Robust Asymptotic Entanglement under Multipartite Collective Dephasing

Abstract

We derive an analytic solution for the ensemble-averaged collective dephasing dynamics of N noninteracting atoms in a fluctuating homogeneous external field. The obtained Kraus map is used to specify families of states whose entanglement properties are preserved at all times under arbitrary field orientations, even for states undergoing incoherent evolution. Our results apply to arbitrary spectral distributions of the field fluctuations.PACS numbers: 03.65. Yz, 03.65.Ud, 03.67.Mn Control of the coherent evolution of quantum systems in noisy environments [1] is one of the crucial prerequisites for exploiting nontrivial quantum effects in composite systems of increasing complexity. Whether in the context of controlled molecular reactions [2], of many-particle quantum dynamics [3], or of quantum computers and simulators [4], uncontrolled fluctuations and noise are detrimental to most purposes of optimal control. Various strategies may be followed to counteract the harmful influence of the environment: shielding the system degrees of freedom [5], correcting environmentinduced errors [6], exploiting basins of attraction in dissipative systems [8], or compensating dissipation-e.g., by coherent dynamics [7], dynamical decoupling [9], or periodic measurements [10]. Such approaches can effectively reduce the environmental effects and can enhance coherence times, but a perfect protection of the quantities of interest is generally not possible.By restricting to superposition states within a decoherencefree subspace, initially entangled states can be shielded completely from collective noise sources, hence protecting their entanglement at all times [11,12]. However, such subspaces are rather fragile to small perturbations, which limit their applicability in the context of dynamical processes [13]. Here, we identify conditions that ensure complete preservation of arbitrary degrees of entanglement, even for states that are not invariant under an incoherent time evolution. Specifically, we consider an important class of environment-induced fluctuations, which are frequently encountered in state-of-the-art experiments [14][15][16]: they manifest in intensity fluctuations of spatially homogeneous experimental control fields, giving rise to an effective dephasing process. We show how control of the external field's orientation can lead to the complete preservation of entanglement in bipartite-as well as multipartitesettings, for arbitrary spectral characteristics of the control field fluctuations. We further identify families of states exhibiting time-invariant entanglement for arbitrary orientations of the external field.To set the stage, let us consider a collection of N noninteracting atomic two-level systems with identical energy splitting ω controlled, e.g., by a homogeneous magnetic field. Integration over the unavoidable fluctuations of the latter's strength will induce a probability distribution p(ω) of the characteristic energy splitting, and the N-atom quantum state at time t therefore needs to be described by the st...

Generating and protecting correlated quantum states under collective dephasing

“…Those obtained from the |Y ñ  states are then categorized as 'robust', while those from |F ñ  are 'fragile'. In fact, as we have shown above, only the Werner state (59) is robust under collective dephasing in any field direction, while the other Werner-like (64) states are robust only when a specific direction is chosen [15]. The preservation of entanglement within decoherence-free subspaces [46][47][48]-which in the present case, due to the absence of a Hamiltonian evolution, coincide with the above-mentioned time-invariant subspacesis a rather obvious phenomenon: if the state does not evolve in time, then its properties are naturally also conserved.…”

“…Characterizing the intensity fluctuations with the probability distribution ( ) w p , the collective dephasing dynamics is described by In [15], the above integral was solved analytically without further assumptions, and in the following we will recall the resulting solution and some of its properties.…”

“…In [15] the map (22) was shown to conserve the trace of the β matrix for bipartite systems, as defined in (5). This integral of motion can, in fact, be understood as the manifestation of the more general conservation of angular momentum in the special case of N=2.…”

“…For example, the generalized Werner states, i.e., those states that are invariant under Ä U N -operations [18,50], are the fixed points of the collective dephasing map for any number of qubits [15]. In [15], the decay and the time-invariant preservation of multipartite entanglement properties was also provided. A compelling explanation for the mechanism that enables this phenomenon in a multipartite case is, however, presently unavailable.…”

Cooperative Effects in Closely Packed Quantum Emitters with Collective Dephasing

“…There, particular attention was paid to the existence and robustness of so called decoherence free sub-spaces (DFS) under collective dephasing [25]. Moreover, recent studies [28,29] have shown that collective dephasing could also be used as a resource to generate strong but separable correlations. We note that, due to their promise as a general passive strategy to protect quantum resources from noise, the study of DFS continues to be an active area of research, see [30] for a recent review on the theoretical aspects and [31] for experimental implementations.…”