Non-Markovian Dynamics of Qubit Systems: Quantum-State Diffusion Equations Versus Master Equations

Abstract

In this review we discuss recent progress in the theory of open quantum systems based on non-Markovian quantum state diffusion and master equations. In particular, we show that an exact master equation for an open quantum system consisting of a few qubits can be explicitly constructed by using the corresponding non-Markovian quantum state diffusion equation. The exact master equation arises naturally from the quantum decoherence dynamics of qubit systems collectively interacting with a colored noise. We illustrate our general theoretical formalism by the explicit construction of a three-qubit system coupled to a non-Markovian bosonic environment. This exact qubit master equation accurately characterizes the time evolution of the qubit system in various parameter domains, and paves the way for investigation of the memory effect of an open quantum system in a non-Markovian regime without any approximation.

Appendices

Appendix 1

Here we supply a proof of the Novikov theorem. To make the proof more generic, we calculate the term ${\fancyscript{M}}(z_{\tau } P_{t} )$, where $\tau$ and $t$ are two independent time indexes. In this, ${\fancyscript{M}}(z_{t} P_{t} )$ is the limit case in which $\tau = t$. By the definition of ensemble average in (25.6), we have [8]

$$ {\fancyscript{M}}(z_{\tau } P_{t} ) = \int \frac{{d^{2} z}}{\pi }e^{{ - |z|^{2} }} z_{\tau } P_{t} . $$

(25.57)

where $|z|^{2} = \sum\nolimits_{k} |z_{k} |^{2}$ and $d^{2} z = d^{2} z_{1} d^{2} z_{2} \cdots$. With the definition of

$z_{\tau } = i\sum\nolimits_{k} g_{k}^{*} z_{k} e^{{ - i\omega_{k} \tau }}$, we have

$$ {\fancyscript{M}}(z_{\tau } P_{t} ) = \int \frac{{d^{2} z_{1} }}{\pi }\frac{{d^{2} z_{2} }}{\pi } \cdots \prod\limits_{n} e^{{ - |z_{n} |^{2} }} \left( {i\sum\limits_{k} g_{k}^{*} z_{k} e^{{ - i\omega_{k} \tau }} } \right)P_{t} . $$

Since all $z_{k}$ are independent to each other, the above integration can be simplified as

$$ {\fancyscript{M}}(z_{\tau } P_{t} ) = i\sum\limits_{k} g_{k}^{*} e^{{ - i\omega_{k} \tau }} \left( {\prod\limits_{n \ne k} \int \frac{{d^{2} z_{n} }}{\pi }e^{{ - |z_{n} |^{2} }} } \right)\int \frac{{dz_{k} dz_{k}^{*} }}{\pi }e^{{ - |z_{k} |^{2} }} z_{k} P_{t} . $$

Integrating by parts, then we have,

$$ \begin{aligned} & \int \frac{{dz_{k} dz_{k}^{*} }}{\pi }e^{{ - |z_{k} |^{2} }} z_{k} P_{t} \\ = & \int \frac{{dz_{k} dz_{k}^{*} }}{\pi }\left( { - \frac{\partial }{{\partial z_{k}^{*} }}e^{{ - |z_{k} |^{2} }} } \right)P_{t} \\ = & \int \frac{{dz_{k} dz_{k}^{*} }}{\pi }\left[ {\left( { - \frac{\partial }{{\partial z_{k}^{*} }}e^{{ - |z_{k} |^{2} }} P_{t} } \right) + e^{{ - |z_{k} |^{2} }} \frac{\partial }{{\partial z_{k}^{*} }}P_{t} } \right] \\ = & \int \frac{{dz_{k} dz_{k}^{*} }}{\pi }e^{{ - |z_{k} |^{2} }} \frac{\partial }{{\partial z_{k}^{*} }}P_{t} . \\ \end{aligned} $$

Then

$$ {\fancyscript{M}}(z_{\tau } P_{t} ) = i\sum\limits_{k} g_{k}^{*} e^{{ - i\omega_{k} \tau }} \int \frac{{d^{2} z}}{\pi }e^{{ - |z|^{2} }} \frac{\partial }{{\partial z_{k}^{*} }}P_{t} . $$

Using the functional derivative chain rule,

$$ \begin{aligned} {\fancyscript{M}}(z_{\tau } P_{t} ) & = i\sum\limits_{k} g_{k}^{*} e^{{ - i\omega_{k} \tau }} \int \frac{{d^{2} z}}{\pi }e^{{ - |z|^{2} }} \int\limits_{0}^{t} ds\frac{{\partial z_{s}^{*} }}{{\partial z_{k}^{*} }}\frac{\delta }{{\delta z_{s}^{*} }}P_{t} \\ & = \int \frac{{d^{2} z}}{\pi }e^{{ - |z|^{2} }} \int\limits_{0}^{t} ds\alpha (\tau ,s)O(t,s,z^{*} )P_{t} , \\ {\fancyscript{M}}(z_{\tau } P_{t} ) & = \int\limits_{0}^{t} ds\alpha (\tau ,s){\fancyscript{M}}[O(t,s,z^{*} )P_{t} ]. \\ \end{aligned} $$

Now we have the Novikov theorem,

$$ \begin{aligned} {\fancyscript{M}}(z_{\tau } P_{t} ) & = \int\limits_{0}^{t} ds{\fancyscript{M}}(z_{\tau } z_{s}^{*} ){\fancyscript{M}}[O(t,s,z^{*} )P_{t} ], \\ {\fancyscript{M}}(z_{\tau }^{*} P_{t} ) & = \int\limits_{0}^{t} ds{\fancyscript{M}}(z_{\tau }^{*} z_{s} ){\fancyscript{M}}[P_{t} O^{\dag } (t,s,z)]. \\ \end{aligned} $$

In the limit $\tau = t$, we obtain

$$ \begin{aligned} {\fancyscript{M}}(z_{t} P_{t} ) & = {\fancyscript{M}}(\bar{O} (t,z^{*} )P_{t} ), \\ {\fancyscript{M}}(z_{t}^{*} P_{t} ) & = {\fancyscript{M}}(P_{t} \bar{O}^{\dag } (t,z)). \\ \end{aligned} $$

(25.58)

Appendix 2

Inserting the expansion series of $O$ operator (25.27) into the $O$ operator evolution (25.26), we have

$$ \partial_{t} O(t,s) = \partial_{t} O_{0} (t,s) + z_{t}^{*} O_{1} (t,s,t) + \int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} \partial_{t} O_{1} (t,s,_{1} ) + \cdots , $$

(25.59)

for the left hand side. Furthermore, the right hand side of (25.26) can be expanded as

$$ \begin{aligned} & \quad [ - iH_{sys} + Lz_{t}^{*} - L^{\dag } \bar{O} ,{\kern 1pt} O] - L^{\dag } \frac{{\delta \bar{O} }}{{\delta z_{s}^{*} }} \\ & = [ - iH_{sys} + Lz_{t}^{*} - L^{\dag } \bar{O}_{0} ,{\kern 1pt} O_{0} ] - L^{\dag } \frac{\delta }{{\delta z_{s}^{*} }}\int\limits_{0}^{t} d\tau \alpha (t,\tau )\int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} O_{1} (t,\tau ,s_{1} ) \\ & \quad + [ - iH_{sys} + Lz_{t}^{*} ,{\kern 1pt} \int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} O_{1} ] - [L^{\dag } \bar{O}_{0} ,{\kern 1pt} \int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} O_{1} ] - [L^{\dag } \int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} \bar{O}_{1} ,\,\bar{O}_{0} ] \\ & \quad - L^{\dag } \frac{\delta }{{\delta z_{s}^{*} }}\int\limits_{0}^{t} d\tau \alpha (t,\tau )\int\limits_{0}^{t} ds_{1} \int\limits_{0}^{t} ds_{2} z_{{s_{1} }}^{*} z_{{s_{2} }}^{*} O_{2} (t,\tau ,s_{1} ,s_{2} ) \\ & \quad + \cdots . \\ \end{aligned} $$

(25.60)

By the definition $\bar{O} = \int_{0}^{t} ds\alpha (t,s)O(t,s,z^{*} )$, we can calculate the terms

$$ \begin{aligned} & \frac{\delta }{{\delta z_{s}^{*} }}\int\limits_{0}^{t} d\tau \alpha (t,\tau )\int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} O_{1} (t,\tau ,s_{1} ) \\ = & \int\limits_{0}^{t} d\tau \alpha (t,\tau )\int\limits_{0}^{t} ds_{1} \delta (s,s_{1} )O_{1} (t,\tau ,s_{1} ) \\ = & \int\limits_{0}^{t} d\tau \alpha (t,\tau )O_{1} (t,\tau ,s) = \bar{O}_{1} (t,s), \\ \end{aligned} $$

and

$$ \begin{aligned} & \quad \frac{\delta }{{\delta z_{s}^{*} }}\int\limits_{0}^{t} d\tau \alpha (t,\tau )\int\limits_{0}^{t} ds_{1} \int\limits_{0}^{t} ds_{2} z_{{s_{1} }}^{*} z_{{s_{2} }}^{*} O_{2} (t,\tau ,s_{1} ,s_{2} ) \\ & = \int\limits_{0}^{t} d\tau \alpha (t,\tau )\int\limits_{0}^{t} ds_{1} \int\limits_{0}^{t} ds_{2} z_{{s_{1} }}^{*} \delta (s,s_{2} )O_{2} (t,\tau ,s_{1} ,s_{2} ) \\ & + \int\limits_{0}^{t} d\tau \alpha (t,\tau )\int\limits_{0}^{t} ds_{1} \int\limits_{0}^{t} ds_{2} z_{{s_{2} }}^{*} \delta (s,s_{1} )O_{2} (t,\tau ,s_{1} ,s_{2} ) \\ & = \int\limits_{0}^{t} d\tau \alpha (t,\tau )\int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} O_{2} (t,\tau ,s_{1} ,s) + \int\limits_{0}^{t} d\tau \alpha (t,\tau )\int\limits_{0}^{t} ds_{2} z_{{s_{2} }}^{*} O_{2} (t,\tau ,s,s_{2} ) \\ & = \int\limits_{0}^{t} d\tau \alpha (t,\tau )\int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} \left( {O_{2} (t,\tau ,s_{1} ,s) + O_{2} (t,\tau ,s,s_{1} )} \right) \\ & = \int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} \left( {\bar{O}_{2} (t,s_{1} ,s) + \bar{O}_{2} (t,s,s_{1} )} \right). \\ \end{aligned} $$

Equating the two sides for each order of noise $z^{*}$, we obtain a set of dynamical equations for the $O_{n}$ $(n = 1,2, \ldots )$. For the non-noise term, we have

$$ \partial_{t} O_{0} = [ - iH_{sys} - L^{\dag } \bar{O}_{0} ,{\kern 1pt} O_{0} ] - L^{\dag } \bar{O}_{1} . $$

For the first-order noise terms, we have

$$ \begin{aligned} & \int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} \partial_{t} O_{1} \\ = & \int\limits_{0}^{t} ds_{1} z_{{s_{1} }}^{*} \left\{ {[ - iH_{sys} L^{\dag } \bar{O}_{0} ,{\kern 1pt} O_{1} ] - [L^{\dag } \bar{O}_{1} ,{\kern 1pt} O_{0} ] - L^{\dag } \left( {\bar{O}_{2} (t,s_{1} ,s) + \bar{O}_{2} (t,s,s_{1} )} \right)} \right\}, \\ \end{aligned} $$

and the evolution equation for $O_{1}$ is obtained as

$$ \partial_{t} O_{1} = [ - iH_{sys} - L^{\dag } \bar{O}_{0} ,{\kern 1pt} O_{1} ] - [L^{\dag } \bar{O}_{1} ,{\kern 1pt} O_{0} ] - L^{\dag } \left( {\bar{O}_{2} (t,s_{1} ,s) + \bar{O}_{2} (t,s,s_{1} )} \right). $$

Similarly, the set of coupled dynamical equations for all $O_{n}$ can be determined sequentially. For the terms containing $z_{t}^{*}$, the boundary conditions can be obtained as

$$ \begin{aligned} O_{1} (t,s,t) & = [L,{\kern 1pt} O_{0} (t,s)], \\ O_{2} (t,s,s_{1} ,t) + O_{2} (t,s,t,s_{1} ) & = [L,O_{1} (t,s,s_{1} )], \\ etc. & \\ \end{aligned} $$

Appendix 3

In order to explicitly derive the $R(t)$ for the three-qubit system model, we need to calculate two terms ${\fancyscript{M}}\{ z_{{s_{1} }} P_{t} \}$ and ${\fancyscript{M}}\{ z_{{s_{1} }} z_{{s_{3} }} P_{t} \}$. Since the term ${\fancyscript{M}}\{ z_{{s_{1} }} z_{{s_{3} }} P_{t} \}$ contains second order of noise, it can be evaluated by using Novikov’s theorem twice (25.18).

$$ \begin{aligned} {\fancyscript{M}}\{ z_{{s_{1} }} P_{t} \} & = \int\limits_{0}^{t} ds_{2} \alpha (s_{1} ,s_{2} ){\fancyscript{M}}\{ O(t,s_{2} )P_{t} \} \\ & = \int\limits_{0}^{t} ds_{2} \alpha (s_{1} ,s_{2} )\left[ {O_{0} (t,s_{2} )\rho_{t} + \int\limits_{0}^{t} ds_{3} O_{1} (t,s_{2} ,s_{3} ){\fancyscript{M}}\{ z_{{s_{3} }}^{*} P_{t} \} } \right] \\ & \quad + \int\limits_{0}^{t} ds_{2} \alpha (s_{1} ,s_{2} )\int\limits_{0}^{t} ds_{3} \int\limits_{0}^{t} ds_{5} O_{2} (t,s_{2} ,s_{3} ,s_{5} ){\fancyscript{M}}\{ z_{{s_{3} }}^{*} z_{{s_{5} }}^{*} P_{t} \} , \\ {\fancyscript{M}}\{ z_{{s_{1} }} z_{{s_{3} }} P_{t} \} & = \int\limits_{0}^{t} ds_{2} \alpha (s_{1} ,s_{2} ){\fancyscript{M}}\{ z_{{s_{3} }} O(t,s_{2} )P_{t} \} \\ & = \int\limits_{0}^{t} ds_{2} \int\limits_{0}^{t} ds_{4} \alpha (s_{1} ,s_{2} )\alpha (s_{3} ,s_{4} ){\fancyscript{M}}\left\{ \frac{{\delta O(t,s_{2} )}}{{\delta z_{{s_{4} }}^{*} }}P_{t} \right\} \\ & \quad + \int\limits_{0}^{t} ds_{2} \int\limits_{0}^{t} ds_{4} \alpha (s_{1} ,s_{2} )\alpha (s_{3} ,s_{4} ){\fancyscript{M}}\{ O(t,s_{2} )O(t,s_{4} )P_{t} \} . \\ \end{aligned} $$

After eliminating the zero terms by the “forbidden conditions”, $R(t)$ can be explicitly shown as (25.54).