In ion trapping experiments, the Lamb Dicke regime (or Lamb Dicke limit) is a quantum regime in which the coupling (induced by an external light field) between the ion's internal qubit's states and its motional states is sufficiently small so that transitions that change the motional quantum number by more than one, are strongly suppressed.
Contents
- Relation between Lamb Dicke parameter and Lamb Dicke regime
- Mathematical background
- Meaning of Lamb Dicke regime
- References
This condition is quantitively expressed by the inequality
where
Relation between Lamb Dicke parameter and Lamb Dicke regime
Considering the ion's motion along the direction of the static trapping potential of an ion trap (the axial motion in
where
is the spread of the zero-point wavefunction,
where
The Lamb-Dicke parameter actually is defined as
Upon absorption or emission of a photon with momentum
The square of the Lamb Dicke parameter then is given by
Hence the Lamb Dicke parameter
Mathematical background
In ion trapping experiments, laser fields are used to couple the internal state of an ion with its motional state. The mechanical recoil of the ion upon absorption or emission of a photon is described by the operators
If the condition for the Lamb-Dicke regime is met, a Taylor expansion is possible,
and it is readily seen that transitions between motional states, which change the motional quantum number
Meaning of Lamb Dicke regime
In the Lamb Dicke regime spontaneous decay occurs predominantly at the frequency of the qubit's internal transition (carrier frequency) and therefore does not affect the ion's motional state most of the time. This is a necessary requirement for resolved sideband cooling to work efficiently.
Reaching the Lamb Dicke regime is a requirement for many of the schemes used to perform coherent operations on ions. It therefore establishes the upper limit on the temperature of ions in order for these methods to create entanglement. During manipulations on ions with laser pulses, the ions cannot be laser cooled. They must therefore be initially cooled down to a temperature such that they stay in the Lamb Dicke regime during the entire manipulation process that creates entanglement.