Thermal-difference states of light: Quantum states of heralded photons

We introduce thermal-difference states (TDSs), a three-parameter family of single-mode non-Gaussian bosonic states whose density operator is a weighted difference of two thermal states. We show that the states of “heralded photons” generated via parametric down-conversion (PDC) are precisely those among the TDSs that are nonclassical, meaning they have a negative $P$ function. The three parameters correspond in that context to the initial brightness of PDC and the transmittances, characterizing the linear loss in the signal and the idler channels. At low initial brightness and unit transmittances, the heralded photon state is known to be a single-photon state. We explore the influence of brightness and linear loss on the heralded state of the signal mode. In particular, we analyze the influence of the initial brightness and the loss on the state nonclassicality by computing several measures of nonclassicality, such as the negative volume of the Wigner function, the sum of quantum Fisher information for two quadratures, and the ordering sensitivity, introduced recently by us [S. De Bièvre, D. B. Horoshko, G. Patera, and M. I. Kolobov, Phys. Rev. Lett. 122, 080402 (2019)]. We finally argue that the TDSs provide benchmark states for the analysis of a variety of properties of single-mode bosonic states.

Figure 1

Schematic representation of the production of heralded photons by means of PDC. NC—nonlinear crystal, where a two-mode squeezed state is generated with the rate $\xi$. PD—photodetector, whose click conditions (heralds) a quasi-single-photon state in mode A.

Figure 2

Parameter space of the thermal-difference states. Each point inside the cube corresponds to a positive density operator. The green plane $d=p$ is the border, above which the state corresponds to some value of the physical parameters $(\xi ,\eta ,\mu )$ of Fig. 1. The edge denominated “single-photon” is multivalued and corresponds to a state $(1-\mu )|0\rangle \langle 0|+\mu |1\rangle \langle 1|$, where $\mu$ determines the angle at which the edge is approached. The edge denominated “photon-added thermal” is also multivalued, see Sec. 3d.

Figure 3

(a) $P$ function, (b) Wigner function, and (c) photon number distribution for the TDS generated in a photon-heralding experiment with $\xi =\eta =\mu =0.5$. This corresponds to the parameters $(q,p,d)=(0.33,0.43,0.86)$. The negativity of the $P$ function establishes the nonclassicality of this conditionally prepared state. Note that the Wigner function is positive for this set of parameters. The vacuum component ($n=0$) appears in the photon number distribution due to loss in the signal channel ($\mu <1$).

Figure 4

Contour plot of the nonclassicality of TDSs (measured by OS) for $\eta =1$ and ranges of $\xi$ and $\mu$ as shown. One notices that the maximum value for OS, $S_{\mathrm{o}}=3$, is reached at $\mu =1, \xi \to 0$, corresponding to the single-photon state. OS decreases with growing $\xi$ and/or diminishing $\mu$. However, at low values of $\mu <0.3$, OS grows again and has the second local maximum $S_{\mathrm{o}}=1$ at $\mu \to 0, \xi \to 0$, corresponding to the vacuum state.

Figure 5

Contour plot of the nonclassicality of TDS (measured by OS) for $\xi =0.05$ and ranges of $\eta$ and $\mu$ as shown. One notices that the OS is highly sensitive to losses in the signal channel and is almost insensitive to the losses in the idler one.

Figure 6

Three witnesses of nonclassicality of TDSs (serving also as measures of nonclassicality when they detect it) as functions of the brightness. Decreasing curves show a trade-off between the nonclassicality and the brightness, taking place at any level of losses. We also see that OS goes below 1 at higher values of brightness than the sum of QFI, meaning it is a better witness for this family of states.

Figure 7

Comparison of three nonclassicality witnesses for TDS with $\eta =1$ and $\xi =0.01$. The value for the Wigner negative volume is scaled to other two witnesses and is given by $3N_{\mathrm{W}}\left(\rho \right)/N_{\mathrm{W}}\left(\right|1\rangle \langle 1\left|\right)$. In the region $0.5<\mu \le 1$, all three witnesses detect the nonclassicality of the state, ascribing to it different measures. In the region $0<\mu \le 0.5$, the OS and the sum of QFI may be growing with the increase of loss. However, in this region they ascribe no measure to the state.