Could you please elaborate on what you mean by keeping the respondent anonymous and still being able to contact them for stage two. I'm assuming you'll need to ask them for some contact detail, that would make anonymity difficult. On the other hand, if you have a mailing list, you could just send them the stage 2 at a later date. So, it's a little unclear what the problem is.

I agree, I think some more detail would be helpful here. At first glance however, I think there are multiple ways to go about this.
I would ensure that both projects (T1 and T2) are made to be anonymous so no panelist data is collected. However at the end of each survey, you could set a hidden question to trigger back to the panelist- T1 = Complete or something like that. This way you know who in the panel completed or did not complete each survey, but you did not tie that PII to the actual survey responses.
Assiming, of course, this meets the requirements you're setting in your question.

Kate
Thanks a lot for your response, in fact, I am following the document https://www.qualtrics.com/support/survey-platform/common-use-cases-rc/pulling-data-into-a-second-survey-longitudinal-surveys/
Let's assume that I have a list of the respondents of my sample and send the survey in T1, then I developed the second survey but I am not sure about are only the questions that I check for embedded (Longin ID) are those that are pulled to the second survey? when I download the results of the second survey, how does the data look like, is both surveys for the same ID are displayed together or they will be downloaded separately?
Please guide if you have the experience of building such longitudinal survey

2 different "projects" (T1 and T2) will result in 2 different data files. There are some ways around this. You can build one project to dynamically Show T1 then T2 using the survey flow.
Or you could use Contact Triggers to put everything from T1 and append it to the contact info- but I think that might violate your want to keep everything anonymous.
I usually just merge files in SPSS when everything is done. I would do some testing to make sure that Anonymizing the data gives you consistent Contact IDs so you can match responses, however.

@Kate thanks a lot I will keep on trying hope it works.

Thanks for the thread-I've had this question periodically so it is helpful to know that there are several possible solutions. RolaC , please keep in touch on this thread so that you can tell us what you end up doing, and how it works!

I followed the steps in
Pulling Data Into a Second Survey (Longitudinal Surveys)but the second survey asks for login Id , how can respondents skip or take automatically the second survey

Put your authenticator in a branch with the true-false condition set to true. This way, whether they fail or pass, they can continue the survey.
Alternatively, you can ask them at the beginning if they have taken survey-1 and branch to the authenticator only when they have indicated yes.
In both these options, those who have filled survey 1, will have their details pre-filled while those in survey 2 will find them empty.