is a quantum superposition of the values 0 and 1 with no classical analog. By combining K > 1 qubits, more complex states can be created. We can either produce separable states, as | Ψ 1 > | 1 Ψ 2 > …| 2 Ψ K > K, where the qubits are independent from one another, or entangled states, as
— 1 –(| 0 √ 2 > | 1 0 > …| 2 0 > K +| 1 > | 1 1 > …| 2 1 > K). This
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is a quantum superposition of the values 0 and 1 with no classical analog. By combining K > 1 qubits, more complex states can be created. We can either produce separable states, as | Ψ 1 > | 1 Ψ 2 > …| 2 Ψ K > K, where the qubits are independent from one another, or entangled states, as
— 1 –(| 0 √ 2 > | 1 0 > …| 2 0 > K +| 1 > | 1 1 > …| 2 1 > K). This
state, analogous to a Schrödinger cat states, has no classical counterpart. While for separable states it has been shown that the QFI scales as ≤K, for entangled states we have that ≤K 2, so such states can lead up to a quadratic enhancement of precision. This is also the maximal possible precision enhancement that can occur in quantum metrology, and it is called the Heisenberg limit. Importantly, this enhancement occurs without increasing the resources which are, in the present case, the number of qubits. We now see how these results manifest in the context of quantum optics.
USING QUANTUM OPTICS FOR QUANTUM METROLOGY We now detail the specific case of time precision limits in quantum optics. The free evolution leads to a time dependent phase to photon states, and temporal delays can be easily implemented in interferometers by creating a path length difference between two propagation modes. The recombination of the propagation modes leads to interference, which enables the measurement of this delay. The QFI associated with the estimation of temporal delays can then be calculated from the free propagation Hamiltonian and the
Figure 1. Experimental setup for investigating the metrological performance of the Hong- Ou-Mandel( HOM) experiment, showing the generation, joint spectral amplitude( JSA) engineering, and HOM interferometer stages.
state used as a probe. For a single frequency mode field, the free propagation Hamiltonian is given by ħωn^ where ω is the field’ s frequency and n^ an operator counting the number of photons of the field. However, it takes a slightly more complex form in the general case, when the field can be decomposed into different frequency modes. In this case, we can still obtain an intuitive expression for the QFI, which reads, for a single mode state,
= 4( ∆ 2 ωn – + ∆ 2 n^ω— 2), where ∆ 2 ω is the frequency variance of the mode,— ω is average frequency, n – is the average number of photons, and ∆ 2 n^ the photon number variance. In general, the total photon number is not fixed, and paradigmatic states, as Gaussian states( squeezed and coherent states), consist of superpositions of different photon numbers. The expression of the QFI for a single mode is elegant, for treating symmetrically modal( frequency variables) and statistical( photon number variance) properties of the field.
In early quantum optics metrology, only single-mode states were considered. Coherent states, which are classical-like states, are an example of a single-mode state. They are a special case of Gaussian states for which the photon number follows a Poissonian distribution. The QFI of a coherent state is given by = 4n –— ω 2. We see that the average photon number appears
here as the analog to the number of qubits K, and we recover the known classical result that the precision in time estimation is limited by the inverse of the frequency for monochromatic fields. Photons in a coherent state behave as independent particles, explaining the observed analogy with the separable states of qubit systems described earlier.
If we fix the average field’ s energy as the available resource( i. e., ħn – ω—), how can we obtain the quadratic scaling in precision observed in qubit-based quantum metrology? It has been shown [ 1 ] that the optimal metrological strategy concentrates all resources into a single mode. In this case, the frequency variance appears as a classical-like quantity, and doesn’ t play a role in the scaling of precision enhancement. However, some non-classical states can enable ∆ 2 n^ = n – 2 Again, the average photon number plays the same role as the number of qubits, leading to a quadratic precision enhancement— the Heisenberg limit.
The previous results assume that modal properties( frequency) and statistical properties( photon number) are independent. However, this is not always the case. A key example is photon pairs, commonly used in Bell inequality experiments [ 2 ]. These states exhibit mode and particle entanglement and cannot be described as single-mode states. Frequency modes can also be entangled for photon pairs [ 3 ], and in this case, the QFI must be modified to account for mode correlations.
To explore this scenario, let us consider a system with K photons and K propagation modes. Each mode can host a single photon with its own frequency distribution. The free evolution Hamiltonian now depends on both propagation direction and frequency distribution in each mode. For a single propagation mode, the QFI follows from the single-mode expression: i = 4∆ 2 ω^i( where we have used the general expression for the QFI of a single mode, labled i, in the case where n – i = 1, one photon per
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