# What is the simplest model of a quantum measurement on a 2 level system

What is the simplest physical system which can be used to model the quantum measurement of a 2 level system?

For example, can the following, spin coupled to a harmonic bath, be used to model a measurement of the 2 level system's z polarisation?

$$\hat{H}=\sum_{j=1}^n\frac{\hat{p}^2_j}{2}+\frac{1}{2}\omega_j^2\bigg(\hat q_j+\frac{\hat\sigma_z c_j}{\omega_j^2}\bigg)^2$$

with a spectral density

$$J(\omega) = \frac{\pi}{2}\sum_{j=1}^n \frac{c_j^2}{\omega_j}\delta(\omega-\omega_j)=\frac{\eta\gamma\Omega^2\omega}{(\omega^2-\Omega^2)^2+\gamma^2\omega^2}$$

my intuition is yes (provided certain conditions on the choice of parameters in $$J(\omega)$$ and possibly the manner in which the infinite bath limit is approached), references to papers discussing this would be appreciated.

Edit

By model a quantum measurement I mean that there exists some initial pure state density operator $$\hat\rho^2(0)=\hat\rho(0)=\hat\rho_s\otimes\hat\rho_b$$ (where $$\hat\rho_s$$ and $$\hat\rho_b$$ are density operators in the system and bath spaces respectively), such that the long time dynamics leads to "collapse" of the spin system

$$\lim_{t\to\infty}\mathrm{tr}_b[\hat\rho(t)]=|0\rangle\langle0|\mathrm{tr}[\hat\rho_s|0\rangle\langle0|]+|1\rangle\langle1|\mathrm{tr}[\hat\rho_s|1\rangle\langle1|]$$

where $$\mathrm{tr}_b[\dots]$$ denotes a trace over the bath degrees of freedom, $$|0\rangle$$ corresponds to spin up and $$|1\rangle$$ to spin down. Furthermore, that there exists some pointer variable e.g. in my example possibly $$y=\frac{\sum_jc_jq_j}{\sqrt{\sum_jc^2_j}}$$ such that at long time the reduced density matrix for the spin and pointer are also in a mixed state with (possibly perfect) correlation between the pointer state and the spin state, i.e. I would imagine this corresponding to something like

$$\lim_{t\to\infty}\hat\rho_{sp}(t)=\lim_{t\to\infty}\mathrm{tr}_{\tilde b}[\hat\rho(t)]=\hat\rho^{(0)}_p|0\rangle\langle0|\mathrm{tr}[\hat\rho_s|0\rangle\langle0|]+\hat\rho^{(1)}_p|1\rangle\langle1|\mathrm{tr}[\hat\rho_s|1\rangle\langle1|]$$

where $$\mathrm{tr}_{\tilde b}[\dots]$$ denotes a trace over the bath degrees of freedom excluding the pointer, and $$\hat\rho_p^{(i)}$$ are operators in the pointer space, corresponding to being a sharply localised state indicating whether the spin is up or down i.e. with something like the following properties

$$\mathrm{tr}[|0\rangle\langle0|\hat\rho_p^{(0)}\hat{y}]=-\frac{\sqrt{\eta}}{\Omega}$$

$$\mathrm{tr}[|1\rangle\langle1|\hat\rho_p^{(1)}\hat{y}]=\frac{\sqrt{\eta}}{\Omega}$$

$$\mathrm{tr}[|0\rangle\langle0|\hat\rho_p^{(0)}\hat{y}^2]=\mathrm{tr}[|1\rangle\langle1|\hat\rho_p^{(1)}\hat{y}^2]\approx\frac{\eta}{\Omega^2}$$

$$\hat\rho_p^{(0)}\hat\rho_p^{(1)}\approx0$$

Edit:2 For example does the following constitute an idealised physical model of measurement on the z polarisation of a spin:

Define the total Hamiltonian as

$$\hat{H}=\sum_{j=1}^n\sum_{k=1}^{n_j}\frac{\hat{p}^2_{jk}}{2}+\frac{1}{2}\omega_{jk}^2\bigg(\hat q_{jk}+\frac{\hat\sigma_z c_{jk}}{\omega_{jk}^2}\bigg)^2$$

where $$\omega_{jk}=\omega_{j}$$ and $$\sum_k c_{jk}^2 = c^2_j$$, this choice will become clear later, and a spectral density given by

$$J(\omega) = \frac{\pi}{2}\sum_{j=1}^n\sum_{k=1}^{n_j} \frac{c_{jk}^2}{\omega_{jk}}\delta(\omega-\omega_{jk})=\frac{\eta\gamma\Omega^2\omega}{(\omega^2-\Omega^2)^2+\gamma^2\omega^2}$$

with parameters chosen such that the collective coordinate $$y=\frac{\sum_{j,k}c_{jk}q_{jk}}{\sqrt{\sum_{j,k}c^2_{jk}}}$$ is 'classical' $$\Omega\ll k_BT$$, the two equilibrium positions are well separated $$\eta\gg\hbar\Omega$$, and for the sake of simplicity is in the moderately over-damped regime $$\gamma=4\Omega$$.

Now we can consider the form of the initial density $$\hat\rho(0)=\hat\rho_s\otimes\hat\rho_b$$, which we define to be

$$\hat\rho_b=\prod_{j=1}^{n}\prod_{k=1}^{n_j} \hat{\rho}_{jk}$$

with $$\hat\rho_{jk}=|\alpha_{jk}\rangle\langle\alpha_{jk}|$$ where

$$\hat{H}_{jk}|\alpha_{jk}\rangle=\bigg(\frac{\hat{p}^2_{jk}}{2}+\frac{1}{2}\omega_{jk}^2\hat q_{jk}^2\bigg)|\alpha_{jk}\rangle=E_{\alpha_{jk}}|\alpha_{jk}\rangle$$

is an eigenstate of $$\hat{H}_{jk}$$ with eigenvalue $$E_{\alpha_{jk}}$$. We can define $$\rho_{jk}$$ by randomly sampling each $$E_{\alpha_{jk}}$$ from the Boltzmann distribution at temperature $$T$$. Now taking the limit that each $$n_j\to\infty$$ followed by $$n\to\infty$$, it follows that the dynamics of the pointer $$y$$ and the 2 level system are equivalent to that of the original Hamiltonian given at the very top but with an initial density

$$\hat\rho(0)=\frac{\hat\rho_s\otimes e^{-\beta \sum_{j=1}^n\frac{\hat{p}^2_j}{2}+\frac{1}{2}\omega_j^2\hat q_j^2}}{\mathrm{tr}[\hat\rho_s\otimes e^{-\beta \sum_{j=1}^n\frac{\hat{p}^2_j}{2}+\frac{1}{2}\omega_j^2\hat q_j^2}]}$$

I believe this is then a relatively straightforward problem to solve, and should lead to dynamics of the form described in the first edit.

Any comments as to errors I have made would be appreciated, also feel free to ignore all but the first line of the post.

• Can you please give details on $n$ and the $c_j$'s? Commented Mar 28, 2019 at 0:00
• @ZeroTheHero the behaviour of the spin system is fully characterised by the spectral density, note $n\to\infty$ as $J(\omega)$ is continuous. I am not sure if the way the limit is taken will affect the answer to my question, I sense it might.
– J.L.
Commented Mar 28, 2019 at 6:57
• It is not clear to me what you mean by "model of a quantum measurement". Would you mind detailing what you expect from such a model, or maybe use more common terminology? Commented Mar 28, 2019 at 9:19
• @StéphaneRollandin I have now added a section describing what I think I should mean by "model of a quantum measurement"
– J.L.
Commented Mar 28, 2019 at 10:59