Laplace differential privacy arises as the trade-off function corresponding to distinguishing between two Laplace distributions with unit scale parameter and locations differing by \(\mu\).
Without loss of generality, the trade-off function is therefore,
$$L_\mu := T\left(\text{Lap}(0, 1), \text{Lap}(\mu, 1)\right) \quad\text{for}\quad \mu \ge 0.$$
The most natural way to satisfy \(\mu\)-Laplace DP is by adding Laplace noise to construct the randomised algorithm.
This is the canonical noise mechanism used in classical \(\varepsilon\)-differential privacy.
Let \(\theta(S)\) be the statistic of the data \(S\) which is to be released.
Then the Laplace mechanism is defined to be
$$M(S) := \theta(S) + \eta$$
where \(\eta \sim \text{Lap}(0, \Delta(\theta) / \mu)\) and,
$$\Delta(\theta) := \sup_{S, S'} |\theta(S) - \theta(S')|$$
the supremum being taken over neighbouring data sets.
The randomised algorithm \(M(\cdot)\) is then a \(\mu\)-Laplace DP release of \(\theta(S)\).
In the classical regime, this corresponds to the Laplace mechanism which satisfies \((\varepsilon=\mu)\)-differential privacy (Dwork et al., 2006).
More generally, any mechanism \(M(\cdot)\) satisfies \(\mu\)-Laplace DP if,
$$T\left(M(S), M(S')\right) \ge L_\mu$$
for all neighbouring data sets \(S, S'\).
In the f-differential privacy framework, the canonical noise mechanism is Gaussian (see gdp()), but \(\mu\)-Laplace DP does arise as the trade-off function in the limit of the group privacy of \(\varepsilon\)-DP as the group size goes to infinity (see Proposition 7, Dong et al., 2022).