Quantum measurements of atoms using cavity QED

Generalized quantum measurements are an important extension of projective or von Neumann measurements in that they can be used to describe any measurement that can be implemented on a quantum system. We describe how to realize two nonstandard quantum measurements using cavity QED. The first measurement optimally and unambiguously distinguishes between two nonorthogonal quantum states. The second example is a measurement that demonstrates superadditive quantum coding gain. The experimental tools used are single-atom unitary operations effected by Ramsey pulses and two-atom Tavis-Cummings interactions. We show how the superadditive quantum coding gain is affected by errors in the field-ionization detection of atoms and that even with rather high levels of experimental imperfections, a reasonable amount of superadditivity can still be seen. To date, these types of measurements have been realized only on photons. It would be of great interest to have realizations using other physical systems. This is for fundamental reasons but also since quantum coding gain in general increases with code word length, and a realization using atoms could be more easily scaled than existing realizations using photons.

Figure 1

Geometrical representation of the IDP measurement. The initial two-dimensional Hilbert space, where $|{\psi }_1\rangle$ and $|{\psi }_2\rangle$ live, is spanned by $\{|1\rangle$, $|2\rangle \}$. This is extended to three dimensions by adding an extra state $|3\rangle$ which is orthogonal to each of the two initial basis states. The measurement consists of a unitary operation $\mathrm{Û}$, followed by a standard projective measurement in the $\{|1\rangle$, $|2\rangle$, $|3\rangle \}$ basis. The transformed states $\mathrm{Û}|{\psi }_1\rangle$ and $\mathrm{Û}|{\psi }_2\rangle$ are orthogonal to $|2\rangle$ and $|1\rangle$, respectively.

Figure 2

Schematic diagram of proposed cavity QED implementation of the POM for superadditive decoding. This figure shows the scheme for the unitary operation performed before the final projective measurement by field-ionization detectors. Atom mirrors have been implemented using electric, magnetic, and light-induced forces (see Refs. [43, 44] and references therein). If required, it could be realized in this scheme also by using an additional Tavis-Cummings operation to swap the state from one atom to that of a fresh one traveling perpendicularly. The superscripts on the operators indicate the order of their application to the two-atom state. These operations are defined in Eqs. (31), (32), and (34).

Figure 3

Plot of the superadditive coding gain as a function of $\delta$, where $\delta$ is the delay $t_d(s)$ as a percentage of the time $t_2$ $(s)$ spent by both atoms in the second cavity. The same delay is applied in both cavities.

Figure 4

Effects of detection errors when ${\mathrm{Û}}_{\mathrm{sa}}$ is used to realize the superadditive measurement. (a) Plot of the difference $G_0$ (bits) between the length-two coding mutual information and the ideal single channel capacity, $G_0=I_2/2(p,q)-C_1(p=0,q=0)$. (b) Plot of the actual superadditive coding gain $G_0$ (bits), $G=I_2/2(p,q)-C_1(p,q)$. $G_0$ and $G$ are plotted as functions of the probabilities, $p$ and $q$, of misidentifying states $|0\rangle$ and $|1\rangle$ of a two-level atom respectively.

Figure 5

Effects of detection errors when ${\mathrm{Û}}_{\mathrm{sa}}^{\prime }$ is used to implement the superadditive measurement. (a) Plot of the difference $G_0^{\prime }$ (bits) between the length-two coding mutual information and the ideal single channel capacity, $G_0^{\prime }=I_2^{\prime }/2(p,q)-C_1(p=0,q=0)$. (b) Plot of the actual superadditive coding gain $G^{\prime }$ (bits) for our proposed scheme, where $G^{\prime }=I_2^{\prime }/2(p,q)-C_1(p,q)$. $G_0^{\prime }$ and $G^{\prime }$ are plotted as functions of the probabilities, $p$ and $q$, of misidentifying states $|0\rangle$ and $|1\rangle$ of a two-level atom respectively.