to scale towards fundamental quantum limits.
We propose an architecture for achieving high-fidelity deterministic quantum logic gates on dual-rail encoded photonic qubits by letting photons interact with a two-level emitter (TLE) inside an optical cavity. The photon wave packets that define the qubit are preserved after the interaction due to a quantum control process that actively loads and unloads the photons from the cavity and dynamically alters their effective coupling to the TLE. The controls rely on nonlinear wave mixing between cavity modes enhanced by strong externally modulated electromagnetic fields or on AC Stark shifts of the TLE transition energy. We numerically investigate the effect of imperfections in terms of loss and dephasing of the TLE as well as control field miscalibration. Our results suggest that III-V quantum dots in GaAs membranes is a promising platform for photonic quantum information processing.
It is well-known that the precision of a phase measurement with a Mach-Zehnder interferometer employing strong (macroscopic) classic light can be greatly enhanced with the addition of a weak (microscopic) light field in a non-classical state. The resulting precision is much greater than that possible with either the macroscopic classical or microscopic quantum states alone. In the context of quantifying non-classicality, the amount by which a non-classical state can enhance precision in this way has been termed its "metrological power". Given the technological difficulty of producing high-amplitude non-classical states of light, this use of non-classical light is likely to provide a technological advantage much sooner than the Heisenberg scaling employing much stronger non-classical states. To date, the enhancement provided by weak nonclassical states has been calculated only for specific measurement configurations. Here we are able to optimize over all measurement configurations to obtain the maximum enhancement that can be achieved by any single or multi-mode nonclassical state together with strong classical states, for local and distributed quantum metrology employing any linear or nonlinear single-mode unitary transformation. Our analysis reveals that the quantum Fisher information for \textitquadrature displacement sensing is the sole property that determines the maximum achievable enhancement in all of these different scenarios, providing a unified quantification of the metrological power. It also reveals that the Mach-Zehnder interferometer is an optimal network for phase sensing for an arbitrary single-mode nonclassical input state, and how the Mach-Zehnder interferometer can be extended to make optimal use of any multi-mode nonclassical state for metrology.
Networking superconducting quantum computers is a longstanding challenge in quantum science. The typical approach has been to cascade transducers: converting to optical frequencies at the transmitter and to microwave frequencies at the receiver. However, the small microwave-optical coupling and added noise have proven formidable obstacles. Instead, we propose optical networking via heralding end-to-end entanglement with one detected photon and teleportation. In contrast to cascaded direct transduction, our scheme absorbs the low optical-microwave coupling efficiency into the heralding step, thus breaking the rate-fidelity trade-off. Moreover, this technique unifies and simplifies entanglement generation between superconducting devices and other physical modalities in quantum networks.
We introduce a new technique for the simulation of dissipative quantum systems. This method is composed of an approximate decomposition of the Lindblad equation into a Kraus map, from which one can define an ensemble of wavefunctions. Using principal component analysis, this ensemble can be truncated to a manageable size without sacrificing numerical accuracy. We term this method \emphEnsemble Rank Truncation (ERT), and find that in the regime of weak coupling, this method is able to outperform existing wavefunction Monte-Carlo methods by an order of magnitude in both accuracy and speed. We also explore the possibility of combining ERT with approximate techniques for simulating large systems (such as Matrix Product States (MPS)), and show that in many cases this approach will be more efficient than directly expressing the density matrix in its MPS form. We expect the ERT technique to be of practical interest when simulating dissipative systems for quantum information, metrology and thermodynamics.
Atom-like quantum systems in solids have been proposed as a compact alternative for atomic clocks, but realizing the potential of solid-state technology will requires an architecture design which overcomes traditional limitations such as magnetic and temperature-induced systematics. Here, we propose a solution to this problem: a `solid-state spin clock' that hybridizes a microwave resonator with a magnetic-field-insensitive spin transition within the ground state of the diamond nitrogen-vacancy center. Detailed numerical and analytical modeling of this `polariton-stabilized' spin clock (PSSC) indicates a potential fractional frequency instability below $10^{-13}$ at 1 second measurement time, assuming present-day experimental parameters. This stability would represent a significant improvement over the state-of-the-art in miniaturized atomic vapor clocks.
Recent progress in nonlinear optical materials and microresonators has brought quantum computing with bulk optical nonlinearities into the realm of possibility. This platform is of great interest, not only because photonics is an obvious choice for quantum networks, but also because it may be the only feasible route to quantum information processing at room temperature. We introduce a paradigm for room-temperature photonic quantum logic that significantly simplifies the realization of various quantum circuits, and in particular, of error correction. It uses only the strongest available bulk nonlinearity, namely the $\chi^{(2)}$ nonlinear susceptibility. The key element is a three-mode resonator that implements programmable bosonic quantum logic gates. We show that just two of these elements suffice for a complete, compact error-correction circuit on a bosonic code, without the need for measurement or feed-forward control. An extrapolation of current progress in nonlinear optical materials and photonic circuits indicates that such circuitry should be achievable within the next decade.
Most future quantum devices, including quantum computers, require control that is broadband, meaning that the rate of change of the time-dependent Hamiltonian is as fast or faster than the dynamics it generates. In many areas of quantum physics, including quantum technology, one must include dissipation and decoherence induced by the environment. While Markovian master equations provide the only really efficient way to model these effects, these master equations are derived for constant Hamiltonians (or those with a discrete set of well-defined frequencies). In 2006, Alicky, Lidar, and Zanardi [Phys. Rev. A 73, 052311 (2006)] provided detailed qualitative arguments that Markovian master equations could not describe systems under broadband control. Despite apparently broad acceptance of these arguments, such master equations are routinely used to model precisely these systems. This odd state of affairs is likely due to a lack of quantitative results. Here we perform exact simulations of two- and three-level systems coupled to an oscillator bath to obtain quantitative results. Although we confirm that in general Markovian master equations cannot predict the effects of damping under broadband control, we find that there is a widely applicable regime in which they can. Master equations are accurate for weak damping if both the Rabi frequencies and bandwidth of the control are significantly smaller than the system's transition frequencies. They also remain accurate if the bandwidth of control is as large as the frequency of the driven transition so long as this bandwidth does not overlap other transitions. Master equations are thus able to provide accurate descriptions of many quantum information processing protocols for atomic systems.
Here we present an expanded analysis of a model for the manipulation and control of observables in a strongly correlated, many-body system, which was first presented in [McCaul et al., eprint: arXiv:1911.05006]. A field-free, non-linear equation of motion for controlling the expectation value of an essentially arbitrary observable is derived, together with rigorous constraints that determine the limits of controllability. We show that these constraints arise from the physically reasonable assumptions that the system will undergo unitary time evolution, and has enough degrees of freedom for the electrons to be mobile. Furthermore, we give examples of multiple solutions to generating target observable trajectories when the constraints are violated. Ehrenfest theorems are used to further refine the model, and provide a check on the validity of numerical simulations. Finally, the experimental feasibility of implementing the control fields generated by this model is discussed.
We present a framework to control and track the observables of a general solid state system driven by an incident laser field. The main result is a non-linear equation of motion for tracking an observable, together with a constraint on the size of expectations which may be reproduced via tracking. Among other applications, this model provides a potential route to the design of laser fields which cause photo-induced superconductivity in materials above their critical temperature. As a first test, the strategy is used to make the expectation value of the current conform to an arbitrary function under a range of model parameters. Additionally, using two reference spectra for materials in the conducting and insulating regimes respectively, the tracking algorithm is used to make each material mimic the optical spectrum of the other.
The nonclassical properties of quantum states are of tremendous interest due to their potential applications in future technologies. It has recently been realized that the concept of a "resource theory" is a powerful approach to quantifying and understanding nonclassicality. To realize the potential of this approach one must first find resource theoretic measures of nonclassicality that are "operational", meaning that they also quantify the ability of quantum states to provide enhanced performance for specific tasks, such as precision sensing. Here we achieve a significant milestone in this endeavor by presenting the first such operational resource theoretic measure. In addition to satisfying the requirements of a resource measure, it has the closest possible relationship to the quantum-enhancement provided by a non-classical state for measuring phase-space displacement: it is equal to this enhancement for pure states, and is a tight upper bound on it for mixed states. We also show that a lower-bound on this measure can be obtained experimentally using a simple Mach-Zehnder interferometer.
We propose an architecture for a high-fidelity deterministic controlled-phase gate between two photonic qubits using bulk optical nonlinearities in near-term feasible photonic integrated circuits. The gate is enabled by converting travelling continuous-mode photons into stationary cavity modes using strong classical control fields that dynamically change the cavity-waveguide coupling rate. This process limits the fidelity degrading noise pointed out by Shapiro [J. Shapiro, Phys. Rev. A, 73, 2006] and Gea-Banacloche [J. Gea-Banacloche, Phys. Rev. A, 81, 2010]. We show that high-fidelity gates can be achieved with self-phase modulation in $\chi^{\scriptscriptstyle(3)}$ materials as well as second-harmonic generation in $\chi^{\scriptscriptstyle(2)}$ materials. The gate fidelity asymptotically approaches unity with increasing storage time for a fixed duration of the incident photon wave packet. Further, dynamically coupled cavities enable a trade-off between errors due to loss and wave packet distortions since loss does not affect the ability to emit wave packets with the same shape as the incoming photons. Our numerical results show that gates with $99\%$ fidelity are feasible with near-term improvements in cavity loss using LiNbO$_3$ or GaAs.
In trapped-ion quantum information processing, interactions between spins (qubits) are mediated by collective modes of motion of an ion crystal. While there are many different experimental strategies to design such interactions, they all face both technical and fundamental limitations to the achievable coherent interaction strength. In general, obtaining strong interactions and fast gates is an ongoing challenge. Here, we extend previous work [Phys. Rev. Lett. 112, 030501 (2019)] and present a general strategy for enhancing the interaction strengths in trapped-ion systems via parametric amplification of the ions' motion. Specifically, we propose a stroboscopic protocol using alternating applications of parametric amplification and spin-motion coupling. In comparison with the previous work, we show that the current protocol can lead to larger enhancements in the coherent interaction that increase exponentially with the gate time.
Realistic models of quantum systems must include dissipative interactions with an environment. For weakly-damped systems the Lindblad-form Markovian master equation is invaluable for this task due to its tractability and efficiency. This equation only applies, however, when the frequencies of any subset of the system's transitions are either equal (degenerate), or their differences are much greater than the transitions' linewidths (far-detuned). Outside of these two regimes the only available efficient description has been the Bloch-Redfield (B-R) master equation, the efficacy of which has long been controversial due to its failure to guarantee the positivity of the density matrix. The ability to efficiently simulate weakly-damped systems across all regimes is becoming increasingly important, especially in the area of quantum technologies. Here we solve this long-standing problem. We discover that a condition on the slope of the spectral density is sufficient to derive a Lindblad form master equation that is accurate for all regimes. We further show that this condition is necessary for weakly-damped systems to be described by the B-R equation or indeed any Markovian master equation. We thus obtain a replacement for the B-R equation over its entire domain of applicability that is no less accurate, simpler in structure, completely positive, allows simulation by efficient quantum trajectory methods, and unifies the previous Lindblad master equations. We also show via exact simulations that the new master equation can describe systems in which slowly-varying transition frequencies cross each other during the evolution. System identification tools, developed in systems engineering, play an important role in our analysis. We expect these tools to prove useful in other areas of physics involving complex systems.
We study theoretically the interaction between two photons in a nonlinear cavity. The photons are loaded into the cavity via a method we propose here, in which the input/output coupling of the cavity is effectively controlled via a tunable coupling to a second cavity mode that is itself strongly output-coupled. Incoming photon wave packets can be loaded into the cavity with high fidelity when the timescale of the control is smaller than the duration of the wave packets. Dynamically coupled cavities can be used to avoid limitations in the photon-photon interaction time set by the delay-bandwidth product of passive cavities. Additionally, they enable the elimination of wave packet distortions caused by dispersive cavity transmission and reflection. We consider three kinds of nonlinearities, those arising from $\chi^{\scriptscriptstyle(2)}$ and $\chi^{\scriptscriptstyle(3)}$ materials and that due to an interaction with a two-level emitter. To analyze the input and output of few-photon wave packets we use a Schrödinger-picture formalism in which travelling-wave fields are discretized into infinitesimal time-bins. We suggest that dynamically coupled cavities provide a very useful tool for improving the performance of quantum devices relying on cavity-enhanced light-matter interactions such as single-photon sources and atom-like quantum memories with photon interfaces. As an example, we present simulation results showing that high fidelity two-qubit entangling gates may be constructed using any of the considered nonlinear interactions.
While relatively easy to engineer, static transverse coupling between a qubit and a cavity mode satisfies the criteria for a quantum non-demolition (QND) measurement only if the coupling between the qubit and cavity is much less than their mutual detuning. This can put significant limits on the speed of the measurement, requiring trade-offs in the circuit design between coupling, detuning, and decoherence introduced by the cavity mode. Here, we study a circuit in which the qubit-cavity and the cavity-feedline coupling can be turned on and off, which helps to isolate the qubit. We do not rely on the rotating-wave or dispersive approximations, but solve the full transverse interaction between the qubit and the cavity mode. We show that by carefully choosing the detuning and interaction time, we can exploit a recurrence in the qubit-cavity dynamics in a way that makes it possible to perform very fast, high fidelity, QND measurements. Here, the qubit measurement is performed more like a gate operation between the qubit and the cavity, where the cavity state can be amplified, squeezed, and released in a time-sequenced fashion. In addition, we also show that the non-demolition property of the off-resonant approximation breaks down much faster than its dispersive property, suggesting that many of the dispersive measurements to date have been implemented outside the QND regime.
Trapped ions offer a pristine platform for quantum computation and simulation, but improving their coherence remains a crucial challenge. Here, we propose and analyze a new strategy to enhance the coherent interactions in trapped ion systems via parametric amplification of the ions' motion--by squeezing the collective motional modes (phonons), the spin-spin interactions they mediate can be significantly enhanced. We illustrate the power of this approach by showing how it can enhance collective spin states useful for quantum metrology, and how it can improve the speed and fidelity of two-qubit gates in multi-ion systems, important ingredients for scalable trapped ion quantum computation. Our results are also directly relevant to numerous other physical platforms in which spin interactions are mediated by bosons.
We propose and experimentally demonstrate a simple and efficient scheme for photonic communication between two remote superconducting modules. Each module consists of a random access quantum information processor with eight-qubit multimode memory and a single flux tunable transmon. The two processor chips are connected through a one-meter long coaxial cable that is coupled to a dedicated "communication" resonator on each chip. The two communication resonators hybridize with a mode of the cable to form a dark "communication mode" that is highly immune to decay in the coaxial cable. We modulate the transmon frequency via a parametric drive to generate sideband interactions between the transmon and the communication mode. We demonstrate bidirectional single-photon transfer with a success probability exceeding 60 %, and generate an entangled Bell pair with a fidelity of 79.3 $\pm$ 0.3 %.
Input-output theory is invaluable for treating superconducting and photonic circuits connected by transmission lines or waveguides. However, this theory cannot in general handle situations in which retro-reflections from circuit components or configurations of beam-splitters create loops for the traveling-wave fields that connect the systems. Here, building upon the network-contraction theory of Gough and James [Commun. Math. Phys. 287, 1109 (2009)], we provide a compact and powerful method to treat any circuit that contains such loops so long as the effective cavities formed by the loops are sufficiently weak. Essentially all present-day on-chip superconducting and photonic circuits will satisfy this weakness condition so long as the reflectors that form the loops are not especially highly reflecting. As an example we analyze the problem of transmitting entanglement between two qubits connected by a transmission line with imperfect circulators, a problem for which the new method is essential. We obtain a full solution for the optimal receiver given that the sender employs a simple turn on/turn off. This solution shows that near-perfect transmission is possible even with significant retro-reflections.
The wave-function Monte-Carlo method, also referred to as the use of "quantum-jump trajectories", allows efficient simulation of open systems by independently tracking the evolution of many pure-state "trajectories". This method is ideally suited to simulation by modern, highly parallel computers. Here we show that Krotov's method of numerical optimal control, unlike others, can be modified in a simple way, so that it becomes fully parallel in the pure states without losing its effectiveness. This provides a highly efficient method for finding optimal control protocols for open quantum systems and networks. We apply this method to the problem of generating entangled states in a network consisting of systems coupled in a unidirectional chain. We show that due to the existence of a dark-state subspace in the network, nearly-optimal control protocols can be found for this problem by using only a single pure-state trajectory in the optimization, further increasing the efficiency.
We discuss a general method of model selection from experimentally recorded time-trace data. This method can be used to distinguish between quantum and classical dynamical models. It can be used in post-selection as well as for real-time analysis, and offers an alternative to statistical tests based on state-reconstruction methods. We examine the conditions that optimize quantum hypothesis testing, maximizing one's ability to discriminate between classical and quantum models. We set upper limits on the temperature and lower limits on the measurement efficiencies required to explore these differences, using a novel experiment in levitated optomechanical systems as an example.
Fermi's golden rule applies to a situation in which a single quantum state $|\psi\rangle$ is coupled to a near-continuum. This "quasi-continuum coupling" structure results in a rate equation for the population of $|\psi\rangle$. Here we show that the coupling of a quantum system to the standard model of a thermal environment, a bath of harmonic oscillators, can be decomposed into a "cascade" made up of the quasi-continuum coupling structures of Fermi's golden rule. This clarifies the connection between the physics of the golden rule and that of a thermal bath, and provides a non-rigorous but physically intuitive derivation of the Markovian master equation directly from the former. The exact solution to the Hamiltonian of the golden rule, known as the Bixon-Jortner model, generalized for an asymmetric spectrum, provides a window on how the evolution induced by the bath deviates from the master equation as one moves outside the Markovian regime. Our analysis also reveals the relationship between the oscillator bath and the "random matrix model" (RMT) of a thermal bath. We show that the cascade structure is the one essential difference between the two models, and the lack of it prevents the RMT from generating transition rates that are independent of the initial state of the system. We suggest that the cascade structure is one of the generic elements of thermalizing many-body systems.
We derive a bound on the ability of a linear optical network to estimate a linear combination of independent phase shifts by using an arbitrary non-classical but unentangled input state, thereby elucidating the quantum resources required to obtain the Heisenberg limit with a multi-port interferometer. Our bound reveals that while linear networks can generate highly entangled states, they cannot effectively combine quantum resources that are well distributed across multiple modes for the purposes of metrology: in this sense linear networks endowed with well-distributed quantum resources behave classically. Conversely, our bound shows that linear networks can achieve the Heisenberg limit for distributed metrology when the input photons are hoarded in a small number of input modes, and we present an explicit scheme for doing so. Our results also have implications for measures of non-classicality.
This paper proposes an efficient method for the simultaneous estimation of the state of a quantum system and the classical parameters that govern its evolution. This hybrid approach benefits from efficient numerical methods for the integration of stochastic master equations for the quantum system, and efficient parameter estimation methods from classical signal processing. The classical techniques use Sequential Monte Carlo (SMC) methods, which aim to optimize the selection of points within the parameter space, conditioned by the measurement data obtained. We illustrate these methods using a specific example, an SMC sampler applied to a nonlinear system, the Duffing oscillator, where the evolution of the quantum state of the oscillator and three Hamiltonian parameters are estimated simultaneously.
In this mainly pedagogical article, we discuss under what circumstances measurements play a special role in quantum processes. In particular we discuss the following facts which appear to be a common area of confusion: i) from a fundamental point of view measurements play no special role whatsoever: all dynamics that can be generated by measurements can be generated by unitary processes (for which post-selection is no exception), ii) from a purely physical point of view, measurements are not "outside" of quantum mechanics, iii) the only difference between the abilities of measurement-based protocols and unitary circuits for quantum computing comes from practical (technology dependent) constraints. We emphasize the importance of distinguishing between differences that are i) fundamental but without physical import, ii) fundamental and possess physical import, iii) are not fundamental but have practical import. We also emphasize the importance of separating theoretical and experimental elements of measurement, primarily projection and amplification, which are physically very different. Note that since we are concerned with facts regarding physical processes, this article has little if anything to do with interpretations of quantum mechanics.
A candidate for converting quantum information from microwave to optical frequencies is the use of a single atom that interacts with a superconducting microwave resonator on one hand and an optical cavity on the other. The large electric dipole moments and microwave transition frequencies possessed by Rydberg states allow them to couple strongly to superconducting devices. Lasers can then be used to connect a Rydberg transition to an optical transition to realize the conversion. Since the fundamental source of noise in this process is spontaneous emission from the atomic levels, the resulting control problem involves choosing the pulse shapes of the driving lasers so as to maximize the transfer rate while minimizing this loss. Here we consider the concrete example of a cesium atom, along with two specific choices for the levels to be used in the conversion cycle. Under the assumption that spontaneous emission is the only significant source of errors, we use numerical optimization to determine the likely rates for reliable quantum communication that could be achieved with this device. These rates are on the order of a few Mega-qubits per second.
It is known that placing a mechanical oscillator in a superposition of coherent states allows, in theory, a measurement of a linear force whose sensitivity increases with the amplitude of the mechanical oscillations, a uniquely quantum effect. Further, entangled versions of these states across a network of $n$ mechanical oscillators enables a measurement whose sensitivity increases linearly with $n$, thus improving over the classical scaling by $\sqrt{n}$. One of the key challenges in exploiting this effect is processing the signal so that it can be readily measured; linear processing is insufficient. Here we show that a Kerr oscillator will not only create the necessary states, but also perform the required processing, transforming the quantum phase imprinted by the force signal into a shift in amplitude measurable with homodyne detection. This allows us to design a relatively simple quantum electro-mechanical circuit that can demonstrate the core quantum effect at the heart of this scheme, amplitude-dependent force sensitivity. We derive analytic expressions for the performance of the circuit, including thermal mechanical noise and photon loss. We discuss the experimental challenges in implementing the scheme with near-term technology.
A fundamental requirement for the emergence of classical behavior from an underlying quantum description is that certain observed quantum systems make a transition to chaotic dynamics as their action is increased relative to $\hbar$. While experiments have demonstrated some aspects of this transition, the emergence of quantum trajectories with a positive Lyapunov exponent has never been observed directly. Here, we remove a major obstacle to achieving this goal by showing that, for the Duffing oscillator, the transition to a positive Lyapunov exponent can be resolved clearly from observed trajectories even with measurement efficiencies as low as 20%. We also find that the positive Lyapunov exponent is robust to highly mixed, low purity states and to variations in the parameters of the system.
Von Neumann's classic "multiplexing" method is unique in achieving high-threshold fault-tolerant classical computation (FTCC), but has several significant barriers to implementation: i) the extremely complex circuits required by randomized connections, ii) the difficulty of calculating its performance in practical regimes of both code size and logical error rate, and iii) the (perceived) need for large code sizes. Here we present numerical results indicating that the third assertion is false, and introduce a novel scheme that eliminates the two remaining problems while retaining a threshold very close to von Neumann's ideal of 1/6. We present a simple, highly ordered wiring structure that vastly reduces the circuit complexity, demonstrates that randomization is unnecessary, and provides a feasible method to calculate the performance. This in turn allows us to show that the scheme requires only moderate code sizes, vastly outperforms concatenation schemes, and under a standard error model a unitary implementation realizes universal FTCC with an accuracy threshold of p < 5.5%, in which p is the error probability for 3-qubit gates. FTCC is a key component in realizing measurement-free protocols for quantum information processing. In view of this we use our scheme to show that all-unitary quantum circuits can reproduce any measurement-based feedback process in which the asymptotic error probabilities for the measurement and feedback are 0.51p and 1.51p, respectively.
We consider a measurement of the position of a spot painted on the surface of a trapped nano-optomechanical sphere. The measurement extracts information about the position of the spot and in doing so measures a combination of the orientation and position of the sphere. The quantum back-action of the measurement entangles and correlates these two degrees of freedom. Such a measurement is not available for atoms or ions, and provides a mechanism to probe the quantum mechanical properties of trapped optomechanical spheres. In performing simulations of this measurement process we also test a numerical method introduced recently by Rouchon and collaborators for solving stochastic master equations. This method guarantees the positivity of the density matrix when the Lindblad operators for all simultaneous continuous measurements are mutually commuting. We show that it is both simpler and far more e?fficient than previous methods.
We compare the performance of continuous coherent feedback, implemented using an ideal single-qubit controller, to that of continuous measurement-based feedback for the task of controlling the state of a single qubit. Here the basic dynamical resource is the ability to couple the system to a traveling-wave field (for example, a transmission line) via a system observable, and the fundamental limitation is the maximum rate that is available for this coupling. We focus on the question of the best achievable control given ideal controllers. To obtain a fair comparison we acknowledge that the amplification involved in measurement-based control allows the controller to use macroscopic fields to apply feedback forces to the system, so it is natural to allow these feedback forces to be much larger than the mesoscopic coupling to the transmission line that mediates both the measurement for measurement-based control and the coupling to the mesoscopic controller for coherent control. Interestingly our numerical results indicate that under the above platform for comparison, coherent feedback is able to exactly match the performance of measurement-based feedback given ideal controllers. We also discuss various properties of, and control mechanisms for, coherent feedback networks.
Here we consider the speed at which quantum information can be transferred between the nodes of a linear network. Because such nodes are linear oscillators, this speed is also important in the cooling and state preparation of mechanical oscillators, as well as frequency conversion. We show that if there is no restriction on the size of the linear coupling between two oscillators, then there exist control protocols that will swap their respective states with high fidelity within a time much less than a single oscillation period. Standard gradient search methods fail to find these fast protocols. We were able to do so by augmenting standard search methods with a path-tracing technique, demonstrating that this technique has remarkable power to solve time-optimal control problems, as well as confirming the highly challenging nature of these problems. As a further demonstration of the power of path-tracing, first introduced by Moore-Tibbets et al. [Phys. Rev. A 86, 062309 (2012)], we apply it to the generation of entanglement in a linear network.
We show that the ability to make direct measurements of momentum, in addition to the usual direct measurements of position, allows a simple configuration of two identical mechanical oscillators to be used for broadband back-action-free force metrology. This would eliminate the need for an optical reference oscillator in the scheme of Tsang and Caves [Phys. Rev. Lett. 105, 123601 (2010)], along with its associated disadvantages. We also show that if one is restricted to position measurements alone then two copies of the same two-oscillator configuration can be used for narrow-band back-action-free force metrology.
We determine the minimum energy required to control the evolution of any mesoscopic quantum system in the presence of arbitrary Markovian noise processes. This result provides the mesoscopic equivalent of the fundamental cost of refrigeration, sets the minimum power consumption of mesoscopic devices that operate out of equilibrium, and allows one to calculate the efficiency of any control protocol, whether it be open-loop or feedback control. As examples we calculate the energy cost of maintaining a qubit in the ground state, the efficiency of resolved-sideband cooling of nano-mechanical resonators, and discuss the energy cost of quantum information processing.
Most methods of optimal control cannot obtain accurate time-optimal protocols. The quantum brachistochrone equation is an exception, and has the potential to provide accurate time-optimal protocols for essentially any quantum control problem. So far this potential has not been realized, however, due to the inadequacy of conventional numerical methods to solve it. Here, using differential geometry, we reformulate the quantum brachistochrone curves as geodesics on the unitary group. With this identification we are able to obtain a numerical method that efficiently solves the brachistochrone problem. We apply it to two examples demonstrating its power.
The control of individual quantum systems is now a reality in a variety of physical settings. Feedback control is an important class of control methods because of its ability to reduce the effects of noise. In this review we give an introductory overview of the various ways in which feedback may be implemented in quantum systems, the theoretical methods that are currently used to treat it, the experiments in which it has been demonstrated to-date, and its applications. In the last few years there has been rapid experimental progress in the ability to realize quantum measurement and control of mesoscopic systems. We expect that the next few years will see further rapid advances in the precision and sophistication of feedback control protocols realized in the laboratory.
Feedback control of quantum systems via continuous measurement involves complex nonlinear dynamics. Except in very special cases, even for a single qubit optimal feedback protocols are unknown. Not even do intuitive candidates exist for choosing the measurement basis, which is the primary non-trivial ingredient in the feedback control of a qubit. Here we present a series of arguments that suggest a particular form for the optimal protocol for a broad class of noise sources in the regime of good control. This regime is defined as that in which the control is strong enough to keep the system close to the desired state. With the assumption of this form the remaining parameters can be determined via a numerical search. The result is a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. We conjecture that this protocol is optimal to leading order in the small parameters that define this regime. The protocol can be described relatively simply, and as a notable feature contains a discontinuity as a function of the feedback strength.
Quantum input-output theory plays a very important role for analyzing the dynamics of quantum systems, especially large-scale quantum networks. As an extension of the input-output formalism of Gardiner and Collet, we develop a new approach based on the quantum version of the Volterra series which can be used to analyze nonlinear quantum input-output dynamics. By this approach, we can ignore the internal dynamics of the quantum input-output system and represent the system dynamics by a series of kernel functions. This approach has the great advantage of modelling weak-nonlinear quantum networks. In our approach, the number of parameters, represented by the kernel functions, used to describe the input-output response of a weak-nonlinear quantum network, increases linearly with the scale of the quantum network, not exponentially as usual. Additionally, our approach can be used to formulate the quantum network with both nonlinear and nonconservative components, e.g., quantum amplifiers, which cannot be modelled by the existing methods, such as the Hudson-Parthasarathy model and the quantum transfer function model. We apply our general method to several examples, including Kerr cavities, optomechanical transducers, and a particular coherent feedback system with a nonlinear component and a quantum amplifier in the feedback loop. This approach provides a powerful way to the modelling and control of nonlinear quantum networks.
We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling --- an example of coherent feedback control --- and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to the fact that both cooling methods are restricted to use only a linear interaction with the resonator. We compare the performance of the two methods on an equal footing --- that is, for the same interaction strength --- and confirm that coherent feedback is able to make much better use of the linear interaction than measurement-based feedback. We find that this performance gap is caused not by the back-action noise of the measurement but by the projection noise. We also obtain simple expressions for the maximal cooling that can be obtained by both methods in this regime, optimized over the interaction strength.
The laws of thermodynamics apply equally well to quantum systems as to classical systems, and because of this quantum effects do not change the fundamental thermodynamic efficiency of isothermal refrigerators or engines. We show that, despite this fact, quantum mechanics permits measurement-based feedback control protocols that are more thermodynamically efficient than their classical counterparts. As part of our analysis we perform a detailed accounting of the thermodynamics of unitary feedback control, and elucidate the sources of inefficiency in measurement-based and coherent feedback.
The existence of a decoherence-free subspace/subsystem (DFS) requires that the noise possesses a symmetry. In this work we consider noise models in which perturbations break this symmetry, so that the DFS for the unperturbed model experiences noise. We ask whether in this case there exist subspaces/subsystems that have less noise than the original DFS. We develop a numerical method to search for such minimal noise subsystems and apply it to a number of examples. For the examples we examine, we find that if the perturbation is local noise then there is no better subspace/subsystem than the original DFS. We also show that if the noise model remains collective, but is perturbed in a way that breaks the symmetry, then the minimal noise subsystem is distinct from the original DFS, and improves upon it.
It is well-known that the nonlinear coupling between a mechanical oscillator and a superconducting oscillator or optical cavity can be used to generate a Kerr-nonlinearity for the cavity mode. We show that the strength of this Kerr-nonlinearity, as well as the effect of the photon-pressure force can be enormously increased by modulating the strength of the nonlinear coupling. We present an electromechanical circuit in which this enhancement can be readily realized.
The subject of controlling quantum systems is not new, but concepts that have been introduced in the last decade and a half, especially that of coherent feedback, suggest new questions that broaden and deepen the field. Here we provide a concise overview and definition of quantum feedback control, both coherent and measurement-based, and discuss its relationship to "standard" time-dependent control; there is a sense in which the latter subsumes the rest. There are many open questions within quantum control and its subfields, and we highlight and discuss some of them here. These questions are of theoretical as well as practical interest: the answers will help to determine the relative power of the different methods of control, and the limits to our ability to control quantum systems imposed by available resources.
In this work, inspired by the study of semidefinite programming for block-diagonalizing matrix *-algebras, we propose an algorithm that can find the algebraic structure of decoherence-free subspaces (DFS's) for a given noisy quantum channel. We prove that this algorithm will work for all cases with probability one, and it is more efficient than the algorithm proposed by Holbrook, Kribs, and Laflamme [Quant. Inf. Proc. 80, 381 (2003)]. In fact, our results reveal that this previous algorithm only works for special cases. As an application, we discuss how this method can be applied to increase the efficiency of an optimization procedure for finding an approximate DFS.
The Wigner function is a useful tool for exploring the transition between quantum and classical dynamics, as well as the behavior of quantum chaotic systems. Evolving the Wigner function for open systems has proved challenging however; a variety of methods have been devised but suffer from being cumbersome and resource intensive. Here we present an efficient fast-Fourier method for evolving the Wigner function, that has a complexity of $O(N\log N)$ where $N$ is the size of the array storing the Wigner function. The efficiency, stability, and simplicity of this method allows us to simulate open system dynamics previously thought to be prohibitively expensive. As a demonstration we simulate the dynamics of both one-particle and two-particle systems under various environmental interactions. For a single particle we also compare the resulting evolution with that of the classical Fokker-Planck and Koopman-von Neumann equations, and show that the environmental interactions induce the quantum-to-classical transition as expected. In the case of two interacting particles we show that an environment interacting with one of the particles leads to the loss of coherence of the other.
We show that when the speed of control is bounded, there is a widely applicable minimal-time control problem for which a coherent feedback protocol is optimal, and is faster than all measurement-based feedback protocols, where the latter are defined in a strict sense. The superiority of the coherent protocol is due to the fact that it can exploit a geodesic path in Hilbert space, a path that measurement-based protocols cannot follow.
We observe that multi-body interactions, unlike two-body interactions, can implement any unitary operation on an encoded system in such a way that the evolution is uninterrupted by noise that the encoding is designed to protect against. Such "error-transparent" evolution is distinct from that usually considered in quantum computing, as the latter is merely correctable. We prove that the minimum body-ness required to protect i) a qubit from a single type of Pauli error, ii) a target qubit from a controller with such errors, iii) a single qubit from all errors, is 3-body, 4-body, and 5-body respectively. We also discuss applications to computing, coherent-feedback control, and quantum metrology. Finally we evaluate the performance of error-transparent evolution for some examples using numerical simulations.
We show that lattice systems, such as the Bose-Hubbard model, can be simulated on a single nano- or micro-mechanical resonator, by exploiting its many modes. The on-site Hamiltonians are engineered by coupling the mechanical modes to the modes of a pair of optical or stripline resonators, and the connections between the lattice sites are engineered in a similar way. The lattice network structure is encoded in the frequency components of the fields driving the resonators. This three-resonator configuration also allows universal quantum computing on the nano-resonator.
Quantum input-output response analysis is a useful method for modeling the dynamics of complex quantum networks, such as those for communication or quantum control via cascade connections. Non-Markovian effects have not yet been studied in such networks. Here we extend the Markovian input-output network formalism developed in optical systems to non-Markovian cascaded networks which can be used, e.g., to analyze the input-output response of mesoscopic quantum networks. We use this formalism to explore the behavior of superconducting qubit networks, where we examine the effect of finite cavity bandwidths. We also discuss its application to open- and closed-loop control networks, and show how these networks create effective Hamiltonians for the controlled system.
The energy cost of measurement is an interesting fundamental question, and may have profound implications for quantum technologies. In the context of Maxwell's demon, it is often stated that measurement has no minimum energy cost, while information has a work value, even though these statements can appear contradictory. However, as we elucidate, these statements do no refer to the cost paid by the measuring device. Here we show that it is only when a measuring device has access to a zero temperature reservoir - that is, never - that the measurement requires no energy. All real measuring devices pay the cost that a heat engine pays to obtain the work value of the information they acquire.
Cooling of a quantum system is limited by the size of the control forces that are available (the "speed" of control). We consider the most general cooling process, albeit restricted to the regime in which the thermodynamics of the system is preserved (weak coupling). Within this regime, we further focus on the most useful control regime, in which a large cooling factor, and good ground-state cooling can be achieved. We present a control protocol for cooling, and give clear structural arguments, as well as strong numerical evidence, that this protocol is globally optimal. From this we obtain simple expressions for the limit to cooling that is imposed by the speed of control.
A central question of dynamics, largely open in the quantum case, is to what extent it erases a system's memory of its initial properties. Here we present a simple statistically solvable quantum model describing this memory loss across an integrability-chaos transition under a perturbation obeying no selection rules. From the perspective of quantum localization-delocalization on the lattice of quantum numbers, we are dealing with a situation where every lattice site is coupled to every other site with the same strength, on average. The model also rigorously justifies a similar set of relationships recently proposed in the context of two short-range-interacting ultracold atoms in a harmonic waveguide. Application of our model to an ensemble of uncorrelated impurities on a rectangular lattice gives good agreement with ab initio numerics.
The recent discovery that for large Hilbert spaces, almost all (that is, typical) Hamiltonians have eigenstates that place small subsystems in thermal equilibrium, has shed much light on the origins of irreversibility and thermalization. Here we give numerical evidence that many-body lattice systems generically approach typicality as the number of subsystems is increased, and thus provide further support for the eigenstate thermalization hypothesis. Our results indicate that the deviation of many-body systems from typicality decreases exponentially with the number of systems. Further, by averaging over a number of randomly-selected nearest-neighbor interactions, we obtain a power-law for the atypicality as a function of the Hilbert space dimension, distinct from the power-law possessed by random Hamiltonians.
We present a theoretical analysis of methods to synthesize entangled states of two superconducting resonators. These methods use experimentally demonstrated interactions of resonators with artificial atoms, and offer efficient routes to generate nonclassical states. We analyze physical implementations, energy level structure, and the effects of decoherence through detailed dynamical simulations.
The present state-of-the-art in cooling mechanical resonators is a version of "sideband" cooling. Here we present a method that uses the same configuration as sideband cooling --- coupling the resonator to be cooled to a second microwave (or optical) auxiliary resonator --- but will cool significantly colder. This is achieved by varying the strength of the coupling between the two resonators over a time on the order of the period of the mechanical resonator. As part of our analysis, we also obtain a method for fast, high-fidelity quantum information-transfer between resonators.
We show that continuous real-time feedback can be used to track, control, and protect a mesoscopic superposition of two spatially separated wave-packets. The feedback protocol is enabled by an approximate state-estimator, and requires two continuous measurements, performed simultaneously. For nanomechanical and superconducting resonators, both measurements can be implemented by coupling the resonators to superconducting qubits.
We present a generalization of continuous position measurements that accounts for a spatially inhomogeneous measurement strength. This describes many real measurement scenarios, in which the rate at which information is extracted about position has itself a spatial profile, and includes measurements that detect if a particle has crossed from one region into another. We show that such measurements can be described, in their averaged behavior, as stochastically fluctuating potentials of vanishing time average. Reasonable constraints restrict the form of the measurement to have degenerate outcomes, which tend to drive the system to spatial superposition states. We present the results of quantum-trajectory simulations for measurements with a step-function profile (a "which-way" measurement) and a Gaussian profile. We find that the particle can coherently reflect from the measurement region in both cases, despite the stochastic nature of the measurement back-action. In addition, we explore the connection to the quantum Zeno effect, where we find that the reflection probability tends to unity as the measurement strength increases. Finally, we discuss two physical realizations of a spatially varying position measurement using atoms.
We show that well-separated quantum superposition states, measurements of strongly nonlinear observables, and quantum dynamics driven by anomalous diffusion can all be achieved for single atoms or molecules by imaging spontaneous photons that they emit via resonance florescence. To generate anomalous diffusion we introduce continuous measurements driven by Lévy processes, and prove a number of results regarding their properties. In particular we present strong evidence that the only stable Lévy density that can realize a strictly continuous measurement is the Gaussian.
Rapid-purification by feedback --- specifically, reducing the mean impurity faster than by measurement alone --- can be achieved by making the eigenbasis of the density matrix to be unbiased relative to the measurement basis. Here we further examine the protocol introduced by Combes and Jacobs [Phys.Rev.Lett. \bf 96, 010504 (2006)] involving continuous measurement of the observable $J_z$ for a $D$-dimensional system. We rigorously re-derive the lower bound $(2/3)(D+1)$ on the achievable speed-up factor, and also an upper bound, namely $D^2/2$, for all feedback protocols that use measurements in unbiased bases. Finally we extend our results to $n$ independent measurements on a register of $n$ qubits, and derive an upper bound on the achievable speed-up factor that scales linearly with $n$.
We present a method to synthesize an arbitrary quantum state of two superconducting resonators. This state-synthesis algorithm utilizes a coherent interaction of each resonator with a tunable artificial atom to create entangled quantum superpositions of photon number (Fock) states in the resonators. We theoretically analyze this approach, showing that it can efficiently synthesize NOON states, with large photon numbers, using existing technology.
Sideband cooling is a technique that potentially allows mechanical resonators to be prepared in their ground states, important for future applications in quantum technologies. Tian has recently shown that side-band cooling can be implemented by modulating the coupling between a nano-resonator and a superconducting oscillator, a process of frequency conversion [L. Tian, PRB 79, 193407 (2009)]. While side-band cooling is usually treated in the steady-state regime, the effective resonant coupling will also generate near perfect state-swapping from the superconductor to the mechanical resonator. We perform numerical simulations of this system, examining the ground-state cooling achieved by the state-swapping. Further, we show that the superconducting oscillator can be used to control the amplitude and phase of the resonator, while simultaneously cooling it, and thus act as a coherent "quantum feedback controller".
There is presently considerable interest in accurately simulating the evolution of open systems for which Markovian master equations fail. Examples are systems that are time-dependent and/or strongly damped. A number of elegant methods have now been devised to do this, but all use a bath consisting of a continuum of harmonic oscillators. While this bath is clearly appropriate for, e.g., systems coupled to the EM field, it is not so clear that it is a good model for generic many-body systems. Here we explore a different approach to exactly simulating open-systems: using a finite bath chosen to have certain key properties of thermalizing many-body systems. To explore the numerical resources required by this method to approximate an open system coupled to an infinite bath, we simulate a weakly damped system and compare to the evolution given by the relevant Markovian master equation. We obtain the Markovian evolution with reasonable accuracy by using an additional averaging procedure, and elucidate how the typicality of the bath generates the correct thermal steady-state via the process of "eigenstate thermalization".
In this paper we consider the problem of tracking the state of a quantum system via a continuous measurement. If the system Hamiltonian is known precisely, this merely requires integrating the appropriate stochastic master equation. However, even a small error in the assumed Hamiltonian can render this approach useless. The natural answer to this problem is to include the parameters of the Hamiltonian as part of the estimation problem, and the full Bayesian solution to this task provides a state-estimate that is robust against uncertainties. However, this approach requires considerable computational overhead. Here we consider a single qubit in which the Hamiltonian contains a single unknown parameter. We show that classical frequency estimation techniques greatly reduce the computational overhead associated with Bayesian estimation and provide accurate estimates for the qubit frequency
Wave-function Monte Carlo methods are an important tool for simulating quantum systems, but the standard method cannot be used to simulate decoherence in continuously measured systems. Here we present a new Monte Carlo method for such systems. This was used to perform the simulations of a continuously measured nano-resonator in [Phys. Rev. Lett. 102, 057208 (2009)].
Braneshop, and
the US National Science Foundation.
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