The excitation of atomic and molecular systems by propagating light in a two-photon state within the Wigner-Weisskopf approximation has been described using stochastic tools. The problem of a stochastic evolution of the quantum system, depending on the results of the measurement of the output field, was formulated and solved making use of the model of repeated interactions and measurement. We defined the discrete in-time interaction between the quantum system and its environment being the electromagnetic field approximated by a chain or chains of harmonic oscillators. We determined analytical formulae for quantum trajectories associated with one-dimensional and two-dimensional counting processes, corresponding respectively to unidirectional or bidirectional input field prepared in the two-photon states. We derived the formulae for the exclusive probability densities of photon counts that allow us to completely characterize the photon statistics of the output field.
Finally, we showed how to apply the quantum trajectories to obtain the formula for the probability of the two-photon absorption for a three-level atom in a ladder configuration. The paper also includes a discussion on the optimal two-photon state that maximizes the two-photon absorption probability.
Kocsis et al. (Science, Reports 3 June 2011, p. 1170) state that the experimentally deduced average photon trajectories are identical to the particle trajectories of Bohm's quantum mechanics. No supporting evidence, however, was provided. The photon trajectories presented in their report do not converge to high probability regions, a familiar and necessary behavior of Bohm trajectories. We reanalyze their data and calculations, conclude that the average photon trajectories do indeed agree with Bohm, and discuss possible interpretations of this result.
Recent work using spontaneous parametric down-conversion (SPDC) has made possible investigations of the Einstein-Podolsky-Rosen paradox in its original (position-momentum) form. We propose an experiment that uses SPDC photon pairs to measure through which slit a photon passes while simultaneously observing double-slit interference.
Using the relativistic quantum Hamilton-Jacobi equation within the framework of the equivalence postulate, and grounding one self on both relativistic and quantum Lagrangians, we construct a Lagrangian of relativistic quantum system in one dimension and derive a third order equation of motion representing a first integral of the relativistic quantum Newton's law. Then, we investigate the free particle case and establish the photon's trajectories.
A century after its conception, quantum mechanics still hold surprises that contradict many "common sense" notions. The contradiction is especially sharp in case one consider trajectories of truly quantum objects such as single photons. From a classical point of view, trajectories are well defined for particles, but not for waves. The wave-particle duality forces a breakdown of this dichotomy and quantum mechanics resolves this in a remarkable way: Trajectories can be well defined, but they are utterly different from classical trajectories. Here, we give an operational definition to the trajec…
We derived quantum trajectories for a system interacting with the environment prepared in a continuous mode single photon state as the limit of discrete filtering model with an environment defined as series of independent qubits prepared initially in the entangled state being an analogue of a continuous mode state. The environment qubits interact with the quantum system and they are subsequently measured. The initial correlation between the bath qubits is the source of the non-Markovianity. The conditional evolutions of the quantum system for limit of the continuous in time observations togeth…
Recently, a scheme based on the method of weak measurements to register the trajectories of photons passing through a nested Mach-Zehnder interferometer was proposed [L. Vaidman, Phys. Rev. A \textbf{87}, 052104 (2013)] and then realized [A. Danan, D. Farfurnik, S. Bar-Ad, et al., Phys. Rev. Lett. \textbf{111}, 240402 (2013)]. Interpreting the results of the experiment, the authors concluded that "the photons do not always follow continuous trajectories". It is shown in this work that these results can be easily and clearly explained in terms of traditional classical electrodynamics or quantum…
Through the constant potential and the linear potential, we establish the existence of nodes for the relativistic quantum trajectories as the same way as for the quantum trajectories. We establish the purely relativistic limit $(\hbar \to 0)$ for these trajectories, and link the nodes to de Broglie's wavelength.
We consider the open two-site Bose-Hubbard dimer, a well-known quantum mechanical model that has been realised recently for photons in two coupled photonic crystal nanocavities. The system is described by a Lindblad master equation which, for large numbers of photons, gives rise to a limiting semiclassical model in the form of a four-dimensional vector field. From the situation where both sites trap the same amount of photons under symmetric pumping, one encounters a transition that involves symmetry breaking, the creation of periodic oscillations and multistability as the pump strength is inc…
We derive stochastic master equation for a quantum system interacting with an environment prepared in a continuous-mode $N$-photon state. To determine the conditional evolution of the quantum system depending on continuous in time measurements of the output field the model of repeated interactions and measurements is applied. The environment is defined as an infinite chain of harmonic oscillators which do not interact between themselves and they are prepared initially in an entangled state being a discrete analogue of a continuous-mode $N$-photon state. We provide not only the quantum trajecto…
In [JETP Lett. 105(3), 152 (2017)], a clear and comprehensive analysis of the paradoxical results of experiment [Phys. Rev. Lett. 111, 240402 (2013)] was carried out on the basis of the classical wave theory of light, which presupposes the continuity of possible of light paths. It was shown that the paradoxical results of the experiment are due not to the discontinuity of the trajectories of light, as claimed in [Phys. Rev. Lett. 111, 240402 (2013)], but to the used way of detecting the path of photons. The experiment modification proposed in [JETP Lett. 105(3), 152 (2017)] allows us to elimin…
We investigate two different kinds of quantum trajectories for a nonlinear photon resonator subject to two-photon pumping, a configuration recently studied for the generation of photonic Schroedinger cat states. In the absence of feedback control and in the strong-driving limit, the steady-state density matrix is a statistical mixture of two states with equal weight. While along a single photon-counting trajectory the systems intermittently switches between an odd and an even cat state, we show that upon homodyne detection the situation is different. Indeed, homodyne quantum trajectories revea…
In a recent paper Mahler {\em et al.} have argued that the experiments of Kocsis {\em et al.} provide experimental evidence for Bohmian mechanics. Unfortunately these experiments used relativistic, zero rest mass photons whereas Bohmian mechanics is based on non-relativistic Schrödinger particles having non-zero rest mass. The experimental results can be accounted for in terms of a different approach based on the electromagnetic field that was already outlined by Bohm in an appendix of the second of his 1952 papers. A subsequent development of this approach by Bohm, Hiley, Holland, Kaloyerou, …
Bohmian mechanics is a nonlocal hidden-variable interpretation of quantum theory which predicts that particles follow deterministic trajectories in spacetime. Historically, the study of Bohmian trajectories has mainly been restricted to nonrelativistic regimes due to the widely held belief that the theory is incompatible with special relativity. Here we derive expressions for the relativistic velocity and spacetime trajectories of photons in a Michelson-Sagnac-type interferometer. The trajectories satisfy quantum-mechanical continuity and the relativistic velocity addition rule. Our new veloci…
In the Hong--Ou--Mandel interferometric scheme, two identical photons that illuminate a balanced beam splitter always leave through the same exit port. Similar effects have been predicted and (partially) experimentally confirmed for multi-photon Fock-number states. In the limit of large photon numbers, the output distribution follows a $(1-x^2)^{-1/2}$ law, where $x$ is the normalized imbalance in the output photon numbers at the two output ports. We derive an analytical formula that is also valid for imbalanced input photon numbers with a large total number of photons, and focus on the extent…
We study the multi-channel quantum routing of the single photons in a waveguide-emitter system. The channels are composed by the waveguides and are connected by intermediate two-level emitters. By adjusting the intermediate emitters, the output channels of the input single photons can be controlled. This is demonstrated for the cases of one output channel, two output channels and the generic N output channels. The results show that the multi-channel quantum routing of single photons can be well achieved in present system. This sheds light on the experimental realization of quantum routing of s…
The question of the representation of quantum stationary partially polarized waves as random superpositions of different polarization ellipses is addressed. To this end, the Bohmian formulation of quantum mechanics is considered and extended to quantum optical polarization. As is shown, this approach properly combines definite time-evolving trajectories with rigorous stationary quantum distributions via the topology displayed by the associated phase field.
We propose a waveguide-cavity coupled system to achieve the routing of photons by the phases of other photons. Our router has four input ports and four output ports. The transport of the coherent-state photons injected through any input port can be controlled by the phases of the coherent-state photons injected through other input ports. This control can be achieved when the mean numbers of the routed and control photons are small enough and require no additional control fields. Therefore, the all-optical routing of photons can be achieved at the single-photon level.
A distribution function approach is applied to describe the dynamics of the laser beam in the Earth atmosphere. Using a formal solution of the kinetic equation for the distribution function, we have developed an iterative scheme for calculation of the scintillation index ($σ^2$). The problem reduces to obtaining the photon trajectories and their correlations. Bringing together theoretical calculations and many-fold computer integrations, the value of $σ^2$ is obtained. It is shown that a considerable growth of $σ^2$ in the range of a moderate turbulence is due to the correlations of different …
We provide a systematic treatment of $N$-photon transport in a waveguide coupled to a local system, using the input-output formalism. The main result of the paper is a general connection between the $N$-photon S matrix and the Green functions of the local system. We also show that the computation can be significantly simplified, by exploiting the connectedness structure of both the S matrix and the Green function, and by computing the Green function using an effective Hamiltonian that involves only the degrees of freedom of the local system. We illustrate our formalism by computing $N$-photon …
The problem of photon adding and subtracting is studied, using conditional output measurement on a beam splitter. It is shown that for various classes of states the corresponding photon-added and -subtracted states can be prepared. Analytical results are presented, with special emphasis on photon-added and -subtracted squeezed vacuum states, which are found to represent two different types of Schroedinger-cat-like states. Effects of realistic photocounting and Fock-state preparation are discussed.
Recently, Kocsis et al. reported the observation of "average trajectories of single photons" in a two-slit interference experiment [Science 332, 1170 (2011)]. This was possible by using the quantum weak-measurements method, which implies averaging over many events, i.e., in fact, a multi-photon limit of classical linear optics. We give a classical-optics interpretation to this experiment and other related problems. It appears that weak measurements of the local momentum of photons made by Kocsis et al. represent measurements of the Poynting vector in an optical field. We consider both the real…
A method for absolute calibration of a photon-number resolving detector producing analog signals as the output is developed using a twin beam. The method gives both analog-to-digital conversion parameters and quantum detection efficiency for the photon fields. Characteristics of the used twin beam are also obtained. A simplified variant of the method applicable to fields with high signal to noise ratios and suitable for more intense twin beams is suggested.
Both cavity QED and photons are promising candidates for quantum information processing. We consider a combination of both candidates with a single photon going through spatially separate cavities to entangle the atomic qubits, based on the input-output process of the cavities. We present a general expression for the input-output process regarding the low-Q cavity confining a single atom, which works in a wide range of parameters. Focusing on low-Q cavity case, we propose some schemes for quantum information processing with Faraday rotation using single photons, which is much different from th…
Bohmian mechanics allows to visualize and understand the quantum-mechanical behavior of massive particles in terms of trajectories. As shown by Bialynicki-Birula, Electromagnetism also admits a hydrodynamical formulation when the existence of a wave function for photons (properly defined) is assumed. This formulation thus provides an alternative interpretation of optical phenomena in terms of photon trajectories, whose flow yields a pictorial view of the evolution of the electromagnetic energy density in configuration space. This trajectory-based theoretical framework is considered here to stu…
We report a comprehensive approach to analysing continuous-output photon detectors. We employ principal component analysis to maximise the information extracted, followed by a novel noise-tolerant parameterised approach to the tomography of PNRDs. We further propose a measure for rigorously quantifying a detector's photon-number-resolving capability. Our approach applies to all detectors with continuous-output signals. We illustrate our methods by applying them to experimental data obtained from a transition-edge sensor (TES) detector.
High fidelity microwave photon counting is an important tool for various areas from background radiation analysis in astronomy to the implementation of circuit QED architectures for the realization of a scalable quantum information processor. In this work we describe a microwave photon counter coupled to a semi-infinite transmission line. We employ input-output theory to examine a continuously driven transmission line as well as traveling photon wave packets. Using analytic and numerical methods, we calculate the conditions on the system parameters necessary to optimize measurement and achieve…
Quantum trajectories describe the stochastic evolution of an open quantum system conditioned on continuous monitoring of its output, such as by an ideal photodetector. In practice an experimenter has access to an output filtered through various electronic devices, rather than the microscopic states of the detector. This introduces several imperfections into the measurement process, of which only inefficiency has previously been incorporated into quantum trajectory theory. However, all electronic devices have finite bandwidths, and the consequent delay in conveying the output signal to the obse…
Interpretations of quantum mechanics (QM), or proposals for underlying theories, that attempt to present a definite realist picture, such as Bohmian mechanics, require strong non-local effects. Naively, these effects would violate causality and contradict special relativity. However if the theory agrees with QM the violation cannot be observed directly. Here, we demonstrate experimentally such an effect: we steer the velocity and trajectory of a Bohmian particle using a remote measurement. We use a pair of photons and entangle the spatial transverse position of one with the polarization of the…
The validity of the superposition principle and of Born's rule are well-accepted tenants of quantum mechanics. Surprisingly, it has recently been predicted that the intensity pattern formed in a three-slit experiment is seemingly in contradiction with the predictions of the most conventional form of the superposition principle when exotic looped trajectories are taken into account. However, the probability of observing such paths is typically very small and thus rendering them extremely difficult to measure. In this work, we confirm the validity of Born's rule and present the first experimenta…
The properties of continuous variable teleportation of single photon states are investigated. The output state is different from the input state due to the non-maximal entanglement in the EPR beams. The photon statistics of the teleportation output are determined and the correlation between the field information beta obtained in the teleportation process and the change in photon number is discussed. The results of the output photon statistics are applied to the transmission of a qbit encoded in the polarization of a single photon.
Single-photon devices at microwave frequencies are important for applications in quantum information processing and communication in the microwave regime. In this work, we describe a proposal of a multi-output single-photon device. We consider two superconducting resonators coupled to a gap-tunable qubit via both its longitudinal and transverse degrees of freedom. Thus, this qubit-resonator coupling differs from the coupling in standard circuit quantum-electrodynamic systems described by the Jaynes-Cummings model. We demonstrate that an effective quadratic coupling between one of the normal mo…
Quantum mechanics does not provide a clear answer to the question: What was the past of a photon which went through an interferometer? Various welcher weg measurements, delayed-choice which-path experiments and weak-measurements of photons in interferometers presented the past of a photon as a trajectory or a set of trajectories. We have carried out experimental weak measurements of the paths of photons going through a nested Mach-Zehnder interferometer which show a different picture: the past of a photon is not a set of continuous trajectories. The photons tell us that they have been in the p…
There has been rapid development of systems that yield strong interactions between freely propagating photons in one dimension via controlled coupling to quantum emitters. This raises interesting possibilities such as quantum information processing with photons or quantum many-body states of light, but treating such systems generally remains a difficult task theoretically. Here, we describe a novel technique in which the dynamics and correlations of a few photons can be exactly calculated, based upon knowledge of the initial photonic state and the solution of the reduced effective dynamics of …
We propose a simple and efficient method for generating metrologically useful quantum entanglement in an ensemble of spin-1 atoms that interacts with a high-finesse optical cavity mode. It requires straightforward preparation of $N$ atoms in the $m_F=0$ sublevel, tailoring of the atom-field interaction to give an effective Tavis-Cummings model for the collective spin-1 ensemble, and a photon counting measurement on the cavity output field. The photon number provides a projective measurement of the collective spin length $S$, which, for the chosen initial state, is heavily weighted around value…
Based on constructive interference in Sagnac waveguide loop, an efficient scheme is proposed for selective frequency conversion and multifrequency modes W entanglement via input-output formalism. We can adjust the probability amplitudes of output photons by choosing parameter values properly. The tunable probability amplitude will lead to the generation of output photon with a selectable frequency and W photonic entanglement of different frequencies modes in a wide range of parameter values. Our calculations show the present scheme is robust to the deviation of parameters and spontaneous decay…
The polarization analysis of quantized probe light transmitted through an atomic ensemble has been used to prepare entangled collective atomic states. In a "balanced" detection configuration, where the difference signal from two detection ports is analyzed, the continuous monitoring of a component of the Stokes field vector provides a means for conditional projective measurements on the atomic system. Here, we make use of classical driving fields, in the pulsed regime, and of an "unbalanced" detection setup (single detector) where the effective photon number of scattered photons is the detecte…
We derive quantum trajectories (also known as stochastic master equations) that describe an arbitrary quantum system probed by a propagating wave packet of light prepared in a continuous-mode Fock state. We consider three detection schemes of the output light: photon counting, homodyne detection, and heterodyne detection. We generalize to input field states that are superpositions and or mixtures of Fock states and illustrate the formalism with several examples.
In this work we present a very simple approach to input-output relations in optical cavities, limiting ourselves to one- and two-photon states of the field. After field quantization, we derive the non-unitary transformation between {\em Inside} and {\em Outside} annihilation and creation operators. Then we express the most general two-photon state generated by {\em Inside} creation operators, through base states generated by {\em Outside} creation operators. After renormalization of coefficients of inside two-photon state, we calculate the outside photon-number probability distribution in a ge…
Photon counting measurements are analyzed for obtaining a classical phase parameter in linear Mach Zehnder interferometer (MZI), by the use of phase estimation theories. The detailed analysis is made for four cases: a) Coherent states inserted into the interferometer. b) Fock number state inserted in one input port of the interferometer and the vacuum into the other input port. c) Coherent state inserted into one input port of the interferometer and squeezed-vacuum state in the other input port. d) Exchanging the first beam-splitter (BS1) of a MZI by a non-linear system which inserts a NOON st…
We develop quantum-optical input-output theory for resonators with arbitrary coupling strength, and for input fields whose spectrum can be wider than the cavity free-spectral range, while ensuring that the field-operator commutator relations in space-time variables are correct. The cavity-field commutator exhibits a series of space-time echoes, representing causal connections of certain space-time points by light propagation. We apply the theory to two-photon wave-packet shaping by cavity reflection, which displays a remarkable illustration of dispersion cancellation. We also show that the the…
State preparation via conditional output measurement on a beam splitter is studied, assuming the signal mode is mixed with a mode prepared in a Fock state and photon numbers are measured in one of the output channels. It is shown that the mode in the other output channel is prepared in either a photon-subtracted or a photon-added Jacobi polynomial state, depending upon the difference between the number of photons in the input Fock state and the number of photons in the output Fock state onto which it is projected. The properties of the conditional output states are studied for coherent and squ…
Conditional preparation of photon number states from a continuous-wave nondegenerate optical parametric oscillator is investigated. We derive the phase space Wigner function for the output state conditioned on photo detection events that are not necessarily simultaneous, and we maximize its overlap with the desired photon number state by choosing the optimal temporal output state mode function. We present a detailed numerical analysis for the case of two-photon state generation from a parametric oscillator driven with an arbitrary intensity below threshold, and in the low intensity limit, we p…
We study the unconventional photon blockade, recently proposed for a coupled-cavity system, in presence of input and output quantum fields. Mixing of the input or output channels still allows strong photon antibunching of the output field, but for optimal values of the system parameters that differ substantially from those that maximize antibunching of the intracavity field. This result shows that the specific input-output geometry in a photonic system determines the optimal design in view of single-photon device operation. We provide a compact analytical formula that allows finding the optima…
We show that conditional output measurement on a beam splitter may be used to produce photon-added states for a large class of signal-mode quantum states, such as thermal states, coherent states, squeezed states, displaced photon-number states, and coherent phase states. Combining a mode prepared in such a state and a mode prepared in a photon-number state, the state of the mode in one of the output channels of the beam splitter ``collapses'' to a photon-added state, provided that no photons are detected in the other output channel. We present analytical and numerical results, with special emp…
It is problematic to interpret the quantum jumps of an atom interacting with thermal light in terms of counts at detectors monitoring the atom's inputs and outputs. As an alternative, we develop an interpretation based on a self-consistency argument. We include one mode of the thermal field in the system Hamiltonian and describe its interaction with the atom by an entangled quantum state while assuming that the other modes induce quantum jumps in the usual fashion. In the weak-coupling limit, the photon number expectation of the selected mode is also seen to execute quantum jumps, although mor…
Low noise single-photon sources are a critical element for quantum technologies. We present a heralded single-photon source with an extremely low level of residual background photons, by implementing low-jitter detectors and electronics and a fast custom-made pulse generator controlling an optical shutter (a LiNbO3 waveguide optical switch) on the output of the source. This source has a second-order autocorrelation g^{(2)}(0)=0.005(7), and an "Output Noise Factor" (defined as the ratio of the number of noise photons to total photons at the source output channel) of 0.25(1)%. These are the best…
We extend the input-output formalism of quantum optics to analyze few-photon transport in waveguides with an embedded qubit. We provide explicit analytical derivations for one and two-photon scattering matrix elements based on operator equations in the Heisenberg picture.
