The longer the path… the shorter the travel time?

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By Danielle St. Jean, for the second season of the New Science Communicators Series

“On a harsh desert evening, Baal Shem Tov, an 18th century Polish rabbi, was travelling with his new students. Having ridden at a full canter all day, the horses gasped for air yet their hooves continued to make an energetic beat on the forest floor amidst the creaking of leather and the harsh breathing of tired men. The sky in the west was beginning to show the first soft smears of twilight, and the sun was quickly falling to the horizon.

One of the students turned to Baal Shem Tov and cried out that they were miles from any city. The roads were not a safe place to camp, for the nights were filled with violence, especially against Jewish men. The Sabbath was quick approaching, and yet the rabbi called out to his men not to worry. He pulled back on the reins of his horse. A misty cloud obscured the sky, the sun, and the earth, only to evaporate to reveal a changed landscape, and the town in front of them. The students nodded to one another, ‘Our master has performed kefitzat ha-derekh: the Shortening of the Way.’”

– an account of a popular Chassidic folklore story

Many Hassidic folklore stories involve miraculous feats performed by rabbis. The miracle of kefitzat ha-derekh, in particular, influenced chemical physicist Eli Pollak, a professor at the Weizmann Institute of Science, to recently begin to unravel the mystery of time in the quantum world.

He recalls that his interest in time in quantum mechanics began as he read a paper published in Science by researchers at the University of Alberta,  which considered the time distribution of the transition path for the folding and unfolding of proteins.  Protein folding is the process that amino acids become intricately packaged so that they presume the shape that will determine the very function of the protein. Essential to all biochemical processes inside of the human body, the complexity of protein folding and unfolding is still a mystery in a lot of ways, and often has transient steps along the way. It is very useful to be able to calculate how long it takes for a protein to transition, as this can be used to help determine those transient steps.

The paper published in Science contained intriguing results about the transition path time for the unfolding and folding process. The results of the paper precisely fitted known models values for the transition path time; however, their fit was established on barrier height parameters much smaller than previously expected for the folding and unfolding process. Barrier heights are the amount of energy it takes for the proteins to transition through the intermediate steps and complete the folding process. Their results were a conundrum – their models fit the experimental data, but the explanation of why these results made sense was not there.

Eli then began to consider how quantum mechanics could perhaps provide an explanation for the mysterious shortening of this important barrier height. He wrestled with these thoughts, but eventually concluded that the quantum effect on the protein folding could not have a great impact on the barrier height. This sparked a question in his mind: in other situations, how does the curious realm of quantum mechanics affect time distributions?

One of the strange abilities of quantum particles is quantum tunneling, a phenomenon where a particle tunnels through a solid barrier that normally it could never surmount. Eli pondered whether quantum mechanics could have unexpected effects on the transition time of tunneling particles. He then postulated it might have something to do with the quantum mechanics that allowed the particles to jump over a high barrier in a short amount of time.

To investigate this, he designed a simulation to predict how quantum particles tunnel through a barrier. He carefully chose the mathematics and parameters so that the calculations could be done without any approximations or numerical methods. In quantum mechanics, solutions must often be calculated using computers and numerical algorithms, creating a small uncertainty in the result.   Computers are powerful and precise, but  there is always an uncertainty associated with how a computer stores numbers. In many circumstances this uncertainty is negligible, but the calculations that Eli was considering required astonishing precision for the results to be accurate. By designing a simulation that could be run entirely without numerical computer calculations, he eliminated this source of uncertainty.

Eli was shocked that when he made his calculations, he discovered that for some quantum systems, the longer the path was, the shorter the transition time . It seemed almost as though the quantum particles saw a great barrier to cross, reined in their horses, and performed a great kefitzat ha-derek.

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