Dynamics of decoherence: Universal scaling of the decoherence factor

Sei Suzuki, Tanay Nag, and Amit Dutta

Phys. Rev. A 93, 012112 – Published 13 January 2016

Abstract

We study the time dependence of the decoherence factor (DF) of a qubit globally coupled to an environmental spin system (ESS) which is driven across the quantum critical point (QCP) by varying a parameter of its Hamiltonian in time $t$ as $1 - t / τ$ or $- t / τ$ , to which the qubit is coupled starting at the time $t \to - \infty$ ; here $τ$ denotes the inverse quenching rate. In the limit of weak coupling we analyze the time evolution of the DF in the vicinity of the QCP (chosen to be at $t = 0$ ) and define three quantities, namely, the generalized fidelity susceptibility $χ_{F} (τ)$ (defined right at the QCP), and the decay constants $α_{1} (τ)$ and $α_{2} (τ)$ which dictate the decay of the DF at a small but finite $t (> 0)$ . Using a dimensional analysis argument based on the Kibble-Zurek healing length, we show that $χ_{F} (τ)$ as well as $α_{1} (τ)$ and $α_{2} (τ)$ indeed satisfy universal power-law scaling relations with $τ$ and the exponents are solely determined by the spatial dimensionality of the ESS and the exponents associated with its QCP. Remarkably, using the numerical t-DMRG method, these scaling relations are shown to be valid in both the situations when the ESS is integrable and nonintegrable and also for both linear and nonlinear variation of the parameter. Furthermore, when an integrable ESS is quenched far away from the QCP, there is a predominant Gaussian decay of the DF with a decay constant which also satisfies a universal scaling relation.

Received 18 September 2015

DOI:https://doi.org/10.1103/PhysRevA.93.012112

Physics Subject Headings (PhySH)

Open quantum systems & decoherence

Critical phenomena

Quantum Information, Science & Technology

Authors & Affiliations

Sei Suzuki¹, Tanay Nag², and Amit Dutta²

¹Department of Liberal Arts, Saitama Medical University, Moroyama, Saitama 350-0495, Japan
²Department of Physics, Indian Institute of Technology, Kanpur 208 016, India

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Vol. 93, Iss. 1 — January 2016

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Images

Figure 1
The schematic phase diagram of the transverse XY chain given in Eq. (1) in the $(h - γ)$ plane with different quenching paths denoted by arrows. The Ising transition line at $h = \pm 1$ for arbitrary $γ$ and the anisotropic transition line at $γ = 0$ (with $| h | < 1$ ) meet at two multicritical points (MCPs) $h = \pm 1$ and $γ = 0$ . When the transverse field $h = 1 - t / τ$ is quenched (e.g., setting $γ = 1$ ) with $t$ from a large positive value to a value close to the QCP $(t > 0)$ , one finds $χ_{F} (τ) \sim τ^{1 / 2}, α_{1} (τ) \sim τ^{0}$ , and $α_{2} (τ) \sim τ^{- 1 / 2}$ in the limit of small $δ$ ; identical scaling relations are obtained when $γ$ is quenched across the anisotropic critical line. Finally, when $γ$ is quenched linearly across the MCP with $h = 1$ , one finds $χ_{F} (τ) \sim τ^{1 / 3}, α_{1} (τ) \sim τ^{- 1 / 3}$ , and $α_{2} \sim τ^{- 1}$ . As shown in the text, all these scaling relations are predicted by a universal scaling form given in Eq. (4).
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Figure 2
Time evolution of the decoherence factor for the transverse Ising chain (TIC) as obtained using the t-DMRG method is plotted as a function of time. The transverse field is driven as $h = 1 - t / τ$ . The scaling of $χ_{F} (τ), α_{1} (τ)$ , and $α_{2} (τ)$ match nicely with the theoretically predicted scaling relations given in Eq. (6). Here we have $- (1 / L) ln D (t)$ , with dimension of 1/length and $t$ has dimension of time. This is true for Figs. 4 and 5.
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Figure 3
(a) Scaling of $χ_{F}, α_{1}$ , and $α_{2}$ with respect to $τ$ for the TIC [Eq. (1) with $J_{y} = 0]$ with a nonlinear ramp $(r = 2)$ of $h$ . We find a very good agreement with analytically predicted scaling in Eq. (7). (b) Scalings of $χ_{L}, α_{1}$ , and $α_{2}$ for the quadratic ramp $(r = 2)$ of the longitudinal field in the nonintegrable TIC given in (8) with $h = 1$ ; the numerical results as obtained using t-DMRG are in excellent agreement with the predictions as in Eq. (10).
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Figure 4
Early time evolution of the decoherence factor for the nonintegrable TIC in the longitudinal field (8). The transverse field is fixed at $h = 1$ and the longitudinal field is ramped as $h_{L} = - t / τ$ . Inset shows scaling of $χ_{L}, α_{1}$ , and $α_{2}$ . Results are in good agreement with the prediction (9).
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Figure 5
(a) Early time evolution of the decoherence factor for quenching the parameter $γ$ as $- t / τ$ of the Hamiltonian (1) with the transverse field fixed at $h = 1$ (see Fig. 1); the inset shows scaling of $χ_{L}, α_{1}$ , and $α_{2}$ . Results are in good agreement with the prediction in Eq. (A24). (b) Numerical results confirm the scaling for the nonlinear case given in Eq. (A25).
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