Optimal estimation of matter-field coupling strength in the dipole approximation

This paper is devoted to the study of the Bayesian-inference approach in the context of estimating the dipole coupling strength in matter-field interactions. In particular, we consider the simplest model of a two-level system interacting with a single mode of the radiation field. Our estimation strategy is based on the emerging state of the two-level system, whereas we determine both the minimum mean-square error and maximum likelihood estimators for uniform and Gaussian prior probability density functions. In the case of the maximum likelihood estimator, we develop a mathematical method which extends the already existing approaches to the variational problem of the average cost function. We demonstrate that long interaction times, large initial mean photon numbers, and nonzero detuning between two-level system transition and the frequency of the electromagnetic field mode have a deleterious effect on the optimality of the estimation scenario. We also present several cases where the estimation process is inconclusive, despite many ideal conditions being met.

Figure 1

Schematic representation of a quantum estimation scenario based on cavity QED. The atoms (gray dots) implementing the two-level systems are captured from a background gas by a magneto-optical trap and loaded into an optical conveyor belt [19]. The atoms move with the help of the conveyor belt into and out of the cavity and toward a detector. The transition frequency of the atom is ${\omega }_{e\leftrightarrow g}$. Further details about the scheme are in the text.

Figure 2

The eigenvalues of ${\widehat{M}}_{\text{min}}/g_0$ as a function of $g_0{\tau }_c$ in the case of the Gaussian prior PDF, with mean $g_0$ and variance ${\sigma }^2/g_0^2=1$. In the case $\alpha =0$, the two initial eigenvalues are 0 and $g_0$. When $\left|\alpha \right|>0$, the eigenvalues are plotted from ${\tau }_c=0^{+}$, at which time they are equal to $g_0$. For large values of ${\tau }_c$, the eigenvalues tend to the same value $g_0$. We set $\gamma {\tau }_f=0$, such that no spontaneous decay occurs.

Figure 3

Figures obtained for a Gaussian prior PDF, with mean $g_0$ and variance ${\sigma }^2/g_0^2=1$. We set $\gamma {\tau }_f=0$, such that no spontaneous decay occurs. In panel (a), each curve has a global minimum that decreases and shifts to larger values of $g_0{\tau }_c$ with increasing $\left|\alpha \right|$. At the time, when $\overline{C}$ attains its minimum, panel (b) displays biased average estimators, where the mean value $g_0$ of the prior PDF is depicted by a vertical line. The curves in panel (c) characterizing the accuracy of the estimation scenario have to be considered together with the appropriate curves in panel (b) in order to obtain a more complete information abut the optimal MMSE estimator. (a) The average minimum cost of error ${\overline{C}}_{\min }/g_0^2$ as a function of $g_0{\tau }_c$. (b) The average estimator $E\left[\tilde{g}\right|g]/g_0$ as a function of $g/g_0$. (c) The lower bound of the mean–squared error as a function of $g/g_0$.

Figure 4

The eigenvalues of ${\widehat{M}}_{\text{min}}/g_0$ as a function of $g_0{\tau }_c$ in the case of the uniform prior PDF with mean $g_0$ and variance ${\sigma }^2/g_0^2=1$. In the case $\alpha =0$, the two initial eigenvalues are 0 and $g_0$. The eigenvalues are plotted from ${\tau }_c=0^{+}$ for all $\left|\alpha \right|>0$ and their starting values are $g_0$. For large values of $g_0{\tau }_c$, the eigenvalues tend to the same value $g_0$, but slower than in Fig. 2. We set $\gamma {\tau }_f=0$.

Figure 5

Figures obtained for a uniform prior PDF. The parameters are set to the same value as in Fig. 3. The curves display very similar properties to those corresponding in Fig. 3. (a) The average minimum cost of error ${\overline{C}}_{\min }/g_0^2$ as a function of $g_0{\tau }_c$. (b) The average estimator $E\left[\tilde{g}\right|g]/g_0$ as a function of $g/g_0$. (c) The lower bound of the mean–squared error as a function of $g/g_0$.

Figure 6

The average minimum cost of error ${\overline{C}}_{\text{min}}/g_0^2$ reached at ${\tau }_c={\tau }_c^{*}$ as a function of the detuning $\mathrm{\Delta }$. The lowest average minimum cost of error is at resonance $\mathrm{\Delta }=0$. We set $\gamma {\tau }_f=0$, $\alpha =0$, and ${\sigma }^2/g_0^2=1$.

Figure 7

The average minimum cost of error ${\overline{C}}_{\text{min}}/g_0^2$ reached at ${\tau }_c={\tau }_c^{*}$ as a function of $\gamma {\tau }_f$. The minimum is reached when $\gamma {\tau }_f=0$ and approaches its limit value ${\sigma }^2/g_0^2=1$ with increasing $\gamma {\tau }_f$. We set $\mathrm{\Delta }=0$ and $\alpha =0$.

Figure 8

The average minimum cost of error ${\overline{C}}_{\text{min}}/g_0^2$ as a function of $g_0{\tau }_c$. The blue (solid) line represents the ideal case: $\gamma =\kappa =0$. The two experimental curves refer to the strong coupling regime ($\gamma /g_0=0.014$, $\kappa /g_0=0.246$) of Colombe et al. [25] and to the intermediate coupling regime ($\gamma /g_0=0.6$, $\kappa /g_0=0.6$) of Ritter et al. [26]. Top: Gaussian prior PDF with mean $g_0$ and variance ${\sigma }^2/g_0^2=1$. Bottom: Uniform prior PDFwith mean $g_0$ and variance ${\sigma }^2/g_0^2=1$. We set $\gamma {\tau }_f=0$, such that no spontaneous decay occurs during the time when the TLS system reaches the detectors.

Figure 9

Figures obtained in the case of the Gaussian prior PDF. We set ${\sigma }^2/g_0^2=1$. We see that spontaneous decay reduces the average cost function (a). In panel (b), we set $g=g_0$. We see there is a jump at $g_0{\tau }_c=\pi /2$ due to the properties of $c_{max}$ defined in Eqs. (44) and (45). In panel (c), the interaction time is ${\tau }_c=\pi /\left(4g_0\right)$. (a) The average maximum cost function ${\overline{C}}_{\max }{g}_0$ as a function of $g_0{\tau }_c$. (b) The average estimator $E\left[\tilde{g}\right|g_0]/g_0$ as a function of $g_0{\tau }_c$. (c) The lower bound of the mean–squared error as a function of $g/g_0$.