Because the Soviet invasion occurred right alongside the dropping of the bombs and that makes me wonder how it was so coordinated.
Only July 24, 1945, Truman cryptically remarked to Stalin at the Potsdam Conference that the United States has a new weapon of unusual power and they planned to use it. He didn't elaborate and Stalin didn't press it, leading Truman to think that Stalin didn't understand. But Stalin had spies in the project and knew that they were planning to test their first atomic bomb soon, and so inferred that the test had gone well.
The US didn't give the Soviets any more information than that. They also didn't know when the Soviets planned to declare war on Japan; they inferred it would happen in mid-August 1945. So the plan was to use the bomb as soon as possible after the Potsdam Conference ended (August 3rd), with a hoping of maybe ending the war before the Soviets got into it. Because by that point they had come to understand that land that the Soviets "liberated" tended to become thralls of the Soviet state, and they'd prefer the Soviets to have as little influence in the Far East as was possible.
Weather delays pushed the first use of the bomb back to August 6th, the Hiroshima attack. After receiving word of this, Stalin sulked for a day, fearing that the Japanese might surrender too soon for his purposes. But once it was clear to him that that wasn't probably happening, he called up his forces on the Eastern Front and demanded they proceed with the invasion immediately, bumping up their own schedule by several days. This occurred at midnight, Transbaikal time, on August 8th/9th, shortly before the attack on Nagasaki. Even after the Japanese indicated their willingness to surrender, but before the surrender documents were signed, the Soviets pushed into Manchuria, the Korean peninsula, Sakhalin, and the Kuril Islands.
All of which is to say: they were coordinated, in a way, but out of competition, not cooperation. For lots of details, see Hasegawa's Racing the Enemy, which is all about this competition.
I believe a large part of the answer depends on what we understand by ‘tell’. The US did not officially inform the Soviets in advance that concrete plans were set in motion to drop the atomic bombs on the Japanese mainland, certainly not in the same way the US had to seek permission from the United Kingdom prior to the bombing (as was dictated by the 1943 Quebec Agreement).
However, is it arguable that the Americans had provided certain intimations which, when understood in the context of the available information the Soviet regime had at the time, could be interpreted as plans to unleash the atomic bomb on Japan. It is unclear to what extent or how sincerely the US intended to communicate their possession of the bomb, and indeed official circles in Washington were divided as to whether Stalin should be informed. At the Potsdam Conference of July 1945, Truman told Stalin personally that the US had a new type of weapon of ‘unusual destructive force’. Secretary of State James F Byrnes, who was present at the conference during this exchange, recalled in his 1947 memoirs that Truman had also said that the US ‘planned to use it very soon unless Japan surrendered’. Stalin was not surprised and reportedly showed little interest--although Truman did not specifically mention it was the atomic bomb, Stalin likely knew what the American president was referring to, since Soviet spies had penetrated the Anglo-American nuclear programmes reportedly as early as 1941, and so Stalin would have been well aware of the bomb’s development. Throughout the war there were at least three separate and successful Soviet attempts to penetrate security at the Los Alamos laboratory where the bomb was being assembled.
Shortly after, the Allies in the Pacific War (meaning only the US, Britain, and China, since Stalin had not yet declared war on Japan) issued the Potsdam Declaration calling on the Japanese to surrender or face ‘prompt and utter destruction’. It is debatable whether the final clause was included as a veiled warning about the atomic bomb, but in any case there was no specific mention of the bomb either at the conference or within the declaration itself, which was kept ambiguous.
Stalin had agreed at the Tehran Conference of 1943 that the Soviets would enter the war against Japan after Germany was defeated. This agreement was further solidified at the Yalta Conference in 1945, during which Stalin confirmed a more specific timeline: the Soviets would declare war on Japan within three months of the end of the war in Europe. The US had in fact expected Moscow’s intervention to be sometime in mid-August, but Soviet forces moved their schedule up to August 8. The reason for this shift is debatable, but the general historical consensus argues that it was most certainly linked to the bombing of Hiroshima. Having significant forces in the Far East would bolster Stalin’s negotiating hand against the Western allies in the same way the preponderance of Soviet forces in Eastern Europe certainly did, and so there was renewed urgency to strengthen the Soviet military position before the Japanese formally surrendered (It is unclear when it became certain that the Japanese were committed to surrendering. Some argue that it was set in stone after the bombing of Hiroshima, but others suggest the Japanese government had maintained hopes, however optimistic, that the Soviets might help mediate a conditional surrender and that these hopes faded only after the Soviet declaration of war). Furthermore, in The Cold War (2005), John Lewis Gaddis notes that while the prevailing view in Washington and London had initially been that Soviet assistance would be vital in securing victory over Japan, the successful testing of the American atomic bomb and Western discomfort at the Soviets' demonstrated unilateralism in occupied Germany and Eastern Europe had caused a reversal of that official view. The Soviets eventually managed to occupy Manchuria before the Japanese surrender (which later became an important base for Mao Zedong’s communist forces during the Chinese Civil War, to whom the Soviets also handed over the bulk of Imperial Japanese weapons they had captured in the region), but scholars like Sergey Radchenko argue that the Soviets would likely also have occupied significant parts of Japan were they not deterred by Americans’ use of the bomb. Stalin interpreted the bombing as an anti-Soviet move, intended to intimidate and deter the Soviet Union from making further strategic gains in the Far East and to give the US greater bargaining power in setting the terms of the post-war settlement. Barton Berstein writes the Soviet decision was ‘probably because of the Hiroshima bombing'. Gaddis reports that Stalin lamented: ‘A-bomb blackmail is the American policy’. The Soviet invasion of Manchuria had an approximate timeline officially since 1943, so in that sense it might be said to be coordinated with the rest of the Allies, but certainly in August 1945 they reacted to the bombing of Japan by pushing that timeline forward.
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