The IF frequency of 100 MHz limits the achievable bandwidth for a real signal centered on it to be 100 MHz, so any pulse out of the modulator would be limited in rise/fall time as given by that bandwidth.  


Given the 1 GHz sampling rate, and ideal IF frequency if we must have a real IF that would maximize the achievable rise / fall time would be $f_s/4$ or 250 MHz. Or even better would be a direct RF approach with two DACs and an analog modulator that can support an RF output at the desired 2 GHz directly with carrier nulling techniques used (as commonly done) to null the inevitable carrier feedthrough with such an approach. With that a maximum possible single-sided pulse bandwidth at baseband would be 500 MHz (plus margin for filtering).   

This then leaves the modulator bandwidth as the limit to the achievable rise/fall time where there are actual possible tricks we can play. If the power and dynamic range allow, and if the driving signal has the excess bandwidth, you can overshoot your maximum value to achieve the faster rise time desired.  

A good approximation of the achievable rise/fall time is 

$$t_r = \frac{0.35}{BW}$$

Where $t_r$ is the 10%/90% rise or fall time, and BW is the single sided bandwidth in Hz of the equivalent baseband signal. This is exact for a first order (single pole) system but works as a good approximation for any system with a dominant single pole and gives us a rough idea for higher order systems as well as to what the settling time will be.


Note that with a simple example of a first order system (or any system with a dominant pole) the time step response is given as:

$$y(t) = A(1 - e^{t/\tau})$$ 

Where $A$ is the intended final value at pulse peak and $\tau$ is the time constant of the system with a BW in Hz given as $1/(2\pi \tau$). If we solve this for 10% and 90% of $A$ we would get the relationship given above for bandwidth and rise/fall time.  

From this we see with the OP's use of a 200 MHz Modulator (as well as the IF as noted) will limit the maximum single-sided bandwidth (or equivalent baseband bandwidth). If we assume this is the RF bandwidth, then the real passband signal is limited to be 100 MHz. However it is possible that this is the baseband modulation bandwidth (not clear from the post). Without doing any "tricks", the 10% - 90% rise/fall time for a 100 MHz BW is 3.5 ns, or 1.75 ns for 200 MHz BW (in which case if a 1.75 ns rise/fall time was acceptable, raising the IF to 250 MHz would be the way to go). The phase transition would also follow a similar settling time. 

However here comes the trick: if we have the power and dynamic range and ability to sink and source with that full power needed, we can instead overshoot the intended final value! Note that the rise time is independent of our final value $A$. So if we were able to (if our power and dynamic range allows) instead create a pulse that were to rise to a much higher value than $A$ on its own, we will reach $A$ in a much shorter time. When we do reach $A$ in that case, we pull the system low (within the same bandwidth constraint) flattening the top of the pulse. The signal at the input to the modulator would appear as a sharply rising peak with much higher bandwidth, which assuming is still in the linear operation of the modulator would be filtered by the modulator to be a faster rising pulse at the output ultimately limited by the 1 GHz  sampling spectrum to not have aliasing artifacts. 

This is exactly what occurs in a PID loop as I explain further at [this post][1], where we are able to achieve rise/fall times beyond what the physical plant would allow based on its own time constant (and nicely so since it is under loop control, so the overshoot generated in the control signal is exactly what is needed to achieve the desired output signal and transition times.) A PID rise/fall time is limited by the second lowest pole in the plant rather than it's lowest pole.      

On a practical level I would not attempt this but wanted to show the possible "trick" that does exist. Instead my priority would be to look for a faster modulator solution, direct modulation at 2 GHz, or raise the IF bandwidth and potentially the sampling rate to achieve the target rise/fall times and phase transition time most simply.


  [1]: https://dsp.stackexchange.com/questions/64310/model-necessary-for-pid-control/64693#64693