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arXiv:1407.2075

Quantum phase transition in a dissipative two-qubit system

Hang Zheng, Zhiguo Lü, Yang Zhao

Published Jul 8, 2014• 1 citation
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Abstract

By means of a unitary transformation, we study quantum phase transitions in the ground state of a two-qubit system interacting with a dissipative reservoir. First, the ground state phase diagram is analyzed in the presence of the Ohmic and sub-Ohmic bath using an analytic ground state wave function which takes into account the competition between intrasite tunneling and intersite correlation. The quantum critical point is determined as the transition point from non-degenerate to degenerate ground state and our calculated critical coupling strength $\alpha_c$ agrees with that from the numerical renormalization group method. Moreover, by computing the entanglement entropy between the qubits and the bath in the ground state, we explore the nature of the quantum phase transition between the delocalized and localized states.

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By means of a unitary transformation, we study quantum phase transitions in the ground state of a two-qubit system interacting with a dissipative reservoir. First, the ground state phase diagram is analyzed in the presence of the Ohmic and sub-Ohmic bath using an analytic ground state wave function which takes into account the competition between intrasite tunneling and intersite correlation. The quantum critical point is determined as the transition point from non-degenerate to degenerate ground state and our calculated critical coupling strength αc\alpha_cαc​ agrees with that from the numerical renormalization group method. Moreover, by computing the entanglement entropy between the qubits and the bath in the ground state, we explore the nature of the quantum phase transition between the delocalized and localized states.
By means of a unitary transformation, we propose an ansatz to study quantum phase transitions in the ground state of a two-qubit system interacting with a dissipative reservoir. First, the ground-state phase diagram is analyzed in the presence of the Ohmic and sub-Ohmic bath using an analytic ground-state wave function that takes into account the competition between intrasite tunneling and intersite correlation. The quantum critical point is determined as the transition point from a nondegenerate to a degenerate ground state, and our calculated critical coupling strength α(c) agrees with that from the numerical renormalization-group method. Moreover, by computing the entanglement entropy between the qubits and the bath as well as the qubit-qubit correlation function in the ground state, we explore the nature of the quantum phase transition between the delocalized and localized states.
The ground state and the spectral structure of lower-lying excited states of a dissipative two-level system coupled to a sub-Ohmic bath (s = 1/2) with nonzero bias have been studied using the unitary transformation method. By calculating the ground-state entanglement entropy, the ground-state average of σ(z)(G), and the static susceptibility of the two-level system, we explore the nature of the transition (crossover) between the delocalized and localized state of the two-level system. Furthermore, we calculate the time-dependent expectation (σ(z)(t)) and the time evolution of the entanglement entropy to show that, when the system undergoes a transition (crossover) from the delocalized to the localized state, the time evolution of the two-level system changes from coherent to decoherent dynamics.
The decoherence of a two-state system coupled with a sub-Ohmic bath is investigated theoretically by means of the perturbation approach based on a unitary transformation. It is shown that the decoherence depends strongly and sensitively on the structure of environment. Nonadiabatic effect is treated through the introduction of a function ${\ensuremath{\xi}}_{k}$ which depends on the boson frequency and renormalized tunneling. The results are as follows: (i) the nonequilibrium correlation function P(t)P(t)P(t), the dynamical susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$, and the equilibrium correlation function C(t)C(t)C(t) are analytically obtained for $s\ensuremath{\leqslant}1$; (ii) the phase diagram of thermodynamic transition shows the delocalized-localized transition point ${\ensuremath{\alpha}}_{l}$ which agrees with exact results and numerical data from the numerical renormalization group; (iii) the dynamical transition point ${\ensuremath{\alpha}}_{c}$ between coherent and incoherent phase is explicitly given. A crossover from the coherent oscillation to incoherent relaxation appears with increasing coupling (for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the coherent dynamics disappear); (iv) the Shiba's relation and sum rule are exactly satisfied when $\ensuremath{\alpha}\ensuremath{\leqslant}{\ensuremath{\alpha}}_{c}$; (v) an underdamping-overdamping transition point ${\ensuremath{\alpha}}_{c}^{*}$ exists in the function $S(\ensuremath{\omega})$. Consequently, the dynamical phase diagrams in both ohmic and sub-Ohmic case are mapped out. For $\ensuremath{\Delta}⪡{\ensuremath{\omega}}_{c}$, the critical couplings (${\ensuremath{\alpha}}_{l},{\ensuremath{\alpha}}_{c}$, and ${\ensuremath{\alpha}}_{c}^{*}$) are proportional to ${\ensuremath{\Delta}}^{1\ensuremath{-}s}$.
By employing the spin-boson model in a dense limit of environmental modes, quantum entanglement and correlation of sub-Ohmic and Ohmic baths for dissipative quantum phase transitions are numerically investigated based on the variational principle. With several measures borrowed from quantum information theory, three different types of singularities are found for the first-order, second-order, and Kosterlitz-Thouless phase transitions, respectively, and the values of transition points and critical exponents are accurately determined. Besides, the scaling form of the quantum discord in the Ohmic case is identified, quite different from that in the sub-Ohmic regime. In a two-spin model, two distinct behaviors of the quantum discord are uncovered: one is related to the quantum entanglement between two spins and the other is decided by the correlation function in the position space rather than the entanglement.
We investigate the behavior of an NNN-component quantum rotor coupled to a bosonic dissipative bath having a sub-Ohmic spectral density $J(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$ with s<1s<1s<1. With increasing dissipation strength, this system undergoes a quantum phase transition from a delocalized phase to a localized phase. We determine the exact critical behavior of this transition in the large-NNN limit. For 1>s>1/21>s>1/21>s>1/2, we find nontrivial critical behavior corresponding to an interacting renormalization group fixed point, while we find mean-field behavior for s<1/2s<1/2s<1/2. The results agree with those of the corresponding long-range interacting classical model. The quantum-to-classical mapping is therefore valid for the sub-Ohmic rotor model.
Quantum phase transitions (QPTs) in the spin-boson model with/without the rotating-wave approximation (RWA) are systematically investigated through variational calculations using a sub-Ohmic bath with high spectral density. Four cases involving different system-environment interactions are examined, where transition points and critical exponents are accurately determined across varying tunneling strengths. Contrary to prior work, a rich phase diagram is revealed in the tunneling-coupling plane even at the low spectral exponent s<1/2s<1/2s<1/2, with a novel U(1)-symmetric phase being identified. As coupling increases, a multi-stage QPT sequence arises for the tunneling 0<Δ<Δ∗=0.074(1)0<Δ< Δ^*=0.074(1)0<Δ<Δ∗=0.074(1), whereas a single transition occurs beyond this range. Furthermore, an odd-parity phase is found to emerge even under the positive tunneling, exhibiting distinct characteristics relative to the prototype model.