Using the principles of the ETH - Approach to Quantum Mechanics we study fluorescence and the phenomenon of ``quantum jumps'' in idealized models of atoms coupled to the quantized electromagnetic field. In a limiting regime where the orbital motion of the atoms is neglected and the velocity of light tends to infinity we derive explicit non-linear stochastic differential equations describing the effective time evolution of states of individual atoms. These equations give rise to a measure on state-trajectories with quantum jumps which is a quantum-mechanical analogue of the Wiener measure of Brownian motion. Our results amount to a derivation of the fundamental randomness in the quantum-mechanical description of microscopic systems from basic principles in the context of some simple models.
In quantum mechanical experiments one distinguishes between the state of an experimental system and an observable measured in it. Heuristically, the distinction between states and observables is also suggested in scattering theory or when one expresses causality. We explain how this distinction can be made also mathematically. The result is a theory with asymmetric time evolution and for which decaying states are exactly unified with resonances. A consequence of the asymmetric time evolution is a beginning of time. The meaning of this beginning of time can be understood by identifying it in da…
We describe a nano-electromechnical system that exhibits the "retroactive" quantum jumps discovered by Mabuchi and Wiseman [Phys. Rev. Lett. 81, 4620 (1998)]. This system consists of a Cooper-pair box coupled to a nano-mechanical resonator, in which the latter is continuously monitored by a single-electron transistor or quantum point-contact. Further, we show that these kinds of jumps, and the jumps that emerge in a continuous quantum non-demolition measurement are one and the same phenomena. We also consider manipulating the jumps by applying feedback control to the Cooper-pair box.
We describe an optomechanical system in which the mean phonon number of a single mechanical mode conditionally displaces the amplitude of the optical field. Using homodyne detection of the output field we establish the conditions under which phonon number quantum jumps can be inferred from the measurement record: both the cavity damping rate and the measurement rate of the phonon number must be much greater than the thermalization rate of the mechanical mode. We present simulations of the conditional dynamics of the measured system using the stochastic master equation. In the good-measurement …
Small quantum systems can now be continuously monitored experimentally which allows for the reconstruction of quantum trajectories. A peculiar feature of these trajectories is the emergence of jumps between the eigenstates of the observable which is measured. Using the Stochastic Master Equation (SME) formalism for continuous quantum measurements, we show that the density matrix of a system indeed shows a jumpy behavior when it is subjected to a tight measurement (even if the noise in the SME is Gaussian). We are able to compute the jump rates analytically for any system evolution, i.e. any Li…
The dynamics of the next quantum jump for a qubit [two level system] coupled to a readout resonator [damped driven harmonic oscillator] is calculated. A quantum mechanical treatment of readout resonator reveals non exponential short time behavior which could facilitate detection of the state of the qubit faster than the resonator lifetime.
We consider the time-inhomogeneous Markovian jump process introduced by John S. Bell [Phys.Rep. 137, 49] for a lattice quantum field theory, which runs on the associated configuration space. Its jump rates, tailored to give the process the quantum distribution $|Ψ_t|^2$ at all times $t$, typically exhibit singularities. We establish the existence of a unique such process for all times, under suitable assumptions on the Hamiltonian or the initial state vector $Ψ_0$. The proof of non-explosion takes advantage of the special role of the $|Ψ_t|^2$ distribution.
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…
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).
It is shown that in spite of a generally accepted concept, there exist nondiffeomorphic solutions to the Cauchy problem in nonempty spacetime, which implies the necessity for canonical complementary conditions. It is nonlocal quantum jumps that provide a canonical global structure of spacetime manifold and, by the same token, the canonical complementary conditions.
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 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.
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.
The universe, as a closed system, is for all time in a state with a determinate value of energy, i.e., in an eigenstate of the Hamiltonian. That is the principle of cosmic energy determinacy. The Hamiltonian depends on cosmic time through metric. Therefore there are confluence and branch points of energy levels. At branch points, quantum jumps must happen to prevent the violation of energy determinacy. Thus quantum jumps are a reaction against the propensity of the universe dynamics to that violation. On the basis of this idea, an internally consistent quantum jump dynamics is developed.
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.
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…
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…
Time-evolution of open, dissipative quantum system is a stochastic process that consists of a series of quantum jumps that take place at random times. Between quantum jumps quantum system idles for some time in a particular quantum state. Based on superfermion/superboson formalism and general Kossakowski- Lindblad master equation for an open quantum system we develop a method to compute distribution of waiting times between quantum jumps. We illustrate the use of the theory by computing waiting time distribution for model Fermi-system in thermodynamic equilibrium.
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.
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.
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.
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 study the dynamics of a driven non-Hermitian superconducting qubit which is perturbed by quantum jumps between energy levels, a purely quantum effect with no classical correspondence. The quantum jumps mix the qubit states leading to decoherence. We observe that this decoherence rate is enhanced near the exceptional point, owing to the cube-root topology of the non-Hermitian eigenenergies. Together with the effect of non-Hermitian gain/loss, quantum jumps can also lead to a breakdown of adiabatic evolution under the slow-driving limit. Our study shows the critical role of quantum jumps in g…
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…
This paper is a proposal for the generation of many-body entangled state in atomic and mechanical systems. Application of strong Rydberg dressing interaction and fast bifurcation scheme in an ultra-stable Cs lattice clock in this proposal results to a breakthrough in the formation of large cat states. By detailed study of the de-coherence effects using Quantum Jump Monte Carlo approach and considering obstacles in dense ensembles namely level mixing, molecule formation and collective decoherence this proposal predicts the formation of 700 atoms cat state. Dipole coupling between Rydberg atoms …
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.
Recently, spin-selective radical pair reactions have been studied using concepts from quantum measurement theory. In this Article, we show that the approach taken by Kominis (Physical Review E, 83, 2011, 056118) leads to erroneous results due to a problematic treatment of quantum jumps. Correct consideration of quantum jumps leads to either the traditional master equation or the Jones-Hore master equation.
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 study an open quantum system of atoms with long-range Rydberg interaction, laser driving, and spontaneous emission. Over time, the system occasionally jumps between a state of low Rydberg population and a state of high Rydberg population. The jumps are inherently collective and in fact exist only for a large number of atoms. We explain how entanglement and quantum measurement enable the jumps, which are otherwise classically forbidden.
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 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.
This paper is a sequel to the series of papers [gr-qc/9409010, gr-qc/9505034, gr-qc/9603022, gr-qc/9609035, gr-qc/9609046, gr-qc/9704033, gr-qc/9704038, gr-qc/9708014, gr-qc/9802016]. The problem of the meaning of objective a priori probability for individual random trials without repetition is considered. A sequence of such trials, namely quantum jumps, is realized in indeterministic dynamics of the universe. A hidden selector for the quantum jumps is constructed.
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.
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…
We consider Markovian open quantum systems subject to stochastic resetting, which means that the dissipative time evolution is reset at randomly distributed times to the initial state. We show that the ensuing dynamics is non-Markovian and has the form of a generalized Lindblad equation. Interestingly, the statistics of quantum-jumps can be exactly derived. This is achieved by combining techniques from the thermodynamics of quantum-jump trajectories with the renewal structure of the resetting dynamics. We consider as an application of our analysis a driven two-level and an intermittent three-l…
For classical nonequilibrium systems, the separation of the total entropy production into the adiabatic and nonadiabatic contributions is useful for understanding irreversibility in nonequilibrium thermodynamics. In this article, we formulate quantum analogs for driven open quantum systems describable by quantum jump trajectories by applying a quantum stochastic thermodynamics. Our main conclusions are based on a quantum formulation of the local detailed balance condition.
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…
This brief note in response to Wojciech Zurek's article "Quantum Darwinism, classical reality, and the randomness of quantum jumps" (Physics Today, October 2014, page 44) points out extant rebuttals in the literature to some of the author's key claims.
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…
We show the continuity property of spectral gaps and complete Logarithmic constants in terms of the jump operators of Lindblad generators in finite dimensional setting. Our method is based on the bimodule structure of the derivation space and the technique developed in [Paulsen09]. Using the same trick, we also show the continuity of the $g^2(0)$ constant used to distinguish quantum and classical lights in quantum optics.
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 …
