This paper gives a constructive answer to the question whether photon states can contain or not, and to what extent, the readings of rulers and clocks. The paper first shows explicitly that, along with the momentum representation, there is room in the one photon Hilbert space for an alternative position representation. This is made possible by the existence of a self-adjoint, involutive, position operator conjugate to the momentum operator (due to M. Hawton). Position and momenta are shown to satisfy the Heisenberg-Weyl quantization rules in the helicity basis, which is analyzed anew from this point of view. The paper then turns to the photon's time of arrival. By picking an appropriate photon Hamiltonian - using Maxwell equations as the photon Schroedinger equation - a conjugate time of arrival operator is built. Its interpretation, including the probability densities for the instant of arrival (at arbitrary points of 3-D space) of photon states with different helicities coming from arbitrary places, is discussed.
We have recently constructed a photon position operator with commuting components. This was long thought to be impossible, but our position eigenvectors have a vortex structure like twisted light. Thus they are not spherically symmetric and the position operator does not transform as a vector, so that previous non-existence arguments do not apply. We find two classes of position eigenvectors and obtain photon wave functions by projection onto the bases of position eigenkets that they define, following the usual rules of quantum mechanics. The hermitian position operator, r0, leads to a Landau-…
In this paper, an experiment of quantum diffraction of position-momentum entangled photons from a straight sharp edge is presented. Path of a single photon of an entangled pair is partially blocked by a sharp edge whereas the other photon is detected at a stationary location without revealing the which-path information of the other photon. Quantum diffraction pattern of the sharp edge is revealed only in the correlated conditional detection of spatially separated photons and no diffraction pattern is formed in local detections of individual photons. Theoretical analysis of the quantum diffract…
We analyse the correlation properties of light in non-linear wave-guides and transmission lines, predict the position-momentum realization of EPR paradox for photon pairs in Kerr-type non-linear photonic circuits, and we show how two-photon entangled states can be generated and detected.
In this article, we show that in the level of quantum mechanics, a photon position operator with commuting components can be obtained in a more natural way; in the level of quantum field theory, the photon position operator corresponds to the center of the photon number. It is most interesting for us to show that, a photon inside a waveguide can be localized in the same sense that a massive particle can be localized in free space, and just as that it is impossible to localize a massive particle with a greater precision than its Compton wavelength, one can also arrival at a similar conclusion: …
The quantum interference of entangled photons forms a key phenomenon underlying various quantum-optical technologies. It is known that the quantum interference patterns of entangled photon pairs can be reconstructed classically by the time-reversal method; however, the time-reversal method has been applied only to time-frequency-entangled two-photon systems in previous experiments. Here, for the first time, we apply the time-reversal method to the position-wavevector-entangled two-photon systems: the two-photon Young interferometer and the two-photon beam focusing system. We experimentally dem…
The measurement of the position-momentum EPR correlations of a two-photon state is important for many quantum information applications ranging from quantum key distribution to coincidence imaging. However, all the existing techniques for measuring the position-momentum EPR correlations involve coincidence detection and thus suffer from issues that result in less accurate measurements. In this letter, we propose and demonstrate an experimental scheme that does not require coincidence detection for measuring the EPR correlations. Our technique works for two-photon states that are pure, irrespect…
We show that the position operator with commuting components proposed by M. Hawton [M. Hawton, Phys. Rev. A {\bf 59}, 954 (1999)] and developed in subsequent papers, including the recent ones, does not have the properties required for a photon position operator. Depending on the exact interpretation of the results it either concerns a triplet of massless spin zero particles rather than the photon, or does not have the required covariance properties under space rotations.
The positive operator valued measure (POVM) for a photon counting array detector is derived and found to equal photon flux density integrated over pixel area and measurement time. Since photon flux density equals number density multiplied by the speed of light, this justifies theoretically the observation that a photon counting array provides a coarse grained measurement of photon position. The POVM obtained here can be written as a set of projectors onto a basis of localized states, consistent with the description of photon position in a recent quantum imaging proposal [M. Tsang, Phys. Rev. L…
One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all o…
We investigate the utility of Einstein-Podolsky-Rosen correlations of the position and momentum of photon pairs from parametric down-conversion in the implementation of a secure quantum key distribution protocol. We show that security is guaranteed by the entanglement between downconverted pairs, and can be checked by either direct comparison of Alice and Bob's measurement results or evaluation of an inequality of the sort proposed by Mancini et al. (Phys. Rev. Lett. 88, 120401 (2002)).
In biorthogonal quantum mechanics, the eigenvectors of a quasi-Hermitian operator and those of its adjoint are biorthogonal and complete and the probability for a transition from a quantum state to any one of these eigenvectors is positive definite. We apply this formalism to the long standing problem of the position observable in quantum field theory. The dual bases are positive and negative frequency one-particle states created by the field operator and its conjugate and biorthogonality is a consequence of their commutation relations. In these biorthogonal bases the position operator is cova…
We extend a procedure for construction of the photon position operators with transverse eigenvectors and commuting components [Phys. Rev. A 59, 954 (1999)] to body rotations described by three Euler angles. The axial angle can be made a function of the two polar angles, and different choices of the functional dependence are analogous to different gauges of a magnetic field. Symmetries broken by a choice of gauge are re-established by transformations within the gauge group. The approach allows several previous proposals to be related. Because of the coupling of the photon momentum and spin, our…
It is proposed in this paper that without a measurement, the wave function of a system periodically transits to a bound energy eigenfunction or the complementary wave function that is orthogonal to all the bound energy eigenfunctions. Applying this assumption to analyze results of position measurement shows that when the wave function of the entangled incident object and a position detector is a linear combination of the bound energy eigenfunctions of the entangled system prior to collapses, the detection rate essentially equals the probability flux of the incident object entering the detector…
We show that the cylindrical symmetry of the eigenvectors of the photon position operator with commuting components, x, reflects the E(2) symmetry of the photon little group. The eigenvectors of x form a basis of localized states that have definite angular momentum, J, parallel to their common axis of symmetry. This basis is well suited to the description of "twisted light" that has been the subject of many recent experiments and calculations. Rotation of the axis of symmetry of this basis results in the observed Berry phase displacement. We prove that {x1,x2,J3} is a realization of the two di…
Channel position finding is the task of determining the location of a single target channel amongst an ensemble of background channels. It has many potential applications, including quantum sensing, quantum reading and quantum spectroscopy. In particular, it could allow for simple detection protocols to be extended to ones of measurement, for example, target ranging with quantum illumination. The use of quantum states and entanglement in such protocols have shown to yield quantum advantages over their optimal classical counterparts. Here we consider quantum channel position finding using sourc…
It has very recently been suggested that asymmetric coupling of electromagnetic fields to thermal reservoirs under nonequilibrium conditions can produce unexpected oscillatory behavior in the local photon statistics in layered structures. Better understanding of the predicted phenomena could enable useful applications related to thermometry, noise filtering, and enhancing optical interactions. In this work we briefly review the field quantization and study the local steady state temperature distributions in optical cavities formed of lossless and lossy media to show that also local field tempe…
We revisit the back-action of emitted photons on the motion of the relative position of two cold atoms. We show that photon recoil resulting from the spontaneous emission can induce the localization of the relative position of the two atoms through the entanglement between the spatial motion of individual atoms and their emitted photons. The result provides a more realistic model for the analysis of the environment-induced localization of a macroscopic object.
The binary (one-bit-per-photon) encoding that most existing quantum key distribution (QKD) protocols employ puts a fundamental limit on their achievable key rates, especially under high channel loss conditions associated with long-distance fiber-optic or satellite-to-ground links. Inspired by the pulse-position-modulation (PPM) approach to photon-starved classical communications, we design and demonstrate the first PPM-QKD, whose security against collective attacks is established through continuous-variable entanglement measurements that also enable a novel decoy-state protocol performed conve…
Different from the usual conclusion that the separation of the photon angular momentum into orbital and spin parts is physically meaningless, the orbital and spin angular momenta are demonstrated in the first-quantization framework not to satisfy the standard commutation relation. It is shown on the basis of the transversality condition that the spin is aligned with the propagation direction so that only the helicity can be the intrinsic degree of freedom. It is also shown on the same basis that only in the so-called intrinsic reference system does the helicity behave intrinsic. The intrinsic …
A first quantized free photon is a complex massless vector field $A=(A^μ)$ whose field strength satisfies Maxwell's equations in vacuum. We construct the Hilbert space $\mathscr{H}$ of the photon by endowing the vector space of the fields $A$ in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in $\mathscr{H}$, determine the operators representing the momentum and the helicity of the photon, and introduce a chira…
Einstein-Podolsky-Rosen (EPR) entanglement introduced in 1935 deals with two particles that are entangled in their positions and momenta. Here we report the first experimental demonstration of EPR position-momentum entanglement of narrowband photon pairs generated from cold atoms. By using two-photon quantum ghost imaging and ghost interference, we demonstrate explicitly that the narrowband photon pairs violate the separability criterion, confirming EPR entanglement. We further demonstrate continuous variable EPR steering for positions and momenta of the two photons. Our new source of EPR-enta…
It is shown that the photon position operator $\hat{\vec{X}}$ with commuting components can be written in the momentum representation as $\hat{\vec{X}}=i \hat{\vec{D}}$, where $\hat{\vec{D}}$ is a flat connection in the tangent bundle $T(\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\})$ over $\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\}$ equipped with the Cartesian structure. Moreover, $\hat{\vec{D}}$ is such that the tangent $2$-planes orthogonal to the momentum are parallelly propagated with respect to $\hat{\vec{D}}$ and, also, $\hat{\vec{D}}$ is a…
Witnessing continuous-variable Bell nonlocality is a challenging endeavor, but Bell himself showed how one might demonstrate this nonlocality. Though Bell nearly showed a violation using the CHSH inequality with sign-binned position-momentum statistics of entangled pairs of particles measured at different times, his demonstration is subject to approximations not realizable in a laboratory setting. Moreover, he doesn't give a quantitative estimation of the maximum achievable violation for the wavefunction he considers. In this article, we show how his strategy can be reimagined using the transv…
Quantum imaging with undetected photons (QIUP) has recently emerged as a new powerful imaging tool. Exploiting the spatial entanglement of photon pairs, it allows decoupling of the sensing and detection wavelengths, facilitating imaging in otherwise challenging spectral regions with mature silicon-based detection technology. All existing implementations of QIUP have so far utilised the momentum correlations within the biphoton state. Here, for the first time, we implement and examine theoretically and numerically the complementary scenario - utilising the tight position correlations formed wit…
We propose a procedure for defining all single-photon observables in terms of Positive-Operator Valued Measures (POVMs), in particular spin and position. We identify the suppression of $0$-helicity photon states as a projection from an extended Hilbert space onto the photon Hilbert space. We show that all single-photon observables are in general described by POVMs, obtained by applying this projection to opportune Projection-Valued Measures (PVMs), defined on the extended Hilbert space. The POVMs associated to momentum and helicity reduce to PVMs, unlike those associated to position and spin, …
We report on a momentum-position realization of the EPR paradox using direct detection in the near and far fields of the photons emitted by collinear type-II phase-matched parametric downconversion. Using this approach we achieved a measured two-photon momentum-position variance product of $0.01\hbar^2$, which dramatically violates the bounds for the EPR and separability criteria.
Optical low-coherence reflectometry is capable of unambiguously measuring positions of stacked, partially reflective layers in a sample object. It relies on the low coherence of the light source and the absolute distances are obtained from the position reading of a mechanical motor stage. We show how to exploit the simultaneous high and low coherence properties of energy-time entangled photon pairs to directly calibrates the position scale of an OLCR scan with a reference laser wavelength. In experiment, a precision of 1.6\,nm and good linearity is demonstrated.
We demonstrate a scalable readout scheme for an infrared single-photon pulse-position camera consisting of WSi superconducting nanowire single-photon detectors (SNSPDs). For an N x N array, only 2 x N wires are required to obtain the position of a detection event. As a proof-of-principle, we show results from a 2 x 2 array.
We report an experiment in which two-photon interference occurs between degenerate single photons that never meet. The two photons travel in opposite directions through our fibre-optic interferometer and interference occurs when the photons reach two different, spatially separated, 2-by-2 couplers at the same time. We show that this experiment is analogous to the conventional Franson-type entanglement experiment where the photons are entangled in position and time. We measure wavefunction overlaps for the two photons as high as 94 $\pm$ 3%.
The realization of Karl Popper's EPR-like experiment by Shih and Kim (published 1999) produced the result that Popper hoped for: no ``action at a distance'' on one photon of an entangled pair when a measurement is made on the other photon. This experimental result is interpretable in local realistic terms: each photon has a definite position and transverse momentum most of the time; the position measurement on one photon (localization within a slit) disturbs the transverse momentum of that photon in a non-predictable way in accordance with the uncertainty principle; however, there is no effect…
The realization of Karl Popper's experiment by Shih and Kim (published 1999) produced the result that Popper hoped for: no ``action at a distance'' on one photon of an entangled pair when a measurement is made on the other photon. This experimental result is interpretable in local realistic terms: each photon has a definite position and transverse momentum most of the time; the position measurement on one photon (localization in a slit) disturbs the transverse momentum of that photon in a non-predictable way in accordance with the uncertainty principle; however, there is no effect on the other…
An entangled pair of photons (1 and 2) are emitted to opposite directions. A narrow slit is placed in the path of photon 1 to provide precise knowledge of its position on the $y$ axis and this also determines the precise $y$ position of its twin, photon 2, due to quantum entanglement. Is photon 2 going to experience a greater uncertainty in momentum, i.e., a greater $Δp_{y}$, due to the precise knowledge of its position $y$? The experimental data shows $ΔyΔp_{y}<\hbar $ for photon 2. Can this recent realization of the historical thought experiment of Karl Popper signal a violation of the uncer…
Encoding information in the position of single photons has no known limits, given infinite resources. Using a heralded single-photon source and a Spatial Light Modulator (SLM), we steer single photons to specific positions in a virtual grid on a large-area spatially resolving photon-counting detector (ICCD). We experimentally demonstrate selective addressing any location (symbol) in a 9072 size grid (alphabet) to achieve 10.5 bit of mutual information between the sender and receiver per detected photon. Our results set the stage for very-high-dimensional quantum information processing.
This manuscript presents a method to convert position correlation of photon-pairs into polarization entanglement. This is achieved by individually manipulating the polarization state of photons generated in different parts of a non-linear medium and putting them in coherent superposition. This concept is experimentally demonstrated using photon-pairs produced by spontaneous parametric down conversion (SPDC). The method was used to implement a compact source producing an observed photon-pair rate of 120,000/s/mW with an entanglement fidelity of 0.99. This method can be extended to any photon-pa…
In spontaneous parametric down-conversion photons are known to be created coherently and with equal probability over the entire length of the crystal. Then, there is no particular position in the crystal where a photon pair is created. We make the seemingly contradictory observation that we can control the time delay with the crystal position along the propagation direction. We resolve this contradiction by showing that the spatio-temporal correlations critically affect the temporal properties of the pair of photons, when using a finite detector size. We expect this to have important implicati…
This comment refutes the claim made by A. Jadczyk and A.M. Schlichtinger in arXiv:2203.14555 that the photon position operator with commuting components proposed in 1999 does not have the properties required for a photon position operator.
An experimental proposal is presented in which dark port post-selection together with weak measurements are used to enlarge the radiation pressure effect of a single photon on a mechanical oscillator placed in the middle of a Fabry-Perot cavity and initialized in the ground state. By preparing and post-selecting the photon (the system) in two quasi orthogonal states, the weak value of the radiation force operator can lie outside the eigenvalue spectrum, producing a large shift on the wave function of the mechanical oscillator (the measuring device) in the position representation. Consequently,…
Since its publication, Aharonov and Vaidman's three-box paradox has undergone three major advances: i). A non-counterfactual scheme by the same authors in 2003 with strong rather than weak measurements for verifying the particle's subtle presence in two boxes. ii) A realization of the latter by Okamoto and Takeuchi in 2016. iii) A dynamic version by Aharonov et al. in 2017, with disappearance and reappearance of the particle. We now combine these advances together. Using photonic quantum routers the particle acts like a quantum "shutter." It is initially split between Boxes A, B and C, the lat…
In light-pulsed atom interferometry, the phase accumulated by atoms depends on the effective wave vector of the absorbed photons. In this work, we proposed a theory model to analyses the effective wave vector of photons in structured light. As for monochromatic optical field, a transverse confinement could lead to diffraction. We put forward that in light-atom interaction, the atom wave function could also provide a transverse confinement thus affect the effective wave vector of the absorbed photons. We calculated the relative shift of the photon effective wave vector when an atom with a Gauss…
We have recently developed a position-dependent quantization scheme for describing the ladder and effective photon-number operators associated with the electric field to analyze quantum optical energy transfer in lossy and dispersive dielectrics [Phys. Rev. A, 89, 033831 (2014)]. While having a simple connection to the thermal balance of the system, these operators only described the electric field and its coupling to lossy dielectric bodies. Here we extend this field quantization scheme to include the magnetic field and thus to enable description of the total electromagnetic field and discuss…
Although non-commutativity of a certain set of quantum operators (e.g., creation/annihilation operators and Pauli spin operators) has been shown experimentally in recent years, the commu- tation relation for the position and the momentum operators has not been directly demonstrated to date. In this paper, we propose and analyze an experimental scheme for directly observing the non-commutativity of the position and the momentum operators using single-photon quantum in- terference. While the scheme is studied for the single-photon state as the input quantum state, the analysis applies equally to…
Since the first quantum ghost imaging (QGI) experiment in 1995, many QGI schemes have been put forward. However, the position-position or momentum-momentum correlation required in these QGI schemes cannot be distributed over optical fibers, which limits their large geographical applications. In this paper, we propose and demonstrate a scheme for long distance QGI utilizing frequency correlated photon pairs. In this scheme, the frequency correlation is transformed to the correlation between the illuminating position of one photon and the arrival time of the other photon, by which QGI can be rea…
We study the single-photon transport in a single-mode waveguide, in which two-level atoms in spatial superpositions are embedded. We find that the transmission of the photon can be used as a non-destructive probe for coherence of the superposition. The protocol proposed is shown to be independent of the distance between the wells in which the atoms are trapped. Furthermore, we show that by looking at the transmission in a suitable regime, a maximally entangled state can be post-selected from an unknown superposition.
We present the first robust implementation of a coined quantum walk over five steps using only passive optical elements. By employing a fiber network loop we keep the amount of required resources constant as the walker's position Hilbert space is increased. We observed a non-Gaussian distribution of the walker's final position, thus characterizing a faster spread of the photon wave-packet in comparison to the classical random walk. The walk is realized for many different coin settings and initial states, which opens the way for the implementation of a quantum walk-based search algorithm.
We demonstrate experimentally that it is possible to prepare and detect photon pairs created by spontaneous parametric down-conversion which exhibit simultaneous position-momentum and polarization correlations that are adequate to implement a four-dimensional key distribution protocol.
We demonstrate a novel approach of violating position dependent Bell inequalities by photons emitted via independent photon sources in free space. We trace this violation back to path entanglement created a posteriori by the selection of modes due to the process of detection.
