Photons cannot be localized in a sharply defined region. The expectation value of their energy density and the photon number density can only be approximately localized, leaving an exponential tail. We show that one may sharply localize either electric or magnetic (but not both) footprints of photons and only momentarily. In the course of time evolution this localization is immediately destroyed. However, the coherent states, like their classical counterparts, can be localized without any limitations. The main tool in our analysis is a set of space-dependent photon creation and annihilation operators defined without any reference to the mode decomposition.
Photons cannot be localized in a sharply defined region. The expectation value of their energy density and the photon number density can only be approximately localized, leaving an exponential tail. We show that one may sharply localize either electric or magnetic (but not both) footprints of photons and only momentarily. In the course of time evolution this localization is immediately destroyed. However, the coherent states, like their classical counterparts, can be localized without any limitations. The main tool in our analysis is a set of space-dependent photon creation and annihilation operators defined without any reference to the mode decomposition.
At the end of the preceding chapter, we showed that the photon number states | n 〉 have a uniform phase distribution over the range 0 to 2π. Essentially, then, there is no well-defined phase for these states and, as we have already shown, the expectation value of the field operator for a number state vanishes. It is frequently suggested (see, for example, Sakurai) that the classical limit of the quantized field is the limit in which the number of photons becomes very large such that the number operator becomes a continuous variable. However, this cannot be the whole story since the mean field 〈 n | Ê x | n 〉 = 0 no matter how large the value of n . We know that at a fixed point in space a classical field oscillates sinusoidally in time. Clearly this does not happen for the expectation value of the field operator for a number state. In this chapter we present a set of states, the coherent states, which do give rise to a sensible classical limit; and, in fact, these states are the “most classical” quantum states of a harmonic oscillator, as we shall see. Eigenstates of the annihilation operator and minimum uncertainty states In order to have a non-zero expectation value of the electric field operator or, equivalently, of the annihilation and creation operators, we are required to have a superposition of number states differing only by ±1.
This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.
States whose local expectation values are identical to those of vacuum outside of a given region of localization are known as strictly localized states. It is believed that states generated on-demand need to be strictly localized to satisfy causality requirements. Single particle states, however, cannot be strictly localized. In this case, to conform with observation, there should exist strictly localized states which are close enough to single particle states to be indistinguishable from them by standards of contemporary experimental precision. A class of photonic states has been constructed which is fully consistent with the above considerations in that they are strictly localized, causally evolving, and generally closer to single photons than experimental precision can distinguish. We call these eta-states. The author's work represents a versatile method of constructing this class of states in a fully physical, three dimensional electromagnetic field. This is to our best knowledge the first such state constructed. Furthermore, by repeated application of the operator which excites the state from vacuum, states close to n-photons follow readily. Several relevant properties of eta-states are derived analytically, such as their energy density expectation value, number expectation value, and their coefficients in a so-called pulse mode basis. Their fidelity, the coefficient corresponding to a single particle component, is studied analytically and numerically. General bounds on the fidelity of an arbitrary strictly localized state to a given single photon are also presented. The fidelity can be made arbitrarily close to 1, but is shown to decrease as the region of localization decreases relative to the carrier frequency of the pulse. This indicates the following: One may falsify the hypothesis that on-demand states must be strictly localized by observing fidelities exceeding this limit. On the other hand, the bounds may pose a real practical limit to the single-photon approximation in experiments and potential photonics applications in the future. The author's work also includes a potential candidate for a localized Dirac fermion state following a similar method to the photonic eta-state, with the notable feature that it is a superposition between a fermion and an antifermion.
The cavity-mode quantization conjecture of Chapter 2 is replaced by local commutation relations — which are independent of the size and shape of the cavity — between field-operator components. This step eliminates the previous dependence on the classical boundary conditions at the ideal cavity wall. The cavity annihilation and creation operators are respectively replaced by the positive- and negative-frequency parts of the vector potential. A simple ad hoc model provides similar results for quantized fields in a passive, linear dielectric. It is shown that the total electromagnetic angular momentum cannot, in general, be expressed as the sum of well defined orbital- and spin-parts. The chapter ends with a discussion of localizability for photons, in which it is shown that there is no photon position operator, no position-space photon wave function, and no local photon number operator.
Space-time description of pulsed laser radiation by means of coherent states is presented. The corresponding displacement operator contains space dependent annihilation and creation operators, not the "standard" operators corresponding to mode decomposition of the electromagnetic field. This allowed for a direct description of space and time dependence of the electromagnetic field of the pulse. The main characteristics of the pulse is its profile in the wave-vector space. It determines space-time structure of expectation values of the electromagnetic field carried by the pulse, and the distribution of photon modes as well. Determination of the latter required transition from the space-time, or field description to the photon mode approach. Expectation values of the electric field of a Gaussian pulse propagating in the z-direction and linearly polarized in the x-direction are presented graphically in the form of contour plots in the x-z plane. Apart of the dominant x-component electric field has a small longitudinal component in the z-direction. The dominant component of the magnetic field in the y-direction is acompannied by small components in z- and x-directions. The plane wave limit of the pulse can be implemented by shrinking transverse width of the profile to zero (transverse spatial extension goes to infinity), while keeping the photon density constant. One can then retrieve standard plane wave form of the pulse also in the dipole approximation.
The theoretical concept of photons is very important for the interpretation of quantum optics experiments. Since Eugene P. Wigner published the important work on the concept of particle localization in relativistic quantum mechanics, it has long been argued in the theoretical literature that photons cannot be exactly located in free space.1 The argument claims that localized photons necessarily have an energy density that drops off as a power law of distance. If this is true, one should seriously examine the experimental consequences of such a claim.