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arXiv:quant-ph/9708042 · Physical Review A

Dissipative dynamics in a quantum register

Paolo Zanardi A

Published Dec 1, 1997• 54 citations
arXivHTMLPDF

Abstract

A model for a quantum register dissipatively coupled with a bosonic thermal bath is studied. The register consists of $N$ quantum bits (qubits) the bath is described by ${N}_{b}$ bosonic modes. The register-bath coupling is chosen in such a way that the total number of excitations is conserved. The Hilbert space splits, allowing the study of the dynamics separately in each sector. Assuming that the coupling with the bath is the same for all qubits, the excitation sectors have a further decomposition according the irreducible representations of the su(2) spin algebra. The stability against environment-generated noise of the information encoded in a quantum state of the register depends on its SU(2) symmetry content. At zero temperature we find that states belonging to the vacuum symmetry sector have, for long time, vanishing fidelity, whereas each lowest spin vector is decoupled from the bath and therefore is decoherence free. Numerical results are shown in the one-excitation space in the case of qubit-dependent bath-system coupling.

DOI
10.1103/physreva.56.4445

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A model for a quantum register dissipatively coupled with a bosonic thermal bath is studied. The register consists of NNN quantum bits (qubits) the bath is described by Nb{N}_{b}Nb​ bosonic modes. The register-bath coupling is chosen in such a way that the total number of excitations is conserved. The Hilbert space splits, allowing the study of the dynamics separately in each sector. Assuming that the coupling with the bath is the same for all qubits, the excitation sectors have a further decomposition according the irreducible representations of the su(2) spin algebra. The stability against environment-generated noise of the information encoded in a quantum state of the register depends on its SU(2) symmetry content. At zero temperature we find that states belonging to the vacuum symmetry sector have, for long time, vanishing fidelity, whereas each lowest spin vector is decoupled from the bath and therefore is decoherence free. Numerical results are shown in the one-excitation space in the case of qubit-dependent bath-system coupling.
A model for a quantum register R\mathcal{R}R made of NNN replicas of a ddd-dimensional quantum system (cell) coupled with the environment is studied by means of a Born-Markov master equation (ME). Dissipation and decoherence are discussed in various cases in which a subdecoherent encoding can be rigorously found. For the quantum-bit case (d=2)(d=2)(d=2) we have solved, for small N,N,N, the ME by numerical direct integration and studied, as a function of the coherence length ${\ensuremath{\xi}}_{c}$ of the bath, fidelity and decoherence rates of states of the register. For large enough ${\ensuremath{\xi}}_{c}$ the singlet states of the global su(2) pseudo spin algebra of the register (noiseless at ${\ensuremath{\xi}}_{c}=\ensuremath{\infty})$ are shown to have a much smaller decoherence rates than the rest of the Hilbert space.
We investigate the effect of dissipation in a Bose-Josephson junction (BJJ) coupled to baths of bosons at its two sites. Apart from the dynamical transition due to repulsive interactions, the BJJ undergoes a quantum phase transition by increasing the coupling strength with the bath modes. We analyze this system by mapping to an equivalent spin model coupled to the bosonic modes. The excitation energies and fluctuation of number imbalance are obtained within a Holstein-Primakoff approximation, which exhibits vanishing of the energy gap and enhanced quantum fluctuations at the critical point. We study the dynamics of BJJ using a time-dependent variational method and analyze stability of different types of steady states. As a special case we study in detail the phase space dynamics of BJJ coupled to a single mode, which reveals diffusive and incoherent behavior with increasing coupling to the bath mode. The dynamical steady states corresponding to the π oscillation and self-trapped state become unstable when their oscillation frequencies are in resonance with the bath modes. We study the Josephson dynamics in the presence of an Ohmic bath with Gaussian noise to incorporate the thermal fluctuations and obtain the Josephson oscillation frequency and damping analytically. We also observe the transition to the symmetry-broken state for strong coupling as well as decay of π oscillation and a self-trapped state to the ground state due to dissipation. Variation of the phase fluctuation with temperature of the bath shows similar behavior as observed in experiment. Finally we discuss the experimental setup to study the observable effects of dissipation in BJJ.
The dynamical evolution of a quantum register of arbitrary length coupled to an environment of arbitrary coherence length is predicted within a relevant model of decoherence. The results are reported for quantum bits (qubits) coupling individually to different environments (``independent decoherence'') and qubits interacting collectively with the same reservoir (``collective decoherence''). In both cases, explicit decoherence functions are derived for any number of qubits. The decay of the coherences of the register is shown to strongly depend on the input states: We show that this sensitivity is a characteristic of both types of coupling (collective and independent) and not only of the collective coupling, as has been reported previously. A nontrivial behavior (``recoherence'') is found in the decay of the off-diagonal elements of the reduced density matrix in the specific situation of independent decoherence. Our results lead to the identification of decoherence-free states in the collective decoherence limit. These states belong to subspaces of the system's Hilbert space that do not get entangled with the environment, making them ideal elements for the engineering of ``noiseless'' quantum codes. We also discuss the relations between decoherence of the quantum register and computational complexity based on the dynamical results obtained for the register density matrix.
In order to eliminate disturbing effects of decoherence, encoding of quantum information in decoherence-free subspaces has been suggested. We analyze the benefits of this concept for a quantum register that is realized in a spin chain in contact with a common bosonic bath. Within a dissipationless model, we provide explicit analytical results for the average fidelity of plain and encoded quantum registers. For the investigation of dissipative spin-boson couplings, we employ a master equation of Bloch-Redfield type.
We investigate the effect of dissipation in a Bose Josephson junction (BJJ) coupled to bath of bosons at two sites as an application of the generalized spin-boson model. Apart from the dynamical transition due to repulsive interactions, the BJJ undergoes a quantum phase transition by increasing the coupling strength with the bath modes. We analyze this system by mapping to an equivalent spin model coupled to the bosonic modes. The excitation energies and fluctuation of number imbalance are obtained within Holstein-Primakoff approximation, which exhibits vanishing of energy gap and enhanced quantum fluctuations at the critical coupling. We study the dynamics of BJJ using time-dependent variational method and analyze stability of different types of steady states. As a special case, we study in details the phase space dynamics of BJJ coupled to a single-mode, which reveals diffusive and incoherent behaviour with increasing coupling to the bath mode. The dynamical steady states corresponding to the Pi-oscillation and self-trapped state become unstable in the region where their oscillation frequencies are in resonance with the bath modes. We investigate the time evolution of number imbalance and relative phase in presence of Ohmic bath with Gaussian noise to incorporate thermal fluctuations. Apart from the damping of Josephson oscillations and transition to symmetry broken state for strong coupling, we observe the decay of Pi-oscillation and self-trapped state to the ground state as a result of dissipation. Variation of phase fluctuation with temperature of the bath shows similar behaviour as observed in the experiments.
In this paper we study a model quantum register RRR made of NNN replicas (cells) of a given finite-dimensional quantum system S. Assuming that all cells are coupled with a common environment with equal strength we show that, for NNN large enough, in the Hilbert space of RRR there exists a linear subspace CN{C}_{N}CN​ which is dynamically decoupled from the environment. The states in CN{C}_{N}CN​ evolve unitarily and are therefore decoherence-dissipation free. The space CN{C}_{N}CN​ realizes a noiseless quantum code in which information can be stored, in principle, for an arbitrarily long time without being affected by errors.