Mode invisibility as a quantum nondemolition measurement of coherent light

We exploit geometric properties of quantum states of light in optical cavities to carry out quantum nondemolition measurements. We generalize the “mode-invisibility” method to obtain information about the Wigner function of a squeezed coherent state in a nondestructive way. We also simplify the application of this nondemolition technique to measure single-photon and few-photon states.

Figure 1

Measurement setup to detect the properties of an unknown state of light trapped in a cavity using a known control light state in another cavity as a reference. A detector moving at constant $v$ is sent into the interferometer to probe the unknown light state on resonance, but in a nondestructive manner using the mode-invisibility technique.

Figure 2

Phase acquired by an atom crossing a cavity with a squeezed coherent field trapped inside, from left to right. Left: as a function of the relative phase difference $\Psi$ between the coherent operator and the squeezed operator; middle: as a function of the amplitude $r$ of the squeeze operator $S(r,\varphi )$; right: as a function of the amplitude $\left|\alpha \right|$ of the coherent state operator $D\left(\alpha \right)$. Each figure shows the sensitivity of the phase to the state parameters. The physically realizable squeeze parameter range is shown in the inset in the middle figure. We see that in that regime reasonable sensitivity to the phase is attained. However, for large squeeze amplitudes $r>5$, the phase value ceases to be sensitive to the squeeze amplitude.

Figure 3

Phase acquired by an atom that interacts with a coherent state as a function of the coherent state amplitude $\left|\alpha \right|$. The inset shows an exponential curve for small values of $\left|\alpha \right|$. For large values, the curve plateaus, making it increasingly difficult to gain information about $\left|\alpha \right|$.

Figure 4

Phase acquired by an atom that interacts with a coherent state as a function of the amplitude of the squeeze operator $r$. The inset shows an exponential curve for small values of $r$. The curve eventually plateaus, making it increasingly difficult difficult to gain information about $r$.

Figure 5

Visibility factor for the cases plotted in Fig. 2. Left figure- as a function of the relative phase difference $\Psi$ between the coherent operator and squeezed operator; middle figure- as a function of amplitude $r$ of the squeeze operator $S(r,\varphi )$; right figure- as a function of the amplitude $\left|\alpha \right|$ of the coherent state operator $D\left(\alpha \right)$.

Figure 6

Phase resolution to distinguish the relative phase difference acquired by an atom that interacts with two different squeezed coherent states. Here we plot the relative phase difference between $\Psi$ and $\Psi +\delta \Psi$ for each squeezed coherent state. Different curves with $\delta \Psi =0.5\pi ,0.4\pi ,0.3\pi ,0.2\pi$, and $0.1\pi$ are shown for the values of $\left|\alpha \right|=r=1$.

Figure 7

Phase resolution required to distinguish between a coherent state with amplitude $\left|\alpha \right|$ and another coherent state with amplitude $|\alpha +\delta \alpha |$. We show the plots for values of $\delta \alpha =1,2,3,4$, and 5, respectively.

Figure 8

Phase resolution required to distinguish between a squeezed vacuum state with amplitude $r+\delta r$ and another squeezed vacuum state with amplitude $r$. The difference curves are shown for $\delta r=1,2,3,4$, and 5, respectively. We see that we have more than enough resolution to differentiate between the amplitudes of two squeezed coherent states.

Figure 9

Phase resolution required to distinguish between a Fock state of unknown number $n$ of photons and another with $n+m$ photons, using a coherent state as a reference. Here we show the plots for different values of $m=5,10,15,20,25$, respectively.

Figure 10

Excitation transition probability of an atom as a function of the switching parameter $\epsilon$. We can see that the value of $P_e\left(\epsilon \right)$ is small for the given values of $\epsilon$.