Entangled Fock states for robust quantum optical metrology, imaging, and sensing

We propose a class of path-entangled photon Fock states for robust quantum optical metrology, imaging, and sensing in the presence of loss. We model propagation loss with beam splitters and derive a reduced density-matrix formalism from which we examine how photon loss affects coherence. It is shown that particular entangled number states, which contain a special superposition of photons in both arms of a Mach-Zehnder interferometer, are resilient to environmental decoherence. We demonstrate an order of magnitude greater visibility with loss than possible with path-entangled $\mid N,0⟩+\mid 0,N⟩$ states. We also show that the effectiveness of a detection scheme is related to super-resolution visibility.

Figure 1

(Color online) Interferometer with loss modeled by beam splitters in both arms. The reflectance of the beam splitters determines how many photons one lost. An accumulated unknown phase $\varphi$ is obtained due to a path length difference between the arms. The unitary operator for the phase shift is given by $\widehat{U}=\exp \left(i{\widehat{b}}^{†}\widehat{b}\varphi \right)$. A simple proof shows that this operator commutes with the beam-splitter operation. The placement of the beam splitter before the phase shift has been acquired, therefore, leads to the same result.

Figure 2

(Color online) Phase sensitivity $\delta \varphi$ for a ∣20∷10⟩ $\mid m\colon\colon m^{\prime }⟩$ state (curved black solid line) versus a ∣10∷0⟩ $N00N$ state (curved blue dashed line) undergoing loss. Loss is 40% in the long arm and zero in the delay arm. Bottom black solid line is the Heisenberg limit for ∣20∷10⟩, $1∕(m+m^{\prime })$. The red solid line is the Heisenberg limit for the ∣10∷0⟩ $N00N$ state and lossless limit for ∣20∷10⟩, $1∕(m-m^{\prime })$. The black dotted line is the shot-noise limit for ∣20∷10⟩, while the blue dashed line is the shot-noise limit for ∣10∷0⟩. The $N00N$ state is no longer below its shot-noise limit while the minimum phase sensitivity for the $\mid m\colon\colon m^{\prime }⟩$ state ∣20∷10⟩ is at its respective shot-noise limit.

Figure 3

(Color online) Phase resolution for a ∣20∷10⟩ $\mid m\colon\colon m^{\prime }⟩$ state (curved black solid line) versus a ∣10∷0⟩ $N00N$ state (curved blue dashed line) undergoing $3\mathrm{dB}$ of loss. Loss is $3\mathrm{dB}$ in the long arm, zero in the delay arm. The amplitude of $⟨\widehat{A}⟩$, and hence the visibility of the super-resolving sub-Rayleigh fringes, for a ∣20∷10⟩ state is 41%, while the ∣10∷0⟩ $N00N$ state visibility is 3.1%.

Figure 4

(Color online) (a) Visibility for ∣10∷0⟩ as a function of loss in the delay and long arms, $L_a$ and $L_b$, respectively. Contour lines represent the value of the visibility. (b) Same as (a) for the state ∣20∷10⟩.