[Paper Review] Quantum fluctuation theorem for dissipative cyclotron motion

G. S. Agarwal, Sushanta Dattagupta|arXiv (Cornell University)|Jan 21, 2016

Quantum and electron transport phenomena3 citations

TL;DR

This paper establishes a quantum fluctuation theorem for dissipative cyclotron motion of a charged particle in a magnetic field, using a quantum Langevin equation derived from a strongly coupled system-bath Hamiltonian. It shows the quantum work operator follows a Gaussian distribution and derives a temperature- and frequency-dependent parameter 𝛼 in the fluctuation theorem 𝑝(𝑊)/𝑝(−𝑊) = exp(𝛼𝑊), revealing unique low-temperature quantum signatures such as zero-point energy dominance at T=0K.

ABSTRACT

We examine the fluctuation theorems which traditionally have been studied for classical systems and enquire if they can be extended to the quantum domain, especially at low temperatures. The example chosen is that of a problem which has proven to be of great interest in the context of Landau diamagnetism viz. the quantized motion of an electron in a magnetic field and in a dissipative environment. It is established from first principles that the quantum work operator W has a Gaussian distribution even though the system under consideration has a four dimensional phase space. The parameter $α$ in the fluctuation theorem $p(W )/p(-W) = exp(αW)$ depends on the system dynamics and has characteristic quantum features, especially at low temperatures. Certain unique low temperature signatures are also discussed.

Motivation & Objective

• To extend classical fluctuation theorems to quantum systems with non-equilibrium dynamics.
• To analyze quantum features of work statistics in a dissipative, two-dimensional cyclotron system under time-dependent driving.
• To derive and examine the quantum fluctuation theorem for a system with four-dimensional phase space and non-Markovian noise.
• To identify unique low-temperature quantum signatures in the work distribution and fluctuation parameter 𝛼.
• To establish the validity of the fluctuation theorem in the quantum regime using exact solutions of the quantum Langevin equation.

Proposed method

• Derive a quantum Langevin equation from a strongly coupled system-bath Hamiltonian with Ohmic dissipation.
• Use the Schrödinger picture with a density operator formalism to describe the system's non-equilibrium dynamics.
• Solve the quantum Langevin equation exactly for cyclotron motion under a time-dependent electric field.
• Compute the average work ⟨𝑊⟩ and work variance 𝜎² using the system's response to external driving.
• Apply the fluctuation theorem relation 𝑝(𝑊)/𝑝(−𝑊) = exp(𝛼𝑊) and determine the parameter 𝛼 from the work distribution.
• Analyze the short-time and low-temperature limits to extract quantum signatures, including the role of zero-point energy.

Experimental results

Research questions

• RQ1 How does the quantum fluctuation theorem 𝑝(𝑊)/𝑝(−𝑊) = exp(𝛼𝑊) manifest in a dissipative, two-dimensional quantum cyclotron system with four-dimensional phase space?
• RQ2 What is the form of the parameter 𝛼 in the quantum fluctuation theorem, and how does it depend on temperature and cyclotron frequency?
• RQ3 What unique quantum signatures emerge in the work distribution at low temperatures, particularly at T=0K?
• RQ4 How do non-Markovian, colored noise effects influence the work statistics in the short-time limit?
• RQ5 To what extent does the zero-point energy ℏ𝜔𝑐 replace thermal energy 2𝑘𝐵𝑇 in the fluctuation theorem at zero temperature?

Key findings

• The quantum work operator 𝑊 has a Gaussian distribution even in a four-dimensional phase space, confirming the validity of the fluctuation theorem in this quantum context.
• The parameter 𝛼 in the fluctuation theorem is given by 𝛼 = tanh(𝛽ℏ𝜔𝑐/2)/(𝛽ℏ𝜔𝑐/2), which depends on the cyclotron frequency and temperature, showing non-classical behavior.
• At zero temperature, the fluctuation parameter satisfies ⟨𝑊²⟩ = ℏ𝜔𝑐⟨𝑊⟩, indicating that zero-point energy replaces thermal energy in the work statistics.
• In the short-time limit, the work variance 𝜎² ∝ 𝑓₀²𝑡⁴, and in the low-temperature limit, 𝜎² → (ℏ𝛾/16𝑚𝜋)𝑓₀²𝑡⁴𝜔𝐷, showing strong quantum effects.
• The high-temperature limit recovers the classical result, while the 𝛺 = 𝜔𝑐 case at T=0K reveals a distinct quantum regime where ℏ𝜔𝑐 dominates over 𝑘𝐵𝑇.
• The fluctuation theorem remains valid in the non-Markovian regime due to colored noise, with significant quantum effects observable on timescales comparable to ℏ/𝑘𝐵𝑇.