Quantum signaling in cavity QED

We consider quantum signaling between two-level quantum systems in a cavity in the perturbative regime of the earliest possible arrival times of the signal. We present two main results: First, we find that, perhaps surprisingly, the analog of amplitude modulated signaling (Alice using her energy eigenstates $\left|g\right.〉,\left|e\right.〉$, as in the Fermi problem) is generally suboptimal for communication, namely, e.g., phase-modulated signaling (Alice using, e.g., $\{\left|+\right.〉,\left|-\right.〉\}$ states) overcomes the quantum noise already at a lower order in perturbation theory. Second, we study the effect of mode truncations that are commonly used in cavity QED on the modeling of the communication between two-level atoms. We show that, on general grounds, namely for causality to be preserved, the UV cutoff must scale at least polynomially with the desired accuracy of the predictions.

Figure 1

Leading-order contribution $P_2$ to the single-detector excitation probability (18) for a detector at $x_b$ in a cavity of length $L=10$. The detector is resonant to the fourth field mode ($n=4$). The contribution $P_2$ is periodic with a periodicity of $T_{\mathrm{per}}=2L$. The number of valleys in every period is equal to $n$. For the calculation, a cutoff of $N_C=100$ modes was used. (All plotted quantities are dimensionless.)

Figure 2

Numerical values of the $O\left({\lambda }^4\right)$ coefficients defined in (19), (20), and (18) for the quantum channel in the Fermi problem. The length of the cavity is $L=10$ and the distance between the two detectors is $|x_{\mathcal{A}}-x_{\mathcal{B}}|=2$. (All plotted quantities are dimensionless.)

Figure 3

Numerical values of the lowest-order contribution $A_4$ from (19) to the signaling term in the Fermi problem for two detectors separated by $|x_A-x_B|=5$ for different switching times $T$, depending on the number of modes below the cutoff $N_C$. The biggest contribution is acquired around the resonance mode number $n=15$. For $N_C>n$, the results oscillate around a limiting value which is approached for higher cutoffs. For the lowest switching time, the detectors are spacelike separated during the interaction with the field, hence no signaling is possible. (All plotted quantities are dimensionless.)

Figure 4

The signaling term $A_4$ from (19) in the Fermi problem for two detectors at a distance of $|x_{\mathcal{A}}-x_{\mathcal{B}}|=2$ for different switching times $T$ for increasing cutoffs at $N_C$. The dashed line indicates the light cone. In general, the values of $A_4$ inside the light cone grow towards the light cone. Hence, to check the level of causality violation for a specific cutoff, the value on the light cone, i.e., for a switching time $T=|x_{\mathcal{A}}-x_{\mathcal{B}}|$, is relevant. (All plotted quantities are dimensionless.)

Figure 5

The plot shows ${max}_{N\ge N_C}\left|A_4\left(N\right)\right|$, the maximum value of the signaling term $A_4$ from (19) on the light cone $|A_4(T=|x_{\mathcal{A}}-x_{\mathcal{B}}\left|\right)|$ obtained for any higher cutoff, i.e., for any number of modes $N\ge N_C$ larger than or equal to $N_C$. For different $n$, the modes have been normalized as explained in the text. The values of $|A_4|$ on the light cone, i.e., for $T=|x_{\mathcal{A}}-x_{\mathcal{B}}|$, approach zero following a power law for large numbers of modes $N_C$. (All plotted quantities are dimensionless.)

Figure 6

Plot of $|C_2+D_2^{*}|$ as defined in (21), (22), the lowest-order contribution to the detection probability in (33), for different cutoffs $N_C$. The detectors are located at a distance of $|x_A-x_B|=2$. The dashed line indicates the light cone. The use of $|\pm \rangle$ states enhances signaling by two orders of magnitude in the coupling constant as compared to the use of energy eigenstates. (All plotted quantities are dimensionless.)

Figure 7

The plot shows ${max}_{N\ge N_C}|C_2\left(N\right)+D_2^{*}\left(N\right)|$ on the light cone, i.e., for $T=|x_A-x_B|=5$. Analogous to Fig. 5, also here the values for different $n$ have been normalized in order to be able to compare them better. Again, we observe a power-law decay for high cutoffs $N_C$. (All plotted quantities are dimensionless.)