Trade-off between coherence and heat in a non-Markovian dephasing dynamics

1. Problem Statement

The paper addresses a fundamental open question in quantum thermodynamics: How does quantum coherence influence thermodynamic behavior, specifically heat dissipation, in the absence of energy exchange?

• Context: In standard open quantum systems, decoherence often accompanies thermalization via energy exchange. However, in pure dephasing dynamics, the system's internal energy is conserved (the system Hamiltonian commutes with the interaction), and thermalization is inhibited.
• The Paradox: While the system's energy remains constant, recent theories suggest that non-trivial heat dissipation can still occur due to the redistribution of energy involving the interaction term.
• The Gap: It remains unclear how this "trivial" energy-conserving process relates to the loss of quantum coherence and whether a direct thermodynamic signature of decoherence exists, particularly in finite, non-Markovian environments where information backflow occurs.

2. Methodology

The authors employ a combination of analytical derivations and numerical simulations within a specific physical model.

A. Theoretical Framework

• Pure Dephasing Model: They consider an open quantum system ($S$) interacting with a thermal environment ($E$) under the condition $[\hat{H}_S, \hat{H}_I] = 0$. This ensures the system's populations (diagonal elements) remain constant while off-diagonal elements (coherence) decay.
• Two-Point Measurement (TPM) Scheme: Heat and work are defined using the TPM approach, where projective energy measurements are performed at $t=0$ and $t=t_f$.
• Bernardo's Reformulated First Law: The authors utilize a modified first law of quantum thermodynamics proposed by Bernardo, which decomposes the change in internal energy into work ($d\bar{W}$), heat ($d\bar{Q}$), and coherent energy ($d\bar{C}$).
• Non-Markovianity Quantification: They use the Breuer-Laine-Piilo (BLP) measure, based on the trace distance, to quantify information backflow from the environment to the system.

B. Physical Model

• System: A central qubit initially in a superposition state ($|0\rangle + |1\rangle$).
• Environment: A finite Ising-like spin chain of $N$ spin-1/2 particles arranged in a ring.
• Hamiltonian:
  • Environment: $\hat{H}_E = -J_z \sum \sigma_j^z \sigma_{j+1}^z - h_z \sum \sigma_j^z$.
  • Interaction: $\hat{H}_I = |0\rangle\langle 0| \otimes g_0 \sum \sigma_j^x + |1\rangle\langle 1| \otimes g_1 \sum \sigma_j^y$.
• Regime: Field-dominated weak coupling ($|h_z| \gg |J_z|, g$), ensuring the spin chain remains nearly polarized and thermal excitations are suppressed.

3. Key Contributions

A. Analytical Equivalence: Heat = Coherent Energy

The authors prove analytically that in the pure dephasing regime:

1. The mean heat dissipated into the environment ($\langle Q \rangle$), defined via the TPM scheme, is exactly equal to the coherent energy contribution ($C(t)$) appearing in Bernardo's reformulated first law.
2. Since the system's internal energy is constant and no classical work is done on the environment (time-independent Hamiltonian), the entire change in the environment's energy is driven by the quantum coherence dynamics.
3. Conclusion: Heat dissipation in pure dephasing is not an energetic byproduct of population relaxation but is a direct thermodynamic signature of the loss of quantum coherence.

B. Numerical Demonstration of Trade-off

Using the Ising spin chain model, the authors numerically demonstrate a temporal trade-off between coherence and heat:

• Coherence Decay: When the $l_1$-norm of coherence ($|\Gamma(t)|$) decreases (decoherence), the mean heat dissipated ($\langle Q \rangle$) increases.
• Coherence Revival: When coherence is restored (revivals), the mean heat decreases.
• Mechanism: This oscillatory behavior is driven by the finite size of the environment, which causes information to flow back from the environment to the system (non-Markovianity).

4. Key Results

1. Non-Trivial Heat in Energy-Conserving Systems: Even though the total energy of the composite system is conserved and the system's internal energy is constant, non-zero heat is dissipated into the environment. This heat arises solely from the interaction term's energy redistribution.
2. Oscillatory Dynamics: In finite environments, coherence does not decay monotonically to zero. Instead, it exhibits oscillations with revivals (returning to unity). These revivals correspond to information backflow.
3. Inverse Correlation: The numerical results (Fig. 5) show a precise inverse correlation:
   • Maxima of $\langle Q \rangle$ coincide with minima of $|\Gamma(t)|$ (maximum decoherence).
   • Minima of $\langle Q \rangle$ coincide with maxima of $|\Gamma(t)|$ (coherence revival).
4. Non-Markovian Signature: The periods where the trace distance increases (indicating information backflow) align perfectly with coherence revivals and heat reduction. This confirms that the reversibility of the unitary dynamics allows the environment to "return" the energy associated with the lost coherence.

5. Significance and Implications

• Thermodynamic Signature of Decoherence: The paper establishes that heat dissipation is the thermodynamic cost of decoherence in pure dephasing. Even without energy exchange between system levels, the process of losing quantum information to the environment generates heat.
• Revisiting the First Law: It validates the necessity of the "coherent energy" term in the first law of quantum thermodynamics. Without this term, the energy balance in pure dephasing would appear paradoxical (zero work, zero classical heat, yet non-zero energy change in the environment).
• Finite vs. Infinite Environments: The study highlights the crucial role of finite environments. In infinite (Markovian) baths, coherence decays irreversibly, and heat would likely accumulate monotonically. In finite (non-Markovian) baths, the reversibility of the dynamics leads to a dynamic exchange where heat and coherence oscillate.
• Future Directions: The work suggests new avenues for investigating entropy production in spin-echo protocols and the energetic costs of dynamically controlled coupling/decoupling in quantum thermodynamic cycles.

In summary, the paper bridges quantum information theory and thermodynamics by proving that coherence loss is intrinsically linked to heat generation, even in scenarios where the system's energy is conserved, and demonstrates that this relationship is modulated by non-Markovian information backflow in finite environments.