Energy Decay of a Dissipative Harmonic Oscillator

This example provides an impression of how to use the qdyn library in order to model the energy decay of an harmonic oscillator interacting with a surrounding bath. In order to model the decay, the Surrogate Hamiltonian method is used. Its idea is based on modeling the infinite dimensional bath Hilbert space by a small number of representative two-level systems (spins) which generate (up to a certain time) the same dynamics as the full bath. The advantage of the Surrogate Hamiltonian compared to other open quantum system methods is that it’s not restricted to weak coupling between open system and environment or a factorizing initial state (as, for instance, required for Lindblad master equations). Thus, it can easily be used for modeling non-Markovian dynamics.

Model

The full Hamiltonian for the harmonic oscillator interacting with a spin bath, i.e., the Hamiltonian for open system and environment, reads [10, 11].

\[\op{H} = \frac{\op{P}^{2}}{2 m} + \frac{m \omega^{2}}{2} \op{Q}^{2} + \sum_{k} \epsilon_{k} \op{n}^{(k)} + \op{Q} \sum_{k} d_{k} \op{\sigma}_{x}^{(k)}\,.\]

The harmonic oscillator is initially in an excited state \(\Ket{\Psi_{HO}(t=0)} = \op{Q} \Ket{\Psi_{g}}\) (with \(\Ket{\Psi_{g}}\) the ground state an harmonic oscillator), while the spin bath is assumed to be in its ground state \(\Ket{\Psi_{B}(t=0)} = \Ket{00...00}\). The joint state is factorized and reads

\[\Ket{\Psi(0)} = \Ket{\Psi_{HO}(0)} \otimes \Ket{\Psi_{B}(0)}\,.\]

For \(t>0\) the state of the harmonic oscillator decays into its ground state, i.e., its energy decays as it is transferred into the environment.

Running the Example

In order to run the example you must execute

./configure

in the main folder of the qdyn library in order to generate the Makefile. Afterwards, the program must be compiled by executing

make

This generates the executable of the program. By executing

./hamos_surrogate

you can run the program. Upon execution, the program will read the config file config in order to get all necessary parameters for harmonic oscillator and spin bath. Note that some of the parameters are not directly read from the config file but are calculated during execution of the program. The source code for the program is entirely contained in hamos_surrogate.f90 and for details about implementation and structure refer to this file as all parts are documented.

Results

For \(m=1\), \(\omega=1\) and a linear sampling of the spin energies \(\epsilon_{k}\) between \(E_{min}=0\) and \(E_{max}=1.5\) the following dynamics of the energy decay can be obtained for different total spin numbers.

energy

The coupling constants \(d_{k}\) between harmonic oscillator and the \(k\)’th spin of the environment have been calculated using

\[d_{k} = \sqrt{J(\epsilon_{k}) / \rho(\epsilon_{k})}\,, \qquad J(\epsilon_{k}) = \eta \epsilon {k} e^{- \epsilon_{k} / \epsilon_{c}}, \qquad \rho(\epsilon_{k}) = \frac{1}{\epsilon_{k+1} - \epsilon_{k}}\,.\]

Bibliography

• [10] Baer and Kosloff, J. Chem. Phys. 106, 8862 (1997)

• [11] C. P. Koch, Quantum dissipative dynamics with a Surrogate Hamiltonian. The method and applications, Ph.D. thesis, Humboldt University Berlin (2002)