The Vacuum Field and Einstein's Stimulated Emission Theory

Chris Spencer
Boston University
EC 570: Lasers and Applications

Introduction

Hedrick Casimir first introduced the idea of the vacuum field in 1948. Interest started when Casimir and Polder earlier in 1948 described the interaction between q perfectly conducting plate and an atom by using the Van Der Waals-London forces, then correcting it for retardation effects in the limit of large distances. Then the next issue was to show the effects between the interaction between two perfectly conducting plates using this Van Der Waals retarded interaction. It is useful to understand the properties of the paper that lead to the Casimir effect. Casimir considers a cubic cavity of a volume \(L^3\) bounded by perfectly conducting square plate with side L be parallel to the XY face and investigates the situation where plate is at a small distance from XY face and situation where the distance between is a distance \(\frac{L}{2}\) [1]. Looking at the resonant frequencies of this system, \( \frac{1}{2} \sum \hbar \omega \), the sum being over all the resonant frequencies, notice the problem that this goes to infinity and is not physical. The differences between the frequencies is well defined for the two situations. Labeling the situations of small interaction and long interaction, differences are \[\frac{1}{2}(\sum \hbar \omega )_I -\frac{1}{2}(\sum \hbar \omega)_{II}\] and the value given is the interaction between the plate and the XY face. Cavity will have dimensions \[0 \leq X \leq L, 0 \leq Y \leq L,0 \leq Z \leq a\] with the wavenumbers given by \[k_i=\frac{ \pi n_i}{L}, k_z=\frac{ \pi n_z}{a}\] where i=x,y and \(a\) is the small distance to be used between plate and XY face. The total K is given by \(K=\sqrt{k_x^2+k_y^2+k_z^2}=\sqrt{x^2+k_z^2}\).
For every \(k_x,k_y,k_z\), there is two be two standing waves unless \(n_i\) is zero, which case there is one standing wave. \(k_x,k_y\) can be regarded as continuous variables for very large L, then in polar coordinates \[\frac{1}{2} \sum \hbar \omega = \frac{ \hbar c L^2 \pi}{\pi^2 2} \sum_{(0)1}^{\infty}\int_{0}^{\infty}\sqrt{\frac{n^2\pi^2}{a^2}+x^2} x dx\] where the notation of \((0)1\) means the n=0 term is to be multiplied by \(\frac{1}{2}\). When \(a\) becomes large, the summation can be turned into an integral. Now the interaction energy between plates for the situations above is \[\frac{1}{2} \sum \hbar \omega = \frac{ \hbar c L^2 \pi}{\pi^2 2} \sum_{(0)1}^{\infty}\int_{0}^{\infty}\sqrt{\frac{n^2\pi^2}{a^2}+x^2} x dx-\int_{0}^{\infty} \int_{0}^{\infty} \sqrt{k_z^2+x^2} x dx(\frac{a dk_z}{\pi})\] For sake of brevity, the result is given for a force per cm^2 \[F=\frac{\hbar c \pi^2}{240 a^4}\] Casimir concluded that there exists a force between the two plates which is independent of material and interpreted as a zero point pressure of electromagnetic waves, this result is now known as the Casimir effect.
This result is hard to understand without a knowledge of quantum field theory or quantum electrodynamics. A brief anecdote on QFT will be given to try to understand the Casimir effect and the vacuum field. Quantum Field Theory states that all the fundamental fields are quantized at each and every point in space. A field can be seen as space being filled with interconnected vibrating balls and springs. Vibrations in a field then propagate and are governed by the appropriate wave equation for the field in question. The second quantization of quantum field theory or canonical quantization requires that each ball and spring combination be quantized. The field at each point in space is a simple harmonic oscillator and its quantization puts a quantum harmonic oscillator at each point. The vacuum we assume to be empty is not so empty after all. As it would be, the vacuum has all the properties a particle may have such as spin, energy but on average these cancel out. If these properties cancel shouldn’t the vacuum be empty? The exception is the vacuum energy. Since the vacuum field’s quantization is a quantum oscillator, we know that the ground state of a quantum harmonic oscillator is \(E_0=\frac{1}{2} \hbar \omega\). For the vacuum field, this is called the zero point energy. This is the same energy Casimir uses when calculating the force between the two plates. [2]

A visualization for the Casimir Effect between two plates

Now with some background and knowledge on the vacuum field, we turn to a new physical model to describe the vacuum field. It should be no surprise that there is still some mystery to the vacuum field and its properties. One of these mysteries is that photonic fields can demonstrate \(terrawatt/cm^2\) power in the ability to push and pull plates. It turns out the vacuum field is responsible for a phenomena called Lamb Shift. According to the dirac equation the S and P orbitals of the hydrogen atom should have the same energies but the interaction between the electron and the vacuum causes an energy shift that causes S and P orbitals to have different energies. So how an these photonic fields have such great power yet the world around us isn’t ionized? Look to Einstein’s “On The Quantum Theory of Radiaiton” for clues [3].
Einstein’s paper on stimulated emission of a two state system leaves us with two clues. The first clue is that blackbody radiation has a statistically equal ability to create excitation and stimulated emission that is proportional to the spectral intensity. In other words, the cross-section for excitation,\(B_{12}\), from a lower state to an upper state is equal to the cross section for stimulated emission,\(B_{21}\), from an upper state to a lower state. Clue 2 says that the spontaneous emission coefficient \(A_{21}\) is related to the stimulated emission coefficient by \[A_{21}=\frac{B_{21} 8 \pi h \nu^3}{c^2}=\frac{1}{\tau_{decay}}\] These clues can be used to find a deeper connection between the vacuum field and stimulated emission. To find the power spectral density that the vacuum electromagnetic contains, look to “Random Electronics” by T.W. Marshall [4]. Marshall describes a distribution function \(Q(x_R,p_R,t)\) where \(X_R\) and \(P_R\) satisfy the equation \[\ddot{x}+\frac{2e^2\omega^2}{3mc^3} \dot{x}+\omega^2x=\frac{e}{m}E(t)\] with the initial values of \(x_R=\dot{X_r}=0\) at \(t=0\) and \(p_r=m\dot{x_R}\). After analyzing the expectation values for \(x_R^2\), Marshall arrives at the conclusion that at \(t=\infty \), the distribution function \[Q(x_R,p_R,\infty)=\frac{1}{\pi \alpha}e^{(\frac{ m\omega x_R^2}{\alpha}-\frac{p_R^2}{m\omega \alpha})}\] where \(\alpha=\frac{\pi e^2}{\gamma m \omega} I(\frac{\omega}{2 \pi})\) where \(\gamma=\frac{e^2 \omega^2}{3 m c^3}\) Marshall explains that the distribution given by \(Q(x_R,p_R,\infty)\) is an ensemble that is identical to that of a quantum harmonic oscillator. This is in agreement with the second quantization given before as long as \(\alpha=\hbar\). Then solving for \[I(\frac{\omega}{2 \pi})=\frac{\omega^3 \hbar}{3 \pi c^3}\] \[I(\nu)=\frac{4 \pi h \nu^3}{3 c^3}\] This result is the spectral field density of the vacuum field density, denoted \(I_{vac}(\nu)\). There is a small difference between Hutchin’s definition of \(I_{vac}=\frac{4 \pi h \nu^3}{c^2}\) and the one given above. The units given by Marshall are in \(joules/(cm)^3/Hz\) , to get the units in terms of watts, multiply by c. Then the 3 given my Masrhall, I believe is from getting the spectral field density for all dimensions (x,y,z), where Hutchin is considering the spectral field density along one direction, in agreement with Faria et al’s given by their equation seven [5]. It should be noted that when integrated \(I_{vac}(\nu)\) over \(4 \pi\) steradians, the result is the power field density.
Experiments have found that wavelengths longer 10 nm, the vacuum field density is about, \(1.8 x 10^{12} watt/(cm)^2\), this is incredibly large. This result leads to pressure that can bring two parallel plates together or force them about, this is the Casimir effect. One should be concerned with the inverse relationship with distance, \(P_{vac}=\frac{-\hbar c \pi^2}{240 d^4}\). At the small scales that we work with in laser systems and micro or nano electronics, this effect can lead to problems. This reinforces the mystery as to why the world around us isn’t ionized with such great vacuum field power. The fields are in fact there, since they can be measured, but there must be some property of the vacuum fields that don’t allow ionization.
Return to clue 2 which, in Einstein’s words, says spontaneous decay of any atom is identical to stimulated emission from a background spectral intensity.