## Abstract

The dual wave–particle nature of light and the degree of polarization are fundamental concepts in quantum physics and optical science, but their exact relation has not been explored within a full vector-light quantum framework that accounts for interferometric polarization modulation. Here, we consider vector-light quantum complementarity in double-pinhole photon interference and derive a general link between the degree of polarization and wave–particle duality of light. The relation leads to an interpretation for the degree of polarization as a measure describing the complementarity strength between photon path predictability and so-called *Stokes visibility*, the latter taking into account both intensity and polarization variations in the observation plane. It also unifies results advanced in classical studies by showing that the degree of polarization can be viewed as the ability of a light beam to exhibit intensity and polarization-state fringes. The framework we establish thus provides novel aspects and deeper insights into the role of the degree of polarization in quantum-light complementarity and photon interference.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

*Wave–particle duality*, a manifestation of complementarity, is a principal physical concept that restricts the coexistence of wave and particle qualities of quantum objects [1–3], in the sense that interferometric “which-path information” (WPI) is complementary to the visibility of intensity fringes [4–8]. While such duality is commonly considered in the context of single-qubit systems, important contributions have indicated that additional physical entities, such as entanglement, play a crucial role in quantifying complementarity in more general quantum configurations [9–11] (similar indications have appeared later in classical studies [12]). In particular, photons may exhibit interference not only in terms of intensity fringes, but also, or solely, via *polarization modulation* (periodic variations in the polarization state) [13–15], a unique property of *vectorial light* that has to date been neglected in complementarity contexts. Only very recently, a general framework concerning complementarity of arbitrary vector-light quantum fields in two-slit interference involving polarization modulation was established, leading to the discovery of a previously unexplored, fundamental aspect of photon wave–particle duality [16]. It was specifically shown that for pure single-photon vectorial light, the *a priori* WPI (path predictability) couples not to the intensity visibility but to a generalized visibility (“total visibility” [16]), hereafter referred to as the *Stokes visibility*, that accounts also for polarization modulation by characterizing the variations of *all four* Stokes parameters in the interference pattern.

The *degree of polarization* is likewise a central notion in optical physics, conventionally regarded as a quantity describing one-point correlations of an optical beam [17,18], and its relation to wave–particle duality of light has lately been examined by various authors [19–26]. However, to our knowledge, all such investigations have dealt with intensity visibility only and excluded the involvement of interferometric polarization modulation. In this work, by using two-point Stokes operators, we consider double-pinhole interference of general vector-light quantum fields taking into account polarization modulation and establish an exact link that connects the degree of polarization, path predictability, and Stokes visibility of a photon. The relationship provides an interpretation for the degree of polarization as a measure of complementarity between the *a priori* WPI and the Stokes visibility. Its general version, valid for arbitrary states of light, further implies that the degree of polarization can be viewed as an intrinsic quantity representing light’s interferometric ability to exhibit intensity and polarization-state variation, thus unifying notions put forward in classical contexts [27,28]. Our work unveils fundamental aspects of the dual wave–particle nature of light and the concept of degree of polarization.

## 2. COHERENCE AND POLARIZATION

To characterize the first-order coherence of a planar quantum-light field, at two space–time points $ {x_1} $ and $ {x_2} $, we introduce the two-point Stokes *operators*

*parameters*[16], i.e., the quantum analogs of the classical parameters [30,31], are obtained as

We quantify the partial coherence and partial polarization of light by the degrees of coherence and polarization. In our case, the recently introduced quantum degree of *vector-light* coherence, $ g({x_1},{x_2}) $, can be written as [16]

*all*components of the quantized electric field at two space–time points. It is invariant under (local) unitary operations and bounded as $ 0 \le g({x_1},{x_2}) \le 1 $. Complete coherence (incoherence), i.e., $ g({x_1},{x_2}) = 1 $ [$ g({x_1},{x_2}) = 0 $], takes places if and only if all field components between $ {x_1} $ and $ {x_2} $ are fully correlated (uncorrelated).

According to the conventional definition [17,18,33], we may express the degree of polarization, $ P(x) $, as

in terms of the normalized Stokes parameters $ {s_j}(x) = {s_j}(x,x) $. The usual degree of polarization in Eq. (5) describes first-order correlations between the orthogonal electric-field components at a single point. Similar to $ g({x_1},{x_2}) $, it obeys $ 0 \le P(x) \le 1 $ and remains unchanged under (local) unitary transformations. We remark, though, that addressing the degree(s) of quantum-light polarization involving higher-order correlations is a subject in constant development [29,34], as is the concept of degree of polarization for general three-dimensional classical light [35]. Nevertheless, as demonstrated below, the quantity $ P(x) $ in Eq. (5) has a well-defined, important physical meaning in vector-light complementarity and interference.## 3. INTENSITY DISTINGUISHABILITY AND STOKES VISIBILITY

Let us now consider double-pinhole interference with the photon polarization taken into account. Two pinholes are located at $ {{\textbf r}_1} $ and $ {{\textbf r}_2} $ in an opaque screen $ {\cal A} $ in the $ xy $ plane, and the emerging light of (angular) frequency $ \omega $ is observed on screen $ {\cal B} $ at position $ {\textbf r} $ in the paraxial regime. To distinguish the light in the openings, we utilize the *intensity distinguishability* (*path predictability* at the single-photon level [4,6,7]) [16]

In the observation plane $ {\cal B} $, the Stokes parameters $ {S_j}({\textbf r}) $, with $ j \in \{ 0, \ldots ,3\} $, take on the forms [13–16]

*photon interference law*, and it fully describes how coherence affects the intensity and polarization-state variations of general vector-light quantum fields in the double-pinhole setup [16].

Due to the oscillations of the Stokes parameters on $ {\cal B} $, we may define for each $ {S_j}({\textbf r}) $ a visibility parameter [13–16]:

*Stokes visibility*(also known as the

*total visibility*[16])

*and*polarization-state variations, i.e., the modulations of

*all four*Stokes parameters on $ {\cal B} $. It shows that $ 0 \le V({\textbf r}) \le 1 $, with the lower limit taking place when $ C({\textbf r}) = 0 $ or $ g({{\textbf r}_1},{{\textbf r}_2}) = 0 $, while the upper limit is reached if and only if $ C({\textbf r}) = 1 $ and $ g({{\textbf r}_1},{{\textbf r}_2}) = 1 $. Specifically, when the field exhibits partial coherence at the openings, in other words $ g({{\textbf r}_1},{{\textbf r}_2}) \gt 0 $, then $ V({\textbf r}) \gt 0 $ and at least one $ {S_j}({\textbf r}) $ is modulated. And conversely, any variation in at least one of the Stokes parameters on $ {\cal B} $ is a signature of partially coherent light at $ {\cal A} $.

## 4. WAVE–PARTICLE DUALITY AND PARTIAL POLARIZATION

It has been shown that any vectorial quantum-light field fulfills the complementarity relation [16]

with the upper bound always saturated when the light is totally coherent in the full vector sense, i.e., $ g({{\textbf r}_1},{{\textbf r}_2}) = 1 $. In such a case, we refer to*strong complementarity*[16]. We note that vector-light quantum fields, in fact, satisfy

*two different*complementarity relations [16]—Eq. (13) being one of them—reflecting two distinct, fundamental facets of wave–particle duality of the photon and having no correspondence in the quantum scalar-light treatment. In particular, in the vector-light framework, the quantity $ {D_0}({{\textbf r}_1},{{\textbf r}_2}) $, which describes the path predictability of the photon [4,6,7], couples to the

*Stokes visibility*$ V({\textbf r}) $, not to the intensity visibility $ {V_0}({\textbf r}) $.

We consider next a uniform light beam, at frequency $ \omega $ and position $ {{\textbf r}_0} $, with two orthogonal polarization modes ($ x $ and $ y $). The light is in an arbitrary (mixed) single-photon state, whose density operator can always be expressed as [5,20]

including the coherent superpositionThe photon beam is divided into $ x $- and $ y $-polarized parts by a polarizing beam splitter (see Appendix A), whereafter the two orthogonal constituents are directed towards openings 1 and 2, respectively, at which arbitrary unitary operations $ {\hat u_1} $ and $ {\hat u_2} $ may be performed (Fig. 1). The unitary operations enable to modify the polarization states at screen $ {\cal A} $ and hence also the intensity visibility $ {v_0}({\textbf r}) $ and/or the polarization contrasts $ {v_1}({\textbf r}),{v_2}({\textbf r}),{v_3}({\textbf r}) $ in observation plane $ {\cal B} $, where the two parts eventually superpose according to the photon interference law (7). Nevertheless, the path predictability $ {d_0}({{\textbf r}_1},{{\textbf r}_2}) $ [Eq. (6)] and the Stokes visibility $ v({\textbf r}) $ [Eq. (11)], which in this case take on the forms (see Appendix A)

Equation (18) indicates that $ \gamma $ does not have any effect on the path predictability, but it contributes to the Stokes visibility. The *a priori* WPI of the photon is thereby independent of whether the state is pure or mixed. The path predictability depends on (partial) polarization via $ {p_x} $ and $ {p_y} $, as these mode probabilities convert to path probabilities once the photon has passed the beam splitter. In particular, $ {p_x} $ ($ {p_y} $) comes to stand for the probability of finding the photon in pinhole 1 (2). For instance, when $ {p_x} \ll {p_y} $, the probability of detecting the photon in opening 1 is much lower than finding it in pinhole 2, leading to high path predictability. Contrarily, for $ {p_x} \approx {p_y} $, the *a priori* WPI is effectively zero.

On combining Eqs. (17) and (18), we now end up with the main result of this work:

Equation (19) forms a fundamental analytical link among the photon path predictability at the openings, the Stokes visibility in the observation plane, and the degree of polarization of the initial single-photon vector-light state. It shows that $ {d_0}({{\textbf r}_1},{{\textbf r}_2}) \le p({{\textbf r}_0}) $ and $ v({\textbf r}) \le p({{\textbf r}_0}) $, i.e., the degree of polarization sets the upper bound for both the*a priori*WPI of a photon and the Stokes visibility. Moreover, Eq. (19) indicates that a variation of $ {d_0}({{\textbf r}_1},{{\textbf r}_2}) $ or $ v({\textbf r}) $ alters its complementary partner so that the sum of their squares strictly equals the square of $ p({{\textbf r}_0}) $. In view of this, one may interpret $ p({{\textbf r}_0}) $ as defining the complementarity strength, or the

*degree of complementarity*, between $ {d_0}({{\textbf r}_1},{{\textbf r}_2}) $ and $ v({\textbf r}) $. Whenever the photon is in a pure state ($ \gamma = 1 $), we have $ p({{\textbf r}_0}) = 1 $ and, consequently, strong complementarity [16].

We emphasize that the Stokes visibility $ v({\textbf r}) $ in Eq. (19) does not, as a rule, coincide with the intensity visibility $ {v_0}({\textbf r}) $, but generally it characterizes the variations of all Stokes parameters at the observation screen [see Eq. (11)], viz., the intensity *and* polarization modulations in the interference pattern. For instance, when $ {\hat u_1} = {\hat u_2} = 1 $, the polarization states at the two pinholes remain orthogonal, resulting automatically in $ {v_0}({\textbf r}) = 0 $. Yet, as long as $ p({{\textbf r}_0}) \gt 0 $ and $ {d_0}({{\textbf r}_1},{{\textbf r}_2}) \lt p({{\textbf r}_0}) $, Eq. (19) shows that $ v({\textbf r}) \gt 0 $ despite the orthogonal polarization states, implying that polarization modulation will occur. Indeed, a photon in a superposition of being $ x $ polarized in one opening and $ y $ polarized in the other, with equal probabilities, leads to $ v({\textbf r}) = 1 $ and maximal polarization modulation, while in this case $ {v_0}({\textbf r}) = 0 $ [16].

## 5. DEGREE OF POLARIZATION AND STOKES VISIBILITY

The single-photon result (19) is actually a special case of the more general complementarity relation (see the Appendix A)

including degree of polarization*arbitrary*quantum state of light, while $ {g_\textit{xy}} = {\rm tr}(\hat \rho \hat a_x^\dagger {\hat a_y})/({\bar n_x}{\bar n_y}{)^{1/2}} $ is the mode correlation coefficient. The annihilation operators $ {\hat a_x} $ and $ {\hat a_y} $ satisfy the standard commutation rules [16,36].

Relation (20) is invariant with respect to $ {\hat u_1} $ and $ {\hat u_2} $ and covers true vector-light fields of *any* quantum state. It shows that all perfectly polarized quantum states [33], i.e., states with $ P({{\textbf r}_0}) = 1 $, obey strong complementarity. We stress that, as with the single-photon case in Eq. (19), Eq. (20) involves the Stokes visibility and is typically no longer valid if only intensity or polarization modulation is considered, highlighting the importance of placing both manifestations of interference on an equal footing when dealing with vector-light fields. The physical content of Eq. (20) is therefore different from formally similar complementarity relations derived for bipartite qubit systems that exclude interferometric polarization modulation [10,11]. Moreover, Eq. (20) is very general in the sense that by replacing quantum expectations with classical statistical averages, it also holds for any classical vector-light field. In the classical context, Eq. (20) should not be mixed with a recently established scalar-light relation {Eq. (10) in [23]} despite their superficial similarity, since the definitions and meanings of the quantities appearing in the equalities—and hence the physical interpretations and validity domains of the relations themselves—are entirely different.

In particular, if the intensity distinguishability (or path predictability for a single photon) is zero, we obtain from Eq. (20) that $ V({\textbf r}) = P({{\textbf r}_0}) $, i.e., the Stokes visibility on $ {\cal B} $ is exactly determined by the degree of polarization at $ {{\textbf r}_0} $. This result unifies the interpretations for the degree of polarization established in classical two-way interferometry, where the degree of polarization has been connected to intensity visibility [27] and to polarization modulation [28]. If the polarization states at the openings are rendered the same, e.g., by rotations, then $ P({{\textbf r}_0}) $ equals the visibility of intensity fringes [27]; if the polarization states are left orthogonal, then $ P({{\textbf r}_0}) $ directly specifies the visibility of polarization-state fringes [28]. In general, employing unitary operations at the openings allows one to adjust the interplay between intensity fringes and polarization contrasts, while leaving the Stokes visibility unchanged and strictly equal to $ P({{\textbf r}_0}) $. Therefore, in addition to its role as a complementarity measure, the degree of polarization can be viewed as an intrinsic quantity that characterizes the ability of light to exhibit intensity *and* polarization-state variation.

## 6. CONCLUSION

We have considered vector-light quantum complementarity in double-pinhole interference and established a fundamental connection among the degree of polarization, path predictability, and Stokes visibility, the latter accounting for intensity and polarization modulation in the observation plane. This exact analytical link leads to an interpretation for the degree of polarization as a measure of complementarity between the *a priori* WPI and the Stokes visibility of a single photon. For general light states, it was shown that the degree of polarization distinguishes as a quantity that describes the interferometric capacity of light to exhibit intensity and polarization-state contrasts, a discovery that unifies results put forward within classical frameworks. We made use, for the first time to our knowledge, of two-point Stokes operators, which are generalizations to the conventional (one-point) Stokes operators utilized in the context of quantum polarization. Our findings thereby reveal fundamental new facets of the role of the degree of polarization in vector-light complementarity and photon wave–particle duality, and identify directions towards future research involving partial polarization and quantum interference.

## APPENDIX A: PROOF OF EQ. (20)

The positive frequency parts of the three electric-field operators, at points $ {{\textbf r}_0} $, $ {{\textbf r}_1} $, and $ {{\textbf r}_2} $, are given by (we omit the time dependence) [36]

To justify Eq. (22), we model the (lossless) beam splitter as a unitary four-port device, involving the six mode operators $ {\hat a_{m\mu }} $ as well as the vacuum modes $ {\hat a_{{\rm v}x}} $ and $ {\hat a_{{\rm vy}}} $, whose action in general can be represented as [36]

Combining Eqs. (21) and (22) subsequently results in Eq. (20), of which relation (19) is a special case, completing our proof.

## Funding

Swedish Cultural Foundation in Finland; Academy of Finland (310511).

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