Zurek spin-bath model ( $H = - Z_{D} \otimes \sum_{k} g_{k} Z_{k}$ )

A device qubit couples diagonally to a finite bath of spins. Conditional bath states acquire opposite phase rotations, and their conditional environment overlap multiplies the device coherence. An initially aligned bath leaves the overlap's modulus at one. Initially equatorial spins yield

z_{N} (t) = \prod_{k} cos (2 g_{k} t)

. Global unitary time evolution remains reversible, and a finite spin-bath coherence recurrence prevents a strict zero long-time limit at finite bath size.

Table of contents
- Random-coupling spin-bath decoherence Zurek spin-bath model
  - Short-time Gaussian spin-bath decoherence Random-coupling spin-bath decoherence
  - Ensemble spin-bath coherence Random-coupling spin-bath decoherence
- Finite spin-bath coherence recurrence Zurek spin-bath model

Random-coupling spin-bath decoherence

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Independent random couplings suppress typical coherence as the bath size increases. This conclusion concerns an ensemble or a specified large-bath limit, and must be distinguished from a pointwise long-time limit for one finite realization. Ensemble spin-bath coherence supplies exact mean and mean-square formulas for a uniform coupling distribution, while short-time Gaussian spin-bath decoherence gives the initial decay scale.

Short-time Gaussian spin-bath decoherence ( $z_{N} (t) ≃ e^{- 2 N g_{*}^{2} t^{2} /3}$ )

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For initially equatorial spins and uniform couplings on

[0, g_{*}]

, expand

lo g cos (2 g_{k} t) = - 2 g_{k}^{2} t^{2} + O (g_{k}^{4} t^{4})

. The strong law of large numbers then gives

\sum g_{k}^{2} ≃ N g_{*}^{2} /3

. On the scale

t = O ((g_{*} N)^{- 1})

, the total fourth-order remainder tends to zero, giving the Gaussian coherence envelope. This approximation concerns early times and does not remove finite spin-bath coherence recurrence.

Ensemble spin-bath coherence ( $E z_{N} (t) = sinc (2 g_{*} t)^{N}$ )

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For independent uniform couplings on

[0, g_{*}]

and equatorial initial bath spins,

E z_{N} (t) = [\frac{s i n ( 2 g _{*} t )}{2 g _{*} t}]^{N}, E ∣ z_{N} (t) ∣^{2} = [\frac{1}{2} + \frac{s i n ( 4 g _{*} t )}{8 g _{*} t}]^{N} .

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At fixed finite

N

, the mean tends to zero and the mean square tends to

2^{- N}

as time grows. At any fixed nonzero time, the mean-square bracket is strictly below one; the Markov inequality proves coherence tends to zero in probability as

N

grows. Individual finite realizations still exhibit finite spin-bath coherence recurrence.

Finite spin-bath coherence recurrence ( $z_{N} (t_{j}) \to 1 for some t_{j} \to \infty$ )

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For any finite set of couplings in

z_{N} (t) = \prod_{k} cos (2 g_{k} t)

, the simultaneous Dirichlet approximation theorem supplies arbitrarily late near-alignment of all phases, or an exact common period. Thus finite bath coherence does not tend to zero as time tends to infinity. This remains true for almost every realization of continuous random couplings; an ensemble mean can decay while each realization has recurrences.

Zurek spin-bath model ( $H = - Z_{D} \otimes \sum_{k} g_{k} Z_{k}$ )

Random-coupling spin-bath decoherence

Short-time Gaussian spin-bath decoherence ( $z_{N} (t) ≃ e^{- 2 N g_{*}^{2} t^{2} /3}$ )

Ensemble spin-bath coherence ( $E z_{N} (t) = sinc (2 g_{*} t)^{N}$ )

Finite spin-bath coherence recurrence ( $z_{N} (t_{j}) \to 1 for some t_{j} \to \infty$ )

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