[Paper Review] Quantum Mechanics with Complex Time : A Comment to the Paper by Rajeev

Kazuyuki Fujii|arXiv (Cornell University)|Feb 15, 2007

Quantum Mechanics and Applications3 references3 citations

TL;DR

This paper proposes a reformulation of quantum mechanics with a complex-valued Hamiltonian for damped harmonic oscillators, showing that such systems can be equivalently described using standard quantum mechanics with complex time. By introducing a complex time variable τ = (1 − iγ/ω₁)t, the dissipative dynamics are mapped to a unitary time evolution under a real Hamiltonian, preserving physical consistency while enabling standard quantization techniques.

ABSTRACT

In (quant-ph/0701141) Rajeev studied quantization of the damped simple harmonic oscillator and introduced a complex-valued Hamiltonian (which is normal). In this note we point out that the quantization is interpreted as a quantum mechanics with {\bf complex time}. We also present a problem on quantization of classical control systems.

Motivation & Objective

• To reinterpret Rajeev's complex Hamiltonian formulation of the damped harmonic oscillator as a quantum theory with complex time.
• To demonstrate that the time evolution of dissipative systems can be recast as unitary evolution in a complex time framework.
• To address the physical consistency of the ground state energy in the presence of damping by enforcing independence from the damping parameter γ.
• To pose a foundational problem in quantizing classical control systems with external classical fields, relevant to quantum computation and decoherence control.

Proposed method

• Transforms the classical damped harmonic oscillator into a linear system using state variables (x, p), leading to a matrix equation dX/dt = AX.
• Diagonalizes the matrix A via a unitary transformation to define complex variables z and z*, which satisfy dz/dt = λ₋z and dz*/dt = λ₊z* with complex eigenvalues.
• Introduces a complex-valued Hamiltonian H̃ = (ω₁ − iγ)zz* that generates the correct classical dynamics via Poisson brackets.
• Applies canonical quantization by mapping z → a† and z* → ħa, yielding a non-Hermitian Hamiltonian H = ħ(ω₁ − iγ)(a†a + ½), corrected to preserve real ground state energy.
• Introduces a complex time τ = (1 − iγ/ω₁)t to map the dissipative dynamics to a standard harmonic oscillator with real Hamiltonian H̃ = ħω₁(a†a + ½) and unitary time evolution.
• Derives time evolution operators e⁻ⁱᵗᴴ and e⁻ⁱᵀᴴ for complex time τ, showing convergence to the ground state in the long-time limit.

Experimental results

Research questions

• RQ1 Can the quantization of a damped harmonic oscillator with a complex Hamiltonian be consistently reinterpreted as a standard quantum theory with complex time?
• RQ2 How does the introduction of complex time preserve unitarity and physical consistency in the presence of damping?
• RQ3 What is the correct treatment of the ground state energy in a damped quantum system to ensure physical consistency in the γ → 0 limit?
• RQ4 How can a classical control field f(t) be incorporated into a quantized dissipative system without quantizing the field itself?
• RQ5 What are the implications of this complex-time formulation for modeling decoherence in quantum computing systems?

Key findings

• The time evolution of the damped oscillator under the complex Hamiltonian leads to a state that asymptotically approaches the ground state |0⟩, with the wavefunction acquiring a phase e⁻ⁱᵗℏω/₂.
• The complex time τ = (1 − iγ/ω₁)t transforms the non-unitary dissipative dynamics into a unitary evolution under a real Hamiltonian H̃ = ħω₁(a†a + ½), restoring standard quantum mechanical structure.
• The energy spectrum is redefined as ħ(ω₁ − iγ)n + ħω/2 for n ≥ 0, ensuring the ground state energy remains real and independent of γ, consistent with the γ = 0 limit.
• The complex time formulation recovers the classical solution x(t) = A e⁻ᵞᵗ sin(ω₁t + θ) through the unitary evolution in τ, confirming consistency with classical damping.
• The time evolution in complex time τ yields |ψ(τ)⟩ = e⁻ⁱᵗℏω/₂ ∑ₙ ψₙ e⁻ⁱᵗℏω₁ₙ |n⟩, showing exponential decay in the amplitude due to the imaginary part of τ.
• The paper identifies a key open problem: how to control a quantized dissipative system driven by a classical external field f(t), relevant to decoherence control in quantum computation.