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# Introduction
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Quantum friction is an external potential added to the Hamiltonian that breaks time-reversal invariance so as to cool the system (decrease its total energy). It may be used to cool fermionic many-body systems with thousands of wavefunctions that must remain orthogonal. It is described in details in:
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Quantum friction is an external potential added to the Hamiltonian that breaks time-reversal invariance so as to cool the system (decrease its total energy). It may be used to cool fermionic many-body systems with thousands of wavefunctions that must remain orthogonal. It is described in detail in:
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* J. E. Alba-Arroyo, D, Pęcak, M, M, Forbes, G, Wlazłowski, _Local Quantum Friction with Pairing: Unitary Dissipation in Large Fermi Systems_, [arXiv:2512.12866](https://arxiv.org/abs/2512.12866)
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* A. Bulgac, M. M. Forbes, K. J. Roche, G. Wlazłowski,
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_Quantum Friction: Cooling Quantum Systems with Unitary Time Evolution_,
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[arXiv:1305.6891](https://arxiv.org/abs/1305.6891)
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The quantum friction potential is given by:
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```math
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V_{\sigma}^{(qf)} = -\alpha \frac{\hbar\,\vec{\nabla}\cdot\vec{j}_\sigma}{\rho_0}
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```
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where $`\rho_0=\frac{k_F^3}{6\pi^2}`$ is reference density. By construction, this potential removes any irrotational currents. Thus it provides a convenient method of removing phonon excitations from the system.
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# Usage
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The quantum friction is controlled via `input` file via tags:
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The quantum friction is controlled via the `input` file via tags:
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```bash
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# --------------- QUANTUM FRICTION ------------------
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# qfalpha 0.0 # alpha parameter for quantum friction term
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# qfstart 0.0 # start time for evolving with quantum friction [eF]
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# qfstop 0.0 # stop time for evolving with quantum friction [eF]
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# qfswitch 0.0 # the friction will be activated and deactivated gradually over this period of time [eF]
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# See: Wiki -> Quantum friction
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# qfalpha 0.0 # alpha parameter for quantum friction term (cooling in the normal channel), default=0
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# qfbeta 0.0 # beta parameter for quantum friction term (cooling in the pairing channel), default=0
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# qfgamma 0.0 # particle control parameter (cooling in the pairing channel), default=0
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# qfNreq 0.0 # requested particle number, meaningful only if qfgamma>0
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# qfstart 0.0 # start time for evolving with quantum friction [eF], default=0
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# qfstop 0.0 # stop time for evolving with quantum friction [eF], default=infinity
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# qfswitch 0.0 # the friction will be activated and deactivated gradually over this period of time [eF], default=1
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```
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For the meaning of these variables, see [arXiv:2512.12866](https://arxiv.org/abs/2512.12866)
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*Notes*:
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* `qfalpha ~ 1` looks to be reasonable choice,
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* too large value of `qfalpha` may lead to instability of the code, typically it manifests via growing of the energy during the evolution.
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* `qfalpha ~ 5-10` provides optimal damping efficiency.
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* `qfbeta ~ 5` provides optimal damping efficiency.
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* `qfgamma ~ 0.1` provides optimal damping efficiency.
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* A too large value of `qfalpha` may lead to instability of the code; typically, it manifests via the growth of the energy during the evolution.
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*
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# Example
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Consider application a time-dependent potential:
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Consider the application a time-dependent potential:
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```math
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V_{\textrm{ext}}(x,y,t)=s(t,t_{\textrm{start}}, t_{\textrm{stop}})\exp\left[-\frac{x^2}{2\sigma_x^2}-\frac{y^2}{2\sigma_y^2}\right]
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```
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to the unitary Fermi gas, being initially in the uniform state, where $`s(t,t_{\textrm{start}}, t_{\textrm{stop}})`$ is (smooth) step function that acquires 1 in time interval $`[t_{\textrm{start}}, t_{\textrm{stop}}]`$, otherwise is 0. Implemamntion of this potential is following (`problem-definition.h`):
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to the unitary Fermi gas, being initially in the uniform state, where $`s(t,t_{\textrm{start}}, t_{\textrm{stop}})`$ is (smooth) step function that acquires 1 in time interval $`[t_{\textrm{start}}, t_{\textrm{stop}}]`$, otherwise is 0. Implementation of this potential is following (`problem-definition.h`):
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```c
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__device__ __host__ inline double switch_function(double t, double T, double alpha)
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{
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| ... | ... | @@ -91,7 +94,7 @@ When executing this code for lattice (`predefines.h`): |
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#define NZ 8
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#define FUNCTIONAL ASLDA
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```
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with input file parmaters:
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with input file parameters:
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```bash
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params0 0.25 # gauss amplitude [eF]
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params1 2.0 # width in x direction
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| ... | ... | @@ -106,7 +109,7 @@ qfstop 250.0 |
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qfswitch 1.0
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# ...
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```
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the resulting evolution of the total energy looks like this:
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The resulting evolution of the total energy looks like this:
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It is observed, that time-dependent potential excites the system. Selected potential does not introduce angular momentum to the system, and only phonons are induced. When evolving this state with the quantum friction (`qfalpha=1.0`) we see that the system returns after some time to its ground state. The origin for energy fluctuations for `qfalpha=0.0` and for $`t\varepsilon_F>50`$ see [here](Regularization schemes of the pairing field). |
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\ No newline at end of file |
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It is observed that the time-dependent potential excites the system. Selected potential does not introduce angular momentum to the system, and only phonons are induced. When evolving this state with quantum friction (`qfalpha=1.0`), the system returns to its ground state after some time. The origin of energy fluctuations for `qfalpha=0.0` and for $`t\varepsilon_F>50`$ see [here](Regularization-schemes-of-the-pairing-field). |
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\ No newline at end of file |
