Numerical study of metastability due to tunneling: The quantum string method

Tiezheng Qian, Weiqing Ren, Jing Shi, Weinan E, Ping Sheng

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13 Scopus citations


We generalize the string method, originally designed for the study of thermally activated rare events, to the calculation of quantum tunneling rates. This generalization is based on the formal analogy between quantum mechanics and statistical mechanics in the path-integral formalism. The quantum string method first locates the minimal action path (MAP), which is a smooth curve connecting two minima of the imaginary-time action in the space of imaginary-time trajectories. From the MAP, the saddle point of the action (called "the bounce") associated with the exponential factor for barrier tunneling probability is obtained and the pre-exponential factor (the ratio of determinants) for the tunneling rate evaluated using stochastic simulation. The quantum string method is implemented to calculate the bounce and rate of tunneling for the Mueller potential in two dimensions. The quantum problem is much more difficult than the thermally activated barrier crossing problem for the same potential. The model calculations show the string method to be an efficient numerical tool for the study of barrier tunneling in higher dimension, from the determination of the saddle point to the computation of the pre-exponential factor.

Original languageEnglish (US)
Pages (from-to)491-502
Number of pages12
JournalPhysica A: Statistical Mechanics and its Applications
Issue number2
StatePublished - Jun 15 2007

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Condensed Matter Physics


  • Barrier tunneling
  • Quantum metastability
  • Rate of decay
  • String method


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