2025-02-01

We have a two-state system, which has a continuous degree of freedom $q$, associated a potential energy function $V(q)$ with two separate minima.

\usepackage{amsmath}

\begin{document}
\begin{tikzpicture}

% Draw the potential curve V(q)
\draw[domain=-2.5:0] plot (\x, {cos(2* \x r)}); 
\draw[domain=0:3] plot (\x, {0.8*cos(1.8* \x r)+0.2}); 

%Labels
\node at (-3,0) {$V(q)$}; 
\node at (0, -1.1) {$q$}; 
\node at (-0.2, 0) {$V_0$};

\node at (-1*3.141/2, -1.3) {$\omega_{-}$};
\node at (3.141/1.8, -1.3) {$\omega_{+}$};

\node at (-1*3.141/2, -1.7) {$-\frac{1}{2}q_0$};
\node at (3.141/1.8, -1.7) {$\frac{1}{2}q_0$};

\node at (-1*3.141/2, -2.2) {$(\sigma_z = -1)$};
\node at (3.141/1.8, -2.2) {$(\sigma_z = 1)$};

\node at (-1*3.141/2+0.4, -0.25) {$\hbar \omega_{-}$};
\node at (3.141/1.8+0.4, -0.1) {$\hbar \omega_{+}$};

\node at (-3.141/2+0.9, -0.55) {$\epsilon $};
% Arrows
\draw[<->] (-1*3.141/2-0.3, -1.1) -- (-1*3.141/2+0.3, -1.1);
\draw[<->] (3.141/1.8-0.3, -1.1)  -- (3.141/1.8+0.3, -1.1) ;

\draw[->] (-2.6, -0.2) -- (-2.6, 0.2);
\draw[<->] (0, 1.0) -- (0, -0.8);

\draw[-] (-1*3.141/2-0.4, -0.7) -- (-1*3.141/2+0.4, -0.7);
\draw[-] (-1*3.141/2-0.7, 0.2) -- (-1*3.141/2+0.7, 0.2);
\draw[<->] (-1*3.141/2, -0.6) -- (-1*3.141/2, 0.1);

\draw[-] (3.141/1.8-0.4, -0.4) -- (3.141/1.8+0.4, -0.4);
\draw[-] (3.141/1.8-0.7, 0.2) -- (3.141/1.8+0.7, 0.2);
\draw[<->] (3.141/1.8, -0.3) -- (3.141/1.8, 0.1);

\draw[<->] (-3.141/2+0.7, -0.4) -- (-3.141/2+0.7, -0.7);

\end{tikzpicture}
\end{document}

This potential energy function, of course, have one barrier $V_0$. Two minima note as ${\omega }_{+}$ and ${\omega }_{-}$

We assume that $V_0$ is much bigger than $\hslash {\omega }_0$. And ${\omega }_0$ is of the order of ${\omega }_{+}$ and ${\omega }_{-}$.

The energy gap between ground state and first excited state is $\hslash {\omega }_{+}$ or $\hslash {\omega }_{-}$

The energy difference of two ground state is $ϵ$, which is much smaller than $\hslash {\omega }_0$

And $k_{B}T\ll \hslash {\omega }_0$, means that system is strictly located in the two-dimension Hilbert Space spanned by two ground states. In one word, here is no tunneling.

Now, we introduce a matrix element $\hslash {\mathrm{\Delta }}_0$, which is exponentially smaller than $\hslash {\omega }_0$, $\hslash {\mathrm{\Delta }}_0\ll \hslash {\omega }_0$.
With condition $\hslash {\omega }_0\ll V(0)$, so that we make sure tunneling process does not involve excited states.

For all systems, degree of freedom $q$ can be or not be geometrical.

I am confused that what's $V(q)$? For a non-geometrical freedom described system, how to draw the curve? We have energy level, how the smooth curve occurred?

Of course, actual systems have many dimension which is more than two. But we can truncate them to be two. These systems can be described by an extended coordinate, as "truncated" two-state systems.

Now, we have isolated hamiltonian terms

And, it is equivalent to a particle of spin-$-\frac{1}{2}$ in a magnetic field, $H=-ϵ\hat{z}+\hslash {\mathrm{\Delta }}_0\hat{x}$,, with eigenvalues $\pm \sqrt{{ϵ}^2+(\hslash {\mathrm{\Delta }}_0{)}^2}$.

We can tell dynamic of ${\sigma }_z$ is sensitive only to the ratio $\frac{\hslash {\mathrm{\Delta }}_0}{ϵ}$.

• ratio is small, system is located in one well
• ratio is large, eigenstates are appreciably delocalized in q

For $ϵ$ is zero, we define $P(t)=P_R-P_L$, P is the probability of finding the system in the right (left) well.

We have $P(t)=\cos {\mathrm{\Delta }}_{0}t$

${\psi }_R,{\psi }_L$ may relate to some symmetry operation, which hamiltonian commuted.
Fluctuating symmetry breaking due to contact with a quantum environment is much important.

Now, the isolated system can be completely described. We know each two-state system interacts with environment.

So, we add coupling term ${\mathrm{\Delta }}_z\hat{\mathrm{\Omega }}$, $\hat{\mathrm{\Omega }}$ is operator of the environment In other word, we add coupling with ${\sigma }_z$ and the environment can "observe" the value of ${\sigma }_z$.

We still have ${\sigma }_x,{\sigma }_y$, why we write ${\sigma }_z$ term only? Here lies many case involved x and y.

The reason is that ${\sigma }_x$ and ${\sigma }_y$ have only non-diagonal matrix elements in the ${\sigma }_z$ representation. Operator with only non-diagonal element do exchange $|0⟩,|1⟩$ ,${\sigma }_z$ do not change basis. Or, we can say ${\sigma }_x,{\sigma }_y$ only change basis, while ${\sigma }_z$ can change eigenvalue.

It's a confusing statement, whatever only add coupling with ${\sigma }_z$ , maybe need python and @Weitang_Li

For tunneling, one change ${\psi }_L$ to ${\psi }_R$. The important interaction must be proportional to the overlap of ${\psi }_L$ and ${\psi }_R$ in the real coordinate space. And the interactions must be of order of $\hslash {\mathrm{\Delta }}_0$, which means that interaction must make tunneling happen.

From smart people's conclusion, we know ${\sigma }_z$ should be coupled to environment. And one points out that system should not only couple one degree of environment. Here comes out a way that we represent environment as a set of harmonic oscillators with a coupling linear in the oscillator coordinates and/or momenta.

So we get a complete hamiltonian $$H_{SB}=-\frac{1}{2}\Delta\sigma_{x}+\frac{1}{2}\epsilon\sigma_{z}+\sum_{\alpha}\left( \frac{1}{2}m_{\alpha}\omega^2_{\alpha} x_{\alpha}+\frac{p^2_{\alpha}}{2m_{\alpha}}\right)+\frac{1}{2}q_{0}\sigma_{z}\sum_{\alpha}C_{\alpha}x_{\alpha}$$
we should note that

• $\mathrm{\Delta }$ is tunneling matrix element, but with limited number oscillators, this should a efficient one. which is renormalized for higher-frequency effects.
• $q_0$ may refer to distance in two-well diagram, also can be a part of coupling,for a intrinsically two-state system $q_0$ is meaningless,
• $C_{\alpha }$ is linear coupling of $\alpha$th oscillator

We now define a spectral function

We say that spectral function is full description of effect of environment.

It is hard to explain clearly, which needs a lot pre-reading. But we can roughly say that equilibrium statistical average over initial states of environment and sum over final states are taken to compress information into spectral function.

One may ask ChatGPT or DeepSeek to get compete answer.

Whatever, now we can define this model with three terms $\mathrm{\Delta },ϵ,J(\omega )$

The form of spectral function can be from equation of motions or microscope knowledge.

One assumption is $J(\omega )$ is a reasonably smooth function of $\omega$.

When it has the form ${\omega }^s$ , $\omega$ up to ${\omega }_c$, ${\omega }_c\gg \mathrm{\Delta }$. Also for a natural two-state system, ${\omega }_c$ can be a cutoff from a truncation procedure.