– 2(| ω 1 > …| ω K > + | ω ' 1 > …| ω ' K >), where ω i≠ ω ' i
LIGHT AND COLOR FOCUS propagation mode). For K independent photons( in independent propagation modes), if we suppose, for simplifying reasons and without loss of generality, that all the variances are the same, the
total QFI is given by =
K
∑ i i = 4∆ 2 ω^ K.
This result resembles the QFI of coherent states and separable qubit states, since the precision scales linearly with the number of photons.
We can now consider a system of maximally entangled photons, both in frequency and in propagation direction. An example of such a state is
1
— √
– 2(| ω 1 > …| ω K > + | ω ' 1 > …| ω ' K >), where ω i≠ ω ' i
for all values of i, that labels auxiliary modes. This state is the generalization of a pair of polarization entangled photons, where each photon is on a different propagation mode. For this state, the frequency variance in each mode is ∆ 2 ω i = 1— 4
( ω i – ω ' i) 2. If the variance is the same for each propagation direction, the QFI becomes = ∆ 2 ωK 2. Then, analogously to the case of entangled qubits, we recovered the quadratic scaling of precision with the number of particles( here, photons instead of qubits). However, now precision is also proportional to the frequency variance for each mode. Interestingly, in this case, we observe at the same time a wavelike behavior of the scaling and a particle-like one.
We have provided a broad picture highlighting how both modal correlations and particle statistics can lead to precision enhancement in quantum optics [ 4 ]. Nevertheless, two natural questions appear: the first one is, how can one engineer frequency entangled states, that permit achieving the Heisenberg precision limit? On the other hand, what is the optimal measurement strategy, for which the Fisher information is equal to the QFI for this type of system? We discuss these two questions in the next section.
QUANTUM METROLOGY USING THE HONG-OU-MANDEL EXPERIMENT Theoretically, the Hong-Ou-Mandel( HOM) interferometer [ 5 ] was shown to be an optimal measurement tool as it allows the quantum Cramér-Rao bound to be reached under ideal conditions of perfect visibility, where photons either perfectly bunch or anti-bunch. Not only it enables reaching the QFI, but also, in principle, the Heisenberg scaling of precision( even though this type of experiment is limited to two photons, precision is proportional to the square of the number of photons). Several experiments have shown the approach to this optimal precision using different two-photon frequency entangled states [ 6 ].
However, in reality, perfect visibility is unattainable due to experimental limitations. Interestingly, the
Figure 2. Scaling of the ratio F ~ V / for different biphoton states with respect to the HOM visibility V. For SC-like states a maximal value of F ~ V / is attained for V = 99.4 %.
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