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Coherent and Incoherent States of the Radiation Field
Abstract
Methods are developed for discussing the photon statistics of arbitrary fields in fully quantum-mechanical terms.
In order to keep the classical limit of quantum electrodynamics plainly in view, extensive use is made of the
coherent states of the field. These states, which reduce the field correlation functions to factorized forms,
are shown to offer a convenient basis for the description of fields of all types. Although they are not orthogonal
to one another, the coherent states form a complete set. It is shown that any quantum state of the field may be
expanded in terms of them in a unique way. Expansions are also developed for arbitrary operators in terms of
products of the coherent state vectors. These expansions are discussed as a general method of representing the
density operator for the field. A particular form is exhibited for the density operator which makes it possible to
carry out many quantum-mechanical calculations by methods resembling those of classical theory. This representation
permits clear insights into the essential distinction between the quantum and classical descriptions of the field.
It leads, in addition, to a simple formulation of a superposition law for photon fields. Detailed discussions are
given of the incoherent fields which are generated by superposing the outputs of many stationary sources.
These fields are all shown to have intimately related properties, some of which have been known for the particular
case of blackbody radiation.
... The theory of intensity correlations from incoherent arXiv:2309.01153v1 [astro-ph.IM] 3 Sep 2023 sources has been studied extensively [1][2][3][4][5][6], and the phenomenon was experimentally demonstrated by Hanbury Brown and Twiss (HBT) via measurements of the angular diameter of distant astronomical sources [7][8][9]. The impact of HBT correlations on mm-wave telescopes is discussed by Padin [10], where an empirical factor is introduced in an attempt to account for the corresponding sensitivity degradation. ...
... Given the thermal detector covariance σ 2 ij in Eq. (11), the (quantum) second-order coherence Γ (2) ij can be defined as ...
... where the factor τ comes from the fact that ⟨∆d i ∆d j ⟩ depends on the integration time τ (or the detector sampling rate 1/τ ). Therefore, the second-order coherence Γ (2) ij represents the system's intrinsic degree of intensity coherence in W 2 · s and is independent of integration time τ . These sampling-rate-independent fluctuations of detected photon power √ τ σ ii are equivalent to the detector's photon noise noise-equivalent power (NEP), as in Ref. [14]. ...
Many modern millimeter and submillimeter (``mm-wave'') telescopes for astronomy are deploying more detectors by increasing detector pixel density, and with the rise of lithographed detector architectures and high-throughput readout techniques, it is becoming increasingly practical to overfill the focal plane. However, when the pixel pitch $p_{\rm pix}$ is small compared to the product of the wavelength $\lambda$ and the focal ratio $F$, or $p_{\mathrm{pix}} \lesssim 1.2 F \lambda$, the Bose term of the photon noise correlates between neighboring detector pixels due to the Hanbury Brown & Twiss (HBT) effect. When this HBT effect is non-negligible, the array-averaged sensitivity scales with detector count $N_{\mathrm{det}}$ less favorably than the uncorrelated limit of $N_{\mathrm{det}}^{-1/2}$. In this paper, we present a general prescription to calculate this HBT correlation based on a quantum optics formalism and extend it to polarization-sensitive detectors. We then estimate the impact of HBT correlations on the sensitivity of a model mm-wave telescope and discuss the implications for focal-plane design.
... In the cases where discussed above properties of operatorsÛ X are violated, we have other representations of density operatorsρ =Â, like the Wigner representation [12], or the quasiprobability representations, like the Husimi quasiprobability representation [29] and Glauber-Sudarshan quasiprobability representation [30,31]. ...
We discuss qubit-state superpositions in the probability representation of quantum mechanics. We study probability distributions describing separable qubit states. We consider entangled states on the example of a system of two qubits (Bell states) using the corresponding superpositions of the wave functions associated with these states. We establish the connection with the properties and structure of entangled probability distributions.
... This approach benefits from readily available equipment and seamless integration into existing optical telecommunications networks [10]. One of the CV QKD protocols, GG02 [11], is widely acclaimed for its security due to the no-cloning theorem of coherent states [12]. Nevertheless, a recent study has delved into the use of squeezed states to achieve even higher levels of security and robustness [13]. ...
The generation of broadband squeezed states of light lies at the heart of high-speed continuous-variable quantum information. Traditionally, optical nonlinear interactions have been employed to produce quadrature-squeezed states. However, the harnessing of electrically pumped semiconductor lasers offers distinctive paradigms to achieve enhanced squeezing performance. We present evidence that quantum dot lasers enable the realization of broadband amplitude-squeezed states at room temperature across a wide frequency range, spanning from 3 GHz to 12 GHz. Our findings are corroborated by a comprehensive stochastic simulation in agreement with the experimental data. The evolution of photonics-based quantum information technologies is currently on the brink of initiating a revolutionary transformation in data processing and communication protocols [1, 2]. A cornerstone within this realm will be the quantum emitter. In recent years, there has been a substantial upsurge in both theoretical and experimental investigations centred around semiconductor quantum dot (QD) nanostructures [3]. A particular emphasis has been placed on self-assembled QDs embedded into microcavities, which facilitate the generation of single photons with high purity and indistinguishability [4-6]. As a result, such sources assume a pivotal role in quantum computing [7, 8] as well as the discrete variables (DV) quantum key distribution (QKD) [9]. In stark contrast to the DV QKD, which requires single-photon sources and detectors, continuous variable (CV) QKD leverages lasers and balanced detection to continuously retrieve the light's quadrature components during key distillation. This approach benefits from readily available equipment and seamless integration into existing optical telecommunications networks [10]. One of the CV QKD protocols, GG02 [11], is widely acclaimed for its security due to the no-cloning theorem of coherent states [12]. Nevertheless, a recent study has delved into the use of squeezed states to achieve even higher levels of security and robustness [13]. This innovative approach strives to completely eliminate information leakage to potential eavesdroppers in a pure-loss channel and to minimize it in a symmetric noisy channel. Within this cutting-edge protocol, information can be exclusively encoded through a Gaussian modulation of amplitude-squeezed states, which are commonly referred to as photon-number squeezed states. These states demonstrate reduced fluctuations in photon number ∆n 2 < n with respect to coherent states, albeit encountering enhanced phase fluctuations due to the minimum-uncertainty principle. Over the past years, squeezed states of light have been frequently generated using χ (2) or χ (3) nonlinear interactions via parametric down-conversion and four-wave mixing [14]. A variety of nonlinear materials have been applied in these processes, including LiNbO 3 (PPLN) [15], KTiOPO 4 (PPKTP) [16], silicon [17], atomic vapour [18], disk resonator [19], and Si 3 N 4 [20]. Recent advancements have also facilitated the transition from traditional benchtop instruments to a more compact single-chip design [21-25]. As opposed to that, Y. Yamamoto et al. [26] initially proposed an alternative to produce amplitude-squeezed states directly with off-the-shelf semiconductor lasers using a "quiet" pump, i.e. a constant-current source. A striking peculiarity of semiconductor lasers is their ability to be pumped by injection current supplied via an electrical circuit. Unlike optical pumping, electrical pumping is not inherently a Poisson point process due to the Coulomb interaction and allows for reducing pump noise below the shot-noise level [26]. Notably, this method can take full advantage of the mature fabrication processes in the semiconductor industry, thereby significantly boosting its feasibility. In other words, its efficacy hinges on the improved performance of recently developed semiconductor lasers, characterized by their compact footprint, ultralow intensity noise and narrow frequency linewidth. While subsequent experiments involving various types of laser diodes have gained widespread interest in this domain, including commercial quantum well (QW) lasers [27, 28], vertical-cavity surface-emitting lasers [29], and semiconductor microcavity lasers [30], the observed bandwidth has, until now, remained relatively limited. The broadest achieved bandwidth has reached 1.1 GHz with a QW transverse junction stripe laser operating at a cryogenic temperature of 77 K [31]. This limitation indeed poses strict constraints on the practical implementation of room-temperature conditions and hinders the realization of high-speed quantum communications. While a recent study did anticipate the theoretical potential of producing broadband amplitude-squeezed states in interband cascade lasers [32], it is worth noting that, prior to this Letter, no experimental demonstration of such phenomenon had been presented.
... Quantum coherence is vital for some applications that cannot be achieved using classical mechanics, serving as a fundamental component for quantum correlations [1,2], a valuable resource for quantum computation protocols [3][4][5][6][7], and a pivotal factor in some emerging fields such as quantum biology [8,9], quantum thermodynamics [10], and quantum metrology [11,12]. Despite its known significance and extensive studies in the field of quantum optics [13][14][15], quantum coherence has recently been adopted as a valuable resource based on the tools of quantum information theory [16,17]. Taking inspiration from the well-established conditions used for entanglement monotones, Baumgratz and colleagues proposed a set of constraints to qualify a coherence measure as a monotone [16]. ...
We propose a setup for studying the quantum coherence properties of a quadrupolar nucleus using the nuclear magnetic resonance platforms. We consider powder samples labeled with ²³ Na and oriented with respect to the static magnetic field. By using the l 1 -norm of coherence, we examine the quantum coherence in the Zeeman basis at thermal equilibrium. Non-zero coherence is found to result from the strong nuclear quadrupolar spin interactions. It is also shown that higher coherence is created as the quadrupolar interaction coefficient increases. We also discuss the stability of coherence in a possible measurement process in order to use it as a potential resource in any quantum computation protocol.
... Readers familiar with the rudiments of quantum optics will recognise this as the Glauber Pfunction for a thermal state of a mode with mean photon numbern = (e β ω − 1) −1 . Indeed we can write the density operator for this state in the form [8,20] ...
We present and discuss two exact, but perhaps unfamiliar, representations of thermal radiation. The first has the form of a superposition of the quantum vacuum and a stochastic classical field and the second is the pure-state thermofield representation introduced by Takahashi and Umezawa. It is interesting that the former is, essentially, the opposite of Planck’s original conception of blackbody radiation.
... Two-mode squeezed state (TMSS), called entangled modes, squeezes light across two spatial locations and directions. The correlation between the quadrature amplitudes for FWM signal and idler photons can be used to demonstrate the EPR paradox [90]. ...
Next-generation quantum technologies herald a revolution across computation, communication, and sensing. The development of integrated photonics enables a new era of monolithic integration, with single-photon sources (SPS) enabling new computational paradigms, encryption backed by the no-cloning theorem and measurement precision beyond the standard quantum limit. Developing state-of-the-art integrated quantum sources consists of design, simulation, and benchmarking. This thesis explores integrated quantum sources based on nonlinear optical effects such as four-wave mixing (FWM) enhanced using microring resonators (MRRs). Building on the classical theoretical framework of MRRs, this work seeks to deepen an understanding of how source design influences their KPIs. A parameterised coupling model was developed to investigate the relationship between resonator coupling and FoMs through a simulated transmission function and validated by an experimental characterisation. The main findings of this work suggest that slightly over-coupled microring resonators, with as low a propagation loss as possible, make brighter SPSs. A bright future lies ahead for integrated sources built on silicon nitride.
... We also consider the dynamic fluorescence spectrum of the system as measured from the end of the waveguide, defined as [42,44] ...
Atomic arrays can exhibit collective light emission when the transition wavelength exceeds their lattice spacing. Subradiant states take advantage of this phenomenon to drastically reduce their overall decay rate, allowing for long-lived states in dissipative open systems. We build on previous work to investigate whether or not disorder can further decrease the decay rate of a singly-excited atomic array. More specifically, we consider spatial disorder of varying strengths in a 1D half waveguide and in 1D, 2D, and 3D atomic arrays in free space and analyze the effect on the most subradiant modes. While we confirm that the dilute half waveguide exhibits an analog of Anderson localization, the dense half waveguide and free space systems can be understood through the creation of close-packed, few-body subradiant states similar to those found in the Dicke limit. In general, we find that disorder provides little advantage in generating darker subradiant states in free space on average and will often accelerate decay. However, one could potentially change interatomic spacing within the array to engineer specific subradiant states.
... Paradigmatic examples of CV QKD protocols are those based on Gaussian states and Gaussian measurements [26][27][28][29][30][31][32][33][34][35][36][37]. The main advantage of this all-Gaussian paradigm [23,38,39] is that it is relatively experimentally friendly: coherent states [40][41][42][43], squeezed states [44][45][46][47][48][49], homodyne and heterodyne detection [15,39] are nowadays relatively inexpensive ingredients, especially compared to general quantum states and operations. At the same time, it is still quite powerful in the context of QKD; in fact, it has been shown that any sufficiently entangled Gaussian state can be used, in combination with local Gaussian operations and public communication, to distil a secret key [50][51][52]. ...
We establish fundamental upper bounds on the amount of secret key that can be extracted from quantum Gaussian states by using local Gaussian operations, local classical processing, and public communication. For one-way public communication or when two-way public communication is allowed but Alice and Bob first perform destructive local Gaussian measurements, we prove that the key is bounded by the Rényi-2 Gaussian entanglement of formation EF,2G. The saturation of this inequality for pure Gaussian states provides an operational interpretation of the Rényi-2 entropy of entanglement as the secret key rate of pure Gaussian states accessible with Gaussian operations and one-way communication. In the general setting of two-way communication and arbitrary interactive protocols, we argue that 2EF,2G still serves as an upper bound on the extractable key. We conjecture that the factor of 2 is spurious, suggesting that EF,2G coincides with the secret key rate of Gaussian states under Gaussian measurements and two-way public communication. We use these results to prove a gap between the secret key rates obtainable with arbitrary versus Gaussian operations. This gap is observed for states produced by sending one half of a two-mode squeezed vacuum through a pure loss channel, in the regime of sufficiently low squeezing or sufficiently high transmissivity. Finally, for a wide class of Gaussian states, including all two-mode states, we prove a recently proposed conjecture on the equality between EF,2G and the Gaussian intrinsic entanglement. The unified entanglement quantifier emerging from such an equality is then endowed with a direct operational interpretation as the value of a quantum teleportation game.
In this paper, the time evolution of signals in axion dark matter experiments is considered from a quantum perspective. The aim is to illuminate the specific case of axion experiments and not to contribute new results to the general discussion of the quantum/classical connection. Classically, one expects a signal oscillating with a frequency equal to the axion mass whose amplitude is slowly rising due to the tiny interaction of the axions with ordinary matter. Quantum mechanically the latter time‐scale arises from the small splitting in the energy levels induced by the interaction between the axions and the experiment. It is always present in suitable, sensitive experiments. Signals that oscillate with the axion mass arise from processes changing the axion number. For certain observables such oscillations are absent for special initial quantum states of the axions. Examples are shown where these special states can be affected by the experiment such that a signal oscillating with the axion mass re‐appears. Suitable correlators that feature an oscillation with the axion mass are discussed, and comments on the connection to the classical treatment are provided. Explicitly, the Cosmic Axion Spin Precession Experiment (CASPEr) is studied, but these findings are expected to be adaptable to other axion dark matter searches.
Background: “Quantum biology” (QB) is a promising theoretical approach addressing questions about how living systems are able to unfold dynamics that cannot be solved on a chemical basis or seem to violate some fundamental laws (e.g., thermodynamic yield, morphogenesis, adaptation, autopoiesis, memory, teleology, biosemiotics). Current “quantum” approaches in biology are still very basic and “corpuscular”, as these rely on a semi-classical and approximated view. We review important considerations of theory and experiments of the recent past in the field of condensed matter, water, physics of living systems, and biochemistry to join them by creating a consistent picture applicable for life sciences. Within quantum field theory (QFT), the field (also in the matter field) has the primacy whereby the particle, or “quantum”, is a derivative of it. The phase of the oscillation and not the number of quanta is the most important observable of the system. Thermodynamics of open systems, symmetry breaking, fractals, and quantum electrodynamics (QED) provide a consistent picture of condensed matter, liquid water, and living matter. Coherence, resonance-driven biochemistry, and ion cyclotron resonance (Liboff–Zhadin effect) emerge as crucial hormetic phenomena. We offer a paradigmatic approach when dealing with living systems in order to enrich and ultimately better understand the implications of current research activities in the field of life sciences.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The $n\mathrm{th}$ order function expresses the correlation of values of the fields at $2n$ different points of space and time. Certain values of these functions are measurable by means of $n$-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
The observation by Klauder that in the space of the a = (1/√2) (p + iq) variables, the Feynman integral can be defined in terms of a Gaussian measure, forms the basis of a presentation of the Feynman formulation of nonrelativistic quantum mechanics. The extension of this formulation to the case of a Bose field is sketched.
Methods of calculation with nonlinear functions of quantized boson fields are developed during the discussion of two problems involving multiple boson processes. In the first of these a simple treatment is given of the multiple radiation of photons by classical current distributions, a special case of which, in effect, is the infrared catastrophe. In the second illustration, generalizations of the scalar and pseudoscalar meson theories are considered in which the interaction hamiltonian depends exponentially on the meson field. In the pseudoscalar case such hamiltonians are closely related to the familiar form of pseudovector coupling. Assuming the over-all coupling of the nucleon and meson fields to be weak, calculations are made of the nuclear forces, and of the multiple production of mesons in meson-nucleon and in nucleon-nucleon collisions. In the latter events statistical independence of meson emissions is found to prevail.
DOI:https://doi.org/10.1103/PhysRevLett.10.277
General quantum mechanical methods for the investigation of correlation ; effects were developed, and results for the distribution of the number of photons ; counted in an incoherent beam were presented. Photoionlzation probability was ; examined to discuss photon correlations. Coherent states of the field led to no ; photoionization correlations. (C.E.S.);
In this paper we discuss the electromagnetic field, as perturbed by a prescribed current. All quantities of physical interest in various situations, eigenvalues, eigenfunctions, and transition probabilities, are derived from a general transformation function which is expressed in a non-Hermitian representation. The problems treated are: the determination of the energy-momentum eigenvalues and eigenfunctions for the isolated electromagnetic field, and the energy eigenvalues and eigenfunctions for the field perturbed by a time-independent current; the evaluation of transition probabilities and photon number expectation values for a time-dependent current that departs from zero only within a finite time interval, and for a time-dependent current that assumes non-vanishing time-independent values initially and finally. The results are applied in a discussion of the infrared catastrophe and of the adiabatic theorem. It is shown how the latter can be exploited to give a uniform formulation for all problems requiring the evaluation of transition probabilities or eigenvalue displacements.
The probability of a configuration is given in classical theory by the Boltzmann formula exp[−VhT] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.