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Computer simulation of quantum dynamics in a classical spin environment

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Abstract

In this paper, a formalism for studying the dynamics of quantum systems coupled to classical spin environments is reviewed. The theory is based on generalized antisymmetric brackets and naturally predicts open-path off-diagonal geometric phases in the evolution of the density matrix. It is shown that such geometric phases must also be considered in the quantum–classical Liouville equation for a classical bath with canonical phase space coordinates; this occurs whenever the adiabatics basis is complex (as in the case of a magnetic field coupled to the quantum subsystem). When the quantum subsystem is weakly coupled to the spin environment, non-adiabatic transitions can be neglected and one can construct an effective non-Markovian computer simulation scheme for open quantum system dynamics in classical spin environments. In order to tackle this case, integration algorithms based on the symmetric Trotter factorization of the classical-like spin propagator are derived. Such algorithms are applied to a model comprising a quantum two-level system coupled to a single classical spin in an external magnetic field. Starting from an excited state, the population difference and the coherences of this two-state model are simulated in time while the dynamics of the classical spin is monitored in detail. It is the author’s opinion that the numerical evidence provided in this paper is a first step toward developing the simulation of quantum dynamics in classical spin environments into an effective tool. In turn, the ability to simulate such a dynamics can have a positive impact on various fields, among which, for example, nanoscience.

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Acknowledgments

The author is grateful to G. S. Ezra for many discussions, encouragement and support received in the past few years.

Author information

Authors and Affiliations

  1. School of Chemistry and Physics, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg, 3209, South Africa

    Alessandro Sergi

  2. National Institute of Theoretical Physics (NITheP), Durban, KwaZulu-Natal, South Africa

    Alessandro Sergi

Authors
  1. Alessandro Sergi
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Corresponding author

Correspondence to Alessandro Sergi.

Additional information

Dedicated to Professor Greg Ezra and published as part of the special collection of articles celebrating his 60th birthday.

This work is based upon research supported by the National Research Foundation of South Africa.

Appendices

Appendix 1: Integration algorithm on the (1, 1) surface

In pseudo-code form, the algorithm provided by \(U_{(1,1)}^1(\tau )\) is:

$$\begin{aligned} U_{1,(1,1)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left\{ \frac{3}{2}C_1C_3\frac{\tau }{4} +[C_2+C_1S_x]^{\frac{3}{2}} \right\} ^{\frac{2}{3}}\\ \quad-\frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(142)
$$\begin{aligned} U_{2,(1,1)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ S_x \rightarrow S_x - \frac{\tau }{2} S_y \left( S_z-c_2b\right) \right. , \end{aligned}$$
(143)
$$\begin{aligned} U_{1,(1,1)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left\{ \frac{3}{2}C_1C_3\frac{\tau }{4} +[C_2+C_1S_x]^{\frac{3}{2}} \right\} ^{\frac{2}{3}}\\ \quad- \frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(144)
$$\begin{aligned} U_{(1,1)}^{S_y}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y +\frac{\tau }{2} \left[ \frac{\mu S_x(b+\mu S_z)-\mu (\Omega +S_x)S_z}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad + S_x\left. \left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(145)
$$\begin{aligned} U_{(1,1)}^{S_z}(\tau ):\left\{ \begin{array}{l} S_z \rightarrow \frac{B_2}{B_1} - \frac{1}{B_1}\left[ \left( B_2 - B_1S_z\right) ^{\frac{3}{2}}\right. \\ \quad-\left. \frac{3 B_1 B_3\tau }{2}\right] ^{\frac{2}{3}}, \end{array} \right. \end{aligned}$$
(146)
$$\begin{aligned} U_{(1,1)}^{S_y}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y +\frac{\tau }{2} \left[ \frac{\mu S_x(b+\mu S_z)-\mu (\Omega +S_x)S_z}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad +\left. S_x\left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(147)
$$\begin{aligned} U_{2,(1,1)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ S_x \rightarrow S_x - \frac{\tau }{2} S_y \left( S_z-c_2b\right) \right. ,\end{aligned}$$
(148)
$$\begin{aligned} U_{1,(1,1)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left\{ \frac{3}{2}C_1C_3\frac{\tau }{4} +[C_2+C_1S_x]^{\frac{3}{2}} \right\} ^{\frac{2}{3}}\\ \quad -\frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(149)
$$\begin{aligned} U_{1,(1,1)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow\frac{1}{C_1}\left\{ \frac{3}{2}C_1C_3\frac{\tau }{4} +[C_2+C_1S_x]^{\frac{3}{2}} \right\} ^{\frac{2}{3}}\\ \quad -\frac{C_2}{C_1}. \end{array} \right. \end{aligned}$$
(150)

The algorithm provided by \(U_{(1,1)}^2(\tau )\) is:

$$\begin{aligned} U_{(1,1)}^{S_z}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_z \rightarrow \frac{B_2}{B_1}\\ \quad - \frac{1}{B_1}\left\{ \left[ B_2 - B_1S_z\right] ^{\frac{3}{2}} -\frac{3 B_1 B_3\tau }{4}\right\} ^{\frac{2}{3}}, \end{array} \right. \end{aligned}$$
(151)
$$\begin{aligned} U_{1,(1,1)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left\{ \frac{3}{2}C_1C_3\frac{\tau }{4} +[C_2+C_1S_x]^{\frac{3}{2}}\right\} ^{\frac{2}{3}}\\ \quad -\frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(152)
$$\begin{aligned} U_{2,(1,1)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ S_x \rightarrow S_x - \frac{\tau }{2} S_y \left( S_z-c_2b\right) \right. ,\end{aligned}$$
(153)
$$\begin{aligned} U_{1,(1,1)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left\{ \frac{3}{2}C_1C_3\frac{\tau }{4} +[C_2+C_1S_x]^{\frac{3}{2}}\right\} ^{\frac{2}{3}}\\ \quad -\frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(154)
$$\begin{aligned} U_{(1,1)}^{S_y}(\tau ):\left\{ \begin{array}{l} S_y \rightarrow S_y +\tau \left[ \frac{\mu S_x(b+\mu S_z)-\mu (\Omega +S_x)S_z}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad + \left. S_x\left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(155)
$$\begin{aligned} U_{1,(1,1)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow\frac{1}{C_1}\left\{ \frac{3}{2}C_1C_3\frac{\tau }{4} +[C_2+C_1S_x]^{\frac{3}{2}} \right\} ^{\frac{2}{3}}\\ \quad-{}\frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(156)
$$\begin{aligned} U_{2,(1,1)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ S_x \rightarrow S_x - \frac{\tau }{2} S_y \left( S_z-c_2b\right) \right. , \end{aligned}$$
(157)
$$\begin{aligned} U_{1,(1,1)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left\{ \frac{3}{2}C_1C_3\frac{\tau }{4} +[C_2+C_1S_x]^{\frac{3}{2}} \right\} ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(158)
$$\begin{aligned} U_{(1,1)}^{S_z}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_z \rightarrow \frac{B_2}{B_1} \\ \quad- \frac{1}{B_1}\left\{ \left[ B_2 - B_1S_z\right] ^{\frac{3}{2}} -\frac{3 B_1 B_3\tau }{4}\right\} ^{\frac{2}{3}}. \end{array} \right. \end{aligned}$$
(159)

The algorithm provided by \(U_{(1,1)}^3(\tau )\) is:

$$\begin{aligned} U_{(1,1)}^{S_y}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y +\frac{\tau }{4} \left[ \frac{\mu S_x(b+\mu S_z)-\mu (\Omega +S_x)S_z}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad +\left. S_x\left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(160)
$$\begin{aligned} U_{(1,1)}^{S_z}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_z \rightarrow \frac{B_2}{B_1} \\ \quad- \frac{1}{B_1}\left[ \left( B_2 - B_1S_z\right) ^{\frac{3}{2}} -\frac{3 B_1 B_3\tau }{4}\right] ^{\frac{2}{3}}, \end{array} \right. \end{aligned}$$
(161)
$$\begin{aligned} U_{(1,1)}^{S_y}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y +\frac{\tau }{4} \left[ \frac{\mu S_x(b+\mu S_z)-\mu (\Omega +S_x)S_z}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad+\left. S_x\left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(162)
$$\begin{aligned} U_{1,(1,1)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left\{ \frac{3}{2}C_1C_3\frac{\tau }{2} +[C_2+C_1S_x]^{\frac{3}{2}} \right\} ^{\frac{2}{3}}\\ \quad -\frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(163)
$$\begin{aligned} U_{2,(1,1)}^{S_x}(\tau ):\left\{ S_x \rightarrow S_x - \tau S_y\left( S_z-c_2b\right) \right. ,\end{aligned}$$
(164)
$$\begin{aligned} U_{1,(1,1)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{ll} S_x \rightarrow \frac{1}{C_1}\left\{ \frac{3}{2}C_1C_3\frac{\tau }{2} +[C_2+C_1S_x]^{\frac{3}{2}} \right\} ^{\frac{2}{3}}\\ \quad -\frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(165)
$$\begin{aligned} U_{(1,1)}^{S_y}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y +\frac{\tau }{4} \left[ \frac{\mu S_x(b+\mu S_z) -\mu (\Omega +S_x)S_z}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad + \left. S_x\left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(166)
$$\begin{aligned} U_{(1,1)}^{S_z}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_z \rightarrow \frac{B_2}{B_1} \\ \quad - \frac{1}{B_1}\left[ \left( B_2 - B_1S_z\right) ^{\frac{3}{2}} -\frac{3 B_1 B_3\tau }{4}\right] ^{\frac{2}{3}}, \end{array} \right. \end{aligned}$$
(167)
$$\begin{aligned} U_{(1,1)}^{S_y}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y +\frac{\tau }{4} \left[ \frac{\mu S_x(b+\mu S_z)-\mu (\Omega +S_x)S_z}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad + \left. S_x\left( S_z-c_2b\right) \right] . \end{array} \right. \end{aligned}$$
(168)

Appendix 2: Integration algorithm on the (2, 2) surface

The algorithm provided by \(U_{(2,2)}^1(\tau )\) is

$$\begin{aligned} U_{1,(2,2)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left[ -\frac{3}{2}C_1C_3\frac{\tau }{4} +\left( C_2+C_1S_x\right) ^{\frac{3}{2}} \right] ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(169)
$$\begin{aligned} U_{2,(2,2)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ S_x \rightarrow S_x -\frac{\tau }{2} S_y \left( S_z-c_2b\right) \right. ,\end{aligned}$$
(170)
$$\begin{aligned} U_{1,(2,2)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left[ -\frac{3}{2}C_1C_3\frac{\tau }{4} +\left( C_2+C_1S_x\right) ^{\frac{3}{2}} \right] ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(171)
$$\begin{aligned} U_{(2,2)}^{S_y}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y + \frac{\tau }{2}\left[ \frac{\mu (\Omega +S_x)S_z-\mu S_x(b+\mu S_z)}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad + \left. S_x\left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(172)
$$\begin{aligned} U_{(2,2)}^{S_z}(\tau ):\left\{ \begin{array}{l} S_z \rightarrow \frac{B_2}{B_1} \\ \quad - \frac{1}{B_1}\left[ \left( B_2-B_1S_z\right) ^{\frac{3}{2}} + \frac{3}{2} B_1B_3 \tau \right] ^{\frac{2}{3}}, \end{array} \right. \end{aligned}$$
(173)
$$\begin{aligned} U_{(2,2)}^{S_y}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y + \frac{\tau }{2}\left[ \frac{\mu (\Omega +S_x)S_z-\mu S_x(b+\mu S_z)}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad + \left. S_x\left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(174)
$$\begin{aligned} U_{1,(2,2)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left[ -\frac{3}{2}C_1C_3\frac{\tau }{4} +\left( C_2+C_1S_x\right) ^{\frac{3}{2}}\right] ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(175)
$$\begin{aligned} U_{2,(2,2)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ S_x \rightarrow S_x -\frac{\tau }{2} S_y\left( S_z-c_2b\right) \right. ,\end{aligned}$$
(176)
$$\begin{aligned} U_{1,(2,2)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left[ -\frac{3}{2}C_1C_3\frac{\tau }{4} +\left( C_2+C_1S_x\right) ^{\frac{3}{2}} \right] ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}. \end{array} \right. \end{aligned}$$
(177)

The algorithm provided by \(U_{(2,2)}^2(\tau )\) is

$$\begin{aligned} U_{(2,2)}^{S_z}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_z \rightarrow \frac{B_2}{B_1} \\ \quad - \frac{1}{B_1}\left[ \left( B_2-B_1S_z\right) ^{\frac{3}{2}} + \frac{3}{2} B_1B_3 \frac{\tau }{2}\right] ^{\frac{2}{3}}, \end{array} \right. \end{aligned}$$
(178)
$$\begin{aligned} U_{1,(2,2)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left[ -\frac{3}{2}C_1C_3\frac{\tau }{4} +\left( C_2+C_1S_x\right) ^{\frac{3}{2}} \right] ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}, \end{array} \right. , \end{aligned}$$
(179)
$$\begin{aligned} U_{2,(2,2)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ S_x \rightarrow S_x -\frac{\tau }{2} S_y \left( S_z-c_2b\right) \right. ,\end{aligned}$$
(180)
$$\begin{aligned} U_{1,(2,2)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left[ -\frac{3}{2}C_1C_3\frac{\tau }{4} +\left( C_2+C_1S_x\right) ^{\frac{3}{2}} \right] ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(181)
$$\begin{aligned} U_{(2,2)}^{S_y}(\tau ):\left\{ \begin{array}{l} S_y \rightarrow S_y + \tau \left[ \frac{\mu (\Omega +S_x)S_z-\mu S_x(b+\mu S_z)}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad + \left. S_x\left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(182)
$$\begin{aligned} U_{1,(2,2)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left[ -\frac{3}{2}C_1C_3\frac{\tau }{4} +\left( C_2+C_1S_x\right) ^{\frac{3}{2}} \right] ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(183)
$$\begin{aligned} U_{2,(2,2)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ S_x \rightarrow S_x -\frac{\tau }{2} S_y\left( S_z-c_2b\right) \right. ,\end{aligned}$$
(184)
$$\begin{aligned} U_{1,(2,2)}^{S_x}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left[ -\frac{3}{2}C_1C_3\frac{\tau }{4} +\left( C_2+C_1S_x\right) ^{\frac{3}{2}} \right] ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(185)
$$\begin{aligned} U_{(2,2)}^{S_z}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_z \rightarrow \frac{B_2}{B_1} \\ \quad - \frac{1}{B_1}\left[ \left( B_2-B_1S_z\right) ^{3/2} + \frac{3}{2} B_1B_3 \frac{\tau }{2}\right] ^{2/3}. \end{array} \right. \end{aligned}$$
(186)

The algorithm provided by \(U_{(2,2)}^3(\tau )\) is

$$\begin{aligned} U_{(2,2)}^{S_y}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y + \frac{\tau }{4}\left[ \frac{\mu (\Omega +S_x)S_z-\mu S_x(b+\mu S_z)}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad + \left. S_x\left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(187)
$$\begin{aligned} U_{(2,2)}^{S_z}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_z \rightarrow \frac{B_2}{B_1} \\ \quad - \frac{1}{B_1}\left[ \left( B_2-B_1S_z\right) ^{\frac{3}{2}} + \frac{3}{2} B_1B_3 \frac{\tau }{2}\right] ^{\frac{2}{3}}, \end{array} \right. \end{aligned}$$
(188)
$$\begin{aligned} U_{(2,2)}^{S_y}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y + \frac{\tau }{4}\left[ \frac{\mu (\Omega +S_x)S_z-\mu S_x(b+\mu S_z)}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad + \left. S_x \left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(189)
$$\begin{aligned} U_{1,(2,2)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left[ -\frac{3}{2}C_1C_3\frac{\tau }{2} +\left( C_2+C_1S_x\right) ^{\frac{3}{2}}\right] ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(190)
$$\begin{aligned} U_{2,(2,2)}^{S_x}(\tau ):\left\{ S_x \rightarrow S_x -\tau S_y\left( S_z-c_2b\right) \right. ,\end{aligned}$$
(191)
$$\begin{aligned} U_{1,(2,2)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_x \rightarrow \frac{1}{C_1}\left[ -\frac{3}{2}C_1C_3\frac{\tau }{2} +\left( C_2+C_1S_x\right) ^{\frac{3}{2}} \right] ^{\frac{2}{3}}\\ \quad - \frac{C_2}{C_1}, \end{array} \right. \end{aligned}$$
(192)
$$\begin{aligned} U_{(2,2)}^{S_y}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y + \frac{\tau }{4}\left[ \frac{\mu (\Omega +S_x)S_z-\mu S_x(b+\mu S_z)}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}}\right. \\ \quad + \left. S_x\left( S_z-c_2b\right) \right] , \end{array} \right. \end{aligned}$$
(193)
$$\begin{aligned} U_{(2,2)}^{S_z}\left( \frac{\tau }{2}\right):\left\{ \begin{array}{l} S_z \rightarrow \frac{B_2}{B_1}\\ \quad - \frac{1}{B_1}\left[ \left( B_2-B_1S_z\right) ^{\frac{3}{2}} + \frac{3}{2} B_1B_3 \frac{\tau }{2}\right] ^{\frac{2}{3}}, \end{array} \right. \end{aligned}$$
(194)
$$\begin{aligned} U_{(2,2)}^{S_y}\left( \frac{\tau }{4}\right):\left\{ \begin{array}{l} S_y \rightarrow S_y + \frac{\tau }{4}\left[ \frac{\mu (\Omega +S_x)S_z-\mu S_x(b+\mu S_z)}{\sqrt{C^2+2\mu (\Omega S_x - c_1 b S_z)}} \right. \\ \quad + \left. S_x\left( S_z-c_2b\right) \right] . \end{array} \right. \end{aligned}$$
(195)

Appendix 3: Integration algorithm on the \((1,2)\) surface

The algorithm provided by \(U_{(1,2)}^1(\tau )\) is

$$\begin{aligned} U_{(1,2)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ S_x \rightarrow S_x - \frac{\tau }{2}S_y\left( S_z-c_2b\right) \right. ,\end{aligned}$$
(196)
$$\begin{aligned} U_{(1,2)}^{S_y}(\tau ):\left\{ S_y \rightarrow S_y +\tau S_x\left( S_z-c_2b\right) \right. , \end{aligned}$$
(197)
$$\begin{aligned} U_{(1,2)}^{S_x}\left( \frac{\tau }{2}\right):\left\{ S_x \rightarrow S_x - \frac{\tau }{2}S_y\left( S_z-c_2b\right) \right. . \end{aligned}$$
(198)

The algorithm provided by \(U_{(1,2)}^2(\tau )\) is

$$\begin{aligned} U_{(1,2)}^{S_y}\left( \frac{\tau }{2}\right):\left\{ S_y \rightarrow S_y +\frac{\tau }{2} S_x\left( S_z-c_2b\right) \right. ,\end{aligned}$$
(199)
$$\begin{aligned} U_{(1,2)}^{S_x}(\tau ):\left\{ S_x \rightarrow S_x - \tau S_y\left( S_z-c_2b\right) \right. ,\end{aligned}$$
(200)
$$\begin{aligned} U_{(1,2)}^{S_y}\left( \frac{\tau }{2}\right):\left\{ S_y \rightarrow S_y +\frac{\tau }{2} S_x\left( S_z-c_2b\right) \right. . \end{aligned}$$
(201)

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Sergi, A. Computer simulation of quantum dynamics in a classical spin environment. Theor Chem Acc 133, 1495 (2014). https://doi.org/10.1007/s00214-014-1495-4

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