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Coherence in two-photon down-conversion induced by a laser
Abstract
We discuss the situation in which idler beams from two parametric down-converter crystals are allowed to interfere. We show that, when two mutually coherent signal beams derived from a common laser are injected into the down-converters, the two idler beams can become mutually coherent also. Moreover, the resulting interference pattern can, in principle, have 100% visibility when the number of injected photons per unit down-converter bandwidth is large. This is just the condition for stimulated down-conversion to dominate over spontaneous down-conversion.
... In the cases where stimulated emission is not negligible in Q 2 , the interference visibility is a nonlinear function of the field field transmittance |T| [42,44]. This regime can be achieved using very high gain sources (e.g using very high pump power), or by seeding Q 1 and Q 2 via mode I with a coherent state (a laser beam) with the idler beam wavelength [45,46]. ...
... Note that if one detects one output of BS 2 in coincidence with an output of BS 3 , i.e. by using post-selection, it is possible to observe interference [45,50], and the visibility is V = 2|T|/(|T| 2 + 1). ...
... where φ I is the phase due to propagation of the idler beam from Q 1 to Q 2 , the operatorâ 0 represents vacuum field at the unused port of the beam splitter (object), T(ρ o ) is the amplitude transmission coefficient of the object at a point ρ o that is related to q I by Eq. (45), and |T(ρ o )| 2 + |R(ρ o )| 2 = 1. The quantity, R(ρ o ), can be interpreted the amplitude reflection coefficient at the same point while illuminated from the other side. ...
We present a tutorial on the phenomenon of induced coherence without induced emission, and specifically its application to imaging and metrology. It is based on a striking effect where two nonlinear crystals, by sharing a coherent pump and one of two output beams each, can induce correlations between the other two individual, non-interacting beams. This can be thought of as a type of quantum-erasure effect, where the ``welcher-weg'' (which-way), or in this case ``which-source'' information is erased when the shared beams are aligned. With the correct geometry this effect can allow an object to be imaged using only photons which have never interacted with the object -- in other words the image is formed using undetected photons. Interest in this and related setups has been accelerating in recent years due to a number of desirable properties, mostly centered around the fact that the fields for detection and imaging (since separate) may have different optical properties, entailing significant advantages to various applications. The purpose of this tutorial is to introduce researchers to this area of research, to provide practical tools for setting up experiments as well as understanding the underlying theory, and to also provide a comprehensive overview of the sub-field as a whole.
... Here, we prove that they are consistent with the experimental observable that is the contrast of interference pattern obtained with our ENBS. To characterize the interference of signal photons, we use the approach in the Heisenberg picture as shown in (34,35). The positive (negative)frequency part of the signal electric field operator, ...
... where, for simplicity, we take υ 1 = |υ|e ip and υ 2 = |υ|, consistent with the equal powers of the pump fields incident on the PPLN 1 and PPLN 2 but with a phase change p accrued by the mirror (see Fig. 1), i.e., p = k p x p . As shown in (30,34,35), we assume that the downconversion efficiency is small so that the higher-order terms, except for the lowest-order terms, in υ can be ignored. Note that, although R D is the normal-ordered correlation function ⟨ E D (−) E D ...
... Also shown in Fig. 4 is the a priori visibility V 0 as a function of || and for comparison. In our ENBS system (30), the stimulated downconversion rate can be controlled easily from the same order of SPDC to a much higher level than that of SPDC (34,35). Furthermore, the source is free from the decoherence issue because the spectrum of the quanton is determined by coherent seed beams. ...
To test the principle of complementarity and wave-particle duality quantitatively, we need a quantum composite system that can be controlled by experimental parameters. Here, we demonstrate that a double-path interferometer consisting of two parametric downconversion crystals seeded by coherent idler fields, where the generated coherent signal photons are used for quantum interference and the conjugate idler fields are used for which-path detectors with controllable fidelity, is useful for elucidating the quantitative complementarity. We show that the quanton source purity μ s is tightly bounded by the entanglement E between the quantons and the remaining degrees of freedom by the relation μ s = 1 − E 2 , which is experimentally confirmed. We further prove that the experimental scheme using two stimulated parametric downconversion processes is an ideal tool for investigating and understanding wave-particle duality and Bohr’s complementarity quantitatively.
... In these experiments the generation of the idler field, for example, is stimulated in the nonlinear crystals by an external laser field. If the laser wavelength lies within the emission band of the spontaneously generated idler photons, and the laser intensity is high enough, the stimulated biphoton generation will be much stronger than the spontaneous biphoton generation [6], and high visibilities are the consequence. But then the question of which-path information arises once again. ...
... Consider finally the experiment of Fig. 6, which differs from that of Fig. 1 by the insertion of a filter between BBO1 and BBO2. The counting rate in our model is again proportional to R as given by equation (6) and the approximations it is based on, but now we must account for the effect of the filter on the idler field incident on BBO2. This is straightforward in the stimulated downconversion case where we approximate the stimulating field of the He-Ne laser as a prescribed classical field as above. ...
... In terms of vacuum fields, the interference is a consequence of the fact that we may in effect take the vacuum expectation value a i10 a † i20 = 1 when the idler modes i1 and i2 are identical [Eqs. (6) and (7)]. ...
Coherence can be induced or stimulated in parametric down-conversion using
two or three crystals when, for example, the idler modes of the crystals are
aligned. Previous experiments with induced coherence [Phys. Rev. Lett. 114,
053601 (2015)] focused on which-path information and the role of vacuum fields
in realizing complementarity via reduced visibility in single-photon
interference. Here we describe experiments comparing induced and stimulated
coherence. Different single-photon interference experiments were performed by
blocking one of the pump beams in a three-crystal setup. Each counted photon is
emitted from one of two crystals and which-way information may or not be
available, depending on the setup. Distinctly different results are obtained in
the induced and stimulated cases, especially when a variable transmission
filter is inserted between the crystals. A simplified theoretical model
accounts for all the experimental results and is also used to address the
question of whether the phases of the signal and idler fields in parametric
down-conversion are correlated.
... Our decision to analyze |z i | = 1 for the distinguishability parameters is motivated by the alignment of the FWM1 and FWM2 processes proposed in [1], which is achieved by the use of two mirrors. This alignment analysis involves the same consideration of path indistinguishability that is central to the description of the Wang-Zou-Mandel effect [24][25][26]. ...
... Proof. We first show that e −iH |0⟩ |0⟩ is actually a two mode squeezed state e ζa † 1 a † 2 −ζa1a2 |0⟩ |0⟩ with appropriate ζ ∈ C. Consider (24), (26) defining the covariance matrix of e −iH |0⟩ |0⟩ in Domain 2 at ϕ = π and define the parameters ...
The SU(1,1) interferometer introduced by Yurke, McCall, Klauder is reformulated starting from the Hamiltonian of two identical optical downconversion processes with opposite pump phases. From the four optical modes, two are singled out up to a relative phase by the assumption of exact alignment of the interferometer (i.e., mode indistinguishability). The state of the two resulting modes is parametrized by the nonlinearity g, the relative phase , and a dynamical phase resulting from the interaction time. The optimal operating point for sensing the relative phase (dynamical phase) is found to be () with quantum Fisher information exhibiting Heisenberg scaling (logarithmically modified Heisenberg scaling ). Compared to the predictions of the circuit-based model, we find in that in the Hamiltonian model: 1. the optimal operating points occur for a non-vacuum state inside the interferometer, and 2. measurement of the total photon number operator does not provide an estimate of the relative or dynamical phase with precision that saturates the quantum Cramer-Rao bound, whereas an observable based on weighted shift operators becomes optimal as g increases. The results indicate a first-principles approach for describing general optical quantum sensors containing multiple optical downconversion processes.
... Note that if one detects one output of BS 2 in coincidence with an output of BS 3 , i.e., by using post-selection, it is possible to observe interference [40,41], and the visibility is V = 2|T|/(|T| 2 + 1). ...
... In cases where stimulated emission is not negligible in Q 2 , the interference visibility is a nonlinear function of the field transmittance |T| [44,46]. This regime can be achieved using very high gain sources (e.g., using very high pump power), or by seeding Q 1 and Q 2 via mode I with a coherent state (laser beam) with the idler beam wavelength [40,47]. ...
We present a tutorial on the phenomenon of induced coherence without induced emission, and specifically its application to imaging and metrology. It is based on a striking effect where two nonlinear crystals, by sharing a coherent pump and one or two output beams, can induce coherence between the other two output beams. This can be thought of as a type of quantum-erasure effect, where the “welcher-weg” (which-way), or in this case, “which-source,” information is erased when the shared beams are aligned. With the correct geometry, this effect can allow an object to be imaged using only photons that have never interacted with the object—in other words, the image is formed using undetected photons. Interest in this and related setups has been accelerating in recent years due to a number of desirable properties, mostly centered around the fact that the fields for detection and imaging (since separate) may have different optical properties, entailing significant advantages for various applications. The purpose of this tutorial is to introduce researchers to this area of research, to provide practical tools for setting up experiments as well as understanding the underlying theory, and also to provide a comprehensive overview of the sub-field as a whole.
... Namely, if idler photons coming from NL1 are perfectly aligned (and matched in polarization) with those induced in NL2, an observer behind NL2 cannot distinguish the source of these photons (according to its authors this fundamental point of Ref. [26] was suggested by Z. Y. Ou), and merely this fact is enough to induce a second-order coherence of their signal photon counterparts [26], which can be detected as an interference image by the camera. We note that, in a somewhat different arrangement from that of Ref. [26], the first idea of aligning each of the two idlers with pump waves in a parametric down conversion process was proposed in Ref. [27]. Regarding applications to biological samples, the parameters of the crystals are chosen such that the wavelength of the down-converted idler photons hitting the sample (λ2) is outside of the main absorption peak of water in the region (centered around 1420 nm), while the wavelength of the detected signal beam (λ1) lies in the sensitivity range of the CCD cameras (< 900 nm). ...
... In this way the presence of the seeding laser does not alter the indistinguishability of the photons in the common idler mode. Nevertheless, in this case, the conditions for the interference of signal photons on the camera are ensured also by another effect, namely induced coherence by a laser [27]. In the Appendix, we present the outline of a quantumoptical calculation for the visibility of interference patterns both in the presence and absence of the seeding background (Fig. S5), by generalizing the model of refs. ...
Novel imaging techniques utilizing nondegenerate, correlated photon pairs sparked intense interest during the last couple of years among scientists of the quantum optics community and beyond. It is a key property of such "ghost imaging" or "quantum interference" methods that they use those photons of the correlated pairs for imaging that never interacted with the sample, allowing detection in a spectral range different from that of the illumination of the object. Extensive applications of these techniques in spectroscopy and microscopy are envisioned, however, their limited spatial resolution to date has not yet supported real-life microscopic investigations of tiny biological objects. Here we report a modification of the method based on quantum interference by using a seeding laser and confocal scanning, that allows the improvement of the resolution of imaging with undetected photons by more than an order of magnitude, and we also present examples of application in the microscopy of biological samples.
... Namely, if idler photons coming from NL1 are perfectly aligned (and matched in polarization) with those induced in NL2, an observer behind NL2 cannot distinguish the source of these photons (according to its authors this fundamental point of Ref. [26] was suggested by Z. Y. Ou), and merely this fact is enough to induce a second-order coherence of their signal photon counterparts [26], which can be detected as an interference image by the camera. We note that, in a somewhat different arrangement from that of Ref. [26], the first idea of aligning each of the two idlers with pump waves in a parametric down conversion process was proposed in Ref. [27]. Regarding applications to biological samples, the parameters of the crystals are chosen such that the wavelength of the down-converted idler photons hitting the sample (λ2) is outside of the main absorption peak of water in the region (centered around 1420 nm), while the wavelength of the detected signal beam (λ1) lies in the sensitivity range of the CCD cameras (< 900 nm). ...
... In this way the presence of the seeding laser does not alter the indistinguishability of the photons in the common idler mode. Nevertheless, in this case, the conditions for the interference of signal photons on the camera are ensured also by another effect, namely induced coherence by a laser [27]. In the Appendix, we present the outline of a quantumoptical calculation for the visibility of interference patterns both in the presence and absence of the seeding background (Fig. S5), by generalizing the model of refs. ...
Novel imaging techniques utilizing nondegenerate, correlated photon pairs sparked intense interest during the last couple of years among scientists of the quantum optics community and beyond. It is a key property of such “ghost imaging” or “quantum interference” methods that they use those photons of the correlated pairs for imaging that never interacted with the sample, allowing detection in a spectral range different from that of the illumination of the object. Extensive applications of these techniques in spectroscopy and microscopy are envisioned, however, their limited spatial resolution to date has not yet supported real-life microscopic investigations of tiny biological objects. Here we report a modification of the method based on quantum interference by using a seeding laser and confocal scanning, that allows the improvement of the resolution of imaging with undetected photons by more than an order of magnitude, and we also present examples of application in the microscopy of biological samples.
... In this report, we demonstrate both ultra-high resolution (sub-Doppler-broadened linewidth) and remote-measurement QSUP method using induced one-photon coherence detection, which involves a dual stimulated parametric down-conversion (StPDC) scheme [35][36][37][38]. We show that the frequency resolution is not limited by the PDC-generated signal or idler spectral bandwidth but by the linewidth of an injected seed beam. ...
... Note that the Hamiltonian in Eq. (3) is semiclassical because the pump field is treated as a classical field. In fact, the time-evolution operator described by this Hamilonian creates a two-mode squeezed vacuum state [35]. In the main text, we presented the photon counting rate of the signal beams at the detector and the visibility of the interference pattern produced by the signal fields. ...
Quantum spectroscopy with undetected photons (QSUP) utilizing the quantum entanglement of parametrically down-converted photons has emerged as a new spectroscopic platform. Here, we demonstrate a high-resolution and remote-measurement QSUP, where light-matter interactions and photon detections are performed in spectrally and spatially different regions. A dual-stimulated parametric down-conversion scheme with an optical frequency-comb pump and ultra-narrow coherent seed beam in an idler mode is used to generate path-entangled pairs of the undetected idler and measured frequency-comb signal photons. To demonstrate the frequency resolution of this scheme, a Fabry-Pérot cavity with a narrow bandwidth is used as a sample that modulates the distinguishability of one-photon-added coherent idler beams, which directly affects the interference fringe visibility of the entangled signal photons. We thus anticipate that the remote QSUP whose frequency resolution is determined by the linewidth of the coherent seed laser will enable the development of quantum spectroscopy featuring high resolution.
... In the absence of any other injected signal or idler beams, the two signal beams show no first-order coherence (|g (1) s1,s2 | = 0) [4,5] and thus do not give rise to interference when recombined in a beam splitter [6]. However, if idler i 1 is injected into the second nonlinear crystal and the experimental configuration is designed to make idlers i 1 and i 2 indistinguishable after NLC 2 , the signal photons s 1 and s 2 will show first-order coherence, i.e., |g (1) s1,s2 | = 1. ...
We demonstrate a different scheme to perform optical sectioning of a sample based on the concept of induced coherence [Zou et al., Phys. Rev. Lett. 67, 318 (1991)]. This can be viewed as a different type of optical coherence tomography scheme where the varying reflectivity of the sample along the direction of propagation of an optical beam translates into changes of the degree of first-order coherence between two beams. As a practical advantage the scheme allows probing the sample with one wavelength and measuring photons with another wavelength. In a bio-imaging scenario, this would result in a deeper penetration into the sample because of probing with longer wavelengths, while still using the optimum wavelength for detection. The scheme proposed here could achieve submicron axial resolution by making use of nonlinear parametric sources with broad spectral bandwidth emission.
... This description coincides with the first non-perturbative descriptions of the Zou-Wang-Mandel experiment [30,31], and was later utilized to analyze the ultimate sensitivity of an interferometry scheme based on path identity induced quantum coherence [32]. The perturbative description of this experiment [8,33,34], while convenient in terms of detailed descriptions of temporal, spatial, and polarization structure, is not sufficient to distinguish the two models for quantum radar with unreflected photons that we consider in the present paper. Therefore, maintaining a general positive value of the SPDC nonlinearity g, the QFI for Model 1 described by (15) is calculated from (5) to be ...
Two descriptions are introduced and analyzed for a reflectivity estimation and detection scheme that does not involve measurement of photons scattered by the target. One description, provided by the Hamiltonian dynamics of the full transmitter/receiver optical system, incurs an exponential cost in transmitter intensity for a given estimation sensitivity but is linearly improved with the intensity of the thermal background. The other description, based on optical quantum circuits, exhibits sensitivity around a factor of 1/2 of the optimal entanglement-assisted scheme, but incurs an inverse linear reduction in sensitivity with increasing thermal background. The results have applications for the design of optically active receivers based on combining echo-seeded spontaneous parametric downconversion and induced coherence due to photon indistinguishability.
... The Zou-Wang-Mandel experiment demonstrates that the photocurrent from the signal modes of two coherently pumped parametric downconversion crystals can depend on parameters of a channel applied to the first of the partially aligned idler modes [1,2,3]. This remarkable consequence of photon indistinguishability and the CV entanglement, also known in the literature as quantum-induced coherence by path identity, has influenced a wide range of experimental and theoretical work in non-linear quantum optical phenomena [4]. ...
The Zou-Wang-Mandel (ZWM) effect is a remarkable consequence of photon indistinguishability and continuous-variable entanglement in which an optical phase shift is imprinted on photonic modes associated with optical paths that that do not pass through the phase shift source. By bringing the canonical formalism of continuous-variable Gaussian states to bear on the mode-structure of the ZWM experiment, we show that the physical consequence of implementing optical path identity is a renormalization of quadrature squeezing which governs the entanglement of four effective optical modes. Nonperturbative expressions for the ZWM interference patterns and normalized first-order coherence function are derived. Generalizations to -graph states with more than four modes directly follow from the general method used to analyze the minimal example. We show that a ZWM interferometer with a laser-seeded signal mode, which estimates an idler phase shift by detecting photons that did not propagate through the phase shift, exhibits an optimal sensitivity comparable to that of a laser-seeded SU(1,1) interferometer if path identity is implemented with high fidelity.
... The nonlinear optical process of parametric downconversion (PDC) has been extensively employed to generate quantum states of light structured in the transverse spatial degrees of freedom [1]. In the classical regime, the same process can be operated in the stimulated emission mode (StimPDC) [2,3], providing a convenient platform for the design of quantum optical schemes [4][5][6], and for the study of the interplay between the spatial structures of the interacting light fields in the parametric process [7][8][9][10][11]. In the same way, parametric upconversion plays an important role in a wide variety of applications in quantum and classical optical schemes, as, for instance, frequency conversion of squeezed light fields [12,13] and imaging with visible and invisible light [14,15]. ...
Preparation, control, and measurement of optical vortices are increasingly important, as they play essential roles in both fundamental science and optical technology applications. Spatial light modulation is the main approach behind the control strategies, although there are limitations concerning the controllable wavelength. It is therefore crucial to develop approaches that expand the spectral range of light modulation. Here, we demonstrate the modulation of light by light in nonlinear optical interactions to demonstrate the identification of the topological charge of optical vortices. A triangular-lattice pattern is observed in light beams resulting from the spatial cross modulation between an optical vortex and a triangular shaped beam undergoing parametric interaction. Both up- and downconversion processes are investigated, and the far-field image of the converted beam exhibits a triangular lattice. The number of sites and the lattice orientation are determined by the topological charge of the vortex beam. In the downconversion process, the lattice orientation can also be affected by phase conjugation. The observed cross modulation works for a large variety of spatial field structures. Our results show that modulation of light by light can be used at wavelengths for which solid-state devices are not yet available.
... The nonlinear optical process of parametric down-conversion has been extensively employed to generate quantum states of light structured in the transverse spatial degrees of freedom [1]. In the classical regime, the same process can be operated in the stimulated emission mode (StimPDC) [2,3], providing a convenient platform for the design of quantum optical schemes [4][5][6], and for the study of the interplay between the spatial structures of the interacting light fields in the parametric process [7][8][9][10][11]. In the same way, parametric up-conversion plays an important role in a wide variety of applications in quantum and classical optical schemes, as for instance frequency conversion of squeezed light fields [12,13] and imaging with visible and invisible light [14,15]. ...
A triangular-lattice pattern is observed in light beams resulting from the spatial cross modulation between an optical vortex and a triangular shaped beam undergoing parametric interaction. Both up- and down-conversion processes are investigated, and the far-field image of the converted beam exhibits a triangular lattice. The number of sites and the lattice orientation are determined by the topological charge of the vortex beam. In the down-conversion process, the lattice orientation can also be affected by phase conjugation. The observed cross modulation works for a large variety of spatial field structures, and could replace solid-state devices at wavelengths where they are not yet available.
... The use of OAM modes, especially Laguerre-Gaussian (LG) beams, in nonlinear optical interactions has also been extensively explored [5][6][7][8]. In this paper, we are concerned with the parametric-down-conversion process [9] in the case where one of the down-converted beams, say the signal, is seeded by a laser and gives rise to an idler beam, which is indirectly stimulated [10,11]. We refer to this process as stimulated parametric down-conversion (Stim-PDC). ...
Stimulated parametric down-conversion is a nonlinear optical process in which the orbital angular momentum of light is conserved under most practical conditions. However, there are instances in which the topological charge is not compatible with the remaining modal structure of the stimulated idler beam. As a result, the stimulated down-converted light diffracts and gives rise to Laguerre-Gaussian-like beams with nonzero radial indices. We present a theoretical and experimental analysis demonstrating this fundamental aspect of nonlinear processes with light beams possessing orbital angular momentum.
... Sub-and hyperdiffusive random media [28] have attracted a great deal of interest. Quantum-optical coherence tomography (QOCT) using entangled-photon-sources in a Hong-Ou-Mandel interferometer [29,30] has been demonstrated [31][32][33]. It yielded an improvement of the resolution of a factor of 2 compared to OCT. ...
We present a quantum field theoretical method for the characterization of disordered complex media with short laser pulses in an optical coherence tomography setup (OCT). We solve this scheme of coherent transport in space and time with weighted essentially nonoscillatory methods (WENO). WENO is preferentially used for the determination of highly nonlinear and discontinuous processes including interference effects and phase transitions like Anderson localization of light. The theory determines spatiotemporal characteristics of the scattering mean free path and the transmission cross section that are directly measurable in time-of-flight (ToF) and pump-probe experiments. The results are a measure of the coherence of multiple scattering photons in passive as well as in optically soft random media. Our theoretical results of ToF are instructive in spectral regions where material characteristics such as the scattering mean free path and the diffusion coefficient are methodologically almost insensitive to gain or absorption and to higher-order nonlinear effects. Our method is applicable to OCT and other advanced spectroscopy setups including samples of strongly scattering mono- and polydisperse complex nano- and microresonators.
... This has the potential to induce coherence between the bi-photon states emitted by the two crystals. It can happen even without induced emission in the second crystal that could be the consequence of an idler field at its input [176,177]. When the two signal waves are combined on a beam-splitter, there is no way to distinguish from which crystal either of the signal and idler photons comes from and interference patterns that appear are detectable directly on the intensities measured on the signal side. ...
The production of pairs of entangled photons simply by focusing a laser beam onto a crystal with a non-linear optical response was used to test quantum mechanics and to open new approaches in imaging. The development of the latter was enabled by the emergence of single photon sensitive cameras able to characterize spatial correlations and high-dimensional entanglement. Thereby new techniques emerged such as the ghost imaging of objects - where the quantum correlations between photons reveal the image from photons that have never interacted with the object - or the imaging with undetected photons by using nonlinear interferometers. Additionally, quantum approaches in imaging can also lead to an improvement in the performance of conventional imaging systems. These improvements can be obtained by means of image contrast, resolution enhancement that exceed the classical limit and acquisition of sub-shot noise phase or amplitude images. In this review we discuss the application of quantum states of light for advanced imaging techniques.
... This has the potential to induce coherence between the bi-photon states emitted by the two crystals. It can happen even without induced emission in the second crystal, which could be the consequence of an idler field at its input 173,174 . When the two signal waves are combined on a beam splitter, there is no way to distinguish from which crystal either of the signal or idler photons come from and interference patterns that appear are detectable directly on the intensities measured on the signal side. ...
Using quantum states of light for imaging both reveals quantum phenomena and enables new protocols that result in images that surpass classical limitations. Such systems require both quantum light sources and often the ingenious use of detector technologies.
... Our experimentally measured visibilities are quantitatively describable by theory, i.e., V = |α| 2 /(1 + |α| 2 ), as can be seen in Fig. 3. This indicates that our experiment occurs in a regime that can be only explained using a quantum mechanical description 56,57 . ...
Due to their frequency scaling and long-term coherence, frequency combs at the single-photon level can provide a fascinating platform for developments in quantum technology. Here we demonstrate frequency comb single-photon interferometry in an unheralded manner. We are able to induce coherence by erasing the which-way information of path-entangled photon pairs. The photon pairs are prepared using a dual parametric down-conversion pumped by a highly stable frequency comb laser and an ultra-narrow seed laser. This is conducted at the extremely low-conversion efficiency regime. The unique feature of our quantum interferometer is that the induced one-photon interference of the path-encoded single photons (signal), with multiple frequency components, is observed with a unit visibility without heralding conjugate photons (idler). We demonstrate that quantum information and frequency comb technology can be combined to realize quantum information platforms. We expect this will contribute to the application of quantum information and optical measurements beyond the classical limit.
... The state produced by the stimulated process, within the monochromatic, paraxial, and thin-crystal approximations, is [43] ...
The advanced-wave picture is “... an intuitive treatment of two-photon correlation with the help of the concept of an effective field acting upon one of the two detectors and formed by parametric conversion of the advanced wave emitted by the second detector ...” [A. V. Belinskii and D. N. Klyshko, Sov. Phys. JETP 78, 259 (1994)]. This quote from Belinskii and Klyshko nicely describes the concept of the advanced-wave picture: an intuitive tool for designing and predicting results from coincidence-based two-photon experiments. Up to now, the advanced-wave picture has been considered primarily for the case of an ideal plane-wave pump beam and only for design purposes. Here we study the advanced-wave picture for a structured pump beam and in the context of stimulated emission provoked by an auxiliary input laser beam. This suggests stimulated parametric down-conversion as a useful experimental tool for testing the experimental sets designed with the advanced-wave picture. We present experimental results demonstrating the strategy of designing the experiment with advanced-wave picture and testing with stimulated emission.
... The state produced by the stimulated process, within the monochromatic, paraxial and thin-crystal approximations, is [43] |ψ = |v s (q) |0 + C dq 1 dq 2 v p (q 1 + q 2 )a † (q 1 ) |v s (q) |1; q 2 (15) where 1 and 2 are indices for signal and idler, respectively,v s (q) is the angular spectrum of the stimulating field at z = 0 (at the crystal) and |v s (q) is the corresponding multimode coherent state in the continuus mode representation (p. 565 of Ref. [39]). ...
The advanced-wave picture is "... an intuitive treatment of two-photon correlation with the help of the concept of an effective field acting upon one of the two detectors and formed by parametric conversion of the advanced wave emitted by the second detector ..." [A. V. Belinskii and D. N. Klyshko, JETP 78, 259 (1994)]. This quote from Belinskii and Klyshko nicely describes the concept of the advanced-wave picture; an intuitive tool for designing and predicting results from coincidence-based two-photon experiments. Up to now, the advanced-wave picture has been considered primarily for the case of an ideal plane-wave pump beam and only for design purposes. Here we study the advanced-wave picture for a paraxial pump beam. This suggests stimulated parametric down-conversion as a useful experimental tool for testing the experimental sets designed with the advanced-wave picture. We present experimental results demonstrating the strategy of designing the experiment with advanced-wave picture and testing with stimulated emission.
... In this work we show a classical analogue of the experiment by Lemos et al. [6]. In our experiment depicted in Fig. 1, a pump and a signal laser beams are injected into a non-linear crystal in the forward and in the backward directions, producing classical light (which we call idler) through the process of stimulated down conversion [17]. An object is placed in the path of the amplified signal beam, after its first passage through the crystal. ...
We obtained the phase and intensity images of an object by detecting classical light which never interacted with it. With a double passage of a pump and a signal laser beams through a nonlinear crystal, we observe interference between the two idler beams produced by stimulated parametric down conversion. The object is placed in the amplified signal beam after its first passage through the crystal, and the image is observed in the interference of the generated idler beams. High contrast images can be obtained even for objects with small transmittance coefficient due to the geometry of the interferometer and to the stimulated parametric emission. Like its quantum counterpart, this three-colour imaging concept can be useful when the object must be probed with light at a wavelength for which detectors are not available.
... In the absence of any other injected signal or idler beams, the two signal beams show no first-order coherence (|g (1) s1,s2 | = 0) [4,5] and thus do not give rise to interference when recombined in a beam splitter [6]. However, if idler i 1 is injected into the second nonlinear crystal and the experimental configuration is designed to make idlers i 1 and i 2 indistinguishable after NLC 2 , the signal photons s 1 and s 2 will show first-order coherence, i.e., |g (1) s1,s2 | = 1. ...
We demonstrate an experimental configuration to perform optical sectioning of a sample based on the concept of induced coherence [Zou et al., Phys. Rev. Lett. 67, 3187 (1991)]. This can be viewed as a new type of optical coherence tomography scheme, where different reflectivities of the sample under study translates into changes of the degree of first-order coherence between two beams. The induced coherence tomography scheme proposed here uses frequency-entangled photons embedded in a nonlinear interferometer. As practical advantage the scheme allows probing the sample with one wavelength and measuring light with another wavelength. In a bio-imaging scenario, this would result in a deeper penetration into the sample thanks to probing with longer wavelengths, while still using the optimum wavelength for detection.
... where γ (A,B) is the degree of the first-order coherence between the two modes, |η (A,A) | is the degree of the so-called "anomalous" autocorrelation inside the mode A, and |η (A,B) | is the degree of the anomalous cross correlation between the modes [5]. Figure 1 shows the variation of the the Cauchy-Schwartz parameter χ (Ψ,b) as a function of the coupling strength G Ψ and the number of photonsn in the thermal field the mirror is oscillating. ...
We examine the role of coherences on the creation of multi-mode entanglement in a nanomechanical cavity containing a finite size one-dimensional optical lattice. We show that, in general, the system is composed of three coupled modes and find that the presence of the first-order coherence between two modes of the system is equally effective in destroying entanglement between them. In the case of two modes coupled through an intermediate mode we show that there is no entanglement between the modes when both modes are coupled to the intermediate mode by the parametric interaction. We find that in order to effectively entangle the modes, one of the modes should be coupled to the intermediate mode by a parametric interaction but the other mode should be coupled by the linear-mixing interaction.
... Although there is no direct connection between the one-and two-photon coherences, we find that in the system considered here the modes exhibit an interesting coherence effect [49][50][51]. Namely, the modes can be anticoherent that the one-photon coherence ρ 23 vanishes and at the same time ρ 14 = 0. This is shown in Fig. 2 where we plot the variation of the absolute values of the coherences |ρ 23 | and |ρ 14 | with γ d . ...
The link of two concepts, indistinguishability and entanglement, with the
energy-time uncertainty principle is demonstrated in a system composed of two
strongly coupled bosonic modes. Working in a short interaction limit, we find
that the inclusion of the anti-resonant terms to the coupling Hamiltonian leads
the system to relax to a state which is not the ground state of the system.
This effect occurs passively by just presence of the anti-resonant terms and is
explained in terms of the time-energy uncertainty principle for the simple
reason that at a very short interaction time, the uncertainty in the energy is
of order of the energy of a single excitation, thereby leading to a
distribution of the population among the zero, singly and doubly excited
states. The population distribution, correlations and entanglement are shown to
be substantially depend on whether the modes decay independently or
collectively to an exterior reservoir. In particular, when the modes decay
independently with equal rates, entanglement with the complete
distinguishability of the modes is observed. The modes can be made mutually
coherent if they decay with unequal rates. However, the visibility in the
single-photon interference cannot exceed 50%. When the modes experience
collective damping, they are indistinguishable even if decay with equal rates
and the visibility can, in principle, be as large as unity. We find that this
feature derives from the decay of the system to a pure entangled state rather
than the expected mixed state. When the modes decay with equal rates, the
steady-state values of the density matrix elements are found dependent on their
initial values.
... As discussed in the previous section, the nature of the downconversion process leads to strong relationships between the characteristics of the photons in this pair. This property of the downconversion process has been exploited to generate two-photon entangled states for use in various experimental applications [44,45,2,46,47,30,48]. ...
The novel quantum statistical properties of the two- photon entangled states generated by spontaneous parametric downconversion have been utilized in a variety of fourth-order interferometric configurations. The extent to which the intense light produced by an unneeded parametric amplifier (optical parametric generator) retains these desirable properties is explored in a series of calculations. Common fourth-order interferometric configurations using two-photon entangled states are summarized, with an emphasis on the Hong-Ou-Mandel and Mach-Zehnder interferometers. This summary is followed by a review of recent proposals for the exploitation of entangled states for sub-Rayleigh-limit imaging. The limitations of using parametric downconversion at two-photon levels are discussed and the replacement of two-photon interferometric sources with the multiphoton output of a high-gain optical parametric generator is considered. The output of the Hong-Ou-Mandel interferometer, Mach-Zehnder interferometer, and quantum lithography configurations as a function of single-pass gain is determined, and the interpretation of these results in the context of multiple photon pair contributions to interferometric patterns is presented. The analysis of the high-gain optical parametric generator as a fourth-order interferometric source is then extended to the case of multiple signal and idler output modes. The impact of the system transfer characteristics on the desired interferometric properties is discussed. The initiation of beam filamentation by vacuum fluctuations is considered, and this four-wave mixing process is compared to parametric downconversion as a source for fourth-order interferometric applications. We conclude by contrasting the states produced by high- gain optical parametric generation with coherent states and the states produced by seeded optical parametric amplification as sources for fourth-order interferometric configurations.
... (This is illustrated by so-called "quantum eraser" experiments [194,195,196]). ...
A single 9Be+ ion confined in an rf (Paul) trap may be used to realize two of the simplest quantum systems: the two-level system and the harmonic oscillator. The two-level system is comprised of two, ground-state hyperfine electronic levels. The trapping potential is harmonic, to a high degree of approximation, and so the ion's motion is that of a three-dimensional harmonic oscillator. By coupling the ion's motional and electronic degrees of freedom, we can engineer entanglement between these systems. This allows us to study quantum mechanics, with all its pecularities, in a well-controlled environment. For example, we can study the interactions of superposition states with the ion's environment, leading to a destruction of quantum superpositions. Furthermore, this system, when scaled up to several ions, may allow us to construct a simple ``quantum computer,'' which promises exponential speed-up over any possible classical computer for some computational problems. Towards this goal, we have cooled two, trapped ions to their ground state of collective motion and deterministically created entanglement of their electronic degrees of freedom, with their joint motion serving to transfer this entanglement between them.
... We are particularly interested in the effect of the first-order correlation (first-order interference) on entanglement between two modes which is associated with secondorder correlation functions. It is well known that in the parametric down-conversion, served as a typical source of entanglement, the signal and the idler beams are strongly entangled but always behave as mutually incoherent [24][25][26]. A similar conclusion applies to the correlations between modes of the optomechanical system, where it was demonstrated that the cavity and mechanical modes play the same role as the signal and the idler of a nondegenerate parametric oscillator and the modes behave as mutually incoherent [8,11]. ...
The coherence and correlation properties of effective bosonic modes of a nanomechanical cavity composed of an oscillating mirror and containing an optical lattice of regularly trapped atoms are studied. The system is modeled as a three-mode system: two orthogonal polariton modes, representing the coupled optical lattice and the cavity mode, and one mechanical mode, representing the oscillating mirror. We examine separately the cases of two-mode and three-mode interactions, which are distinguished by suitable tuning of the mechanical mode to the polariton mode frequencies. In the two-mode case, we find that the occurrence of entanglement in the system is highly sensitive to the presence of first-order coherence between the modes. In particular, the creation of the first-order coherence among the polariton and mechanical modes is achieved at the expense of entanglement between them. In the three-mode case, we show that no entanglement is created between the independent polariton modes if both modes are coupled to the mechanical mode by the parametric interaction. There is no entanglement between the polaritons even if the oscillating mirror is damped by a squeezed vacuum field. The interaction creates first-order coherence between the polaritons, and the degree of coherence can, in principle, be as large as unity. This demonstrates that the oscillating mirror can establish first-order coherence between two independent thermal modes. Further analysis shows that two independent thermal modes can be made entangled in the system only when one of the modes is coupled to the intermediate mode by a parametric interaction and the other is coupled by a linear-mixing interaction.
Increasing the interaction path length is a well‐known method for enhancing the sensitivity of the optical detection system. Hollow–core fibers (HCFs) represent a viable alternative to the traditional multi‐path cells offering low optical losses and strong confinement of the optical field. Here, the incorporation of an Antiresonant Hollow–core Fiber (AR‐HCF) section into a nonlinear interferometer, where the AR‐HCF section serves as a gas‐sensing cell operating in the IR range is presented. By exploiting the effect of nonlinear interference, the detection is brought into the more operation‐friendly visible range. The detection of methane (CH4) gas at mid‐IR wavelengths within a half‐meter section of AR‐HCF, with an estimated concentration accuracy of 200 ppm·m is demonstrated. These results represent the combination of two research fields within a single instrument and pave the way for further advancement of quantum‐inspired gas sensing techniques.
The Zou-Wang-Mandel (ZWM) effect is a remarkable consequence of photon indistinguishability and continuous-variable entanglement in which an optical phase shift is imprinted on photonic modes associated with optical paths that do not pass through the phase-shift source. By bringing the canonical formalism of continuous-variable Gaussian states to bear on the mode structure of the ZWM experiment, we show that the physical consequence of implementing optical path identity is a renormalization of quadrature squeezing which governs the entanglement of four effective optical modes. Nonperturbative expressions for the ZWM interference patterns and normalized first-order coherence function are derived. Generalizations to H-graph states with more than four modes directly follow from the general method used to analyze the minimal example. We show that a ZWM interferometer with a laser-seeded signal mode, which estimates an idler phase shift by detecting photons that did not propagate through the phase shift, exhibits an optimal sensitivity comparable to that of a laser-seeded SU(1,1) interferometer if the path identity is implemented with high fidelity.
Interference between two waves is a well-known concept in physics, and its generalization to more than two waves is straightforward. The order of interference is defined as the number of paths that interfere in a manner that cannot be reduced to patterns of a lower order. In practice, second-order interference means that in, say, a triple-slit experiment, the interference pattern when all three slits are open can be predicted from the interference patterns between all possible pairs of slits. Quantum mechanics is often said to only exhibit second-order interference. However, this is only true under specific assumptions, typically single particles undergoing linear evolution. Here we experimentally show that nonlinear evolution can in fact lead to higher-order interference. The higher-order interference in our experiment can be understood using a simple classical or quantum description, namely optical coherent states interacting in a nonlinear medium. Our work shows that nonlinear evolution could open a loophole for experiments attempting to verify Born's rule by ruling out higher-order interference.
Hyperentanglement of photonic light modes is a valuable resource in quantum information processing and quantum communication. Here we propose a protocol using the interference of two optical nonlinearities and control of the heralding (detection) basis in the orbital angular momentum degree of freedom. This setup is capable of generating states which are both maximally entangled and hyperentangled in at least four dimensions. The resultant state in the four-dimensional case is a generalization of the so-called NOON state (a maximally path-entangled state well known in quantum optics). The production of this state is “perfect” (in other words noiseless) at least in the ideal case, excluding experimental imperfections. The presented setup is very versatile, and with control of the detection and pumping protocols a massively large parameter space, of arbitrarily large dimensionality, may be searched for other states of interest. Also, we present a few specific cases to demonstrate the versatility of the system.
To test the principle of complementarity and wave-particle duality quantitatively, we need a quantum composite system that can be controlled by experimental parameters. Here, we demonstrate that a double-path interferometer consisting of two parametric downconversion crystals seeded by coherent idler fields, where the generated coherent signal photons are used for quantum interference and the conjugate idler fields are used for which-path detectors with controllable fidelity, is useful for elucidating the quantitative complementarity. We show that the source purity is tightly bounded by the entanglement measure E by the relation and the visibility V and detector fidelity F determine the coherence of the quantons, i.e., . The quantitative complementarity of the double-path interferometer we developed recently is explained in terms of the quanton-detector entanglement or quanton source purity that are expressed as functions of injected seed photon numbers. We further suggest that the experimental scheme utilizing two stimulated parametric downconversion processes is an ideal tool for investigating and understanding wave-particle duality and complementarity quantitatively.
We present a novel technique of molecular vibration spectroscopy with undetected photons (MSUP) with dual stimulated parametric down-conversion crystals. In our MSUP, quantum coherence between independent signal photons is induced by the path indistinguishability of the conjugate idler single-photon-added coherent states and results in perfect visibility under the same pump and seed beam intensities. If the seed beam intensity is imbalanced by the sample absorption, the modulated interference fringe reveals the absorption spectrum as a function of seed beam frequency. As a proof-of-principle experiment, a rovibrational absorption spectrum of hydrogen cyanide (H13C14N) molecules in a gas cell at approximately 1550 nm is successfully measured by analyzing the single-photon interference fringe of the signal field at 807 nm.
We present a phase-stable and selectable repetition-rate (frep) divider (PSRD) of an optical frequency comb (OFC) by down-picking a train of pulses from the fundamental OFC phase coherently. The PSRD operating at the integer pulse-picking order p uses an acousto-optic modulator driven by a train of phase-synchronized single-cycle wave packets oscillating at the carrier frequency of fc≡frep and the gating frequency of fgp≡frep/p. The demonstrated PSRD can selectively pick a train of pulses with p=2 to 25 from the fundamental OFC with frep=250 MHz at 530 nm. Integrated phase noises from 10 Hz to 25 kHz offset frequency from the optical beat note between each down-picked OFC and the fundamental OFC are measured to be about (59±6) mrad through the whole pulse picking order. The phase-coherent down-picked OFC could be of use for various applications where phase-stable and repetition-rate selectable OFCs are the essential radiation sources.
Interferometric spectroscopy with undetected photons (ISUP) utilizing quantum mechanically path-entangled photon pairs has received considerable attention as an optical-measurement platform. Recently, we carried out a proof-of-concept experiment to show that ISUP can be used to measure the transmission spectrum of a Fabry-Perot resonator. Here, we demonstrate that ISUP with dual stimulated parametric down-conversion (StPDC) processes, which allows us to perform infrared rovibrational spectroscopy with visible photons. In our ISUP method, quantum coherence between two independent signal photons from each StPDC crystal is induced by the indistinguishability of conjugate idler fields and results in high visibility of the signal single-photon interferometry at the nondegenerate wavelength. If the seed-beam intensity is imbalanced due to the sample absorption, the corresponding envelope modulation of the interference fringe as a function of the seed-beam frequency reveals the absorption spectrum of the optical sample. As a proof-of-principle experiment, the full rovibrational transmission spectrum at 1550 nm of the hydrogen cyanide (H13C14N) molecules in a gas cell is measured from the single-photon interference fringe of the signal fields at 807 nm. We thus anticipate that the single-photon ISUP technique with dual StPDC crystals will find broad use for the development of high-resolution atomic and molecular spectroscopy with undetected photons.
We demonstrate a frequency-comb single-photon interferometry for quantum spectroscopy and imaging with undetected photons by utilizing both of optical frequency comb technique and quantum erasing mechanism with path-entangled photon pairs.
Parametric down-conversion is a second-order nonlinear optical process annihilating a pump photon and creating a pair of photons in the signal and idler modes. Then, by using two parametric down-converters and introducing a path indistinguishability for the two generated idler modes, a quantum coherence between two conjugate signal beams can be induced. Such a double spontaneous or stimulated parametric down-conversion scheme has been used to demonstrate quantum spectroscopy and imaging with undetected idler photons via measuring one-photon interference between their correlated signal beams. Recently, we considered another quantum optical measurement scheme utilizing W-type tripartite entangled signal photons that can be generated by employing three spontaneous parametric down-conversion crystals and by inducing coherences or path-indistinguishabilities between their correlated idler beams and between quantum vacuum fields. Here, we consider an extended triple stimulated parametric down-conversion scheme for quantum optical measurement of sample properties with undetected idler and photons. Noting the real effect of vacuum field indistinguishability on the fringe visibility as well as the role of zero-point field energy in the interferometry, we show that this scheme is an ideal and efficient way to create a coherent state of W-type entangled signal photons. We anticipate that this scheme would be of critical use in further developing quantum optical measurements in spectroscopy and microscopy with undetected photons.
As is well known, since the foundation of quantum mechanics, at the first quarter of this century, its interpretation has given rise to endless polemics. The reason of these debates can be rooted in the most uncommon, at least in the field of the so called exact sciences, epistemology proposed by the usual paradigm of quantum mechanics. It was traditional to believe that the facts, the results from the experiments, could be understood in the conceptual framework of spacetime. Furthermore it was supposed that, at least in principle, there would be no limits for the human possibilities in understanding Natura. Plain quantum mechanics categorically denies these assumptions. Therefore it is not surprising to observe that many people still keep trying to check its foundations. It must be stated that no one questions the main empirical predictions of the plain theory. Everybody agrees that quantum mechanics represents, perhaps one of the greatest achievements of mankind. What is under discussion is for one side whether the quantum theory is a complete theory in the sense of Bohr, meaning that the laws of nature were settled once for all. The only possible developments are those horizontal connected mainly with the practical applications of the known fundamental laws. The other point that makes people feel uneasy is the rejection of causality. If it is true that, at the quantum level, space and time are mere auxiliary concepts, without any intrinsic meaning, superfluous commodities, we have been used for too long, then the full consequences of these postulates may in fact be astonishing. It would be the greatest scientific revolution in the history of science.
We studied a situation where coherence between the signal and the idler fields is induced by a laser field. It is predicted that, in the weak-field limit, interference between the idler field and another local oscillator field can be observed in a homodyne measurement when there is one photon in the signal field. It is found that indistinguishability of the photon path is directly related to this interference.
We investigated the phenomenon of induced temporal coherence in optical parametric down conversion. We used a multimode blue-diode-laser pump and a single-mode laser for the injection signal beam. The temporal coherence of the idler beam is known to depend on the coherence function of the pump, the coherence function of the injection beam, the phase-matching condition, and the average photon number of the injection laser beam. Our experimental data show very good agreement with the multimode theory which predicts the changing the spectral shape as a function of the average photon number of the injection beam.
By using two mutually coherent light beams of adjustable intensity to induce downconversions of the signal beams in two similar nonlinear crystals and then allowing the two idlers to interfere, we measure the variation of the induced degree of coherence as a function of inducing intensity. The key parameter N is the average photon occupation number per mode of the inducing field. We show experimentally that the induced degree of coherence is given by N / ( N + 1 ) , as is expected theoretically.
We present ways of realizing quantum cloning via stimulated emission. Universality of the cloning procedure is achieved by choosing systems that have appropriate symmetries. We first discuss a scheme based on certain three-level systems, e.g. atoms in a cavity. Our numerical results show that this scheme approaches optimal cloning for short interaction times. Then we demonstrate that optimal universal cloning can be realized using parametric down-conversion. At the same time, our down-conversion scheme also implements the optimal universal NOT operation. We conclude with some remarks on cloning and superluminal signalling, using our cloner as an illustrative example.
Temporal coherence of the idler photon beam induced by a coherent input signal was studied for non-degenerate optical parametric down conversion. When the optical path difference of the interferometer is much longer than the coherence length of the spontaneous idler radiation, the experimental results show that the first-order temporal coherence of the idler photon can be induced by a coherence input signal and the interference fringe visibility is determined by the intensity of the coherent input signal.
We consider the action of a beam splitter with identical degenerate parametric down converters in each of the outputs and show that with single photons injected into each of the input ports, under certain conditions, a superposition of two- and four-photon maximally entangled states may be produced. Also we describe techniques for verifying the generation of the four-photon component of the state. We then discuss the application of such states to quantum photolithography assuming the existence of substrates capable of absorbing four photons. The two-photon component makes no contribution to the background.
Fourth-order interference patterns of biphotons generated by type-I spontaneous parametric downconversion are examined for beams with partial angular and spectral entanglement, both for degenerate and nondegenerate constituent photons. Two-beam configurations using a single beam-splitter interferometer and a Mach-Zehnder interferometer are explicitly studied, as are four-beam configurations using pairs of beam splitters and Mach-Zehnder interferometers. The interference pattern generally comprises a number of harmonic functions of the path-length difference with frequencies and visibilities that usually depend on direction, and may have a finite or infinite duration (fourth-order coherence length). The relation between the visibility and the spectral indistinguishability of the two beams is established. Certain components of the interference pattern are independent of direction so that they are not washed out by the use of apertures of finite size.
The final technical report on research performed under the ONR contract 'Fundamental Studies of Photon Statistics', emphasizes some of the most interesting results obtained. They deal with non-classical interference, violations of locality, and the determination of the time interval between two photons with femtosecond accuracy. Some possibilities for a new type of optical communication channel were explored. The research resulted in 62 publications and 88 lectures or papers presented at scientific meetings and seminars. These are all listed. Seven graduate students completed Ph.D. degrees with support from the ONR contract. (EMK)
Consider a bipartite entangled system, half of which falls through the event horizon of an evaporating black hole, while the other half remains coherently accessible to experiments in the exterior region. Beyond complete evaporation, the evolution of the quantum state past the Cauchy horizon cannot remain unitary, raising the questions: how can this evolution be described as a quantum map, and how is causality preserved? What are the possible effects of such non-standard quantum evolution maps on the behaviour of the entangled laboratory partner? More generally, the laws of quantum evolution under extreme conditions in remote regions (not just in evaporating black-hole interiors, but possibly near other naked singularities and regions of extreme spacetime structure) remain untested by observation, and might conceivably be non-unitary or even nonlinear, raising the same questions about the evolution of entangled states. The answers to these questions are subtle, and are linked in unexpected ways to the fundamental laws of quantum mechanics. We show that terrestrial experiments can be designed to probe and constrain exactly how the laws of quantum evolution might be altered, either by black-hole evaporation, or by other extreme processes in remote regions possibly governed by unknown physics.
Among interference effects that have been observed using parametric down-conversion light, some experiments show that second and fourth order interference effects can be observed between independent light beams. Also, it has been shown that the presence of mirrors reflecting the pump and the down converted beams can give rise to interference effects. In this paper we show that these mirror effects can be explained alternatively by means of a coherence induced between initially independent beams. Our analysis is based on a longitudinal multimode approach and leads to intensity and coincidence distributions and fringe visibilities. We also present experimental results showing that mirror effects can be observed even using a pump beam with a coherence length much smaller than the distance between crystal and pump mirror.
It is shown that the two-photon phase coherence of parametrically generated photon pairs, which is at the origin of squeezed-light generation, can be directly probed using an intensity-correlation measurement. The resulting intensity correlation leads to a new violation of Bell's inequalities, which could be experimentally tested.
The amplification of electromagnetic fields is analyzed in a quantum-mechanical context by discussing the behavior of a simple theoretical model of the parametric amplifier. The statistical properties of the amplifier fields are described by means of the time-dependent density operator for the system. In doing this, extensive use is made of the coherent states and the P representation of the density operator, which provide a quantum-mechanical description of the fields closely resembling their classical description. Explicit solutions are obtained for the density operator for either of the two field modes for a variety of initial states of the modes. Initial states considered include combinations of coherent states, chaotic mixtures, and n-quantum states. Particular attention is given the behavior of the amplifier fields in the limit of large amplification. The conditions are established under which the amplification process leads in this limit to the existence of a non-negative P representation for the density operator for a single mode of oscillation.
A new connection between autocorrelation and cross correlation functions of the intensities of signal and idler in parametric oscillators is derived, with the same origin and generality as the Manley-Rowe relations and with inserting experimental implications in the quantum domain.
The problem of nonlinear optical sum-frequency generation and second harmonic generation is solved by a method neglecting changes of statistics of generating radiations. It is shown here that the increase of fluctuation levels in subfrequency radiations reduces the total efficiency of the sum-frequency generation, whilst in the degenerate case the second harmonic generation is accelerated with an increasing fluctuation level in the fundamental radiation. Total efficiencies of real nonlinear processes are estimated at the end of the paper.
If two initially coherent light fields with the frequencies omega and 2 omegas, which have been produced by a second harmonic generator, pass a degenerate parametric amplifier, photon-anticorrelation effects can develop. The time-development of the photon distribution for the signal wave is calculated by treating both fields quantum mechanically.
I discuss the photon antibunching effect and point out that there exist many states which exhibit the effect and demonstrate a simple procedure for generating them mathematically. I also suggest here a possible approach to the experimental observation of the effect.
The effect of phase and amplitude fluctuations of the pump mode on the quantum-statistical properties of the signal mode is considered. It is shown that these fluctuations in the laser field tend to decrease the squeezing of the signal field.
We present a quantum-statistical analysis of the steady-state light fields in driven parametric oscillation in a cavity. Using the solution of a Fokker-Planck equation for the generalized P representation of the signal and idler modes, we calculate the mean photon numbers, second-order correlation functions, and intermode correlation function as functions of the driving field. The generalized P distribution describes the signal and idler modes' statistics over the whole range of driving-field strengths except in the region far below the oscillation threshold. The second-order correlation functions are found to violate the Cauchy-Schwartz inequality, a violation allowed because the P distribution is complex. Squeezing is also found in a linear combination of the signal and idler fields.
A two-step nonlinear optical interaction consisting of (i) a spontaneous frequency down-conversion process coupled with (ii) a frequency up-conversion process is treated quantum mechanically in the experimentally observed case of simultaneous collinear phase matching of both processes. The temporal behavior of the output signals is found to depend on which process, (i) or (ii), predominates. The nature of the photon statistics for each of the output modes is found to be "super-Poisson," regardless of which process, (i) or (ii), is dominant, and the zero-point fluctuations characteristic of parametric amplification are exhibited.
Correlation effects in parametric photon-pair production are studied within the framework of a physically realistic model. The analysis, which is fully quantum mechanical, takes into account the finite sizes of the target and the beam cross section, and allows for dispersion and anisotropy in the linear susceptibility. The correlations in position and time at which the two members of a parametrically generated pair may be detected are carefully evaluated. These correlations, which have been measured experimentally, are intrinsically quantum mechanical; i.e., they can be explained by no theory in which the subharmonic fields are described purely by c-number functions. A complete solution, from which field correlation functions of arbitrarily high order may be evaluated, is obtained by a method which at the same time allows for an arbitrary degree of parametric gain. The solution is expressed entirely in terms of a particular two-point field-correlation-function, as evaluated in lowest order in the incident field strength, at points distant from the target. The function in question is found by directly examining the fluctuating currents in the material medium, rather than by eliminating the matter variables at the outset through the introduction of a nonlinear electomagnetic susceptibility.
The relationship between the squeezing and photon-number fluctuations in
the output of a degenerate parametric oscillator is investigated. The
addition of a second driving field at the idler frequency allows the
direction of the squeezing to be changed. The squeezing may appear in
the amplified quadrature. Photon antibunching or bunching may occur
depending on whether the quadrature carrying the coherent excitation is
squeezed.
The influence of intermodal correlations upon the efficiencies of nonlinear quadratic optical processes is treated for special cases of three-mode non-degenerate and degenerate optical processes, when the amplitude of at least one mode is equal to zero at the beginning of the process.
The nonlinear optical sum-frequency generation and the second harmonic generation with coherent and chaotic input radiations are treated in this paper. The short-time (-length) approximation method including the small parameter up to the eighth order is used. Effects of spectral widths of radiations as well as dispersion effects are neglected. Mean photon numbers, intermodal correlations and autocorrelations are calculated in the course of the nonlinear process. The obtained results are discussed from the point of view of the effect of intermodal correlations on nonlinear optical processes.
An analysis of the degenerate parametric amplifier including the quantisation of pump and signal modes is presented. It is shown that the fluctuations in one quadrature of the signal mode may be reduced at most by a factor of two. This is in contradistinction to analyses where the pump field is treated classically which overestimate the reduction in fluctuations possible.
We analyse the nonlinear interaction between a fundamental and second harmonic wave inside a Fabry-Pérot cavity where both modes are driven by a coherent driving field with a definite phase. We find that under certain experimentally accessible conditions bistable operation is possible. In addition it is shown that photon antibunching in the steady state may occur.
Simultaneity in optical photon pairs parametric production, verifying quantum mechanical description of fluorescence
Homodyne detection has been proposed as a means of detecting squeezed coherent radiation. Here the response of a balanced homodyne detector to wideband squeezed coherent states is presented. In order to carry out the analysis the theory of wideband photodetection is reviewed and in order to determine the ultimate performance limits of photoemissive detectors small terms of order Deltaomega/omega0 that are usually neglected, where omega0 is the optical carrier frequency and Deltaomega is the electronics bandwidth, have been kept. It is shown that the ultimate noise reduction that can be achieved in the noise-power spectrum of a homodyne detector, detecting squeezed coherent radiation, is a factor of 2 worse when photoemissive detectors are used instead of power flux detectors.
Wideband calculations of the response of a homodyne detector to the outputs of various four-wave-mixer configurations are presented. It is shown that the noise-power spectrum of the homodyne detector output can exhibit regions where the noise is greatly reduced below the shot-noise level even at frequencies far from dc. Hence, in the detection of noise squeezing via homodyne detectors, 1/f noise and other low-frequency noise sources may be avoided by observing the homodyne detector’s noise power at frequencies far from dc.
A theoretical study is made of the process in which incident pump photons that interact with a nonlinear medium (such as a crystal lacking inversion symmetry) are spontaneously split into lower-frequency signal and idler photons. The down-converted fields are quantized and described by a continuum of modes, a subset of which interacts with each photodetector. It is shown that when two ideal photodetectors are appropriately located so that they receive the conjugate signal and idler photons, then the joint probability of two-photon detection by the two detectors can equal the single-photon detection probability. The time correlation between the two detected photons is shown to be limited either by the resolving time of the detectors, or by the bandwidth of the down-converted light, and to be independent of the coherence time of the pump field or of the length of the nonlinear medium. These conclusions are compared with the results of recent experiments.
A proposed experiment is analyzed theoretically. In the proposed experiment two coherent pump waves fall on two identical nonlinear crystals, down-converted signal and idler beams from the two crystals are mixed by two beam splitters, and the coincidence counting rate for photons leaving the beam splitters is measured. We show that this counting rate depends on the phase difference between the two coherent pump waves, and results from the interference of the vacuum with the down-converted photons. The experiment could be used to look for locality violations along the lines recently proposed by Grangier, Potasek, and Yurke [Phys. Rev. A 38, 3132 (1988)], but without the need for a coherent reference beam for homodyning.
An experiment has been carried out in which two pairs of light beams produced by down conversion in two nonlinear crystals driven by a common pump were mixed by two beam splitters, and the coincidence rate for simultaneous detections of mixed signal and idler photons was measured. It is found that the down-converted light carries information about the phase of the pump field through the entanglement of the down-converted photons with the vacuum.
An exposition is given of the fundamental ideas of the recently opened field of two-particle interferometry, which employs spatially separated, quantum mechanically entangled two-particle states. These ideas are illustrated by a realizable arrangement, in which four beams are selected from the output of a laser-pumped down-converting crystal, with two beams interferometrically combined at one locus and two at another. When phase shifters are placed in these beams, the coincident count rates at the two loci will oscillate as the phases are varied, but the single count rates will not.
- B. R. Mollow