(aws-amplify-alpha): Amplify ProductionBranch not being populated on cdk deploy

### What is the problem?

When using CDK to deploy Amplify, the Production Branch field is not being populated. Production Branch also does not populate after successful build. I cannot find anywhere in the documentation where ProductionBranch can be defined. This is the closest documentation i could find: [Amplify API](https://docs.aws.amazon.com/amplify/latest/APIReference/API_ProductionBranch.html).

I also tried with the aws_amplify package for cdk2.0 as well.


### Reproduction Steps

Deploy Amplify App using GitHub repo as the provider.

Code I am using:
```
import * as cdk from "aws-cdk-lib";
import * as amplify from '@aws-cdk/aws-amplify-alpha';
import { Construct } from "constructs";

export interface CdkInfraStack extends cdk.StackProps {
owner: string;
repository: string;
role: string;
domain: string;
}

export class AmplifyInfraStack extends cdk.Stack {
constructor(scope: Construct, id: string, props: CdkInfraStack) {
super(scope, id, props);

// Amplify Application
const amplifyApp = new amplify.App(this, "next-js-amplify", {

sourceCodeProvider: new amplify.GitHubSourceCodeProvider({
owner: props.owner,
repository: props.repository,
oauthToken: cdk.SecretValue.secretsManager("github/pat/crs-k", {
jsonField: "githubPat",
}),

}),
role: cdk.aws_iam.Role.fromRoleArn(this,'service-role',props.role) // necessary for explictly defining a service role that can create other resources as needed.
});

const main = amplifyApp.addBranch("main", {autoBuild: true});

const domain = amplifyApp.addDomain(props.domain, {
enableAutoSubdomain: true, // subdomains auto registered for branches
autoSubdomainCreationPatterns: ['*'], // regex for branches that should auto registered as subdomains
});
domain.mapRoot(main); // map main branch to domain root
domain.mapSubDomain(main, 'www');
}

}
```

### What did you expect to happen?

Production Branch field to be populated with addBranch or default branch name.

### What actually happened?

Production Branch field was unpopulated.

### CDK CLI Version

2.10.0

### Framework Version

_No response_

### Node.js Version

17.3

### OS

windows & ubuntu

### Language

Typescript

### Language Version

4.5.5

### Other information

_No response_

Reproduce the provided TypeScript stack using the aws-amplify-alpha App, addBranch, and GitHubSourceCodeProvider entry points, then trace how the branch and production-branch settings are represented during deployment. Done means the deployed Amplify application populates Production Branch after CDK deployment or a successful build, with a regression test covering the behavior.

Written by the indexing model from the issue text.