Simulating heat flow through a quantum spin between hot and cold baths
A trick that replaces a warm environment with an equivalent cold one lets a single wave-function simulation reproduce finite-temperature quantum heat flow, including strong-coupling effects.
Source
Matrix Product State Simulations of Non-Equilibrium Steady States and Transient Heat Flows in the Two-Bath Spin-Boson Model at Finite Temperatures
Study at a glance
- Design
- Computational / modelling — Matrix-product-state (1TDVP) wave-function simulations of the Ohmic spin-boson model using the T-TEDOPA mapping of thermal baths onto zero-temperature proxy chains.
- N
- No sample; simulations span a grid of coupling strengths and bath temperatures for single-bath and two-bath models.
- Population
- Model quantum two-level system coupled to bosonic harmonic-oscillator baths
- Outcome
- Spin relaxation and steady-state polarisation, bath mode occupations, entanglement entropy and heat currents between baths
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What they did
The authors simulated a quantum two-level system (a spin) strongly coupled to one or two baths of harmonic oscillators at set temperatures. They used the T-TEDOPA method, which maps a thermal bath onto an extended zero-temperature bath with positive and negative frequencies, and then onto a chain that can be evolved as a matrix product state. They compared spin relaxation, steady states and heat currents against analytical rate equations with polaron renormalisation of the energy gap.
What they found
The final spin polarisation depended on coupling strength, not just temperature, because strong coupling shrinks the effective energy gap (polaron effect); analytical predictions matched once the coupling was rescaled by a constant factor of 0.66. With two baths, the gap renormalisation was non-additive: two equal baths do not simply multiply their individual effects. A net heat current flowed from hot to cold bath in steady state, roughly linear (Fourier-like) for small temperature differences and saturating for large ones.
The limits
What it doesn't show
This is a model calculation with an idealised Ohmic spectral density and hard cut-off, not an experiment on a real device. The number of bath excitations and the entanglement in the simulation grow without bound over time, so long-time steady-state results need care and computing costs rise. The negative-frequency modes are non-physical bookkeeping, so their dynamics describe the simulation rather than the real environment. Agreement with theory relied on an empirically chosen rescaling of the coupling.
Key terms
- Spin-boson model
- A two-level quantum system coupled linearly to a bath of harmonic oscillators; a standard model of dissipation and decoherence.
- Matrix product state
- A compressed way of writing a many-body wave function as a chain of matrices, efficient when entanglement along the chain is limited.
- T-TEDOPA
- A mapping that represents a finite-temperature bath as an extended zero-temperature bath whose couplings encode the temperature, so no thermal sampling is needed.
- Polaron renormalisation
- Strong coupling dresses the system with bath distortions, reducing its effective energy gap.
- Non-additivity
- When the combined effect of two environments is not the simple sum or product of their separate effects.
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Quiz yourself
What does the T-TEDOPA method let the authors avoid?
Common questions
Why is simulating a warm bath normally so hard?
A thermal bath is a statistical mixture of an enormous number of configurations, and averaging many expensive wave-function simulations over them is impractical; T-TEDOPA replaces this with one pure-state simulation.
Why does the steady-state spin polarisation depend on coupling strength?
Strong coupling shrinks the effective energy gap of the spin, so the thermal populations are set by the smaller renormalised gap rather than the bare one.
Does heat flow obey Fourier's law here?
Only approximately, for small temperature differences; for large differences the current through the single spin saturates.
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