Interference effects induced by non-local spatial filtering

S. P. Walborn; P. H. Souto Ribeiro; C. H. Monken

doi:10.1364/OE.19.017308

1. Introduction

The spatial correlations between twin-photons produced in parametric down-conversion have given rise to a large number of experimental and theoretical studies [1]. Various interesting and counter-intuitive features have been revealed, including the observation of ghost interference and ghost images [2–4], the observation of fourth-order quantum images [5], subwavelength interference fringes [6], spatial antibunching [7] and retrieval of a coherent image from incoherent illumination [8]. An early experiment by Souto Ribeiro et al. [9] showed that the visibility of a Young’s interference experiment involving the signal photon could be controlled by appropriate selection of the idler photon. It was observed that the increasing of the spatial filtering on the idler increased the visibility of the interference fringes. This result is adequately explained by the spatial correlation between the down-converted photons using a multi-mode theory [10]. By filtering the idler and detecting in coincidence, one remotely filters the signal photon by post-selection, thus increasing the spatial coherence, and the visibility of the interference pattern. This behavior can also be understood using the advanced wave picture introduced by Klyshko [11, 12], which is a convenient way to visualize two-photon coincidence experiments. In this picture, the idler detector plays the role of a source emitting photons back towards the crystal. These photons are then reflected by the crystal as though it were a mirror, follow the path of the signal photons, and are registered by the signal detector. In this way, the increased spatial filtering of the idler detector is equivalent to the reduction of the size of the source in the advanced wave picture, which increases the spatial coherence. This effect has also been observed for ghost interference [13].

In this paper we explore the influence of the spatial symmetry of two-photon states on their fourth- order coherence properties, considering an interference experiment involving an asymmetric Mach- Zehnder (AMZ) interferometer [14], which has been used in several experiments [15–17] (see Fig. 1). We show that when one pumps the crystal with a Gaussian pump beam, an effect similar to that described above appears: stronger filtering of one of the twin beams increases the visibility of the interference fringes which appear in the coincidence counts. However, when one pumps the crystal with a first-order Hermite-Gaussian beam, counter-intuitive effects appear. Depending on the widths of the signal and idler detectors, one can actually decrease the interference visibility by spatial filtering. We show that this curious effect can be explained by considering the spatial parity correlations between the down-converted photons [18, 19].

Fig. 1 Interference experiment considered here. The orthogonally polarized photons created by type-II SPDC are separated by a polarizing beam splitter (PBS). One photon is subject to an AMZ interferometer. The photons are registered by detectors D ₁ and D ₂ with square detection apertures of width 2a ₁ and 2a ₂, respectively. Here ϕ is an adjustable phase.

Download Full Size | PDF

2. Asymmetric interferometer

Consider the experimental arrangement shown in Fig. 1. We suppose that a sufficiently weak cw pump beam is incident on a nonlinear crystal, producing down-converted photons 1 and 2. One photon then passes through the AMZ interferometer and is registered by detector D ₂, while the other photon propagates directly to detector D ₁. In our choice of coordinate system, the fields propagate along the z-direction, as shown in Fig. 1. Due to the reflections at the 50-50 non-polarizing beam splitters (BS) and the mirrors, a field propagating through the AMZ suffers an even number of reflections when passing through one arm and an odd number of reflections when passing through the other. Under an odd number of reflections, the horizontal (with respect to the laboratory table) coordinate suffers a sign change, that is, in momentum space: q_y → −q_y [20, 21]. We can understand the interference process in the AMZ interferometer as the interference between the field and its own mirror image, due to the asymmetric number of reflections. We will also assume that the path difference between the two arms of the interferometer is much less than the coherence length of the down-converted fields. Therefore, we can relate the photon annihilation operator at the exit port with the operator at the input:

a_{2} (q) = \frac{1}{2} {a_{i n} (q_{x}, q_{y}) - e^{i ϕ} a_{i n} (q_{x}, - q_{y})}

where ϕ is the relative phase difference between optical paths 1 and 2, and q = q_x x̂ + q_y ŷ is the transverse component of the wave vector.

The two-photon detection probability is [22]

P (r_{1}, r_{2}) = {| Ψ (r_{1}, r_{2}) |}^{2},

where the two-photon detection amplitude can be defined as

Ψ (r_{1}, r_{2}) = 〈 vac | E_{2}^{(+)} (r_{2}) E_{1}^{(+)} (r_{1}) | ψ 〉,

and

E_{l}^{(+)} (r_{l})

is the field operator for the mode l = 1, 2 and r _l is the detection position for the down-converted modes l = 1, 2 and |ψ〉 is a two-photon state.

In the paraxial approximation and Fraunhofer regime, $E_{l}^{(+)} (r)$ is proportional to

E_{l}^{(+)} (r) \propto e^{i k z} \int d q a_{l} (q) e^{i q \cdot ρ},

where k is the magnitude of the wave vector k and ρ = x x̂ + y ŷ is the transverse component of the position vector r = (x, y, z). The operator a_l(q) annihilates a photon in mode l with transverse wave vector q.

Using Eqs. (1), (3) and (4), the two-photon detection amplitude, also called the biphoton wave function, in the detection planes (x ₁, y ₁), (x ₂, y ₂) can be calculated:

Ψ_{d} (r_{1}, r_{2}) = \frac{1}{\sqrt{2}} {Ψ (ρ_{1}, ρ_{2}) + e^{i ϕ} Ψ (ρ_{1}, ρ_{2}^{'})},

where ρ′ = x x̂ − y ŷ due to the odd number of reflections in one arm.

The detection probability is then

P_{d} (r_{1}, r_{2}) = P (ρ_{1}, ρ_{2}) + P (ρ_{2}, ρ_{2}^{'}) + Ψ * (ρ_{1}, ρ_{2}) Ψ (ρ_{1}, ρ_{2}^{'}) e^{i ϕ} + Ψ * (ρ_{1}, ρ_{2}^{'}) Ψ (ρ_{1}, ρ_{2}) e^{- i ϕ},

where P(ρ ₁, ρ ₂) is given by Eq. (2). The last two terms in Eq. (6) are responsible for the interference fringes.

Before we proceed, let us briefly summarize the experimental conditions for SPDC that will be assumed here. We suppose that there are narrow bandwidth interference filters in the detection system, which allow us to work in the monochromatic approximation, and that they are centered at the degenerate wavelength λ_c = 2λ_p, where λ_p is the pump beam wavelength. We will also assume the paraxial and thin-crystal approximations, and consider that the polarization state of each down-converted photon pair is well defined. Under these approximations, the two-photon detection amplitude is given by [1, 5]

Ψ (ρ_{1}, ρ_{2}) \propto U (\frac{ρ_{1} + ρ_{2}}{2}) .

Here U is the transverse profile of the pump beam propagated to the detection region. We have assumed that the distance from the crystal to each detector is the same: z ₁ = z ₂ = Z. For notational simplicity, the z-dependence of U has been omitted.

Now suppose that the pump beam is prepared so that the transverse profile U(ρ) is accurately described by a Hermite-Gaussian (HG) mode [23], such that we can define U(ρ) ≡ U_x(x)U_y(y). In this case, the detection probability in Eq. (6) can be factorized as

P_{d} (r_{1}, r_{2}) = P_{d} (x_{1}, x_{2}) P_{d} (y_{1}, y_{2})

where P_d(x ₁, x ₂) ∝ |U_x(x ₁/2 + x ₂/2)|² and

\begin{matrix} P_{d} (y_{1}, y_{2}) \propto & {| U_{y} (\frac{y_{1} + y_{2}}{2}) |}^{2} + {| U_{y} (\frac{y_{1} - y_{2}}{2}) |}^{2} \\ + 2 U_{y} (\frac{y_{1} + y_{2}}{2}) U_{y} (\frac{y_{1} - y_{2}}{2}) cos [ϕ + ξ (y_{1}, y_{2})], \end{matrix}

where ξ (y ₁, y ₂) = ky ₁ y ₂/2R(Z) comes from the quadratic phase of the Hermite-Gaussian pump beam profile U [23]. Here R(Z) is the radius of curvature and k = 2π / λ_p the wave number of the pump beam. We assume that the interferometer is perfectly aligned. Let us suppose that detectors D ₁ and D ₂ are equipped with square apertures of width 2a ₁ and 2a ₂, centered at ρ ₁ = ρ ₂ = 0, respectively. The coincidence count rate is obtained by integrating over the square detection apertures. From Eqs. (8) and (9) it is evident that the transverse spatial dependence of the interference curves depends only on the y coordinates of the detectors, and we notice that the dx ₁ and dx ₂ integrals will not affect the visibility of the interference fringes. With these considerations, we omit the x-dependence of the coincidence count rate, and write

C (a_{1}, a_{2}) \propto \int \int A_{1} (y_{1}) A_{2} (y_{2}) P_{d} (y_{1}, y_{2}) d y_{1} d y_{2},

where A ₁ and A ₂ are functions describing the apertures in the y direction.

2.1. Gaussian pump profile

For simplicity, let us consider that the pump beam is prepared with the beam waist in the detection plane, so that Z = 0. In this case, the radius of curvature R(Z) is infinite, and ξ (y ₁, y ₂) = 0. Let us also assume that the pump beam is given by the usual Gaussian profile: $U (x, y) \propto exp [- (x^{2} + y^{2}) / w_{p}^{2}]$ . Then we have

\begin{matrix} C (a_{1}, a_{2}) \propto & \int \int A_{1} (y_{1}) A_{2} (y_{2}) {e^{- {| y_{1} + y_{2} |}^{2} / w^{2}} + e^{- {| y_{1} - y_{2} |}^{2} / w^{2}} \\ + 2 e^{- y_{1}^{2} / w^{2}} e^{- y_{2}^{2} / w^{2}} cos ϕ} d y_{1} d y_{2}, \end{matrix}

where

w = \sqrt{2} w_{p}

. A straighforward but somewhat lengthy calculation gives

C (a_{1}, a_{2}) \propto I_{g} (a_{1} + a_{2}) + w^{2} π erf (\frac{a_{1}}{w}) erf (\frac{a_{2}}{w}) cos ϕ,

where

\begin{matrix} I_{g} (a_{1}, a_{2}) = & 2 min (a_{1}, a_{2}) w \sqrt{π} erf (\frac{| a_{1} - a_{2} |}{w}) + (a_{1} + a_{2}) w \sqrt{π} erf (\frac{a_{1} + a_{2}}{w}) \\ - (a_{1} + a_{2}) w \sqrt{π} erf (\frac{| a_{1} - a_{2} |}{w}) - w^{2} [e^{- {| a_{1} - a_{2} |}^{2} / w^{2}} - e^{- {(a_{1} + a_{2})}^{2} / w^{2}}] . \end{matrix}

The visibility of the interference fringes is usually defined as

V (a_{1}, a_{2}) = | \frac{C_{max} (a_{1}, a_{2}) - C_{min} (a_{1}, a_{2})}{C_{max} (a_{1}, a_{2}) + C_{min} (a_{1}, a_{2})} | .

Since only the last term in Eq. (12) depends on the phase ϕ and this term is always nonnegative, C _max occurs when ϕ = 0 and C _min occurs when ϕ = π. The visibility of fringes in the coincidence counts, or fourth-order visibility, is then

V (a_{1}, a_{2}) = {w \sqrt{π} erf (\frac{a_{1}}{w}) erf (\frac{a_{2}}{w})} {[I_{g} (a_{1}, a_{2})]}^{- 1} .

We would also like to obtain the visibility of interference fringes in the single counts at detector D ₂. This can be easily done by calculating the single count rate C ₂(a ₂), obtained by letting the width of the aperture A ₁ tend to infinity. Taking the limit, C ₂(a ₂) is

C_{2} (a_{2}) \propto 2 \sqrt{π} w^{2} [\frac{a_{2}}{w} + \frac{\sqrt{π}}{2} erf (\frac{a_{2}}{w}) cos ϕ] .

C ₂(a ₂) has a maximum and minimum at ϕ = 0 and ϕ = π, respectively. Using Eq. (14), the visibility for the interference fringes in the single counts, or second-order visibility, is

V_{2} (a_{2}) = \frac{\sqrt{π}}{2} \frac{w}{a_{2}} erf (\frac{a_{2}}{w}) .

Note that in the limiting case a ₂ → ∞, V ₂ → 0. Eq. (15) gives the visibility as a function of the size of the detection apertures and the beam width w. Figure 2 shows a plot of the fourth-order visibility given by Eq. (15) as a function of a ₂/w for various values of a ₁/w. We have also plotted the second-order visibility V ₂ in Fig. 2. Note that the fourth-order visibility V tends to the second-order visibility V ₂ for large values of a ₁/w.

For usual interference experiments involving quasi-monochromatic fields, the interference visibility is directly related to the spatial coherence of the field. In the coincidence experiment considered here, we can associate a coherence area to the field going through the interferometer as being equal to the detection surface for which the visibility is above some value. A usual lower bound for the visiblity is V ≥ 0.88 [24]. In our analysis considering only one dimension, we have a transverse coherence length instead of a coherence area. For the second-order interference, the coherence length is a _2coh ≈ 0.9w. For the fourth-order interference, spatial filtering of photon 1 extends the transverse coherence length. For example, for a ₁ = w/2, the effective coherence length is a _2coh ≈ 2.25w, and in general, a _2coh increases as a ₁ decreases. This behavior is similar to that observed in experiments using transverse interferometers like a Young’s double slit [9, 13, 25].

Fig. 2 Visibility as a function of a ₂/w for a Gaussian pump beam of width $w_{p} = 1 / \sqrt{2}$ . The solid lines correspond to the fourth-order visibility for various values of a ₁/w. The dark blue dashed line is the second-order visibility.

Download Full Size | PDF

2.2. Hermite-Gaussian HG₀₁ pump profile

Let us now analyze the visibility of the fringes in the AMZ interferometer when the pump profile is a first order Hermite-Gaussian beam, as has been done in a number of experiments [20, 26]. At the position of the beam waist, the Hermite-Gaussian beam HG₀₁ is defined as

{HG}_{01} (x, y) \propto y e^{- (x^{2} + y^{2}) / w_{p}^{2}} .

Using this pump profile, the coincidence count rate Eq. (10) becomes

\begin{matrix} C (a_{1}, a_{2}) \propto & \int \int A_{1} (y_{1}) A_{2} (y_{2}) d y_{1} d y_{2} {2 (y_{1}^{2} - y_{2}^{2}) e^{- y_{1}^{2} / w^{2}} e^{- y_{2}^{2} / w^{2}} cos ϕ \\ + {(y_{1} + y_{2})}^{2} e^{- {| y_{1} + y_{2} |}^{2} / w^{2}} + {(y_{1} - y_{2})}^{2} e^{- {| y_{1} - y_{2} |}^{2} / w^{2}}}, \end{matrix}

where we have used

w = \sqrt{2} w_{p}

. Using the same square apertures as in the case of the Gaussian pump beam, we arrive at

C (a_{1}, a_{2}) \propto \sqrt{π} w^{3} {I_{h g} (a_{1} + a_{2}) + cos ϕ a_{2} e^{- a_{2}^{2} / w^{2}} erf (\frac{a_{1}}{w}) - cos ϕ a_{1} e^{- a_{1}^{2} / w^{2}} erf (\frac{a_{2}}{w})},

where

\begin{matrix} I_{h g} (a_{1}, a_{2}) = & \sqrt{π} w^{3} {(a_{1} + a_{2} - 2 min (a_{1}, a_{2})) [\frac{| a_{1} - a_{2} |}{\sqrt{π} w} e^{- {| a_{1} - a_{2} |}^{2} / w^{2}} - \frac{1}{2} erf (\frac{| a_{1} - a_{2} |}{w})] \\ + (a_{1} + a_{2}) \frac{1}{2} erf (\frac{a_{1} + a_{2}}{w}) - (a_{1} + a_{2}) \frac{a_{1} + a_{2}}{\sqrt{π} w} e^{- {(a_{1} + a_{2})}^{2} / w^{2}}} . \end{matrix}

Defining the visibility as in Eq. (14), we have, for the HG₀₁ pump beam

V (a_{1}, a_{2}) = | a_{2} e^{- a_{2}^{2} / w^{2}} erf (\frac{a_{1}}{w}) - a_{1} e^{- a_{1}^{2} / w^{2}} erf (\frac{a_{2}}{w}) | {I_{h g} (a_{1}, a_{2})}^{- 1}

As in section 2.1, the second-order visibility V ₂(a ₂) can be found by calculating the single count rate C ₂(a ₂) at detector D ₂ as the limit of the coincidence counts when a ₁ tends to infinity. In this case, we obtain

C_{2} (a_{2}) \propto \sqrt{π} w^{4} a_{2} (1 + e^{- a_{2}^{2} / w^{2}} cos ϕ) .

Again, C ₂(a ₂) has a maximum and minimum at ϕ = 0 and ϕ = π, respectively. Using Eq. (14), the second-order visibility is

V_{2} (a_{2}) = e^{- a_{2}^{2} / w^{2}} .

Figure 3 shows a plot of the fourth-order visibility for various values of a ₂/w as a function of a ₁/w. We note that the visibility is symmetric, that is V (a ₁, a ₂) = V (a ₂, a ₁). A striking feature can be observed: for fixed a ₂ (a ₁), the fourth-order visibility can also increase as a ₁ (a ₂) increases. In other words, increased spatial filtering of the non-interfering photon can decrease the visibility of the fourth-order interference, in contrast to what is observed for the Gaussian pump profile. Another interesting point of Eq. (22) is that V (a, a) = 0 for all values of a, as shown in Fig. 3. Therefore, there is always an aperture size for the detector of the non-interfering photon for which no interference is observed. These effects can be seen directly in the visibility in Eq. (22), which is given by the difference of two terms. When a ₁ = a ₂ these two terms are equal, and V = 0. Depending on whether a ₁ > a ₂ or a ₂ > a ₁, the term inside the absolute value is positive or negative. Since V is defined as the absolute value, this accounts for the increase in visibility when a ₁ and a ₂ are different. The dashed line in Fig. 3 represents the second order visibility V ₂ as a function of a ₂/w. Notice that V ₂ is zero only for large values of a ₂/w.

Fig. 3 Visibility as a function of a ₂/w for a Hermite-Gaussian pump beam HG₀₁ of width $w_{p} = 1 / \sqrt{2}$ . The solid lines correspond to the fourth-order visibility for various values of a ₁/w. The dark blue dashed line is the second-order visibility.

Download Full Size | PDF

For the second-order visibility V ₂, the transverse coherence length in the case of the HG₀₁ pump beam is a _2coh ≈ 0.5w. However, for the fourth-order interference, it is not possible to define an effective coherence length as was done when the pump was Gaussian. It would be necessary to define a higher-order coherence “length” or coherence area, depending on the parameters of pump, signal and idler fields. In usual second-order coherence theory, the complex degree of coherence between points r ₁ and r ₂ generally decreases as a function of |r ₁ − r ₂| [24]. The coherence area is then defined as the region in which the complex degree of coherence is above some cut-off value, typically 1/2 or 1/e. For a circularly symmetric coherence function, for example, the coherence area is $π r_{c}^{2}$ , where r_c is the radius for which the coherence is above the cut-off value. In the case of the effective fourth-order coherence length for the HG₀₁ pump profile, it is not possible to make this simple distinction, and a more elaborate definition is necessary. We leave the careful derivation of this definition to future work.

In the above analyses it was assumed that the waist of the pump beam coincided with the detection planes of the down-converted photons, which gave ξ (y ₁, y ₂) = 0 in the detection probability given by Eq. (9). The same qualitative effects will appear even when the waist of the pump laser is not at the detection plane. The propagation of the pump beam in this context appears in two forms. First, due to the divergence of the pump beam, the beam width is no longer given by the waist w but by the z-dependent width w(Z). Thus, replacing w with w(Z) in all of the results obtained above accounts for the divergence of the pump beam. Secondly, since ξ (y ₁, y ₂) = ky ₁ y ₂/2R(Z), the term cos[ϕ + ξ (y ₁, y ₂)] introduces a phase that depends on the spatial coordinates which appears only in the interference term in Eq. (9). We note first that this phase is identical for both the Gaussian and the HG₀₁ pump beam. The overall effect is a reduction in the visibilities obtained in both cases, but the form of the curves in Figs. 2 and 3 do not change. For Z that is much less than the Rayleigh range of the pump beam, ξ (y ₁, y ₂) can be ignored. For example, for w = 1 mm, λ_p = 400 nm and Z = 100 mm, (k/2R(Z)) ≈ 0.01/w ². Hence, the phase variation is slow compared to the variation of the amplitude of the fourth-order field.

3. Parity Correlations

We can understand this counter-intuitive behavior of the visibilitiy in the experiment using the HG ₀₁ mode by decomposing the two-photon state in terms of Hermite-Gaussian modes [18], which illustrates the spatial parity correlations of the down-converted fields. The two-photon state is maximally-entangled in parity for a HG₀₁ pump beam [16–19]. The two-photon quantum state produced by SPDC in this case can be written as:

| ψ 〉 = \frac{1}{\sqrt{2}} ({| E O 〉}_{12} + {| O E 〉}_{12}),

where, for example, |EO〉₁₂ is the (normalized) state of photon pairs such that photon 1 and 2 have even and odd y-indices:

| E O 〉_{12} = \sum_{j even, k odd} C_{j k} | H G_{j} 〉_{1} | H G_{k} 〉_{2},

and |HG_n〉 are single photon states in Hermite-Gaussian modes. The asymmetric interferometer is sensitive to the parity of the input field [14–17], such that even and odd fields produce interference fringes that are out of phase. At the same time, the detectors register the different HG modes with different probabilities, depending on the size of the detectors. An example is illustrated in Fig. 4. The total coincidence count curve (P ₁ + P ₂ in Fig. 4) is composed of two contributions (P ₁ and P ₂) that are out of phase. The visibility depends on the difference in the overall weight of each contribution.

Fig. 4 The coincidence counts (P ₁ + P ₂) are composed of two out of phase contributions, P ₁ and P ₂. The visibility depends on the difference in overall weights.

Download Full Size | PDF

Explicitly, the fringe visibility is given by

V_{h g} = | \frac{C_{O E} (a_{1}, a_{2}) - C_{E O} (a_{1}, a_{2})}{C_{O E} (a_{1}, a_{2}) + C_{E O} (a_{1}, a_{2})} |

where, for example, C_EO(a ₁, a ₂) is the number of coincidence counts from the even-odd term, as a function of the width of the detection apertures. The two-photon state in this case is symmetric, and as a result C_EO(a, b) = C_OE(b, a). Thus, if the detectors are the same size, C_EO(a, a) = C_OE(a, a) and V_hg = 0, as is shown in Fig. 3. Depending on the size of the detectors, it is possible to have C_EO(a ₁, a ₂) > C_OE(a ₁, a ₂) or C_EO(a ₁, a ₂) < C_OE(a ₁, a ₂). Both of these cases result in non-zero visibility. Thus, due to the parity correlations of the down-converted photons, the appearance of interference fringes depends non-trivially on the size of the detector apertures. The parity correlations produce a different effect in the case of the Gaussian pump beam. In this case, the two-photon state in Hermite-Gaussian expansion is [18]

| ψ 〉 = α {| E E 〉}_{12} + β {| O O 〉}_{12},

where α > β. In this case the down-converted photons are produced with the same parity. The visibility for a Gaussian pump beam can be written as

V_{g} = | \frac{C_{E E} (a_{1}, a_{2}) - C_{O O} (a_{1}, a_{2})}{C_{E E} (a_{1}, a_{2}) + C_{O O} (a_{1}, a_{2})} |

When a ₁ or a ₂ is small, C_EE(a ₁, a ₂) ≫ C_OO(a ₁, a ₂), and interference fringes with large visibility are observed. As a ₁ and a ₂ grow, C_EE(a ₁, a ₂) and C_OO(a ₁, a ₂) become more and more similar, and the visibility becomes reduced.

4. Discussion

The use of optical devices, and in particular optical interferometers, illuminated by twin photons and detecting in coincidence has given rise to the observation of several new phenomena and the exploration of fundamental optical concepts from a new perspective. In some cases, it is rather simple to extend concepts that were developed in the context of light intensity measures to the domain of the fourth order correlations measured by the coincidence count rates. One example is the visibility of interference fringes, that can be directly translated from intensity to coincidence.

Coherence area, which is a concept developed in the context of intensity measurements and is related to interference visibility, could also have a direct extension from intensity to coincidence experiments. Our results demonstrate that for a Gaussian pump, one can think of a conditional coherence area in coincidence measurements. However, for a SPDC pumped with a higher-order transverse mode, this simple extension fails. Therefore, our results help in understanding that a more elaborate and complete notion of coherence area should be developed.

From the point of view of practical applications, the results presented here show that all applications of asymmetric interferometers that can be used with twin photons and coincidence counting, should suffer the influence of the correlation between the parity of the different modes. For example, the higher-order transverse spatial modes have been proposed as a method to encode quantum information, as was demonstrated in Ref. [27] with an intense laser. Let us consider a quantum communication experiment, in which single-photon states are prepared by detecting one of the down-converted photons. Information can be encoded into the single photon by preparing the pump beam in the appropriate higher-order transverse mode. Extraction of information encoded in the single-photon can be performed using a higher-order mode sorter, composed of asymmetric interferometers, as was shown in Ref. [14, 28]. This scenario is exactly that considered here, in which the transverse spatial structure of the pump beam plays a non-trivial role in the visibility of fourth-order interference. To achieve proper sorting of the higher-order modes in the quantum communication experiment, and thus proper readout of the encoded information, care must be taken in choosing the aperture size of both detectors. Though the results presented above were derived specifically for parametric down-conversion, they apply to any physical realization of the entangled state of the form Eq. (7).

5. Conclusion

We have shown that the usual notions of spatial coherence do not necessarily apply in the case of photon pairs produced from spontaneous parametric down-conversion. We analyze the dependence of the visibility of interference fringes visibility on the size of the detection apertures when one photon passes through an asymmetric Mach-Zehnder interferometer. We show that, depending on the pump beam profile, quite different behaviors can be observed. For a pump beam with Gaussian profile, remote spatial filtering of the photon that does not pass through the interferometer increases the visibility of interference fringes in coincidence counts. This is similar to previous experiments with transverse interferometers, such as a Young’s double slit [9, 13]. For a first order Hermite-Gaussian pump, on the other hand, the opposite can be observed: increased spatial filtering can reduce the visibility. In fact, the visibility depends non-trivially on the relation between the size of the detection apertures of the down-converted photons. We show that these results can be understood in terms of spatial parity correlations between down-converted field modes.

Acknowledgments

Financial support was provided by Brazilian agencies CNPq, CAPES, FAPERJ, and the Instituto Nacional de Ciência e Tecnologia - Informação Quântica (INCT-IQ).

References and links

1. S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87–139 (2010). [CrossRef]

2. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995). [CrossRef] [PubMed]

3. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995). [CrossRef] [PubMed]

4. R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004). [CrossRef] [PubMed]

5. C. H. Monken, P. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A. 57, 3123–3126 (1998). [CrossRef]

6. E. Fonseca, C. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868–2871 (1999). [CrossRef]

7. W. A. T. Nogueira, S. P. Walborn, S. Pádua, and C. H. Monken, “Experimental observation of spatial antibunching of photons,” Phys. Rev. Lett. 86, 4009–4012 (2001). [CrossRef] [PubMed]

8. A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004). [CrossRef] [PubMed]

9. P. H. S. Ribeiro, S. Pádua, J. C. M. da Silva, and G. A. Barbosa, “Controlling the degree of visibility of young’s fringes with photon coincidence measurements,” Phys. Rev. A 49, 4176–4179 (1994). [CrossRef] [PubMed]

10. G. A. Barbosa, “Quantum images in double-slit experiments with spontaneous down-conversion light,” Phys. Rev. A 54, 4473–4478 (1996). [CrossRef] [PubMed]

11. D. N. Klyshko, “A simple method of preparing pure states of an optical field, of implementing the Einstein-Podolsky-Rosen experiment, and of demonstrating the complementarity principle,” Sov. Phys. Usp. 31, 74–85 (1988). [CrossRef]

12. A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” J. Exp. Theor. Phys. 78, 259–262 (1994).

13. P. H. Souto Ribeiro and G. A. Barbosa, “Direct and ghost interference in double-slit experiments with coincidence measurements,” Phys. Rev. A 54, 3489–3492 (1996). [CrossRef] [PubMed]

14. H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003). [CrossRef]

15. A. N. de Oliveira, S. P. Walborn, and C. H. Monken, “Implementation of the Deutsch algorithm using polarization and transverse modes,” J. Opt. B: Quantum Semiclassical Opt. 7, 288–292 (2005). [CrossRef]

16. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett. 99, 170408 (2007). [CrossRef] [PubMed]

17. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and analysis of entangled photonic qubits in spatial-parity space,” Phys. Rev. Lett. 99, 250502 (2007). [CrossRef]

18. S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A 71, 053812 (2005). [CrossRef]

19. S. P. Walborn and C. H. Monken, “Transverse spatial entanglement in parametric down-conversion,” Phys. Rev. A 76, 062305 (2007). [CrossRef]

20. S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003). [CrossRef] [PubMed]

21. S. P. Walborn, W. A. T. Nogueira, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interferometry,” Mod. Phys. Lett. B 19, 1–19 (2005). [CrossRef]

22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

23. B. E. A. Saleh and M. C. Teich, Fundamental Photonics (Wiley, 1991). [CrossRef]

24. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

25. P. H. Souto Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Control of Young’s fringes visibility by stimulated down-conversion,” Phys. Rev. A 51, 1631–1633 (1995). [CrossRef] [PubMed]

26. S. P. Walborn, W. A. T. Nogueira, S. Pádua, and C. H. Monken, “Optical Bell-state analysis in the coincidence basis,” Europhys. Lett. 62, 161–167 (2003). [CrossRef]

27. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef] [PubMed]

28. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002). [CrossRef] [PubMed]

Interference effects induced by non-local spatial filtering

Abstract

1. Introduction

2. Asymmetric interferometer

2.1. Gaussian pump profile

2.2. Hermite-Gaussian HG₀₁ pump profile

3. Parity Correlations

4. Discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (29)

Optics Express

Abstract

1. Introduction

2. Asymmetric interferometer

2.1. Gaussian pump profile

2.2. Hermite-Gaussian HG01 pump profile

3. Parity Correlations

4. Discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (29)

Optics Express

2.2. Hermite-Gaussian HG₀₁ pump profile