Experimental tomography of NOON states with large photon numbers

We have performed experimental quantum state tomography of NOON states with up to four photons. The measured states are generated by mixing photons from a classical coherent state with down-converted photon pairs. We show that the fidelity between the produced states and the ideal NOON states is high. The fidelity is limited by the overlap of the two-photon down-converted state with any two photons originating from the coherent state, for which we introduce and measure a figure of merit. A second limitation on the fidelity set by the total setup transmission is discussed. We also apply the same tomography procedure for characterizing correlated photon hole states.

Figure 1

Experimental setup for quantum state tomography of a multiphoton state in two modes. (a) Schematic of the setup depicting a prepared beam forming a multiphoton quantum state in two orthogonal polarization modes ${\widehat{a}}_1$ and ${\widehat{a}}_2$, undergoing a linear state transformation introduced by a half wave plate (HWP) and a liquid crystal (LC). The transformed state is photon-number resolved, in one of its polarization modes ${\widehat{b}}_1$. (b) Detailed layout of the setup. A pulsed Ti:Sapphire oscillator with 120 fs FWHM pulse width and 80 MHz repetition rate is doubled using a $2.74$mm LBO crystal to obtain 404nm ultra-violet pulses with maximum power of 225mW. These pulses then pump collinear degenerate type-I SPDC at a wavelength of 808nm using a $1.78$mm long BBO crystal. The SPDC ($\widehat{H}$ polarization) is spatially and temporally overlapped with a CS ($\widehat{V}$ polarization) using a polarizing beam-splitter cube (PBS). A thermally induced drift in the relative phase, ${\varphi }_{cs}$, between $\widehat{H}$ and $\widehat{V}$ polarizations is corrected every few minutes using a LC phase retarder. The coherent light amplitude is adjusted using a variable attenuator. Modes ${\widehat{a}}_1$,${\widehat{a}}_2$ are realized collinearly at the $\pm {45}^{\circ }$ $\widehat{X},\widehat{Y}$ polarizations respectively. These modes then pass through a LC aligned with the $\widehat{X}$, and $\widehat{Y}$ polarizations introducing a relative phase shift $\phi$ between them, and a HWP rotated by $\theta$ in the $\widehat{H}$ and $\widehat{V}$ plane. Modes ${\widehat{b}}_1$, ${\widehat{b}}_2$ are oriented at polarization axes $\widehat{H}$ and $\widehat{V}$ respectively. The spatial and spectral modes are matched using a 2m long PMF and a 3nm FWHM band-pass (BPF) filter centered at 808nm. Photon number resolving detection is performed using multiple single photon counting modules (Perkin Elmer). Additional components: Dichroic mirrors, short-pass filter (SPF).

Figure 2

Two photon overlap measurement of CS and SPDC. 2-fold coincidence measurement of an output port of a beam splitter, $P_{b1}^{\left(2\right)}$, fed by CS in one of its input port and SPDC in the other. Error bars indicate $\pm \sigma$ statistical uncertainty. Solid line is obtained by fitting the data according to Eq. (10) resulting with an overlap of $|\langle {\Psi }_{SPDC}|{\Psi }_{CS}\rangle |=0.970\pm 0.020$ (with $95\%$ confidence bound, see text).

Figure 3

Experimental results: real and imaginary parts of density matrices of (a) NOON states for $N=2,3,4$ (b) correlated photon holes of 3,0 and 0,3 (in colored bars), reconstructed in the photon-number basis of two modes states labeled $|m,n\rangle$, where m and n are the number of photons in polarization modes ${\widehat{a}}_1$ and ${\widehat{a}}_2$ respectively (see Fig. 1). Solid frames represent density matrices of the ideal states.

Figure 4

NOON states fidelity dependence on setup transmission $\eta$. Fidelity of the states generated by our setup to ideal NOON states are shown as a function of the setup transmission $\eta$ for $N=2,3,4$ in blue, red and green respectively. Simulations (solid lines) are shown together with the measurements (squares) in the current setup transmission, $\eta =0.11$. Error bars indicate $\pm \sigma$ statistical uncertainty.