Energies and collapse times of symmetric and symmetry-breaking states of finite systems with a U(1) symmetry

We study quantum systems of volume V, which will exhibit the breaking of a U(1) symmetry in the limit of V \to \infty, when V is large but finite. We estimate the energy difference between the `symmetric ground state' (SGS), which is the lowest-energy state that does not breaks the symmetry, and a `pure phase vacuum' (PPV), which approaches a symmetry-breaking vacuum as V \to \infty. Under some natural postulates on the energy of the SGS, it is shown that PPVs always have a higher energy than the SGS, and we derive a lower bound of the excess energy. We argue that the lower bound is O(V^0), which becomes much larger than the excitation energies of low-lying excited states for a large V. We also discuss the collapse time of PPVs for interacting many bosons. It is shown that the wave function collapses in a microscopic time scale, because PPVs are not energy eigenstates. We show, however, that for PPVs the expectation value of any observable, which is a finite polynomial of boson operators and their derivatives, does not collapse for a macroscopic time scale. In this sense, the collapse time of PPVs is macroscopically long.Comment: In the revised manuscript, Eq. (22), Ref. [8], and Notes [13], [15] and [17] have been adde

Appearance and Stability of Anomalously Fluctuating States in Shor's Factoring Algorithm

We analyze quantum computers which perform Shor's factoring algorithm, paying attention to asymptotic properties as the number L of qubits is increased. Using numerical simulations and a general theory of the stabilities of many-body quantum states, we show the following: Anomalously fluctuating states (AFSs), which have anomalously large fluctuations of additive operators, appear in various stages of the computation. For large L, they decohere at anomalously great rates by weak noises that simulate noises in real systems. Decoherence of some of the AFSs is fatal to the results of the computation, whereas decoherence of some of the other AFSs does not have strong influence on the results of the computation. When such a crucial AFS decoheres, the probability of getting the correct computational result is reduced approximately proportional to L^2. The reduction thus becomes anomalously large with increasing L, even when the coupling constant to the noise is rather small. Therefore, quantum computations should be improved in such a way that all AFSs appearing in the algorithms do not decohere at such great rates in the existing noises.Comment: 11 figures. A few discussions were added in verion 2. Version 3 is the SAME as version 2; only errors during the Web-upload were fixed. Version 4 is the publised version, in which several typos are fixed and the reference list is update

Superconducting pairing of spin polarons in the t - J model

Consiglio Nazionale delle Ricerche (CNR). Biblioteca Centrale / CNR - Consiglio Nazionale delle RichercheSIGLEITItal