User:Chbald/Adiabatic Elimination

In quantum mechanics Adiabatic elimination is a technique used to reduce the number of states in a given quantum system. This can be beneficial in describing a complex system with many states and can lead to analytic solutions in certain regimes. Often times, in large quantum systems, the states evolve and interact on different time scales. Adiabatic elimination takes advantage of the fact that some time scales are much smaller than others. When this is the case the smaller time scales, which come from terms with high frequency, cause the corresponding states to reach a steady state much faster than other states in the system. These states can then be eliminated, under good approximation, simplifying the problem.

Basic Approach
In quantum mechanics, the evolution of the system is usually described by first order differential equation, either from the Schrödinger Equation or the Lindblad equation. This leads to a coupled set of differential equations which, when solved, fully describes the evolution of the system. However, in general, the set of coupled differential equations is difficult to solve. The goal of adiabatic elimination is to simplify this problem by solving for one, or more, of the parameters directly as will be shown. It will be assumed that the differential equations have the following form


 * $$ \frac{d x(t)}{dt} = \alpha x(t) + f(t), $$

where x(t) is a parameter of the system, ƒ(t) is a collection of other parameter, and t is the time of the evolution. The constant α is the frequency with corresponding time scale τ = 1/α. The differential equation has solution


 * $$ x(t) = e^{-\alpha t} \int_0^t dt' e^{\alpha t'} f(t'). $$

As stated previously, adiabatic elimination can be performed on a parameter when its time scale is much smaller than the evolution of the rest of the system. If τ is much smaller then the time scale of the evolution of ƒ(t) then ƒ(t) can be pulled out of the integral since it will be fairly constant over the range of integration. This leads to the solution


 * $$ x(t) \approx \frac{f(t)}{\alpha}, $$

Where $$ e^{-\alpha t} \ll 1 $$ since α is large. Now, x(t) is solved for directly reducing the number of coupled differential equations in the system. This approximation can also be derived directly from the original differential equation. Since it is assumed that x(t) evolves quickly compared to the other time dependent term, ƒ(t), then for a time t >> τ, x(t) will reach a steady state and, therefore, $$ \frac{d x(t)}{dt} \approx 0 $$ leading to the same solution for x(t) as shown before.

Application to three level system


A basic example in which adiabatic elimination is useful is for a three level quantum system, such as the one shown Fig. (1) in the lambda configuration. There are two classical fields that interact independently between ground states one and two and excited state three. The coherent evolution is described by the following Hamiltonian after moving to a rotating frame and making the rotating wave approximation


 * $$ H = -\hbar \delta | 2 \rangle \langle 3 | - \hbar \Delta | 3 \rangle \langle 3 | + \frac{\hbar \Omega_1}{2} \left( | 3\rangle \langle 1 | + | 1 \rangle \langle 3 | \right) + \frac{\hbar \Omega_2}{2} \left( | 3\rangle \langle 2| + | 2 \rangle \langle 3 | \right) $$

Also shown in Fig. (1) is decay form the excited state $$ | 3 \rangle $$ which both decreases the population in excited state and feeds the populations in states one and two. These are included by using the Lindblad master equation such that


 * $$ \frac{d \rho}{d t} = -\frac{i}{\hbar} \left( H_{eff} \rho - \rho H_{eff}^{\dagger} \right) + \Gamma_{1} \rho_{33} | 1 \rangle \langle 1| + \Gamma_{2} \rho_{33} | 2 \rangle \langle 2 |. $$

Where $$ H_{eff} = H - i \hbar \frac{\Gamma}{2} | 3 \rangle \langle 3| $$. The rate Γ = Γ1 + Γ2 is the rate of decay from the excited state. From here, one can write differential equations for the elements of the density matrix ρ. However, the analytical solution to this problem is tedious and does not provide much insight. In certain regimes we can use adiabatic elimination to write a simplified solution. Assuming $$ \Delta, \Gamma \gg \Omega_1, \Omega_2 $$ then the excited state is short lived and, therefore, matrix elements related to the excited states evolve on a time scale τ = 1/Δ which is much smaller then the time scales for the other elements.

Coherent Evolution
The equations of evolution described above account for decoherence. If the rate is weak, $$ \Delta \gg \Gamma \gg \Omega_1, \Omega_2 $$ then the system will follow coherent evolution described by the Scrhödinger equation driven purely by the Hamiltonian H given above. The state of the quantum system at a time t is given by $$ | \psi(t) \rangle = c_1(t) |1 \rangle + c_2(t) | 2 \rangle + c_3(t) | 3 \rangle $$. Solving the Schrödinger equation gives


 * $$ \dot{c}_1(t) = -i \frac{\Omega_1}{2} c_3(t) $$
 * $$ \dot{c}_2(t) = i \delta c_2(t) - i \frac{\Omega_2}{2} c_3(t) $$
 * $$ \dot{c}_3(t) = -i\frac{\Omega_1}{2} c_1(t) - i \frac{\Omega_2}{2} c_2(t) + i \Delta c_3(t) $$

The third equation fits the form discussed in the previous section since $$ \Delta \gg \Omega_1, \Omega_2 $$. Now, c3(t) can be adiabatically eliminated by either the integration method or by setting $$ \frac{d c_3(t)}{dt} = 0. $$ Both lead to


 * $$ c_3(t) \approx \frac{1}{2 \Delta} \left( \Omega_1 c_1(t) + \Omega_2 c_2(t) \right) $$

The evolution for $$ c_1(t) $$ and $$ c_2(t) $$ can be solved for by substituting this equation back into the differential equations above.


 * $$ \dot{c}_1(t) = -i\ \delta_1 c_1(t) -i \frac{\Omega_R}{2} c_2(t) $$
 * $$ \dot{c}_2(t) = -i\frac{\Omega_R}{2} c_1(t) -i \delta_2 c_2(t), $$

where $$ \delta_i = \frac{\Omega_i^2}{4 \Delta} $$ and $$ \Omega_R = \frac{\Omega_1 \Omega_2}{2\Delta} $$ . This is equivalent to coherent evolution of a two level quantum system interacting with a classical field. Therefore, in this limit we can treat a three-state quantum system as a two-state system. This can be useful in several respects. First, Two-state quantum system are well understood and easy to simulate. Also, two-state quantum system, or qubits, are desirable for possible creation of a Quantum computer. Finally, it allows the creation of qubits in atoms between states that might not be connected by an atomic Selection rule.

Optical Pumping
Another limit that is useful to study is when one of the classical fields is turned off and spontaneous emission is strong. Without loss of generality it will be assumed that Ω2 = 0. This is what is commonly referred to as Optical pumping since, as will be seen, the population will be pumped form one state to another. It is assumed no population starts in the excited state so when $$ \Delta \gg \Omega_1 $$ the population in the excited state will remain very small for all times compared to the population in the other states.

Solving the Lindblad equation stated earlier for the elements of the density matrix gives


 * $$ \dot{\rho}_{11}(t) = \Gamma_1 \rho_{33}(t) - i \frac{\Omega_1}{2} \left( \rho_{31}(t) - \rho_{13}(t) \right) $$
 * $$ \dot{\rho}_{22}(t) = \Gamma_2 \rho_{33}(t) $$
 * $$ \dot{\rho}_{33}(t) = -\Gamma \rho_{33}(t) + i \frac{\Omega_1}{2} \left( \rho_{31}(t) - \rho_{13}(t) \right) $$
 * $$ \dot{\rho}_{13}(t) = -i\left( \Delta - i\frac{\Gamma}{2} \right) \rho_{13}(t) - i\frac{\Omega_1}{2}\left( \rho_{33}(t) - \rho_{11}(t) \right) $$
 * $$ \dot{\rho}_{23}(t) = -i\left( \Delta - i\frac{\Gamma}{2} \right) \rho_{13}(t) - i\frac{\Omega_1}{2}\rho_{21}(t) $$
 * $$ \dot{\rho}_{12}(t) = -i \delta \rho_{12}(t) + i \frac{\Omega_1}{2} \rho_{32}(t) $$



Again, the matrix elements ρi,j where i or j = 3 evolve on a time scale much smaller then the other matrix elements. This allows for adiabatic elimination of these variables,


 * $$ \dot{\rho}_{13}(t) \approx 0 \Rightarrow \rho_{13}(t) \approx \frac{\Omega_1/2}{\Delta - i\Gamma/2}\rho_{11}(t), $$
 * $$ \dot{\rho}_{33}(t) \approx 0 \Rightarrow \rho_{33}(t) \approx \frac{i \Omega_1}{2 \Gamma} \left(\rho_{31}(t) - \rho_{13}(t) \right). $$

Combining these equations leaves $$ \dot{\rho}_{11}(t) \approx -\Gamma_2 \frac{s_1}{2} \rho_{11}(t) $$, where $$ s_1 = \frac{\Omega_1^2/2}{\Delta + \Gamma/2} $$ is commonly referred to as the saturation parameter. This leads to the analytic solution for the populations


 * $$ \rho_{11}(t) \approx \rho_{11}(0)e^{-\Gamma_2 s_1 t/2}, $$
 * $$ \rho_{22}(t) \approx 1 - \rho_{11}(0) e^{-\Gamma_2 s_1 t/2}, $$
 * $$ \rho_{33}(t) \approx 0. $$

The original assumption that allows for adiabatic elimination is also stated as $$ s_1 \ll 1 $$. As t → ∞ the population will go to the second state regardless of the initial state. It has not been assumed that the system starts in a pure state just that there is no population in the excited state. A comparison with solutions from adiabatic elimination with numerical solution to the differential equations is shown in Fig. (2). This matches the physical interpretation of the system. The field, Ω1 pumps the population from the first state to the excited state. The excited state then decays to both states, depending on the values of Γ1 and Γ2. However, the population that decays to the first state is repumped back to the excited state and the process repeats. However, the population that decays to the second state has no way to leave and is trapped. This continues until all the population becomes trapped in the second state.

Drawbacks to adiabatic elimination
Aside from applications outside of the regimes discussed above there are several drawbacks to adiabatic elimination which can further limits usefulness:
 * 1)  There is no description of the excited states dynamics
 * 2)  It requires that the population does not start in the excited state
 * 3)  There are infinitely many rotating frames to move into that do not all have the same result from adiabatic elimination

These questions have been addressed more recently in and which give higher level approximations for adiabatic elimination as well as more details on the method.