Abstract
We theoretically investigate quantum interference of two single photons at a lossy asymmetric beam splitter, the most general passive 2×2 optical circuit. The losses in the circuit result in a non-unitary scattering matrix with a non-trivial set of constraints on the elements of the scattering matrix. Our analysis using the noise operator formalism shows that the loss allows tunability of quantum interference to an extent not possible with a lossless beam splitter. Our theoretical studies support the experimental demonstrations of programmable quantum interference in highly multimodal systems such as opaque scattering media and multimode fibers.
© 2016 Optical Society of America
1. Introduction
Multiphoton quantum correlations are crucial for quantum information processing and quantum communication protocols in linear optical networks [1, 2]. Beam splitters form a fundamental component in the implementation of these linear optical networks [3]. They have been realized in a variety of systems including integrated optics, atomic systems, scattering media, multimode fibers, superconducting circuits and plasmonic metamaterials [4–11]. In plasmonic systems, beam splitters have been used to generate coherent perfect absorption in the single-photon regime [12,13] and on-chip two-plasmon interference [10,11]. Inherent losses in optical systems are unavoidable and can arise from dispersive ohmic losses or from imperfect control and collection of light in dielectric scattering media. The effect of losses in beam splitters has attracted a lot of theoretical attention due to the fundamental implications of unavoidable dispersion in dielectric media [14–17]. However, all these studies have dealt with either symmetric (equal reflection-transmission amplitudes for both input arms) or balanced (equal reflection and transmission amplitudes in each arm) beam splitters. In this article, we analyze the most general two-port beam splitter which can be lossy, asymmetric and unbalanced, and find the non-trivial constraints on the matrix elements. We derive general expressions for the probabilities to measure zero, one or two photons in the two outputs when a single photon is injected in each of the two inputs. Further, we comment on the possible measurements of quantum interference through coincidence detection in a Hong-Ou-Mandel-like setup [18]. The presented theoretical analysis establishes that losses allow programmability of quantum interference, which is required in a variety of useful quantum information processing and simulation protocols [19,20].
A general two-port beam splitter or a linear optical network consists of two input ports a1,a2 and two output ports b1,b2 as schematized in Fig. 1(a). The linearity of the beam splitter gives rise to a linear relation between the electric fields, E(bi) = ∑i,j sijE(aj). The complex numbers sij are the elements of a scattering matrix S and correspond to the transmission and reflection coefficients with s11 = t expiϕ11, s22 = τ expiϕ22, s12 = ρ expiϕ12, and s21 = r expiϕ21, where t,τ,r,ρ are positive real numbers. The phases ϕij are not all independent and can be reduced to ϕ1 and ϕ2 which correspond to the phase differences between transmission and reflection at a given input port. This gives the scattering matrix
Without further constraints on the matrix elements, the scattering matrix S need not be unitary. Special cases include the balanced beam splitter where τ = ρ;t = r and the symmetric beam splitter where τ = t;ρ exp(iϕ2) = r exp(iϕ1).
The six parameters in the scattering matrix are required to describe the behavior of the output intensities. Figure 1(b) illustrates the intensities |E|2 at b1 and b2 as the phase of the input coherent field at a1 is varied (with phase at a2 fixed). For a general beam splitter, the amplitudes, intensity offsets and phase offsets at the two output ports, b1 and b2, can be completely free. Of particular interest is the value of the phase α between the output peak intensities, which is related to the phases of the reflection coefficients ϕ1 and ϕ2 as α = ϕ1 + ϕ2. This phase α determines the visibility of quantum interference between two single photons, as discussed in the subsequent sections.
2. Energy constraints
The beam-splitter scattering matrix in Eq. (1) is defined without any constraints on the parameters. However, the physical constraint that the output energy must be less than or equal to the input energy imposes restrictions on the parameters as derived below. Let us consider the scenario where coherent states of light with fields E1 and E2 are incident at input ports a1 and a2 respectively. Energy conservation at a lossy beam splitter imposes the restriction that the total output powers in the arms should be less than or equal to the input,
The two input coherent state fields can be related through a complex number c = |c|e−iδ as E2 = cE1, which gives
As the inequality holds for all values of |c|, it should also hold in the limiting case where the right hand side of Eq. (3) is minimized. This occurs for |c|2 = (1 −t2 − r2)/(1 − τ2 −ρ2). Upon substitution, the inequality becomes
The above inequality can be algebraically manipulated using trigonometric identities into the following form
where θoff = arctan[(tρ cosϕ2 + τr cosϕ1)/(tρ sinϕ2 − τr sinϕ1)]. As the inequality holds for all values of δ, it should hold in the limiting case of the maximum value of the left hand side which occurs when δ +θoff = π/2. Substituting α = ϕ1 + ϕ2 results in the following inequality in terms of the reflection and transmission amplitudesFor the lossless beam splitter, the equality results in α = π. For a symmetric balanced beam splitter, i.e. t = r = τ = ρ and ϕ1 = ϕ2, Eq. (1) reduces to the well-known beam splitter matrix [21]
The inequality in Eq. (6) corresponds to the most general constraint on the parameters of a passive lossy asymmetric beam splitter. For the sake of clarity, we will discuss the case of a lossy symmetric beam splitter with τ = t and ρ = r. In this scenario, the inequality has three parameters
This inequality results in an allowed range of α between . Figure 2 depicts the tuning width Δα as a function of reflectance r2 and transmittance t2. The lossless beam splitters lie on the diagonal line that separates the forbidden and allowed regions. Evidently, lossless beam splitters have Δα = 0, i.e. the phase α between the output arms is fixed and equals π. With increasing losses in the beam splitter, Δα increases and achieves a maximum value of 2π, i.e. complete tunability of α. The beam splitters that exactly satisfy t + r = 1 (red dotted line) correspond to those lossy beam splitters that allow completely programmable operation with maximum transmission or reflection. In the following section, we discuss the effect of this tunability on the quantum interference between two single photons incident at the input ports of the general beam splitter.
3. Quantum interference of two single photons
The quantum-mechanical input-output relation of the lossy asymmetric beam splitter can be written using the scattering matrix in Eq. (1). From this point, we explicitly take into account the frequency dependence that is required to calculate the Hong-Ou-Mandel interference between single photons incident at the input ports.
The operators âi(ω) and are creation-annihilation operators of photons at the input and output ports, respectively. The canonical commutation relations of these operators are satisfied even in the presence of loss.
The introduction of noise operators in Eq. (9), which represent quantum fluctuations, are necessary in the presence of loss as reported earlier [14,22,23]. We assume that the underlying noise process is Gaussian and uncorrelated across frequencies.
The commutation relations of the noise operators can be calculated as the noise sources are independent of the input light, i.e.
which results inTo calculate the effect of the quantum interference, let us suppose that a single photon with frequency ω1 is incident at input a1 and another single photon with frequency ω2 is incident at input a2. The two photons together have a bi-photon amplitude ψ(ω1,ω2) which results in the following input state,
The bi-photon amplitude ψ(ω1,ω2) is normalized as , ensuring that the state vector |Ψ〉 is normalized.
In a lossy beam splitter, there are in total six possible outcomes with either two, one or zero photons at each output port. The probabilities of these outcomes can be represented as expectation values of the number operators for the output ports, defined as
Assuming that detectors have perfect efficiency, the probabilities can be calculated using the Kelley-Kleiner counting formulae [24] and can be grouped into 3 sets:
Of particular interest is the coincidence probability P(11,12) which decreases to zero at a lossless, symmetric balanced beam splitter, which is known as the Hong-Ou-Mandel effect [18].
Under the assumption that coefficients t,r,τ,ρ are frequency independent, the expectation values of the number operators are
where Ioverlap(δt) is the spectral overlap integral of the two single photons at the input ports of the beam splitter, given asUsually, in experimental measurements of quantum interference, the time delay is varied to retrieve the Hong-Ou-Mandel dip in the coincidence rates.
For the case of a symmetric beam splitter as discussed in Fig. 2, the probabilities of different outcomes are
The coincidence probability P(11,12) varies sinusoidally with α. For a lossless and balanced beamsplitter, α = π and the coincidence probability is zero, corresponding to the well-known Hong-Ou-Mandel bunching of photons. However in a lossy beam splitter, the coincidence probability between perfectly indistinguishable photons varies with α from (t2 − r2)2 to (t2 + r2)2, assuming Δα = 2π. Further, it is interesting to note that the probability of photon bunching at the first output port, P(21,02) or the second output port, P(01,22) is independent of α.
Figure 3(a) depicts the maximal coincidence rate maxαP(11,12) which occurs at ,δt = 0 as a function of transmittance t2 and reflectance r2. The cross-sections along the solid lines in Fig. 3(a) are shown in Fig. 3(b) in corresponding colors. The cross-sections correspond to different imbalance ratios t2/r2. A common feature among all the curves is a point of inflexion along the dotted curve and termination on the dashed curve. In the limiting cases of t2/r2 → ∞ or t2/r2 → 0, the two points coincide. The dashed curve corresponds to the coincidence probability in a lossless beam splitter, which varies as (1 − 2t2)2. The dotted line corresponds to the coincidence rate at largest value of t2 that allows full programmability, i.e. Δα = 2π.
4. Hong-Ou-Mandel like interference
In an experiment, the quantum interference can be measured by performing a Hong-Ou-Mandel-like experiment, where the distinguishability of the photons is varied by adding a time delay δt. Let us suppose that the two photons are generated using collinear type-II spontaneous parametric down conversion in a periodically poled potassium titanyl phosphate (PPKTP) under pulsed pumping (the center frequency and Fourier-transformed pulse width of the pump are ωp and τp respectively). The resulting bi-photon amplitude of the idler (ωi) and signal (ωs) photons is [25]
where Λ and L are the poling period and length of crystal, respectively. From the above bi-photon amplitude, the overlap integral Ioverlap(δt) can be calculated, which gives the coincidence probability P(11,12). Figure 4 elucidates the expected Hong-Ou-Mandel-like curve at various values of α for a symmetric balanced beam splitter with t = r = ρ = τ = 1/2. The delay time is normalized to the coherence time Δτc of the single photons generated by the source. For α = π, a Hong-Ou-Mandel like dip (red curve) is evident which slowly evolves into a peak as α approaches 0 or 2π, indicating increased antibunching of photons. The sinusoidal variation of the coincidence probability P(11,12) for perfectly indistinguishable photons, i.e. δt = 0, with the phase α indicates the programmability of quantum interference at these beam splitters.5. Discussion and conclusions
Through the above theoretical analysis of a general two-port circuit, we demonstrated that losses introduced in a beam splitter allow the tunability of α and hence of the quantum interference. We can quantify the programmability of quantum interference by defining the parameter ΔP(11,12) which is the programmable range of coincidence probability, defined as
where, the numerator is the difference between maximum and minimum coincidence probabilities (see Fig. 4) with indistinguishable photons (δt = 0) and the denominator is the coincidence rate with distinguishable photons (δt → ±∞). Figure 5 depicts ΔP(11,12) as a function of transmittance and reflectance with few representative contours shown in red. The lossless beam splitters, which lie on the diagonal separating the allowed and the forbidden regions, show no programmability. Maximal programmability of ΔP(11,12) = 2, is allowed by lossy balanced beam splitters for perfectly indistinguishable photons. The black dashed line in the figure corresponds to t +r = 1. While Δα = 2π in the region t +r < 1, the programmability is not uniform. This arises from the imbalance t2/r2 ≠ 1 in unbalanced beam splitters.Our theoretical calculations explain the recent experimental demonstrations of programmable quantum interference in opaque scattering media and multimode fibers [6, 7]. In these experiments, two-port circuits were constructed using wavefront shaping that selects two modes from an underlying large number of modes [26, 27]. Light that is not directed into the two selected modes due to imperfect control or noise can be modeled as loss. Typical transmission of ∼10% in opaque scattering media ensures the full programmability when a balanced two-port circuit is constructed [28,29].
In summary, we theoretically analyzed the most general passive linear two-port circuit from only energy considerations. We establish the programmability of quantum interference between two single photons in the context of recent experimental demonstrations in massively multichannel linear optical networks. These networks with the envisaged programmability of quantum interference has the potential for large-scale implementation of quantum simulators and programmable quantum logic gates. In this context, the theoretical analysis presented in this article establishes that imperfections or dissipation in optical networks, which are unavoidable in experiments, are not detrimental. In fact, the losses introduce a novel dimensionality to the networks, in that the quantum interference is programmable. The theoretical framework presented here can be extended to model larger programmable multiport devices [30, 31], which are required in a variety of useful quantum information processing and simulation protocols [19,20].
Acknowledgments
We would like to thank Klaus Boller, Allard Mosk, and Willem Vos for discussions. The work was financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) and the Stichting Fundamenteel Onderzoek der Materie (FOM).
References and links
1. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). [CrossRef] [PubMed]
2. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007). [CrossRef]
3. M. Reck, A. Zeilinger, H.J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994). [CrossRef] [PubMed]
4. A. Politi, M.J. Cryan, J.G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646–649 (2008). [CrossRef] [PubMed]
5. R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic Hong-Ou-Mandel experiment,” Nature 520, 66–68 (2015). [CrossRef] [PubMed]
6. T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016). [CrossRef]
7. H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016). [CrossRef]
8. J. R. Petta, H. Lu, and A. C. Gossard, “A coherent beam splitter for electronic spin states,” Science 327, 669–672 (2010). [CrossRef] [PubMed]
9. C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013). [CrossRef]
10. R. W. Heeres, L. P. Kouwenhoven, and V. Zwiller, “Quantum interference in plasmonic circuits,” Nat. Nanotechnol. 8, 719–722 (2013). [CrossRef] [PubMed]
11. J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photon. 8, 317–320 (2014). [CrossRef]
12. T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015). [CrossRef] [PubMed]
13. L. Baldacci, S. Zanotto, G. Biasiol, L. Sorba, and A. Tredicucci, “Interferometric control of absorption in thin plasmonic metamaterials: general two port theory and broadband operation,” Opt. Exp. 23, 9202–9210 (2015). [CrossRef]
14. S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998). [CrossRef]
15. T. Gruner and D.-G. Welsch, “Quantum-optical input-output relations for dispersive and lossy multilayer dielectric plates,” Phys. Rev. A 54, 1661–1677 (1996). [CrossRef] [PubMed]
16. J. Jeffers, “Interference and the lossless lossy beam splitter,” J. Mod. Opt. 47, 1819–1824 (2000). [CrossRef]
17. S. Scheel, L. Knöll, T. Opatrný, and D.-G. Welsch, “Entanglement transformation at absorbing and amplifying four-port devices,” Phys. Rev. A 62, 043803 (2000). [CrossRef]
18. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987). [CrossRef] [PubMed]
19. A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012). [CrossRef]
20. I. A. Walmsley, “Quantum optics: Science and technology in a new light,” Science 348, 525–530 (2015). [CrossRef] [PubMed]
21. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).
22. B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306–4322 (1992). [CrossRef] [PubMed]
23. J. Zmuidzinas, “Thermal noise and correlations in photon detection,” App. Opt. 42, 4989–5008 (2003). [CrossRef]
24. P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964). [CrossRef]
25. W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997). [CrossRef]
26. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007). [CrossRef] [PubMed]
27. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nature Photon. 6, 283–292 (2012). [CrossRef]
28. S. R. Huisman, T. J. Huisman, S. A. Goorden, A. P. Mosk, and P. W. H. Pinkse, “Programming balanced optical beam splitters in white paint,” Opt. Express 22, 8320–8332 (2014). [CrossRef] [PubMed]
29. S. R. Huisman, T. J. Huisman, T. A. W. Wolterink, A. P. Mosk, and P. W. H. Pinkse, “Programmable multiport optical circuits in opaque scattering materials,” Opt. Express 23, 3102–3116 (2015). [CrossRef] [PubMed]
30. J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015). [CrossRef] [PubMed]
31. N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).