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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.

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  • $\begingroup$ Can you please give details on $n$ and the $c_j$'s? $\endgroup$ – ZeroTheHero Mar 28 at 0:00
  • $\begingroup$ @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. $\endgroup$ – J.L. Mar 28 at 6:57
  • $\begingroup$ 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? $\endgroup$ – Stéphane Rollandin Mar 28 at 9:19
  • $\begingroup$ @StéphaneRollandin I have now added a section describing what I think I should mean by "model of a quantum measurement" $\endgroup$ – J.L. Mar 28 at 10:59

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