Emergent quantum probability from full quantum dynamics and the role of energy conservation

The electron-photon system is modeled as two quantum dots each owning single electron, separately coupled to two photonic resonators, shown in Fig. 1(a). The Hamiltonian is described as $H_{s}=\sum_{v={l},{r}}H_{s,v}$. The $v$-th system component is specified as

where $|e_{v}{\rangle}$ and $|g_{v}{\rangle}$ are the excited and ground states with the corresponding occupation energies $E_{v,e}$ and $E_{v,g}$ in the $v$-th quantum dot, $a^{\dagger}_{v}~{}(a_{v})$ is the creating (annihilating) operator of the photonic resonator with the photon energy $\omega_{0,v}$, $\lambda_{v}$ denotes the electron-photon coupling strength. In the following, we set $E_{e,{l}}=E_{e,{r}}=0$ without losing any generality.

Considering $H_{s,v}$ in the subspace $\{|n_{v}-1,e_{v}{\rangle},|n_{v},g_{v}{\rangle}\}$ with $n_{v}$ being the photon excitation number, it is straightforward to find that the $n$-th eigenvalues of the electron-photon model are given by $E_{n_{v},\pm}=\omega_{0,v}n_{v}+\frac{E_{g,v}-\omega_{0,v}}{2}\pm\sqrt{(\frac{\omega_{0,v}+E_{g,v}}{2})^{2}+\lambda^{2}_{v}n_{v}}$, and the eigenvectors are given by

with $\tan\theta_{n_{v}}=2\lambda_{v}\sqrt{n_{v}}/(\omega_{0,v}+E_{g,v})~{}(n_{v}{\geq}1)$. Moreover, we define the effective ground state $|\psi_{0_{v}}{\rangle}=|0,g_{v}{\rangle}$ and the corresponding energy $E_{0,v}=E_{g,v}$. It need note that the real bare ground state actually is $|0,g_{v}{\rangle}$. However, this state will never take participate into the dissipative dynamics of the dot-photon system.

We consider master dynamics of the electron-photon model with weak photon dissipation uweiss2012book ; hpb2002book ; pstegman2022acs

where $\kappa_{v}$ means the dissipation strength of photons from the $v$-th resonator into the corresponding zero-temperature environment (e.g., the $v$-th photon counter). Here, the dissipation of quantum dots is assumed negligible compared to the photon damping strength. Then, we select the initial state of the electron-photon model. There exists one electron in each quantum dot, i.e., $|\phi_{e}(0)=C_{l}|e_{{l}},g_{{r}}{\rangle}+C_{r}|g_{{l}},e_{{r}}{\rangle}$, and resonators are initially both in vacuum before the procedures of the probability readout. Hence, the initial wavefunction is generally described as

where $C_{v}$ is the wavefunction coefficient.

We focus on the individually measuring the quantum probability in the local dots man2002book ; jcorler2020prl . Consequently, the initial reduced density matrix of the $v$-th system component is obtained as

where the Born’s quantum probabilities are expressed as

with ${\overline{v}}={r}$ for $v={l}$ and ${\overline{v}}={l}$ for $v={r}$. Hence, the dissipative dynamics of the $v$-th component is expressed as

with $\rho_{{l}({r})}=\textrm{Tr}_{{r}({l})}\{\rho_{s}\}$, which is also exhibited in Fig. 1(b). Moreover, the dynamics is constraint to the lowest three eigenstates, i.e., $|\psi_{1_{v},+}\rangle$, $|\psi_{1_{v},-}\rangle$, and $|\psi_{0_{v}}{\rangle}$. And it is straightforward to show the inverse relations $|0_{v},e_{v}{\rangle}=(\sin{\frac{\theta_{v}}{2}}|\psi_{1_{v},+}\rangle+\cos{\frac{\theta_{v}}{2}}|\psi_{1_{v},-}\rangle)$ and $|1_{v},g_{v}{\rangle}=(\cos{\frac{\theta_{v}}{2}}|\psi_{1_{v},+}\rangle-\sin{\frac{\theta_{v}}{2}}|\psi_{1_{v},-}\rangle)$, with $\tan{\theta_{v}}={2\lambda_{v}}/{(\omega_{0,v}+E_{g,v})}$.

We first study the probability measurement based on the quantum dots and photonic resonators in the long time evolution limit, i.e., complete measurement. Then, we further analyze the finite time behavior of the emitted photons, i.e., finite time measurement. Finally, we briefly discuss the influence of finite temperature reservoir on measured probability.

When the electron-photon coupling becomes strong and even comparable with the bare frequency of the optical resonator, the counter-rotating terms of the electron-photon interaction should be considered mk2014prb ; bka2016prb ; jclu2019prb . The system then becomes analogous with the seminal quantum Rabi model dbraak2011prl ; afk2019nrp ; pfd2019rmp , is modified to

Due to the inclusion of the counter-rotating terms $(a^{\dagger}_{v}|e_{v}{\rangle}{\langle}g_{v}|+a_{v}|g_{v}{\rangle}{\langle}e_{v}|)$, the particle conservation breaks down, whereas the energy conservation is still valid. Accordingly, we find that the analytical eigensolution of $H_{s,v}$ is hard to obtain until now, except the solution mapped to the special function via Bargmann algebra dbraak2011prl and photonic coherent states qhchen2012pra . Here, we include the master equation Eq. (II.2) to study the influence of strong electron-photon interaction on probability measurement.

With the initial state $|\phi(0)=|0_{l},e_{{l}}\rangle{\otimes}|0_{r},g_{{r}}\rangle$, we first plot Fig. 4(a) to study the short-time dissipative dynamics of average photon number ${\langle}a^{\dagger}_{l}a_{l}{\rangle}(t)$ at strong electron-photon coupling, e.g., $\lambda=0.5\omega_{0}$, by comparing with the counterpart based on the system Hamiltonian at Eq. (II.1). It is interesting to find that the photon number shows the aperiodic oscillation due to the effect of counter-rotating terms, which is significantly distinct from the one with the electron-photon interaction $(a^{\dagger}_{l}|g_{l}{\rangle}{\langle}e_{l}|+a_{l}|e_{l}{\rangle}{\langle}g_{l}|)$. Furthermore, the peak of photon number breaches the unity, which is unavailable in absence of counter-rotating terms.

Next, we study the photon number dynamics at long time evolution in Fig. 4(b). It is shown that by increasing the electron-photon coupling strength, the aperiodicity degree becomes increasingly apparent. Moreover, it is interesting to find that after long time evolution the photon number becomes nonvanishing. This directly demonstrates the breakdown of the complete measurement at Eq. (14) with weak electron-photon interaction. We plot the inset of Fig. 4(b) to quantify the robustness of the photon number. ${\langle}a^{\dagger}_{l}a_{l}{\rangle}(t{\rightarrow}\infty)$ becomes monotonic enhancement with the increase of $\lambda$.

As is known that after long time evolution the system may reach the steady state, which generally may not be the eigenstate, e.g., the ground state at zero temperature. In particular for rather strong electron-photon coupling, by considering the degenerate perturbation theory, the eigenvectors of $H_{s,l}$ at Eq. (III.2) are estimated as

with $|\pm{\rangle}=(|e_{{l}}{\rangle}{\pm}|g_{{l}}{\rangle})/\sqrt{2}$ and $a_{l}|0{\rangle}_{a}=0$. And the corresponding eigenvalues become $E_{n,\pm}\approx\omega_{0}n{\mp}|D_{nn}(2\lambda/\omega_{0})|^{2}E_{g}/2-\lambda^{2}/\omega_{0}+E_{g}/2$, with the coherent state overlap coefficient $D_{nn}=e^{-2(\lambda/\omega_{0})^{2}}\sum^{n}_{r=0}(-1)^{r}n!(2\lambda/\omega_{0})^{2n-2r}/[(n-r)!(n-r)!r!]$ qhchen2008pra . Then, the transition rates between two eigenstate populations via the master equation could be reestablished. Specifically, by including the secular approximation the population dynamics is given by $dP_{n,\pm}/dt{\approx}[\kappa{(n+1)}P_{n+1,\pm}-\kappa{n}P_{n,\pm}]+\kappa(\lambda/\omega_{0})^{2}(P_{n,\mp}-P_{n,\pm})$, with the density matrix elements $P_{n,\pm}={\langle}\psi_{n,\pm}|\rho_{l}|\psi_{n,\pm}{\rangle}$ and the transition coefficients ${\langle}\psi_{n-1,\pm}|a_{l}|\psi_{n,\pm}{\rangle}=\sqrt{n}$ and ${\langle}\psi_{n-1,\pm}|a_{l}|\psi_{n,\mp}{\rangle}=0$. The dissipative dynamics generally results in the steady state

Thus, the corresponding steady-state photon number is given by ${\langle}a^{\dagger}_{l}a_{l}{\rangle}(t{\rightarrow}\infty){\approx}({\lambda}/{\omega_{0}})^{2}$. This partially explain the nonvanishing steady-state value of the average photon number at strong electron-photon coupling in the inset of Fig. 4(b). Therefore, we conclude that strong electron-photon interaction generally breaks the microscopic particle conservation and complete measurement.

We study the effect of finite-temperature reservoir on the quantum measurement of probability. At weak electron-photon coupling $\lambda_{v}/\omega_{0,v}{\ll}1$, the master equation for dissipative dynamics of the electron-photon model can be described as

where the dissipator denotes $\mathcal{L}[a_{v}]\rho_{s}=(2a_{v}\rho_{s}a^{\dagger}_{v}-a^{\dagger}_{v}a_{v}\rho_{s}-\rho_{s}a^{\dagger}_{v}a_{v})/2$. Here, we empirically describe the incoherent coupling between the photon resonator and the bosonic reservoir at the temperature $T_{v}$ with the Bose-Einstein distribution function $\overline{n}_{v}=1/[\exp(\omega_{0,v}/k_{B}T_{v})-1]$ jmfink2010prl ; tbosen2023prl . We select the initial state of the reduced $H_{s,v}$ as

with the partition function $\mathcal{Z}_{v}=1/[1-\exp(-\beta\omega_{0,v})]$ and $\beta_{v}=1/k_{B}T_{v}$.

We investigate the energy flow into the $v$-th reservoir. At low temperature regime, it is numerically found that the dissipative dynamics of the flow is typically contributed by three lowest eigenstates, i.e., $|\psi_{1_{v},\pm}{\rangle}$ and $|\psi_{0_{v}}{\rangle}$. Accordingly, relation between the energy flow and the photon number can be expressed as

with $P_{\pm,v}={\langle}\psi_{1_{v},\pm}|\rho_{v}|\psi_{1_{v},\pm}\rangle$, $P_{0,v}={\langle}\psi_{0_{v}}|\rho_{v}|\psi_{0_{v}}\rangle$, and $\Lambda_{v}=\sqrt{(\omega_{0,v}+E_{g,v})^{2}+4\lambda^{2}_{v}}$. Moreover, the total energy loss of the quantum system is obtained as $\int^{\infty}_{0}J_{v}dt=[{\langle}H_{s,v}{\rangle}(0)-{\langle}H_{s,v}{\rangle}(\infty)]$, which is approximated as $\int^{\infty}_{0}J_{v}dt{\approx}E_{g,v}(1/[1+\exp(-\beta_{v}E_{g,v})]-|C_{v}|^{2})$. Hence, based on the microscopic energy conservation law, we obtain the effective probability measurement by integrating the current at Eq. (23) as

with $\mathcal{Z}^{\prime}_{v}=(1+e^{-\beta_{v}E_{g,v}})$. If the temperature approaches zero, i.e., $\overline{n}_{v}=0$, the effective relation is naturally reduced to the complete probability measurement relation, i.e., $\kappa_{v}\int^{\infty}_{0}dt{\langle}a^{\dagger}_{v}{a}_{v}{\rangle}(t)=|C_{v}|^{2}$ at Eq. (14).

Alternatively, the dynamical equation of the ground state probability is described as

in which the change of the probability is contributed by the photon flow into the reservoir and the backward flow from the reservoir thermal noise. Straightforwardly, the effective probability measurement at Eq. (24) can be recovered from Eq. (25) by considering $P_{0,v}(0){\approx}|C_{\overline{v}}|^{2}/\mathcal{Z}_{v}$ and $P_{0,v}(\infty){\approx}1/\mathcal{Z}_{v}\mathcal{Z}^{\prime}_{v}$.