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The classical Jarzynski equality establishes an exact relation between the stochastic work performed on a system driven out of thermal equilibrium and the free energy difference in a corresponding quasi-static process. This fluctuation theorem bears experimental relevance, as it enables the determination of the free energy difference through the measurement of externally applied work in a nonequilibrium process. In the quantum case, the Jarzynski equality only holds if the measurement procedure of the stochastic work is drastically changed: it is replaced by a so-called two-point measurement (TPM) scheme that requires the knowledge of the initial and final Hamiltonian and therefore lacks the predictive power for the free energy difference that the classical Jarzynski equation is known for. Here, we propose a quantum {{fluctuation theorem}} that is valid for externally measurable quantum work determined during the driving protocol. In contrast to the TPM case, the theorem also applies to open quantum systems and the scenario can be realized without knowing the system Hamiltonian. Our fluctuation theorem comes in the form of an inequality and therefore only yields bounds to the true free energy difference. The inequality is saturated in the quasiclassical case of vanishing energy coherences at the beginning and at the end of the protocol. Thus, there is a clear quantum disadvantage.

We provide a new perspective on quantum dynamical phase transitions (DPTs) by explaining their origin in terms of caustics that form in the Fock space representation of the many-body state over time, using the fully connected transverse field Ising model as an example. In this way we establish a connection between DPTs in a quantum spin system and an everyday natural phenomenon: The dark band between the primary and seconday bows (caustics) in rainbows known as Alexander's dark band. The DPT occurs when the Loschmidt echo crosses the switching line between the evanescent tails of two back-to-back Airy functions that dress neighbouring fold caustics in Fock space and is the time-dependent analogue of what is seen as a function of angle in the sky. The structural stability and universal properties of caustics, as described mathematically by catastrophe theory, explains the generic occurrence of DPTs in the model and suggests that our analysis has wide applicability. Based on our thorough analytical understanding we propose a protocol which can be used to verify the existence of a DPT in a finite system experiment.

Gradient descent algorithms on Riemannian manifolds have been used recently for the optimization of quantum channels. In this contribution, we investigate the influence of various regularization terms added to the cost function of these gradient descent approaches. Motivated by Lasso regularization, we apply penalties for large ranks of the quantum channel, favoring solutions that can be represented by as few Kraus operators as possible. We apply the method to quantum process tomography and a quantum machine learning problem. Suitably regularized models show faster convergence of the optimization as well as better fidelities in the case of process tomography. Applied to quantum classification scenarios, the regularization terms can simplify the classifying quantum channel without degrading the accuracy of the classification, thereby revealing the minimum channel rank needed for the given input data.

We present a fully quantum dynamical treatment of a quantum heat engine and its baths based on the Hierarchy of Pure States (HOPS), an exact and general method for open quantum system dynamics. We show how the change of the bath energy and the interaction energy can be determined within HOPS, for arbitrary coupling strength and smooth time dependence of the modulation protocol. The dynamics of all energetic contributions during the operation can be carefully examined both, in its initial transient phase and also later, in its periodic steady state. A quantum Otto engine with a qubit as inherently nonlinear work medium is studied in a regime where the energy associated with the interaction Hamiltonian plays an important role for the global energy balance and, thus, must not be neglected when calculating its power and efficiency. We confirm that the work required to drive the coupling with the baths depends sensitively on the speed of the modulation protocol. Remarkably, departing from the conventional scheme of well-separated phases by allowing for temporal overlap, we discover that one can even gain energy from the modulation of the bath interactions. We visualize these various work contributions using the analogue of state change diagrams of thermodynamic cycles. We offer a concise, full presentation of HOPS with its extension to bath observables, as it serves as a universal tool for the numerically exact description of general quantum dynamical (thermodynamic) scenarios far from the weak-coupling limit., Comment: 8 figures

In classical stochastic theory, the joint probability distributions of a stochastic process obey by definition the Kolmogorov consistency conditions. Interpreting such a process as a sequence of physical measurements with probabilistic outcomes, these conditions reflect that the measurements do not alter the state of the underlying physical system. Prominently, this assumption has to be abandoned in the context of quantum mechanics, yet there are also classical processes in which measurements influence the measured system. Here, we derive conditions that characterize uniquely classical processes that are probed by a reasonable class of invasive measurements. We then analyse under what circumstances such classical processes can simulate the statistics arising from quantum processes associated with informationally-complete measurements. We expect that our investigation will help build a bridge between two fundamental traits of non-classicality, namely, coherence and contextuality., Comment: 9+10 pages, 3 figures. Comments are welcome!

Non-Markovian processes may arise in physics due to memory effects of environmental degrees of freedom. For quantum non-Markovianity, it is an ongoing debate to clarify whether such memory effects have a verifiable quantum origin, or whether they might equally be modeled by a classical memory. In this contribution, we propose a criterion to test locally for a truly quantum memory. The approach is agnostic with respect to the environment, as it solely depends on the local dynamics of the system of interest. Experimental realizations are particularly easy, as only single-time measurements on the system itself have to be performed. We study memory in a variety of physically motivated examples, both for a time-discrete case, and for time-continuous dynamics. For the latter, we are able to provide an interesting class of non-Markovian master equations with classical memory that allows for a physically measurable quantum trajectory representation.

Approaching the long-time dynamics of non-Markovian open quantum systems presents a challenging task if the bath is strongly coupled. Recent proposals address this problem through a representation of the so-called process tensor in terms of a tensor network. We show that for Gaussian environments highly efficient contraction to matrix product operator (MPO) form can be achieved with infinite MPO evolution methods, leading to significant computational speed-up over existing proposals. The result structurally resembles open system evolution with carefully designed auxiliary degrees of freedom, as in hierarchical or pseudomode methods. Here, however, these degrees of freedom are generated automatically by the MPO evolution algorithm. Moreover, the semi-group form of the resulting propagator enables us to explore steady-state physics, such as phase transitions.

We introduce an exact open system method to describe the dynamics of quantum systems that are strongly coupled to specific types of environments comprising of spins, such as central spin systems. Our theory is similar to the established non-Markovian quantum state diffusion (NMQSD) theory, but for a spin bath instead of a Gaussian bath. The method allows us to represent the time-evolved reduced state of the system as an ensemble average of stochastically evolving pure states. We present a comprehensive theory for arbitrary linear spin environments at both zero and finite temperatures. Furthermore, we introduce a hierarchical expansion method that enables the numerical computation of the time evolution of the stochastic pure states, facilitating a numerical solution of the open system problem in relevant strong coupling regimes.

An important challenge in non-Markovian open quantum systems is to understand what information we gain from continuous measurement of an output field. For example, atoms in multimode cavity QED systems provide an exciting platform to study many-body phenomena in regimes where the atoms are strongly coupled amongst themselves and with the cavity, but the strong coupling makes it complicated to infer the conditioned state of the atoms from the output light. In this work we address this problem, describing the reduced atomic state via a conditioned hierarchy of equations of motion, which provides an exact conditioned reduced description under monitoring (and continuous feedback). We utilise this formalism to study how different monitoring for modes of a multimode cavity affects our information gain for an atomic state, and to improve spin squeezing via measurement and feedback in a strong coupling regime. This work opens opportunities to understand continuous monitoring of non-Markovian open quantum systems, both on a practical and fundamental level.

In this Letter we investigate the concept of quantum work and its measurability from the viewpoint of quantum measurement theory. Very often, quantum work and fluctuation theorems are discussed in the framework of projective two-point measurement (TPM) schemes. According to a well known no-go theorem, there is no work observable which satisfies both (i) an average work condition and (ii) the TPM statistics for diagonal input states. Such projective measurements represent a restrictive class among all possible measurements. It is desirable, both from a theoretical and experimental point of view, to extend the scheme to the general case including suitably designed unsharp measurements. This shifts the focus to the question what information about work and its fluctuations one is able to extract from such generalized measurements. We show that the no-go theorem no longer holds if the observables in a TPM scheme are jointly measurable for any intermediate unitary evolution. We explicitly construct a model with unsharp energy measurements and derive bounds for the visibility that ensure joint measurability. In such an unsharp scenario a single work measurement apparatus can be constructed that allows us to determine the correct average work and to obtain free energy differences with the help of a Jarzynski equality.

We use contemporary quantum computers to experimentally investigate quantum steering ofan open quantum system by measurements on its environment. On three IBMQ processors wedistinguish a qubit as the open system and perform pairwise interactions with multiple environmentancillas, following a collision model approach. Different measurement strategies on the ancillaslead to different state ensembles of the open system, which are reconstructed by employing statetomography. The amount of steering within the resulting assemblages is quantified with the help of asemidefinite program. We successfully observe the presence of quantum steering in our experimentalsimulations, and can discriminate the different performance qualities and noise levels of the selectedquantum devices.

This Article investigates dissipative preparation of entangled non-equilibrium steady states (NESS). We construct a collision model where the open system consists of two qubits which are coupled to heat reservoirs with different temperatures. The baths are modeled by sequences of qubits interacting with the open system. The model can be studied in different dynamical regimes: with and without environmental memory effects. We report that only a certain bath temperature range allows for entangled NESS. Furthermore, we obtain minimal and maximal critical values for the heat current through the system. Surprisingly, quantum memory effects play a crucial role in the long time limit. First, memory effects broaden the parameter region where quantum correlated NESS may be dissipatively prepared and, secondly, they increase the attainable concurrence. Most remarkably, we find a heat current range that does not only allow but guarantees that the NESS is entangled. Thus, the heat current can witness entanglement of non-equilibrium steady states.

Considering two non-interacting qubits in the context of open quantum systems, it is well known that their common environment may act as an entangling agent. In a perturbative regime the influence of the environment on the system dynamics can effectively be described by a unitary and a dissipative contribution. For the two-spin Boson model with (sub-) Ohmic spectral density considered here, the particular unitary contribution (Lamb shift) easily explains the buildup of entanglement between the two qubits. Furthermore it has been argued that in the adiabatic limit, adding the so-called counterterm to the microscopic model compensates the unitary influence of the environment and, thus, inhibits the generation of entanglement. Investigating this assertion is one of the main objectives of the work presented here. Using the hierarchy of pure states (HOPS) method to numerically calculate the exact reduced dynamics, we find and explain that the degree of inhibition crucially depends on the parameter $s$ determining the low frequency power law behavior of the spectral density $J(\omega) \sim \omega^s e^{-\omega/\omega_c}$. Remarkably, we find that for resonant qubits, even in the adiabatic regime (arbitrarily large $\omega_c$), the entanglement dynamics is still influenced by an environmentally induced Hamiltonian interaction. Further, we study the model in detail and present the exact entanglement dynamics for a wide range of coupling strengths, distinguish between resonant and detuned qubits, as well as Ohmic and deep sub-Ohmic environments. Notably, we find that in all cases the asymptotic entanglement does not vanish and conjecture a linear relation between the coupling strength and the asymptotic entanglement measured by means of concurrence. Further we discuss the suitability of various perturbative master equations for obtaining approximate entanglement dynamics., Comment: minor changes due to the reviewing process by the Quantum Journal

In this article we investigate driven dissipative quantum dynamics of an ensemble of two-level systems given by a Markovian master equation with collective and non-collective dissipators. Exploiting the permutation symmetry in our model, we employ a phase space approach for the solution of this equation in terms of a diagonal representation with respect to certain generalized spin coherent states. Remarkably, this allows to interpolate between mean-field theory and finite system size in a formalism independent of Hilbert-space dimension. Moreover, in certain parameter regimes, the evolution equation for the corresponding quasiprobability distribution resembles a Fokker-Planck equation, which can be efficiently solved by stochastic calculus. Then, the dynamics can be seen as classical in the sense that no entanglement between the two-level systems is generated. Our results expose, utilize and promote techniques pioneered in the context of laser theory, which we now apply to problems of current theoretical and experimental interest., Comment: 16 pages, 3 figures

We study dynamical phase transitions (DPT) in the driven and damped Dicke model, realizable for example by a driven atomic ensemble collectively coupled to a damped cavity mode. These DPTs are characterized by non-analyticities of certain observables, primarily the overlap of time evolved and initial state. Even though the dynamics is dissipative, this phenomenon occurs for a wide range of parameters and no fine-tuning is required. Focusing on the state of the 'atoms' in the limit of a bad cavity, we are able to asymptotically evaluate an exact path integral representation of the relevant overlaps. The DPTs then arise by minimization of a certain action function, which is related to the large deviation theory of a classical stochastic process. From a more general viewpoint, in the considered system, non-analyticities emerge generically in a Fock space representation of the state. Finally, we present a scheme which allows a measurement of the DPT in a cavity-QED setup.

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 diffusion. Further, we expand the applicability of non-Markovian quantum jumps and non-Markovian quantum diffusion by using a kernel smoothing technique allowing a significant simplification in their use. Lastly, we demonstrate the applicability of our results by studying a driven dissipative two level atom in a non-Markovian regime using all of the three methods., Comment: 6 pages, 2 figures. Comments are welcome!

We propose a definition of externally measurable quantum work in driven systems. Work is given as a quantum observable on a control device which is forcing the system and can be determined without knowledge of the system Hamiltonian $H_\mathcal{S}$. We argue that quantum work fluctuation theorems which rely on the knowledge of $H_\mathcal{S}$ are of little practical relevance, contrary to their classical counterparts. Using our framework, we derive a fluctuation theorem which is operationally accessible and could in principle be implemented in experiments to determine bounds on free energy differences of unknown systems.

Quantum measurements and the associated state changes are properly described in the language of instruments. We investigate the properties of a time continuous family of instruments associated with the recently introduced family of general Gaussian non-Markovian stochastic Schr\"odinger equations. In this Letter we find that when the covariance matrix for the Gaussian noise satisfies a particular $\delta$-function constraint, the measurement interpretation is possible for a class of models with self-adjoint coupling operator. This class contains, for example the spin-boson and quantum Brownian motion models with colored bath correlation functions. Remarkably, due to quantum memory effects the reduced state, in general, does not have a closed form master equation while the unraveling has a time continuous measurement interpretation., Comment: 5 pages

We present a stochastic matter field equation for an interacting many-body Bose system in equilibrium at ultracold finite temperature. Moreover, the proposed equation can be used for non-equilibrium dynamics on phenomenological grounds. This stochastic differential equation is based on a field phase space representation reflecting the underlying canonical density operator (fixed particle number N). Remarkably, it allows for an efficient numerical implementation. We apply our canonical c-field method to interference experiments with quasi-one dimensional Bose gases. Crucial features of these interference patterns are reproduced very well and also statistical properties in terms of distributions, e.g. for the contrasts, agree well with experimental results., Comment: 14 pages, 7 figures

In the space of quantum channels, we establish the geometry that allows us to make statistical predictions about relative volumes of entanglement breaking channels among all the Gaussian quantum channels. The underlying metric is constructed using the Choi-Jamio{\l}kowski isomorphism between the continuous-variable Gaussian states and channels. This construction involves the Hilbert-Schmidt distance in quantum state space. The volume element of the one-mode Gaussian channels can be expressed in terms of local symplectic invariants. We analytically compute the relative volumes of the one-mode Gaussian entanglement breaking and incompatibility breaking channels. Finally, we show that, when given the purities of the Choi-Jamio{\l}kowski state of the channel, one can determine whether or not such channel is incompatibility breaking.

Trapped ions are among the most promising candidates for performing quantum information processing tasks. Recently, it was demonstrated how the properties of geometric phases can be used to implement an entangling two qubit phase gate with significantly reduced operation time while having a built-in resistance against certain types of errors (Palmero et. al., Phys. Rev. A 95, 022328 (2017)). In this article, we investigate the influence of both quantum and thermal fluctuations on the geometric phase in the Markov regime. We show that additional environmentally induced phases as well as a loss of coherence result from the non-unitary evolution, even at zero temperature. We connect these effects to the associated dynamical and geometrical phases. This suggests a strategy to compensate the detrimental environmental influences and restore some of the properties of the ideal implementation. Our main result is a strategy for zero temperature to construct forces for the geometric phase gate which compensate the dissipative effects and leave the produced phase as well as the final motional state identical to the isolated case. We show that the same strategy helps also at finite temperatures. Furthermore, we examine the effects of dissipation on the fidelity and the robustness of a two qubit phase gate against certain error types., Comment: 14 pages, 6 figures. Comments are welcome!

We address the question of verifying the quantumness of thermal machines. A Szil\'ard engine is truly quantum if its work output cannot be described by a local hidden state (LHS) model, i. e. an objective local statistical ensemble. Quantumness in this scenario is revealed by a steering-type inequality which bounds the classically extractable work. A quantum Maxwell demon can violate that inequality by exploiting quantum correlations between the work medium and the thermal environment. While for a classical Szil\'ard engine an objective description of the medium always exists, any such description can be ruled out by a steering task in a truly quantum case.

The reduced dynamics of an open quantum system obtained from an underlying microscopic Hamiltonian can in general only approximately be described by a time local master equation. The quality of that approximation depends primarily on the coupling strength and the structure of the environment. Various such master equations have been proposed with different aims. Choosing the most suitable one for a specific system is not straight forward. By focusing on the accuracy of the reduced dynamics we provide a thorough assessment for a selection of methods (Redfield Equation, Quantum Optical Master Equation, Coarse-Grained Master Equation, a related dynamical map approach and a partial-secular approximation). Whether or not an approach guarantees positivity we consider secondary, here. We use two qubits coupled to a Lorentzian environment in a spin-boson like fashion modeling a generic situation with various system and bath time scales. We see that, independent of the initial state, the simple Redfield Equation with time dependent coefficients is significantly more accurate than all other methods under consideration. We emphasize that positivity violation in the Redfield formalism becomes relevant only in a regime where any of the perturbative master equations considered here are rendered invalid anyway. This implies that the loss of positivity should in fact be welcomed as an important feature: it indicates the breakdown of the weak coupling assumption. In addition we present the various approaches in a self-contained way and use the behavior of their errors to provide further insight into the range of validity of each method., Comment: minor changes for reasons of a consistent notation

A damped and driven collective spin system is analyzed by using quantum state diffusion. This approach allows for a mostly analytical treatment of the investigated non-equilibrium quantum many body dynamics, which features a phase transition in the thermodynamical limit. The exact results obtained in this work, which are free of any finite size defects, provide a complete understanding of the model. Moreover, the trajectory framework gives an intuitive picture of the two phases occurring, revealing a spontaneously broken symmetry and allowing for a qualitative and quantitative characterization of the phases. We determine exact critical exponents, investigate finite size scaling, and explain a remarkable non-algebraic behaviour at the transition in terms of torus hopping., Comment: 5 pages, 5 figures

We investigate different geometries and invariant measures on the space of mixed Gaussian quan- tum states. We show that when the global purity of the state is held fixed, these measures coincide and it is possible, within this constraint, to define a unique notion of volume on the space of mixed Gaussian states. We then use the so defined measure to study typical non-classical correlations of two mode mixed Gaussian quantum states, in particular entanglement and steerability. We show that under the purity constraint alone, typical values for symplectic invariants can be computed very elegantly, irrespectively of the non-compactness of the underlying state space. Then we consider finite volumes by constraining the purity and energy of the Gaussian state and compute typical values of quantum correlations numerically., Comment: 22 pages, 4 figures

We investigate quantum steering of an open quantum system by measurements on its environment in the framework of collision models. As an example we consider a coherently driven qubit dissipatively coupled to a bath. We construct local non-adaptive and adaptive as well as nonlocal measurement scenarios specifying explicitly the measured observable on the environment. Our approach shows transparently how the conditional evolution of the open system depends on the type of the measurement scenario and the measured observables. These can then be optimized for steering. The nonlocal measurement scenario leads to maximal violation of the used steering inequality at zero temperature. Further, we investigate the robustness of the constructed scenarios against thermal noise. We find generally that steering becomes harder at higher temperatures. Surprisingly, the system can be steered even when bipartite entanglement between the system and individual subenvironments vanishes.

We show that the general and numerically exact Hierarchy of Pure States method (HOPS) is very well applicable to calculate the reduced dynamics of an open quantum system. In particular we focus on environments with a sub-Ohmic spectral density (SD) resulting in an algebraic decay of the bath correlation function (BCF). The universal applicability of HOPS, reaching from weak to strong coupling for zero and non-zero temperature, is demonstrated by solving the spin-boson model for which we find perfect agreement with other methods, each one suitable for a special regime of parameters. The challenges arising in the strong coupling regime are not only reflected in the computational effort needed for the HOPS method to converge but also in the necessity for an importance sampling mechanism, accounted for by the non-linear variant of HOPS. In order to include non-zero temperature effects in the strong coupling regime we found that it is highly favorable for the HOPS method to use the zero temperature BCF and include temperature via a stochastic Hermitian contribution to the system Hamiltonian., Comment: This document is the unedited Author's version of a Submitted Work that was subsequently accepted for publication in the Journal of Chemical Theory and Computation, copyright \c{opyright} American Chemical Society after peer review. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.jctc.7b00751

We derive a family of Gaussian non-Markovian stochastic Schr\"odinger equations for the dynamics of open quantum systems. The different unravelings correspond to different choices of squeezed coherent states, reflecting different measurement schemes on the environment. Consequently, we are able to give a single shot measurement interpretation for the stochastic states and microscopic expressions for the noise correlations of the Gaussian process. By construction, the reduced dynamics of the open system does not depend on the squeezing parameters. They determine the non-hermitian Gaussian correlation, a wide range of which are compatible with the Markov limit. We demonstrate the versatility of our results for quantum information tasks in the non-Markovian regime. In particular, by optimizing the squeezing parameters, we can tailor unravelings for optimal entanglement bounds or for environment-assisted entanglement protection., Comment: 5 pages, 1 figure

We investigate the quantum dynamics of a harmonically trapped particle (e.g. an ion) that is immersed in a Bose--Einstein condensate. The ultracold environment acts as a refrigerator, and thus, the influence on the motion of the ion is dissipative. We study the fully coupled quantum dynamics of particle and Bose gas in a linearized regime, treating the quasi-particle excitations of the gas as a (non-Markovian) environment for the particle dynamics. The density operator of the latter follows a known non-Markovian master equation with a highly non-trivial bath correlation function that we determine and study in detail. The corresponding damping rate and frequency shift of the particle oscillations can be read off. We are able to identify a Quantum Landau criterion for harmonically trapped particles in a superfluid environment: for frequencies $\omega$ well below the chemical potential, the damping rate is strongly suppressed by a power law $\omega^4$. This criterion can be seen as emerging from the classical Landau criterion involving a critical velocity combined with Heisenberg's uncertainty principle for the localized wave packet of the quantum particle. Furthermore, due to the finite size of the Bose gas, after some time we observe memory effects and thus non-Markovian dynamics of the quantum oscillator., Comment: 5 pages, 3 figures, reference added

We present a stochastic projection formalism for the description of quantum dynamics in Bosonic or spin environments. The Schr\"odinger equation in coherent state representation with respect to the environmental degrees of freedom can be reformulated by employing the Feshbach partitioning technique for open quantum systems based on the introduction of suitable non-Hermitian projection operators. In this picture the reduced state of the system can be obtained as a stochastic average over pure state trajectories. The corresponding non-Markovian stochastic Schr\"odinger equations include a memory integral over the past states. In the case of harmonic environments and linear coupling the approach gives a new form of the established non-Markovian quantum state diffusion (NMQSD) stochastic Schr\"odinger equation without functional derivatives. Utilizing spin coherent states, the evolution equation for spin environments resembles the Bosonic case with, however, a non-Gaussian average for the reduced density operator.

The theoretical description of quantum dynamics in an intriguing way does not necessarily imply the underlying dynamics is indeed intriguing. Here we show how a known very interesting master equation with an always negative decay rate [eternal non-Markovianity (ENM)] arises from simple stochastic Schr\"odinger dynamics (random unitary dynamics). Equivalently, it may be seen as arising from a mixture of Markov (semi-group) open system dynamics. Both these approaches lead to a more general family of CPT maps, characterized by a point within a parameter triangle. Our results show how ENM quantum dynamics can be realised easily in the laboratory. Moreover, we find a quantum time-continuously measured (quantum trajectory) realisation of the dynamics of the ENM master equation based on unitary transformations and projective measurements in an extended Hilbert space, guided by a classical Markov process. Furthermore, a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) representation of the dynamics in an extended Hilbert space can be found, with a remarkable property: there is no dynamics in the ancilla state. Finally, analogous constructions for two qubits extend these results from non-CP-divisible to non-P-divisible dynamics., Comment: 14 pages, 2 figures, with supplementary material

We study the applicability of collisional models for non-Markovian dynamics of open quantum systems. By allowing interactions between the separate environmental degrees of freedom in between collisions we are able to construct a collision model that allows to study quantum memory effects in open system dynamics. We also discuss the possibility to embed non-Markovian collision model dynamics into Markovian collision model dynamics in an extended state space. As a concrete example we show how using the proposed class of collision models we can discretely model non- Markovian amplitude damping of a qubit. In the time-continuous limit, we obtain the well-known results for spontaneous decay of a two level system in to a structured zero-temperature reservoir, Comment: 10 pages, 6 figure, comments welcome!

Recent experimental progress using ultracold gases in optical lattices necessitates a quantitative theoretical description for a significant number of bosons. In the present paper, we investigate if time-dependent semiclassical initial value methods, with propagators expressed as integrals over phase space using classical trajectories, is suitable to describe interacting bosons, concentrating on a single mode. Despite the nonlinear contribution from the self-interaction, the corresponding classical dynamics allows for a largely analytical treatment of the semiclassical propagator. We find that the Herman-Kluk (HK) propagator conserves unitarity in the semiclassical limit ($n\to\infty$), but a decay of the norm is seen for low $n$. The frozen Gaussian approximation (FGA, i.e. HK with unit prefactor) is explicitly shown to violate unitarity in the present system for non-vanishing interaction strength, even in the semiclassical limit. Furthermore, we show by evaluating the phase space integral in steepest descent approximation, that the HK propagator reproduces the exact spectrum correctly in the limit $n\to \infty$. An error is, however, incurred in next-to-next-to-leading order (small parameter $1/n$), as seen upon numerical evaluation of the integral or by considering finite $n$ corrections to steepest descent. The FGA, in contrast, is only accurate to lowest order, and an erroneous next-to-leading order term in the energy spectrum was found analytically. Finally, as an example application, we study the dynamics of wave packets by computing the time evolution of the Wigner function. While the often-used truncated Wigner approximation cannot capture any interferences present in the exact quantum mechanical solution (known analytically), we find that the HK approach, despite also using classical information only, reproduces the salient features of the exact solution correctly., Comment: 18 pages, 4 figures; accepted for publication in present form by J. Phys. A: Math. Theor

Surprisingly often decoherence is due to classical fluctuations of ambient fields and may thus be described in terms of random unitary (RU) dynamics. However, there are decoherence channels where such a representation cannot exist. Based on a simple and intuitive geometric measure for the distance of an extremal channel to the convex set of RU channels we are able to characterize the set of true quantum phase-damping channels. Remarkably, using the Caley-Menger determinant, our measure may be assessed directly from the matrix representation of the channel. We find that the channel of maximum quantumness is closely related to a symmetric, informationally-complete positive operator-valued measure (SIC-POVM) on the environment. Our findings are in line with numerical results based on the entanglement of assistance., Comment: 5 pages, 3 figures

We study the Hilbert-Schmidt measure on the manifold of mixed Gaussian states in multi mode continuous variable quantum systems. An analytical expression for the Hilbert-Schmidt volume element is derived. Its corresponding probability measure can be used to study typical properties of Gaussian states. It turns out that although the manifold of Gaussian states is unbounded, an ensemble of Gaussian states distributed according to this measure still has a normalizable distribution of symplectic eigenvalues, from which unitarily invariant properties can be obtained. By contrast, we find that for an ensemble of one-mode Gaussian states based on the Bures measure the corresponding distribution cannot be normalized. As important applications, we determine the distribution and the mean value of von Neumann entropy and purity for the Hilbert-Schmidt measure.

We present novel approaches to the dynamics of an open quantum system coupled linearly to a non-Markovian fermionic or bosonic environment. In the first approach, we obtain a hierarchy of stochastic evolution equations of the diffusion type. For the bosonic case such a hierarchy has been derived and proven suitable for efficient numerical simulations recently [arXiv:1402.4647]. The stochastic fermionic hierarchy derived here contains Grassmannian noise, which makes it difficult to simulate numerically due to its anti-commutative multiplication. Therefore, in our second approach we eliminate the noise by deriving a related hierarchy of density matrices. A similar reformulation of the bosonic hierarchy of pure states to a master equation hierarchy is also presented., Comment: 12 pages

Non-Markovian Quantum State Diffusion (NMQSD) has turned out to be an efficient method to calculate excitonic properties of aggregates composed of organic chromophores, taking into account the coupling of electronic transitions to vibrational modes of the chromophores. NMQSD is an open quantum system approach that incorporates environmental degrees of freedom (the vibrations in our case) in a stochastic way. We show in this paper that for linear optical spectra (absorption, circular dichroism) no stochastics is needed, even for finite temperatures. Thus, the spectra can be obtained by propagating a single trajectory. To this end we map a finite temperature environment to the zero temperature case using the so-called thermofield method. The resulting equations can then be solved efficiently by standard integrators., Comment: 14 pages, 4 figures

We analyse the metric properties of $\textit{conditioned}$ quantum state spaces $\mathcal{M}^{(n\times m)}_{\eta}$. These spaces are the convex sets of $nm \times nm$ density matrices that, when partially traced over $m$ degrees of freedom, respectively yield the given $n\times n$ density matrix $\eta$. For the case $n=2$, the volume of $\mathcal{M}^{(2\times m)}_{\eta}$ equipped with the Hilbert-Schmidt measure is a simple polynomial of the radius of $\eta$ in the Bloch-Ball. Remarkably, the probability $p_{\mathrm{sep}}^{(2\times m)}(\eta)$ to find a separable state in $\mathcal{M}^{(2\times m)}_{\eta}$ is independent of $\eta$ (except for $\eta$ pure). Both these results are proven analytically for the case of the family of $4\times 4$ $X$-states, and thoroughly numerically investigated for the general case. The important implications of these results for the clarification of open problems in quantum theory are pointed out and discussed., Comment: 23 pages, 7 figures

We apply a path integral variational approach to obtain analytical expressions for condensate wave functions of an ultracold, interacting trapped Bose gases. As in many recent experiments, the particles are confined in a 1D or 3D harmonic oscillator trap which is superimposed by a periodic potential in one direction. Based on the first order cumulant expansion with respect to a harmonic trial action, and emplyoing a mean-field approximation, optimal variational parameters are obtained by minimizing an analytical expression for the ground state energy. Our largely analytical results for energy and condensate wave function are in good agreement with fully numerical calculations based on the Gross-Pitaevskii equation.

We study the out-of-equilibrium dynamics of ultracold bosons in a double- and triple-well potential within the Bose-Hubbard model by means of the semiclassical Herman-Kluk propagator and compare the results to the frequently applied "classical dynamics" calculation in terms of the truncated Wigner approximation (TWA). For the double-well system we find the semiclassical results in excellent agreement with the numerically exact ones, while the TWA is not able to reproduce any revivals of the wave function. The triple-well system turns out to be more difficult to handle due to the irregularity of the corresponding classical phase space. Here, deviations of the TWA from the exact dynamics appear even for short times, while better agreement is obtained using the semiclassical approach presented in this article.

We study the dynamics of ultracold Bosons in a double-well potential within the two-mode Bose-Hubbard model by means of semiclassical methods. By applying a WKB quantization we find analytical results for the energy spectrum, which are in excellent agreement with numerical exact results. They are valid in the energy range of plasma oscillations, both in the Rabi and the Josephson regime. Adopting the reflection principle and the Poisson summation formula we derive an analytical expression for the dynamics of the population imbalance depending on the few relevant parameters of the system only. This allows us to discuss its characteristic dynamics, especially the oscillation frequency, and the collapse- and revival time, as a function of the model parameters, leading to a deeper understanding of Josephson physics. We find that our fomulae match previous experimental observations.

We determine the total state dynamics of a dephasing open quantum system using the standard environment of harmonic oscillators. Of particular interest are random unitary approaches to the same reduced dynamics and system-environment correlations in the full model. Concentrating on a model with an at times negative dephasing rate, the issue of "non-Markovianity" will also be addressed. Crucially, given the quantum environment, the appearance of non-Markovian dynamics turns out to be accompanied by a loss of system-environment correlations. Depending on the initial purity of the qubit state, these system-environment correlations may be purely classical over the whole relevant time scale, or there may be intervals of genuine system-environment entanglement. In the latter case, we see no obvious relation between the build-up or decay of these quantum correlations and "Non-Markovianity".

A key lesson of the decoherence program is that information flowing out from an open system is stored in the quantum state of the surroundings. Simultaneously, quantum measurement theory shows that the evolution of any open system when its environment is measured is nonlinear and leads to pure states conditioned on the measurement record. Here we report the discovery of a fundamental relation between measurement and entanglement which is characteristic of this scenario. It takes the form of a scaling law between the amount of entanglement in the conditional state of the system and the probabilities of the experimental outcomes obtained from measuring the state of the environment. Using the scaling, we construct the distribution of entanglement over the ensemble of experimental outcomes for standard models with one open channel and provide rigorous results on finite-time disentanglement in systems coupled to non-Markovian baths. The scaling allows the direct experimental detection and quantification of entanglement in conditional states of a large class of open systems by quantum tomography of the bath., Comment: 12 pages (including supplementary information), 4 figures

Open quantum system dynamics of random unitary type may in principle be fully undone. Closely following the scheme of environment-assisted error correction proposed by Gregoratti and Werner [M. Gregoratti and R. F. Werner, J. Mod. Opt. 50(6), 915-933 (2003)], we explicitly carry out all steps needed to invert a phase-damping error on a single qubit. Furthermore, we extend the scheme to a mixed-state environment. Surprisingly, we find cases for which the uncorrected state is closer to the desired state than any of the corrected ones.

The linear and the nonlinear non-Markovian quantum state diffusion equation (NMQSD) are well known tools for the description of certain non-Markovian open quantum systems. In this work, we systematically investigate whether the normalized linear NMQSD or the nonlinear NMQSD solutions can be generated by means of a time-continuous measurement. By considering any conceivable measurement scheme in the framework of instruments, we derive a necessary criterion for a measurement interpretation of both equations. Concrete examples show that the normalized linear NMQSD solutions are realizable only in the Markovian limit in general. The application of the presented criterion to the nonlinear NMQSD remains an open issue., Comment: 19 pages

We thoroughly investigate vibrational quantum dynamics of dimers attached to He droplets motivated by recent measurements with K-2 [1]. For those femtosecond pump-probe experiments, crucial observed features are not reproduced by gas phase calculations but agreement is found using a description based on dissipative quantum dynamics, as briefly shown in [2]. Here we present a detailed study of the influence of possible effects induced by the droplet. The helium droplet causes electronic decoherence, shifts of potential surfaces, and relaxation of wave packets in attached dimers. Moreover, a realistic description of (stochastic) desorption of dimers off the droplet needs to be taken into account. Step by step we include and study the importance of these effects in our full quantum calculation. This allows us to reproduce and explain all major experimental findings. We find that desorption is fast and occurs already within 2-10 ps after electronic excitation. A further finding is that slow vibrational motion in the ground state can be considered frictionless., Comment: 17 pages, 5 figures