Effect of losses on the performance of an SU(1,1) interferometer

We study the effect of losses on the phase sensitivity of the SU(1,1) interferometer for different configurations. We find that this type of interferometer is robust against losses that result from an inefficient detection system. This type of loss only introduces an overall prefactor to the sensitivity but does not change the $1/n$ scaling, where $n$ is the average number of particles inside the interferometer, characteristic of the Heisenberg limit. In addition, we show that under some conditions the SU(1,1) interferometer with coherent state inputs is also robust against internal losses. These results show that the SU(1,1) interferometer is a viable candidate for experimentally reaching the Heisenberg limit with current technology. Possible implementations of this interferometer using four-wave mixing in atomic vapors or an atom interferometer in a spinor Bose-Einstein condensate are compared.

Figure 1

Schematic drawing of an ideal SU(1,1) interferometer. An SU(1,1) interferometer is similar to a Mach-Zehnder interferometer except that the input and output beam splitters are replaced by parametric processes. The interferometer is typically studied in a balanced configuration, in which the parametric process in the second region is set to undo the interaction in the first region.

Figure 2

Uncertainty in the phase estimation of the ideal SU(1,1) interferometer with no input fields, Eq. (11), as a function of (a) $\phi$ for different photon numbers and (b) the number of spontaneous photons inside the interferometer, $n_s$, for different values of $\phi$.

Figure 3

Uncertainty in the phase estimation of the SU(1,1) interferometer in the presence of internal losses for the case of no inputs, Eq. (22). The contributions from (a) the ideal lossless SU(1,1) interferometer and from (b) the extra term that results from losses are out of phase. (c) Uncertainty when the losses are taken into account. ${\eta }_1=0.9$ and $n_s=5$.

Figure 4

(a) Uncertainty in the phase estimation of the SU(1,1) interferometer in the presence of losses for the configuration with coherent state inputs as a function of the normalized internal number of photons, $n_i/{\left(\right|\alpha |}^2+{\left|\beta \right|}^2)$, for different levels of internal loss $(1-{\eta }_1)$. The red dashed line indicates the Heisenberg limit while the red dotted line indicates the SQL. (b) Level of loss at which the extra noise term in Eq. (24) dominates vs the normalized number of photons inside the interferometer. Note that $n_i/{\left(\right|\alpha |}^2+{\left|\beta \right|}^2)=15$ corresponds to $s=1.7$.