K. Kowalski, J. Rembielinski
It is demonstrated that in contrast to the well-known case with a quantum particle moving freely in a real line, the wave packets corresponding to the coherent states for a free quantum particle on a circle do not spread but develop periodically in time. The discontinuous changes during the course of time in the phase representing the position of a particle can be interpreted as the quantum jumps on a circle.
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We show how a quantum walk can be implemented for the first time in a quantum quincunx created via superconducting circuit quantum electrodynamics (QED), and how interpolation from quantum to random walk is implemented by controllable decoherence using a two resonator system. Direct control over the coin qubit is difficult to achieve in either cavity or circuit QED, but we show that a Hadamard coin flip can be effected via direct driving of the cavity, with the result that the walker jumps between circles in phase space but still exhibits quantum walk behavior over 15 steps.
We introduce jumptime unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jumptime-averaged quantum st…
This work deals with both instantaneous uniform mixing property and temporal standard deviation for continuous-time quantum random walks on circles in order to study their fluctuations comparing with discrete-time quantum random walks, and continuous- and discrete-time classical random walks.
The dynamics of a non-Markovian open quantum system described by a general time-local master equation is studied. The propagation of the density operator is constructed in terms of two processes: (i) deterministic evolution and (ii) evolution of a probability density functional in the projective Hilbert space. The analysis provides a derivation for the jump probabilities used in the recently developed non-Markovian quantum jump (NMQJ) method (Piilo et al 2008 Phys. Rev. Lett. 100 180402).
The recently introduced detected-jump correcting quantum codes are capable of stabilizing qubit-systems against spontaneous decay processes arising from couplings to statistically independent reservoirs. These embedded quantum codes exploit classical information about which qubit has emitted spontaneously and correspond to an active error-correcting code embedded in a passive error-correcting code. The construction of a family of one detected jump-error correcting quantum codes is shown and the optimal redundancy, encoding and recovery as well as general properties of detected jump-error corre…
We propose a variation of the quantum walk on a circle in phase space by conjoining the Hadamard coin flip with simultaneous displacement of the walker's location in phase space and show that this generalization is a proper quantum walk albeit over multiple concentric circles in phase instead of just over one circle. We motivate the conjoining of Hadamard and displacement operations by showing that the Jaynes-Cummings model for coin+walker approximately yields this description in the dispersive limit. The quantum walk signature is evident in the phase distribution of the walker provided that a…
The uncertainty relations for the position and momentum of a quantum particle on a circle are identified minimized by the corresponding coherent states. The sqeezed states in the case of the circular motion are introduced and discussed in the context of the uncertainty relations.
We report the observation of quantum jumps between macroscopic quantum states in a superconducting phase qubit coupled to the two-level systems in the Josephson tunnel junction, and all key features of quantum jumps are confirmed in the experiments. Moreover, quantum jumps can be used to calibrate such two-level systems, which are believed to be one of the main decoherence sources in Josephson devices.
A brief analysis of the compatibility between the quantum jump and the Markov property for quantum systems described by a stochastic evolution scheme is presented.
Effect of quantum jumps on non-Hermitian system
本文通过将量子跃迁视为微扰，推导有效哈密顿量并展开本征态与本征能量，研究了量子跃迁对非厄米系统动力学及非平衡相变的影响，并应用于耗散二能级系统与费米子超流体。
Quantum mechanics on a circle with a finite number of α-uniformly distributed points
本文研究了具有有限个α-均匀分布点的圆上的量子力学，定义了角度和平移算符，利用离散角度表示构造了两类离散角动量算符和厄米哈密顿量，并计算了自由粒子的能级。
The quantum harmonic oscillator on a circle -- fragmentation of the algebraic method
研究圆圈上二次势量子粒子的能谱，发现其非谐振子特性但保留代数性质，探讨代数方法失效的原因及其物理意义。
The Dirac method is used to analyze the classical and quantum dynamics of a particle constrained on a circle. The method of Lagrange multipliers is scrutinized, in particular in relation to the quantization procedure. Ordering problems are tackled and solved by requiring the hermiticity of some operators. The presence of an additional term in the quantum Hamiltonian is discussed.
Backaction evasion measurements of a quadrature component of the light field vacuum necessarily induce quantum jumps in the photon number. The correlation between measurement results and quantum jump events reveals fundamental nonclassical aspects of quantization.
Metrical separability of the spin systems energetic surfaces is shown. The irreversibility of the spin system evolution is considered as the consequence of random quantum jumps on energy surfaces.
Quantum random walks (QRWs) can be used to perform both quantum simulations and quantum algorithms. In order to exploit this potential, quantum walks on different types of graphs must be physically implemented. To this end we design, model and experimentally fabricate, using the femtosecond laser direct-write technique, a 3D tubular waveguide array within glass to implement a photonic quantum walk on a circle. The boundary conditions of a QRW on a circle naturally suggests a 3D waveguide implementation - allowing much simpler device design than what could be achieved using a 2D waveguide archi…
We first use the quantum method to replicate the well-known results of a single atom relaxing, whilst demonstrating the intuitive picture it provides for dissipative dynamics. By use of individual "quantum trajectories", the method allows for simulation of systems inaccessible to ensemble treatments. This is shown by replicating resonance fluorescence, allowing us to concurrently demonstrate the method's facilitation of calculating photon statistics by the creation of discrete photon streams. To analyse these, we solidify the theoretical basis for, and implement, a computational method of calc…
The coherent states for the quantum particle on the circle are introduced. The Bargmann representation within the actual treatment provides the representation of the algebra $[\hat J,U]=U$, where $U$ is unitary, which is a direct consequence of the Heisenberg algebra $[\hat ϕ, \hat J]=i$, but it is more adequate for the study of the circlular motion.
In the preceding Comment (quant-ph/0209032) Trifonov disputes our recently proposed uncertainty relations for a quantum particle on a circle. He states that (i) the quantity $Δ^2(\hatϕ)$ introduced by us representing the uncertainty of the angle is not a proper measure of the position uncertainty and therefore the proposed inequalities can hardly be qualified as a relevant uncertainty relations on a circle; and that (ii) the most suitable uncertainty relations on a circle are those based on the Gram-Robertson matrix. It is shown by examples that both points are erroneous.
We discuss the quantum jump operation in an open system, and show that jump super-operators related to a system under measurement can be derived from the interaction of that system with a quantum measurement apparatus. We give two examples for the interaction of a monochromatic electromagnetic field in a cavity (the system) with 2-level atoms and with a harmonic oscillator (representing two different kinds of detectors). We show that derived quantum jump super-operators have `nonlinear' form which depends on assumptions made about the interaction between the system and the detector. A continuo…
Temporal fluctuations in the Hadamard walk on circles are studied. A temporal standard deviation of probability that a quantum random walker is positive at a given site is introduced to manifest striking differences between quantum and classical random walks. An analytical expression of the temporal standard deviation on a circle with odd sites is shown and its asymptotic behavior is considered for large system size. In contrast with classical random walks, the temporal fluctuation of quantum random walks depends on the position and initial conditions, since temporal standard deviation of the …
With microwave irradiation, the switching current of a Josephson junction coupled to a microscopic two-level system jumps randomly between two discrete states. We modeled the switching process of the coupled system with quantum jump approach that was generally used in quantum optics. The parameters that affect the character of the quantum jumps between macroscopic quantum states are discussed. The results obtained from our theoretical analysis agree well with those of the experiments and provide a clear physical picture for the macroscopic quantum jumps in Josephson junctions coupled with two-…
We present a method to estimate the time scale of quantum jumps from the time correlation of photon pairs generated from a cascade decay in an atomic system, and realize it experimentally in a cold cloud of 87Rb. Taking into account the photodetector response, we find an upper bound for the duration of a quantum jump of 21+/-11 ps.
Deterministic Quantum Jump (DQJ) Method for Weakly Dissipative Systems
DQJ方法通过消除随机采样误差，在弱耗散系统中更高效地模拟量子跃迁。
It was recently argued [Phys. Rev. Lett 108, 220402 (2012)] that the stochastic dynamics of an open quantum system are not inherent to the system, but rather depend on the existence and nature of a distant detector. The proposed experimental tests involved homodyne detection, giving rise to quantum diffusion, and required total efficiencies of well over 50%. Here we prove that for no system is it possible to demonstrate detector-dependence using diffusive-type detection of efficiency less than 50%. However, this no-go theorem does not apply to quantum jumps, and we propose a test involving a q…
Open quantum systems that interact with structured reservoirs exhibit non-Markovian dynamics. We present a quantum jump method for treating the dynamics of such systems. This approach is a generalization of the standard Monte Carlo Wave Function (MCWF) method for Markovian dynamics. The MCWF method identifies decay rates with jump probabilities and fails for non-Markovian systems where the time-dependent rates become temporarily negative. Our non-Markovian quantum jump (NMQJ) approach circumvents this problem and provides an efficient unravelling of the ensemble dynamics.
It is shown by examples that the position uncertainty on a circle, proposed recently by Kowalski and Rembieliński [J. Phys. A 35 (2002) 1405] is not consistent with the state localization. We argue that the relevant uncertainties and uncertainty relations (UR's) on a circle are that based on the Gram-Robertson matrix. Several of these generalized UR's are displayed and related criterions for squeezed states are discussed.
The quantum channel capacity gives the ultimate limit for the rate at which quantum data can be reliably transmitted through a noisy quantum channel. Degradable quantum channels are among the few channels whose quantum capacities are known. Given the quantum capacity of a degradable channel, it remains challenging to find a practical coding scheme which approaches capacity. Here we discuss code designs for the detected-jump channel, a degradable channel with practical relevance describing the physics of spontaneous decay of atoms with detected photon emission. We show that this channel can be …
We show that direct feedback based on quantum jump detection can be used to generate entangled steady states. We present a strategy that is insensitive to detection inefficiencies and robust against errors in the control Hamiltonian. This feedback procedure is also shown to overcome spontaneous emission effects by stabilising states with high degree of entanglement.
Non-Markovian Quantum Jump Method for Driven-Dissipative Two-Level Systems
本文提出了一种改进的非马尔可夫量子跃迁方法，用于高效研究驱动耗散双态系统的动态行为，通过减少计算资源消耗并观察到相干与纠缠的复兴效应。
Resonance fluorescence, consisting of light emission from an atom driven by a classical oscillating field, is well-known to yield a sub-Poissonian photon counting statistics. This occurs when only emitted light is detected, which corresponds to a master equation (ME) unraveling in terms of the canonical jump operator describing spontaneous decay. Formally, an alternative ME unraveling is possible in terms of a shifted jump operator. We show that this shift can result in sub-Poissonian, Poissonian or super-Poissonian quantum jump statistics. This is shown in terms of the Mandel Q parameter in t…
We propose a simple model which describes the statistical properties of quantum jumps in a single-spin measurement using the oscillating cantilever-driven adiabatic reversals technique in magnetic resonance force microscopy. Our computer simulations based on this model predict the average time interval between two consecutive quantum jumps and the correlation time to be proportional to the characteristic time of the magnetic noise and inversely proportional to the square of the magnetic noise amplitude.
We theoretically study the occurrence of quantum jumps in the resonance fluorescence of a trapped atom. Here, the atom is laser cooled in a configuration of level such that the occurrence of a quantum jump is associated to a change of the vibrational center-of-mass motion by one phonon. The statistics of the occurrence of the dark fluorescence period is studied as a function of the physical parameters and the corresponding features in the spectrum of resonance fluorescence are identified. We discuss the information which can be extracted on the atomic motion from the observation of a quantum j…
Photon emission without quantum jumps
本文提出通过局部哈密顿量求解薛定谔方程来描述无量子跃迁的光子发射，替代传统量子跃迁模型，适用于远场干涉效应的量子光学系统。
We define the hitting time for a model of continuous-time open quantum walks in terms of quantum jumps. Our starting point is a master equation in Lindblad form, which can be taken as the quantum analogue of the rate equation for a classical continuous-time Markov chain. The quantum jump method is well known in the quantum optics community and has also been applied to simulate open quantum walks in discrete time. This method however, is well-suited to continuous-time problems. It is shown here that a continuous-time hitting problem is amenable to analysis via quantum jumps: The hitting time ca…
We argue that the participants of the discussion have overlooked an essential circumstance, in view of which Stapp's fifth proposition fails. The circumstance is that though $L$ and $R$ measurements, being causally separated, are not invariantly time ordered, quantum-jump hypersurfaces associated with the measurements are causelikewise ordered. Stapp's fifth proposition is true iff $L$ hypersurface precedes $R$ one; but within the limits of special relativity, it is impossible to determine the causelike order of those hypersurfaces. The entire Stapp's construct is revised.
We find that existing multi-party quantum key agreement (MQKA) protocols designed for fairness of the key are, in fact, unfair. Our analysis shows that these protocols are sensitive to collusive attacks; that is, dishonest participants can collaborate to predetermine the key without being detected. In fact, the transmission structures of the quantum particles in those unfair MQKA protocols, three of which have already been analyzed, have much in common. We call these unfair MQKA protocols circle-type MQKA protocols. Likewise, the transmission structures of the quantum particles in MQKA protoco…
Developing the analysis of the distribution of the particle's position-momentum dot product, the so-called \textit{posmom} $\mathbf{x}% \cdot \mathbf{p}$\textbf{,} to quantum states on a circular circle on two-dimensional Cartesian coordinates, we give its posmometry (introduced recently by Y. A. Bernard and P. M. W. Gill, Posmom: The Unobserved Observable, J. Phys. Chem. Lett. 1\textbf{(}2010\textbf{)}1254) for eigenstates of the free motion on the circle, i.e., $z$-axis component of the angular momentum. The posmom has two parity symmetries, specifically, invariant under two operations $m_{x…
We study the quantum-jump-based feedback control on the entanglement shared between two qubits with one of them subject to decoherence, while the other qubit is under the control. This situation is very relevant to a quantum system consisting of nuclear and electron spins in solid states. The possibility to prolong the coherence time of the dissipative qubit is also explored. Numerical simulations show that the quantum-jump-based feedback control can improve the entanglement between the qubits and prolong the coherence time for the qubit subject directly to decoherence.
We may infer a transition $|n \rangle \to |m \rangle$ between energy eigenstates of an open quantum system by observing the emission of a photon of Bohr frequency $ω_{mn} = (E_n-E_m) / \hbar$. In addition to the "collapses" to the state $|m\rangle$, the measurement must also have brought into existence the pre-measurement state $|n \rangle$. As quantum trajectories are based on past observations, the condition state will jump to $| m \rangle$, but the state $|n\rangle$ does not feature in any essential way. We resolve this paradox by looking at quantum smoothing and derive the time-symmetric m…
It is shown that symmetric configurations of fuzzy spin direction detectors generate, through quantum jumps, IFS fractals on the sphere S^2. The IFS fractals can be also interpreted as resulting from applications of Lorentz boosts to the projective light cone.
Topological extension including quantum jump
本文从拓扑视角研究了具有集体损耗和增益的Su-Schrieffer-Heeger模型，定性分析了主方程中被忽略的量子跃迁项对相变位置的影响，揭示了其在量子系统中的独特作用。
A method for deriving accurate analytic approximations for Markovian open quantum systems was recently introduced in [F. Lucas and K. Hornberger, Phys. Rev. Lett. 110, 240401 (2013)]. Here, we present a detailed derivation of the underlying non-perturbative jump expansion, which involves an adaptive resummation to ensure optimal convergence. Applying this to a set of exemplary master equations, we find that the resummation typically leads to convergence within the lowest two to five orders. Besides facilitating analytic approximations, the optimal jump expansion thus provides a numerical schem…
We solve two long standing problems for stochastic descriptions of open quantum system dynamics. First, we find the classical stochastic processes corresponding to non-Markovian quantum state diffusion and non-Markovian quantum jumps in projective Hilbert space. Second, we show that the diffusive limit of non-Markovian quantum jumps can be taken on the projective Hilbert space in such a way that it coincides with non-Markovian quantum state diffusion. However, the very same limit taken on the Hilbert space leads to a completely new diffusive unraveling, which we call non-Markovian quantum diff…
We describe the absorption by the walls of a quantum electrodynamics cavity as a process during which the elementary excitations (photons) of an internal mode of the cavity exit by tunneling through the cavity walls. We estimate by classical methods the survival time of a photon inside the cavity and the quality factor of its mirrors.
It has been recently proved that a quantum jump may be reversed by a unitary process provided the initial state is restricted by some conditions. The application of such processes for preventing decoherence, for example in quantum computers, was suggested. We shall show that in the situation when the quantum jump is reversible it supplies no information about the initial state additional to the information known beforehand. Therefore the reversibility of this type does not contradict the general statement of quantum measurement theory: a measurement cannot be reversed. As a consequence of this…
A universal set of quantum gates is constructed for the recently developed jump-error correcting quantum codes. These quantum codes are capable of correcting errors arising from the spontaneous decay of distinguishable qubits into statistically independent reservoirs. The proposed universal quantum gates are constructed with the help of Heisenberg- and Ising-type Hamiltonians acting on these physical qubits. This way it is guaranteed that the relevant error correcting code space is not left at any time even during the application of one of these quantum gates. The proposed entanglement gate is…
Quantum metrology exploits quantum correlations in specially prepared entangled or other non-classical states to perform measurements that exceed the standard quantum limit. Typically though, such states are hard to engineer, particularly when larger numbers of resources are desired. As an alternative, this paper aims to establish quantum jump metrology which is based on generalised sequential measurements as a general design principle for quantum metrology and discusses how to exploit open quantum systems to obtain a quantum enhancement. By analysing a simple toy model, we illustrate that par…
Two possible applications of random decoupling are discussed. Whereas so far decoupling methods have been considered merely for quantum memories, here it is demonstrated that random decoupling is also a convenient tool for stabilizing quantum algorithms. Furthermore, a decoupling scheme is presented which involves a random decoupling method compatible with detected-jump error correcting quantum codes. With this combined error correcting strategy it is possible to stabilize quantum information against both spontaneous decay and static imperfections of a qubit-based quantum information processor…
Quantum physics was invented to account for two fundamental features of measurement results -- their discreetness and randomness. Emblematic of these features is Bohr's idea of quantum jumps between two discrete energy levels of an atom. Experimentally, quantum jumps were first observed in an atomic ion driven by a weak deterministic force while under strong continuous energy measurement. The times at which the discontinuous jump transitions occur are reputed to be fundamentally unpredictable. Can there be, despite the indeterminism of quantum physics, a possibility to know if a quantum jump i…
Multi-level quantum systems loose coherence due to quantum jumps or spontaneous decay between their internal levels. Here we propose a way to simulate experimentally a three-level system under quantum jump using a three-mode photonic system. We simulated three different dynamics of spontaneous decay in a three-level atomic system: cascade decay, $Λ$ decay and $V$ decay. With an attenuated light coherent source at the photon level, we prepared a photonic qutrit state encoded in the parallel path of Gaussian modes. By exploring periodical phase modulation in spatial light modulator, the correspo…
Current fluctuations in a non-additive open quantum system: breakdown of the quantum-jump approach
本文研究非加性开放量子系统中电流涨落问题，评估量子跳变方法的适用性，发现完美跳变检测下该方法失效，不完美检测时可复现Landauer-Büttiker形式化结果。
Dynamical quantum jumps were initially conceived by Bohr as objective events associated with the emission of a light quantum by an atom. Since the early 1990s they have come to be understood as being associated rather with the detection of a photon by a measurement device, and that different detection schemes result in different types of jumps (or diffusion). Here we propose experimental tests to rigorously prove the detector-dependence of the stochastic evolution of an individual atom. The tests involve no special preparation of the atom or field, and the required efficiency can be as low as …
Harmonic oscillators with multiple abrupt jumps in their frequencies have been investigated by several authors during the last decades. We investigate the dynamics of a quantum harmonic oscillator with initial frequency $ω_0$, that undergoes a sudden jump to a frequency $ω_1$ and, after a certain time interval, suddenly returns to its initial frequency. Using the Lewis-Riesenfeld method of dynamical invariants, we present expressions for the mean energy value, the mean number of excitations, and the transition probabilities, considering the initial state different from the fundamental. We show…
We discuss in detail how non-Markovian open system dynamics can be described in terms of quantum jumps [J. Piilo et al., Phys. Rev. Lett. 100, 180402 (2008)]. Our results demonstrate that it is possible to have a jump description contained in the physical Hilbert space of the reduced system. The developed non-Markovian quantum jump (NMQJ) approach is a generalization of the Markovian Monte Carlo Wave Function (MCWF) method into the non-Markovian regime. The method conserves both the probabilities in the density matrix and the norms of the state vectors exactly, and sheds new light on non-Marko…
We propose an implementation of a quantum walk on a circle on an optomechanical system by encoding the walker on the phase space of a radiation field and the coin on a two-level state of a mechanical resonator. The dynamics of the system is obtained by applying Suzuki-Trotter decomposition. We numerically show that the system displays typical behaviors of quantum walks, namely, the probability distribution evolves ballistically and the standard deviation of the phase distribution is linearly proportional to the number of steps. We also analyze the effects of decoherence by using the phase damp…
It is problematic to interpret the quantum jumps of an atom interacting with thermal light in terms of counts at detectors monitoring the atom's inputs and outputs. As an alternative, we develop an interpretation based on a self-consistency argument. We include one mode of the thermal field in the system Hamiltonian and describe its interaction with the atom by an entangled quantum state while assuming that the other modes induce quantum jumps in the usual fashion. In the weak-coupling limit, the photon number expectation of the selected mode is also seen to execute quantum jumps, although mor…
The Monte Carlo wave function method or the quantum trajectory/jump approach is a powerful tool to study dissipative dynamics governed by the Markovian master equation, in particular for high-dimensional systems and when it is difficult to simulate directly. In this paper, we extend this method to the non-Markovian case described by the generalized Lindblad master equation. Two examples to illustrate the method are presented and discussed. The results show that the method can correctly reproduce the dissipative dynamics for the system. The difference between this method and the traditional Mar…
Tracing flows of information in our quantum Universe explains why we see the world as classical.
Fermionic Insights into Measurement-Based Quantum Computation: Circle Graph States Are Not Universal Resources
本文探讨了测量基量子计算中圆图状态作为通用资源的有效性，通过图论方法证明圆图状态无法高效实现通用量子计算，为后续研究提供新视角。
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