We investigate the quantum walk on the line when decoherences are introduced either through simultaneous measurements of the chirality and particle position, or as a result of broken links. Both mechanisms drive the system to a classical diffusive behavior. In the case of measurements, we show that the diffusion coefficient is proportional to the variance of the initially localized quantum random walker just before the first measurement. When links between neighboring sites are randomly broken with probability p per unit time, the evolution becomes decoherent after a characteristic time that scales as 1/p. The fact that the quadratic increase of the variance is eventually lost even for very small frequencies of disrupting events, suggests that the implementation of a quantum walk on a real physical system may be severely limited by thermal noise and lattice imperfections.

We analyze in detail the discrete-time quantum walk on the line by separating the quantum evolution equation into Markovian and interference terms. As a result of this separation, it is possible to show analytically that the quadratic increase in the variance of the quantum walker's position with time is a direct consequence of the coherence of the quantum evolution. If the evolution is decoherent, as in the classical case, the variance is shown to increase linearly with time, as expected. Furthermore we show that this system has an evolution operator analogous to that of a resonant quantum kicked rotor. As this rotator may be described through a quantum computational algorithm, one may employ this algorithm to describe the time evolution of the quantum walker.

In this article we explore the quantum properties of a degenerate optical parametric oscillator when it is tuned to the first family of transverse modes at the down-converted frequency. Recently we found [C. Navarrete-Benlloch et al., Phys. Rev. Lett. 100, 203601 (2008)] that above threshold a TEM 10 mode following a random rotation in the transverse plane emerges in this system (we denote it as the bright mode), breaking thus its rotational invariance. Then, owing to the mode orientation being undetermined, we showed that the phase quadrature of the transverse mode orthogonal to this one (denoted as the dark mode) is perfectly squeezed at any pump level and without an increase in the fluctuations on its amplitude quadrature (which seems to contradict the uncertainty principle). In this article we go further in the study of this system and analyze some important features not considered previously. First we show that the apparent violation of the uncertainty principle is just that-"apparent"-as the conjugate pair of the squeezed quadrature is not another quadrature but the orientation of the bright mode (which is completely undetermined in the long term). We also study a homodyne scheme in which the local oscillator is not perfectly matched to the dark mode, as this could be impossible in real experiments due to the random rotation of the mode, showing that even in this case large levels of noise reduction can be obtained (also including the experimentally unavoidable phase fluctuations). Finally, we show that neither the adiabatic elimination of the pump variables nor the linearization of the quantum equations are responsible for the remarkable properties of the dark mode (which we prove analytically and through numerical simulations, respectively), which were simplifying assumptions used in Navarrete-Benlloch et al. [Phys. Rev. Lett. 100, 203601 (2008)]. These studies show that the production of noncritically squeezed light through spontaneous rotational symmetry breaking is a robust phenomenon.

The conditional shift in the evolution operator of a quantum walk generates entanglement between the coin and position degrees of freedom. This entanglement can be quantified by the von Neumann entropy of the reduced density operator (entropy of entanglement). In the long time limit, it converges to a well defined value which depends on the initial state. Exact expressions for the asymptotic (long-time) entanglement are obtained for (i) localized initial conditions and (ii) initial conditions in the position subspace spanned by | ± 1 .

No abstract

We generalize the discrete quantum walk on the line using a time dependent unitary coin operator. We find an analytical relation between the long-time behaviors of the standard deviation and the coin operator. Selecting the coin time sequence allows to obtain a variety of predetermined asymptotic wave-function spreadings: ballistic, sub-ballistic, diffusive, sub-diffusive and localized.

The asymptotic behavior of the quantum walk on the line is investigated focusing on the probability distribution of chirality independently of position. It is shown analytically that this distribution has a long-time limit that is stationary and depends on the initial conditions. This result is unexpected in the context of the unitary evolution of the quantum walk, as it is usually linked to a Markovian process. The asymptotic value of the entanglement between the coin and the position is determined by the chirality distribution. For given asymptotic values of both the entanglement and the chirality distribution it is possible to find the corresponding initial conditions within a particular class of spatially extended Gaussian distributions.

We study the evolution of initially extended distributions in the coined quantum walk on the line by analyzing the dispersion relation of the process and its associated wave equations. This allows us, in particular, to devise an initially extended condition leading to a uniform probability distribution whose width increases linearly with time, with increasing homogeneity. PACS numbers: 03.67.-a, 42.30.KqIntroduction.-The discrete, or coined, quantum walk (QW) [1] is a process originally introduced as the quantum counterpart of the classical random walk (RW). In both cases there is a walker and a coin: at every time step the coin is tossed and the walker moves depending on the toss output. In the RW the walker moves to the right or to the left, while in the QW, as the walker and coin are quantum in nature, coherent superpositions right/left and head/tail happen. This feature endows the QW with outstanding properties, such as making the standard deviation of the position of an initially localized walker grow linearly with time t, unlike the RW in which this growth goes with t 1/2 . This has strong consequences in algorithmics and is one of the reasons why QWs are receiving so much attention from the past decade. However the relevance of QW's is being recognized to go beyond this specific arena and, for example, some simple generalizations of the standard QW have shown unsuspected connections with phenomena such as Anderson localization [2] and quantum chaos [3,4]. Moreover, theoretical and experimental studies evidence that the QW finds applications in outstanding systems, such as Bose-Einstein condensates [5], atoms in optical lattices [6,7], trapped ions [8,9], or optical devices [10][11][12][13][14], just to mention a few. This enhances the relevance of the QW as it can constitute a means for controlling the performance of those systems. Apart from the discrete QW we consider here, continuous versions exist as well [15], whose relationship with the coined QW has been discussed in [16].