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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.

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 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.

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 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.

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.