Collapse and Revival Quantum Beats

Quantum beats occur when there is a coherence established between nondegenerate energy eigenstates of an atom. For example, the linear Zeeman shifts induced in an isolated atomic state by a magnetic field cause Larmor precession of the atomic polarization, with a precession frequency given by the size of the Zeeman splitting (see "Larmor Precession"). When a non-isolated state is subject to a magnetic field, on the other hand, there can be, in addition to the linear Zeeman effect, nonlinear shifts induced due to mixing between the different states. For an atomic state with hyperfine structure, this is the Breit-Rabi splitting (see "Hyperfine structure: Breit-Rabi Diagram"). When the shifts are small relative to the hyperfine splitting, they are described by the linear Zeeman effect plus a small additional quadratic shift, resulting in a low-frequency quantum-beat evolution superimposed on the higher-frequency Larmor precession.
We can model this effect in a single J=2 state (assumed to have a tensor polarizability) by applying an electric field proportional to the magnetic field to simulate the quadratic part of the Zeeman splitting. Here we find the evolution of atomic polarization that is initially stretched along x and then evolves due to z-directed magnetic and electric fields.
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Define the atomic system, consisting of a J=2 state.
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The Hamiltonian for the system subject to z-directed electric and magnetic fields. The electric field is proportional to the magnetic field—the parameter a represents the relative size of the quadratic term.
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The evolution equations.
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An initial state, stretched along the z direction.
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Rotate the initial state so that it points along x.
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Create a set of initial conditions at t=0 corresponding to the rotated state.
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Solve the system of differential equations.
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The time-dependent density matrix.
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To examine the time-dependence of the atomic polarization, we plot the x-component of the atomic orientation (the rank-1 polarization moment).
Extract the x-component of the atomic orientation from the solution for the density matrix.
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We can see that this component of the atomic polarization is proportional to the expectation value of Jx.
The Jx operator.
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The expectation value of Jx, which is proportional to the above expression for the x-component of orientation.
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Plot the orientation as a function of time for particular values of the linear and quadratic splittings.
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The fast oscillation represents the Larmor precession. The amplitude of the fast oscillation is modulated at a slower frequency, determined by the value of the quadratic splitting. Since the quantum beats seem to disappear and then reappear, this phenomenon is known as "collapse and revival."
We can plot the complete polarization state using angular-momentum probability surfaces (see "Angular-momentum Probability Surfaces"). We first plot the evolution over the shortest time scale—a period of the fast evolution.
A series of frames showing a complete cycle of the fast evolution of the atomic polarization.
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We see that the atomic polarization, initially oriented ("pointing") along x, undergoes Larmor precession about the z-axis. This corresponds to the fast oscillation in the plot above.
We now plot the evolution over the longest time scale—a complete period of the slow evolution.
A series of frames showing a complete cycle of the slow evolution of the atomic polarization.
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This series of frames corresponds to the entire range of the orientation plot above. The first frame is the same as in the previous series: atomic polarization initially oriented along . In the third frame, the orientation is gone: the polarization points equally along the and directions, a characteristic of atomic alignment, the rank-2 polarization moment. The orientation subsequently reappears, pointing along , and disappears again before the polarization returns to its initial state. This evolution is responsible for the slow modulation in the orientation plot. (In fact, a J=2 system can support polarization moments up to rank 4, and these higher-order moments are also present. The evolution shown here corresponds to the disappearance and reappearance of the odd-order moments.)
We can make an animation from these plots showing both the fast and slow evolution. Many frames are required, so rather than assembling the animation inside the Mathematica notebook, it is more convenient to export the frames individually as images and join them with a video editor. Here is a link to the exported animation on the AtomicDensityMatrix website. We see the polarization undergo fast Larmor precession about the z-axis, as well as a more gradual change in shape of the surface, corresponding to the quadratic quantum beats. Initially the surface displays a preferred direction (orientation). At an intermediate stage, the orientation has disappeared, and the atomic polarization contains only even-order moments (there is a preferred axis, but no preferred direction). This represents a collapse, from the point of view of the Larmor precession of orientation. The orientation subsequently reappears; this is the revival.