When any two optical fields are superposed, a fringe pattern will be formed if the observation interval is short enough (within the coherent time) and a phase value may be extracted from the fringe and assigned as the phase difference between the fields. When measured against a common reference field, phase can be defined for an optical field. However, when the intensities of the optical fields are low or the observation time is so short that only a few photons are detected, the fringe pattern is unrecognizable due to the particle nature of light. By analyzing the fringe patterns simulated by a quantum Monte Carlo method, we find that there exist uncertainties in determining the phase value. This uncertainty is universal for any optical field with a finite photon number and decreases to zero as the photon number of the field increases to infinity or the classical limit is approached. The existence of this uncertainty in determining the phase value makes it problematic to define a quantum mechanical phase. An interference experiment with coherent state is performed to confirm the theoretical predictions. The implication of the above is that phase is a classical quantity that is associated with a large number of particles and may not have a quantum mechanical analogy.
When any two optical fields are superposed, a fringe pattern will be formed if the observation interval is short enough (within the coherent time) and a phase value may be extracted from the fringe and assigned as the phase difference between the fields. When measured against a common reference field, phase can be defined for an optical field. However, when the intensities of the optical fields are low or the observation time is so short that only a few photons are detected, the fringe pattern is unrecognizable due to the particle nature of light. By analyzing the fringe patterns simulated by a quantum Monte Carlo method, we find that there exist uncertainties in determining the phase value. This uncertainty is universal for any optical field with a finite photon number and decreases to zero as the photon number of the field increases to infinity or the classical limit is approached. The existence of this uncertainty in determining the phase value makes it problematic to define a quantum mechanical phase. An interference experiment with coherent state is performed to confirm the theoretical predictions. The implication of the above is that phase is a classical quantity that is associated with a large number of particles and may not have a quantum mechanical analogy.
We have experimentally determined all of the quantities involved in the uncertainty relation for the phase and photon number of a mode of the electromagnetic field when the field mode is in a coherent state of small average photon number. This is accomplished by determining the quantum state of the field using optical homodyne tomography, which uses measured distributions of electric-field quadrature amplitude to determine the Wigner function and hence the density matrix. The measured state is then used to calculate the uncertainty product for the number and phase, as well as the expectation value of the commutator of the number and phase operators. The experimental results agree with the quantum-mechanical predictions. We also present measured phase- and photon-number distributions for these weak coherent states, as well as their measured complex wave functions.
3.1 Interference of a Single Photon The kinds of study originally conducted by Thomas Young - in particular, the celebrated "double slit" experiment that we briefly examined in Chapter 1 (see Fig. 1) - provide a capacity to reveal far more than the wave nature of light, important as that is. True enough, the observed characteristic fringe pattern, when monochromatic light passes through two closely spaced slits onto a screen, appears to be simply explicable on the basis of waves from each slit undergoing constructive and destructive interference at different positions on the screen. Yet, astonishingly, these patterns of interference are still exhibited when individual photons of a low-intensity input beam reach the screen one at a time, showing that it cannot in fact be a matter of different photons having to interfere with each other. Indeed, Dirac enunciated the general principle that photons can never directly interact with each other. The double slit and other such experiments indicate that each photon interacts in a delocalized, wave-like manner with both slits at the same time. Successive experiments build interference patterns that conform to an exact mathematical distribution formula, but the position at which an individual photon is detected in any particular instance cannot be predicted; this is a quintessential illustration of quantum uncertainty. Using the path integral approach of quantum theory, the phenomenon is more accurately represented by introducing a summation over all possible routes between the source and the screen on which the pattern appears, signifying an entire "lifetime" for each photon involved in the experiment. This relates to the fact that the actual trajectory is unknown without an observation, and thus we only obtain the probabilities of the photon striking the screen at different positions and times; all trajectories (potentialities) must be considered in a superposition. Each route is considered equally probable, and the variation in phase produces quantum interference. The differences in phase of each trajectory, at varying positions on the screen, generates the fringe pattern. An intriguing variation on this experiment involves "marking" the photons so that the actual route of each photon to the screen is known. Remarkably, in such a case, the fringe patterns do not appear because quantum interference cannot occur. A consequence of the observation of the photon route is an effect that, in other quantum mechanical connections, is known as a collapse of the wavefunction. However, it is possible for the fringe patterns to reappear when information on the route of the photons is lost; this is known as a quantum eraser.
Quantum mechanical uncertainty After reviewing the main characteristics of the classical description of light, let us discuss those aspects of the quantization of the electromagnetic field which are of relevance for the analysis of the phenomena we are interested in. It seems a reasonable place to start to make clear the fundamental difference between the classical and the quantum mechanical description of nature; we will come across this difference many times when discussing experiments, and it will often give us a headache. We have to deal with the physical meaning of what is called uncertainty. The starting point of the classical description is the conviction that natural processes have a “factual” character. This means that physical variables such as the position or momentum of a particle have, in each single case, a well defined (in general, time dependent) value. However, it will not always be possible to measure all the appropriate variables (for instance, the instantaneous electric field strength of a radiation field); furthermore under normal circumstances we are able to measure only with a finite precision. Hence the basic credo of classical physics should be given in the following form: we are justified in imagining a world with variables possessing well defined values which are not known precisely (or not known at all). In doing this we are not forming any conclusions that contradict our everyday experiences. This is the fundamental concept of classical statistics: we are satisfied with the use of probability distributions for the variables we are interested in, not from fundamental but purely practical reasons.
The uncertainty principle applied to electromagnetic fields implies that the field phase and photon number cannot simultaneously have arbitrarily narrow distributions. Here we describe an experiment which determines, for the first time, the variances of phase and photon number for a field containing small numbers of photons. The measurements allow a test of the uncertainty relation for the phase and photon number of a mode of the electromagnetic field when the field mode is in a very weak coherent state. The phase of the field is described by the Pegg-Barnett phase operator formalism. [1] The measurements are accomplished by experimentally determining the quantum state of the field by using optical homodyne tomography. [2] This method of state characterization requires that an ensemble of many measurements be performed. [3] In this case, the relevant interpretation of the uncertainty relation is that the state of the field does not have well defined phase and photon number.
We conjecture that optical coherences, i.e., quantum-mechanical coherences between states separated by Bohr frequencies in the optical regime, do not exist in optics experiments. We claim the exact vanishing of optical field amplitudes and atomic dipole expectation values, and we discuss the seemingly contradictory success of assigning finite values to such quantities in theoretical calculations. We show that our conjecture is not at variance with the observed interference between different light sources. The connection to the concept of spontaneous symmetry breaking and the identification of entangled states as pointer basis states is discussed.
The measurement of a phase difference between two optical modes in an interferometer has several important applications in metrology. This measurement can be useful in several contexts, such as in the detection of gravitational waves. The quantum nature of light imposes a limit on how accurately a phase difference can be measured, this limit being known as the Heisenberg limit (HL). On the other hand, photonic quantum metrology studies alternatives to reach the HL through entangled or squeezed quantum states. According to the theory of parameter estimation, the minimum uncertainty possible in the estimation of a phase θ considering classical light scales with $1/\sqrt{\Bar{N}}$, where $\Bar{N}$ is the average number of photons. This dependence is known as the standard quantum limit. However, when considering quantum light sources, there is a significant improvement in the estimation of the phase parameter, as its minimum uncertainty can scale with $1/{\Bar{N}}$, known as HL. In this thesis, starting from the relative phase probability distribution P(ϕ) introduced by Luis and S\'anchez-Soto in 1996, we obtained this distribution for several quantum states of light useful in photonic quantum metrology. We show that, within the numerical precision of our calculations, the Fisher information obtained via the relative phase distribution is equal to the quantum Fisher information for the considered states, such that the average differences between these quantities were on the order of 0,1%. Our results indicate that the relative phase distribution can be used to predict the minimum possible uncertainty in the phase detection process in quantum metrology, since this uncertainty depends on the quantum Fisher information, at least for pure states.