how could japan recover very fast their economy eventhough the two bombs were dropped?

The economy of Japan was in terrible shape. There were shortages of all essentials such as food and housing. The functioning Japanese government had little in the way of financial resources. The tax system was not operative and the public had little to tax. The U.S. government provided food supplies to the Japanese government which sold them to raise funds for its operation. The government dealt with its deficits by printing up MONEY. This of course led to inflation. SCAP called for Japanese authorities to write a new constitution. The old constitution which had been adopted in the Meiji Era was modeled on the Prussian constitution and gave all the real authority to the bureaucracy. The government authorities made inconsequential modifications to the old constitution and MacArthur gave up on them making any real changes. Instead he FORMED a group within SCAP and assigned them to formulating a new Japanese constitution in six days. None of the members of the group had a background in constitutional law.

I suppose the answer to your question is that, firstly, we definitely have to consider the aid by the U.S. after the war. Like, well, Japanese children asking for the American soldiers for chocolates etc, but the real booster in terms of the economic impact was probably due to the korean war.

SCAP officials believed economic development could not only democratize Japan but also prevent the reemergence of militarism, and forfend communism. Military hostilities in the Korean peninsula further boosted the economy in 1950 because the U.S. government paid the Japanese government large sums for "special procurement." These payments amounted to 27% of Japan’s total export trade.1 The United States also insisted that Japan be admitted to GATT as a "temporary member" – over British opposition. During the Korean War, SCAP departed and the Treaty of San Francisco restored sovereignty to the government of Japan.

This ratio quoted above was remarkable. Kindly take a look here too.

Coincidentally one of the initial pushes towards the boom came from the start of the Korean war, and the US paid Japan for military goods and services. Between 1951-53 approximately 60% of all Japanese exports went toward supporting the Korean war. Large companies within Japan made profits for the first time since the end of the war and Japan's GDP soared like never before.

Although the ratio (to the export) described is different, Japanese official text mentioned the effect of Korean war too, so that I think the Korean war was the first booster.

After the war came the period of the production of 三種の神器(translated : "Three sacred treasures", referring to 1 - Television, 2 - Washing machine, 3 - Fridge)

Three signs of affluence in the post-war period were the “three sacred treasures”, a television, a fridge, and a washing machine, and by 1964, 90% of households possessed all “three sacred treasures”. Most families had sought to equip themselves with these luxury items, and in the years that followed, consumer demand increased significantly.3 From 1951 to 1967, for instance, the percentage of paid radio subscriber households rose from 58.6% to 93.4%, while from 1952 to 1970, the percentage of paid television subscriber households rose from 0.01% in 1951 to 94.% in 1970.4 By 1970, 98% of all employee households owned a washing machine, 95% a gas or electric refrigerator, 80% a vacuum cleaner, 77% a camera, and 67% to 70% a television set.

That was when today's Mitsubishi Electronic, Sony, Toshiba Inc, etc, which I think are still famous worldwide today, were born.

Then we saw "Golden Sixties", but kindly read the page I sourced, since I would have to cover the Japanese post-war economic movement entirely here.

Thank you very much.