Influence of losses on the wave-particle duality

We theoretically investigate the influence of losses on the wave-particle duality in a Mach-Zehnder interferometer. Balanced losses (equal loss on the two paths) have no effect on the visibility and predictability (also the duality relation), while unbalanced losses do have influence on them. If the unbalanced losses occur inside the interferometer, the losses have an effect on both the visibility and predictability, while the losses have no effect on the duality relation. If the unbalanced losses occur after the interferometer, the losses have no effect on the visibility, but do have an effect on the predictability (also the wave-particle duality), which can lead to a result that the duality relation exceeds 1, $P^2+V^2>1$. The influence of losses can be eliminated by exchanging the two detectors or the two inputs (one photon and the vacuum) of the interferometer and then averaging the two results. Consequently, we have the visibility, predictability, and duality relation $P^2+V^2\le 1$ independent of the unbalanced losses. The obtained $P$ and $V$ for the unbalanced losses in the two paths do not represent the original predictability and visibility, and the result of $P_D^2+V_D^2>1$ does not mean the “violation” of the original duality.

Figure 1

Losses inside the Mach-Zehnder interferometer. A single input particle is split by a variable beam splitter (VBS) with adjustable reflectivity $R$. The loss in the interferometer is modeled by the beam splitter, with reflectivity $R_1$ ($R_2$). PZT stands for the piezoelectric transducer and M stands for a normal mirror. Then the particle is recombined with a 50:50 beam splitter (BS) and detected by detectors 1 (${\mathrm{D}}_1$) and 2 (${\mathrm{D}}_2$). $a,b,a^{\prime }$ and $b^{\prime }$ stand for the path.

Figure 2

$P^2$ and $V^2$ versus $R$ for $R_1=0.4$ and $R_2=0.6$. The red and blue lines stand for $P^2$ and $V^2$, respectively. The green solid line and black dotted line correspond to $P^2+V^2$ with and without losses, respectively.

Figure 3

Losses after Mach-Zehnder interferometer. The loss rate is $R_1^{\prime }$ ($R_2^{\prime }$). The other elements are same as the case in Fig. 1.

Figure 4

$P_D^2+V_D^2$ as a function of $R_1^{\prime }$ and $R_2^{\prime }$. (a)–(e) correspond to the cases with $R=0.1-0.5$. $P_D^2+V_D^2$ can be larger than 1 for some certain values of $R_1^{\prime }$ and $R_2^{\prime }$.

Figure 5

The red solid line describes $P_D^2+V_D^2$ as a function of $R$ for $R_1^{\prime }=0.4$ and $R_2^{\prime }=0.6$. The dashed line stands for the maximum standard line of duality relation, $P^2+V^2=1$.

Figure 6

Losses inside the MZI. This device is same as Fig. 1 except that the VBS and BS are switched.

Figure 7

$K^2$ and $V_{1\left(2\right)}^2$ versus $R$ for $R_1=0.4$ and $R_2=0.6$.

Figure 8

Losses after the MZI. This device is the same as Fig. 3 except that the VBS and BS are switched.

Figure 9

$K^{\prime 2}$ and $V^{\prime 2}$ versus $R$ for $R_1^{\prime }=0.3$ and $R_2^{\prime }=0.7$.