Fermi's golden rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak perturbation. This transition rate is effectively independent of time and is proportional to the strength of the coupling between the initial and final states of the system as well as the density of states. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.
Historical background
Although the rule is named after Enrico Fermi, the first to obtain the formula was Paul Dirac, as he had twenty years earlier formulated a virtually identical equation, including the three components of a constant, the matrix element of the perturbation and an energy difference. It was given this name because, on account of its importance, Fermi called it "golden rule No. 2".Most uses of the term Fermi's golden rule are referring to "golden rule No. 2", but Fermi's "golden rule No. 1" is of a similar form and considers the probability of indirect transitions per unit time.
The rate and its derivation
Fermi's golden rule describes a system that begins in an eigenstate of an unperturbed Hamiltonian and considers the effect of a perturbing Hamiltonian applied to the system. If is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If is oscillating sinusoidally as a function of time with an angular frequency, the transition is into states with energies that differ by from the energy of the initial state.In both cases, the transition probability per unit of time from the initial state to a set of final states is essentially constant. It is given, to first-order approximation, by
where is the matrix element of the perturbation between the final and initial states, and is the density of states at the energy of the final states. This transition probability is also called "decay probability" and is related to the inverse of the mean lifetime. Thus, the probability of finding the system in state is proportional to.
The standard way to derive the equation is to start with time-dependent perturbation theory and to take the limit for absorption under the assumption that the time of the measurement is much larger than the time needed for the transition.
Statement of the problem
The golden rule is a straightforward consequence of the Schrödinger equation, solved to lowest order in the perturbation of the Hamiltonian. The total Hamiltonian is the sum of an "original" Hamiltonian and a perturbation:. In the interaction picture, we can expand an arbitrary quantum state's time evolution in terms of energy eigenstates of the unperturbed system, with.Discrete spectrum of final states
We first consider the case where the final states are discrete. The expansion of a state in the perturbed system at a time is. The coefficients are yet unknown functions of time yielding the probability amplitudes in the Dirac picture. This state obeys the time-dependent Schrödinger equation:Expanding the Hamiltonian and the state, we see that, to first order,
where and are the stationary eigenvalues and eigenfunctions of.
This equation can be rewritten as a system of differential equations specifying the time evolution of the coefficients :
This equation is exact, but normally cannot be solved in practice.
For a weak constant perturbation that turns on at, we can use perturbation theory. Namely, if, it is evident that, which simply says that the system stays in the initial state.
For states, becomes non-zero due to, and these are assumed to be small due to the weak perturbation. The coefficient which is unity in the unperturbed state, will have a weak contribution from. Hence, one can plug in the zeroth-order form into the above equation to get the first correction for the amplitudes :
whose integral can be expressed as
with, for a state with,, transitioning to a state with.
The probability of transition from the initial state to the final state is given by
It is important to study a periodic perturbation with a given frequency since arbitrary perturbations can be constructed from periodic perturbations of different frequencies. Since must be Hermitian, we must assume, where is a time independent operator. The solution for this case is
This expression is valid only when the denominators in the above expression are non-zero, i.e., for a given initial state with energy, the final state energy must be such that Not only must the denominators be non-zero, but they also must not be small since is supposed to be small.
Consider now the case where the perturbation frequency is such that where is a small quantity. Unlike the previous case, not all terms in the sum over in the above exact equation for matters, but depends only on and vice versa. Thus, omitting all other terms, we can write
The two independent solutions are
where
and the constants and are fixed by the normalization condition.
If the system at is in the state, then the probability of finding the system in the state is given by
which is a periodic function with frequency ; this function varies between and. At the exact resonance, i.e.,, the above formula reduces to
which varies periodically between and, that is to say, the system periodically switches from one state to the other. The situation is different if the final states are in the continuous spectrum.
Continuous spectrum of final states
Since the continuous spectrum lies above the discrete spectrum, and it is clear from the previous section, major role is played by the energies lying near the resonance energy, i.e., when. In this case, it is sufficient to keep only the first term for. Assuming that perturbations are turned on from time, we have thenThe squared modulus of is
Therefore, the transition probability per unit time, for large t, is given by
Note that the delta function in the expression above arises due to the following argument. Defining the time derivative of is, which behaves like a delta function at large .
The constant decay rate of the golden rule follows. As a constant, it underlies the exponential particle decay laws of radioactivity.
Applications
Semiconductors
The Fermi's golden rule can be used for calculating the transition probability rate for an electron that is excited by a photon from the valence band to the conduction band in a direct band-gap semiconductor, and also for when the electron recombines with the hole and emits a photon. Consider a photon of frequency and wavevector, where the light dispersion relation is and is the index of refraction.Using the Coulomb gauge where and, the vector potential of light is given by where the resulting electric field is
For an electron in the valence band, the Hamiltonian is
where is the potential of the crystal, and are the charge and mass of an electron, and is the momentum operator. Here we consider process involving one photon and first order in. The resulting Hamiltonian is
where is the perturbation of light.
From here on we consider vertical optical dipole transition, and thus have transition probability based on time-dependent perturbation theory that
with
where is the light polarization vector. and are the Bloch wavefunction of the initial and final states. Here the transition probability needs to satisfy the energy
conservation given by. From perturbation it is evident that the heart of the calculation lies in the matrix elements shown in the bracket.
For the initial and final states in valence and conduction bands, we have and, respectively and if the operator does not act on the spin, the electron stays in the same spin state and hence we can write the Bloch wavefunction of the initial and final states as
where is the number of unit cells with volume. Calculating using these wavefunctions, and focusing on emission rather than absorption, we are led to the transition rate
where defined as the optical transition dipole moment is qualitatively the expectation value and in this situation takes the form
Finally, we want to know the total transition rate. Hence we need to sum over all possible initial and final states that can satisfy the energy conservation, and take into account spin degeneracy, which after calculation results in
where is the joint valence-conduction density of states. In 3D, this is
but the joint DOS is different for 2D, 1D, and 0D.
We note that in a general way we can express the Fermi's golden rule for semiconductors as
In the same manner, the stationary DC photocurrent with amplitude proportional to the square of the field of light is
where is the relaxation time, and are the
difference of the group velocity and Fermi-Dirac distribution between possible the initial and
final states. Here defines the optical transition dipole. Due to the commutation relation between position and the Hamiltonian, we can also rewrite the transition dipole and photocurrent in terms of position operator matrix using. This effect can only exist in systems with broken inversion symmetry and nonzero components of the photocurrent can be obtained by symmetry arguments.
Scanning tunneling microscopy
In a scanning tunneling microscope, the Fermi's golden rule is used in deriving the tunneling current. It takes the formwhere is the tunneling matrix element.
Quantum optics
When considering energy level transitions between two discrete states, Fermi's golden rule is written aswhere is the density of photon states at a given energy, is the photon energy, and is the angular frequency. This alternative expression relies on the fact that there is a continuum of final states, i.e. the range of allowed photon energies is continuous.