In most applications, evaluators typically exploit the timing of a policy/intervention with one dichotomous treatment variable. The model usually takes the following form,
$$
y_{igt} = \gamma_g + \lambda_t + \delta D_{gt} + \epsilon_{igt},
$$
where the parameters $\gamma_g$ and $\lambda_t$ denote group (state) and time (year) fixed effects, respectively. Here, $D_{gt}$ 'turns on' (i.e., switches from 0 to 1) in the year the treated groups become exposed to the treatment, 0 otherwise. To be specific, the early group switches from 0 to 1 in year 6, while the late group switches from 0 to 1 in year 7. For those never exposed to treatment, the variable is 0 in all years. I presume once you're treated you stay treated. Note that the "always 0" group is the control group.
I am wondering how I would change this so that there are now three treatment groups, untreated, longer treated and less treated.
You cannot include three groups on the right-hand side of your equation. You must drop one group to serve as a reference. The unexposed group is a natural choice; leave it out. By constraining the unexposed to have a value of 0, you now have a baseline history of never receiving a treatment.
Here is the model you're proposing,
$$
y_{igt} = \gamma_g + \lambda_t + \delta_1 E_{gt} + \delta_2 L_{gt} + \epsilon_{igt}
$$
where $E_{gt}$ and $L_{gt}$ denote the early- and late-adopters, respectively. As before, the interaction term is implicit in the coding of each treatment variable. They 'switch on' once a treated group $g$ is treated in year $t$, 0 otherwise. Such a model is estimable.
In my opinion, your proposed setup is more common in settings where evaluators want to assess the independent effect of qualitatively different treatments, not necessarily the same treatment rolled out at different times. You can still proceed with your approach, but be prepared to let your audience know why you're separating out the different timing cohorts. Maybe there's something about when your first exposed that's important to look at independently.
As per your comments (see below), we can assess "common trends" by multiplying the group identifiers with the year dummies. Since you have a clean group of individuals that never experience a treatment, why not run two separate equations, one with a different treatment group. Supppose we have a treatment variable $T_{g}$, which could be your group with the long post-period (i.e., treated early) or your group the short post-period (i.e., treated late). Say, for example, $T_g$ is the group treated in year 6 onward. Why not try estimating the following:
$$
y_{igt} = \gamma T_{g} + \lambda_{1} (T_{g}*\mathbf{I}_{t = 1}) + \lambda_{2} (T_{g}*\mathbf{I}_{t = 2}) + \lambda_{3} (T_{g}*\mathbf{I}_{t = 3}) + \lambda_{4} (T_{g}*\mathbf{I}_{t = 4}) + \lambda_{6} (T_{g}*\mathbf{I}_{t = 6}) + \lambda_{7} (T_{g}*\mathbf{I}_{t = 7}) + \epsilon_{igt},
$$
where we interact $T_g$ with a full series of time indicators. Note how I omitted the year before the exposure as a reference, which is year 5. The $\hat{\lambda}_t$'s show how the effects evolve over time, before and after the exposure of interest. The confidence bands on the four pre-period coefficients should contain zero.
