\n \n

\n Temporal Flow from\n \n Energy–Entanglement\n \n Coupling\n

\n Discovering the quantum origins of time through continuous-variable clock–spin models\n

\n \n \n

\n \n

\n\n \n

Abstract

\n The origin of physical time remains an open question at the foundations of quantum mechanics and quantum gravity. We demonstrate that a stationary, energy-neutral global state can exhibit an unambiguous, observer-dependent flow of time once energy fluctuations are coupled to bipartite entanglement.\n

\n Our \"thermofield-clock\" (TFC) state maximally entangles a harmonic-oscillator clock with a finite-dimensional spin register, creating a unique pathway to understanding temporal emergence.\n

Thermofield-Clock State

\n $$|\\\\Psi_{\\\\text{TFC}}\\\\rangle = Z^{-1/2} \\\\sum_{n=0}^{\\\\infty} e^{-\\\\beta\\\\hbar\\\\omega n/2} |n\\\\rangle_C \\\\otimes |n\\\\rangle_S$$\n

\n Where $Z = (1 - e^{-\\\\beta\\\\hbar\\\\omega})^{-1}$ is the partition function\n

\n\n \n

Key Innovations

∇

Temporal Information Flux

\n A scalar functional linking energy variance to clock entropy, strictly positive under generic laboratory conditions.\n

\n $$\\\\Phi_T(\\\\rho) = \\\\|\\\\nabla_\\\\rho[\\\\sigma_E^2(\\\\rho) + \\\\lambda S_C(\\\\rho)]\\\\|^2$$\n

\n \n

⟲

Modular Flow

\n Tomita–Takesaki theory promotes TIF into a unique automorphism satisfying the KMS condition.\n

\n $$\\\\alpha_\\\\tau(A) = e^{i\\\\tau(K_C + \\\\lambda\\\\hat{H})} A e^{-i\\\\tau(K_C + \\\\lambda\\\\hat{H})}$$\n

\n \n

⚡

Experimental Platform

\n All ingredients available in state-of-the-art circuit-QED devices, providing concrete testing routes.\n

\n 3-D niobium cavity ($\\\\omega \\\\approx 6$ GHz, $Q \\\\approx 10^8$) + transmon qubits\n

\n\n \n

Theoretical Framework

\n \n

Energy-Neutral Hamiltonian

\n We construct a system where the global state remains stationary ($\\\\hat{H}|\\\\Psi_{\\\\text{TFC}}\\\\rangle = 0$), yet internal observers experience temporal flow through energy-entanglement coupling.\n

\n The harmonic oscillator frequency $\\\\omega$ is precisely balanced against qubit splittings $\\\\epsilon_k = \\\\hbar\\\\omega/2$.\n

\n $$\\\\hat{H} = \\\\hbar\\\\omega(a^\\\\dagger a + \\\\frac{1}{2}) - \\\\sum_{k=1}^L \\\\epsilon_k \\\\hat{\\\\sigma}_z^k$$\n

\n Energy-neutral condition: $\\\\hat{H}|\\\\Psi_{\\\\text{TFC}}\\\\rangle = 0$\n

\n \n

Arrow-of-Time Bound

\n $$\\\\frac{dS_C}{d\\\\tau} \\\\geq 2\\\\lambda\\\\sigma_E^2|\\\\tau|$$\n

\n Quantitative link between entropy production and energy fluctuations\n

Emergent Irreversibility

\n Through log-Sobolev inequalities for modular flows, we derive a rigorous lower bound that ties the arrow-of-time to energy fluctuations.\n

\n This constraint determines how slowly internal time can run and is saturated only when $\\\\Phi_T \\\\to 0$.\n

\n \n

Recovery of Schrödinger Dynamics

\n Clock phase states $|\\\\theta\\\\rangle_C = \\\\sum_n e^{in\\\\theta}|n\\\\rangle_C$ furnish an over-complete basis with $\\\\langle\\\\theta|\\\\hat{H}_C|\\\\theta\\\\rangle = \\\\hbar\\\\omega \\\\partial_\\\\theta$.\n

\n Conditioning on $\\\\theta$ produces standard quantum evolution where $t = \\\\theta/\\\\omega$.\n

\n $$\\\\rho_S(\\\\theta) = e^{-i\\\\hat{H}_S\\\\theta/\\\\omega} \\\\rho_S(0) e^{i\\\\hat{H}_S\\\\theta/\\\\omega}$$\n

\n Yields: $i\\\\hbar \\\\partial_t \\\\rho_S(t) = [\\\\hat{H}_S, \\\\rho_S(t)]$\n

\n\n \n

Experimental Implementation

\n \n

Circuit-QED Platform

\n •\n 3-D niobium cavity: $\\\\omega \\\\approx 6$ GHz, $Q \\\\approx 10^8$ serves as harmonic oscillator\n
\n •\n Three transmon qubits: Provide finite-dimensional spin register\n
\n •\n Two-mode squeezing: $r \\\\approx 0.5$ prepares $|\\\\Psi_{\\\\text{TFC}}\\\\rangle$ in 400 ns\n
\n •\n Fast homodyne detection: Selects clock phases for conditioning\n

Measurement Protocol

\n State Preparation:\n 400 ns\n

\n Squeezing Parameter:\n $r = 0.5$\n

\n Dephasing Error:\n < 5%\n

\n Coherence Time:\n ~100 μs\n

\n \n

Energy Variance & Clock Entropy

\n $$\\\\sigma_E^2 = (\\\\hbar\\\\omega)^2 \\\\frac{e^{-\\\\beta\\\\hbar\\\\omega}}{(1-e^{-\\\\beta\\\\hbar\\\\omega})^2}$$\n

\n $$S_C = \\\\frac{\\\\beta\\\\hbar\\\\omega e^{-\\\\beta\\\\hbar\\\\omega}}{1-e^{-\\\\beta\\\\hbar\\\\omega}} - \\\\ln(1-e^{-\\\\beta\\\\hbar\\\\omega})$$\n

\n \n

\n All required coherence and control capabilities are already available in current circuit-QED setups, making this an immediately testable theoretical framework.\n

\n \n

\n\n \n

Broader Implications

\n \n

Foundational Physics

\n →\n Unifies Page-Wootters conditional dynamics with thermal-time hypothesis\n
\n →\n Energy variance becomes the progenitor of dynamics, not just a symptom\n
\n →\n Provides experimental handle on irreversibility in quantum systems\n

\n \n

Future Directions

\n →\n Finite-dimensional clock variants for continuous time studies\n
\n →\n Relativistic extensions on quantum networks\n
\n →\n Interplay with quantum thermodynamic resource theories\n

\n \n

Key Insight

\n \"Time is not fundamental but emerges from the interplay between energy fluctuations and quantum entanglement. Our framework provides the first experimentally accessible platform to test this profound idea.\"\n

\n\n \n \n\n \n\n`;\n\n return new Response(html, {\n headers: {\n 'Content-Type': 'text/html',\n },\n });\n}], ['\\\\\\\$', '\\\\\\\$']],\n displayMath: [['$', '$'], ['\\\\\\\\[', '\\\\\\\\]']],\n macros: {\n bra: [\"\\\\langle{#1}|\", 1],\n ket: [\"|{#1}\\\\rangle\", 1],\n braket: [\"\\\\langle{#1}|{#2}\\\\rangle\", 2],\n Tr: \"\\\\operatorname{Tr}\",\n TFC: \"\\\\text{TFC}\",\n KMS: \"\\\\text{KMS}\"\n }\n },\n chtml: {\n scale: 1.1\n }\n };\n \n \n\n\n \n \n\n \n

\n \n

\n Temporal Flow from\n \n Energy–Entanglement\n \n Coupling\n

\n Discovering the quantum origins of time through continuous-variable clock–spin models\n

\n \n \n

\n \n

\n\n \n

Abstract

\n Our \"thermofield-clock\" (TFC) state maximally entangles a harmonic-oscillator clock with a finite-dimensional spin register, creating a unique pathway to understanding temporal emergence.\n

Thermofield-Clock State

\n $$|\\\\Psi_{\\\\text{TFC}}\\\\rangle = Z^{-1/2} \\\\sum_{n=0}^{\\\\infty} e^{-\\\\beta\\\\hbar\\\\omega n/2} |n\\\\rangle_C \\\\otimes |n\\\\rangle_S$$\n

\n Where $Z = (1 - e^{-\\\\beta\\\\hbar\\\\omega})^{-1}$ is the partition function\n

\n\n \n

Key Innovations

∇

Temporal Information Flux

\n A scalar functional linking energy variance to clock entropy, strictly positive under generic laboratory conditions.\n

\n $$\\\\Phi_T(\\\\rho) = \\\\|\\\\nabla_\\\\rho[\\\\sigma_E^2(\\\\rho) + \\\\lambda S_C(\\\\rho)]\\\\|^2$$\n

\n \n

⟲

Modular Flow

\n Tomita–Takesaki theory promotes TIF into a unique automorphism satisfying the KMS condition.\n

\n $$\\\\alpha_\\\\tau(A) = e^{i\\\\tau(K_C + \\\\lambda\\\\hat{H})} A e^{-i\\\\tau(K_C + \\\\lambda\\\\hat{H})}$$\n

\n \n

⚡

Experimental Platform

\n All ingredients available in state-of-the-art circuit-QED devices, providing concrete testing routes.\n

\n 3-D niobium cavity ($\\\\omega \\\\approx 6$ GHz, $Q \\\\approx 10^8$) + transmon qubits\n

\n\n \n

Theoretical Framework

\n \n

Energy-Neutral Hamiltonian

\n The harmonic oscillator frequency $\\\\omega$ is precisely balanced against qubit splittings $\\\\epsilon_k = \\\\hbar\\\\omega/2$.\n

\n $$\\\\hat{H} = \\\\hbar\\\\omega(a^\\\\dagger a + \\\\frac{1}{2}) - \\\\sum_{k=1}^L \\\\epsilon_k \\\\hat{\\\\sigma}_z^k$$\n

\n Energy-neutral condition: $\\\\hat{H}|\\\\Psi_{\\\\text{TFC}}\\\\rangle = 0$\n

\n \n

Arrow-of-Time Bound

\n $$\\\\frac{dS_C}{d\\\\tau} \\\\geq 2\\\\lambda\\\\sigma_E^2|\\\\tau|$$\n

\n Quantitative link between entropy production and energy fluctuations\n

Emergent Irreversibility

\n Through log-Sobolev inequalities for modular flows, we derive a rigorous lower bound that ties the arrow-of-time to energy fluctuations.\n

\n This constraint determines how slowly internal time can run and is saturated only when $\\\\Phi_T \\\\to 0$.\n

\n \n

Recovery of Schrödinger Dynamics

\n Conditioning on $\\\\theta$ produces standard quantum evolution where $t = \\\\theta/\\\\omega$.\n

\n $$\\\\rho_S(\\\\theta) = e^{-i\\\\hat{H}_S\\\\theta/\\\\omega} \\\\rho_S(0) e^{i\\\\hat{H}_S\\\\theta/\\\\omega}$$\n

\n Yields: $i\\\\hbar \\\\partial_t \\\\rho_S(t) = [\\\\hat{H}_S, \\\\rho_S(t)]$\n

\n\n \n

Experimental Implementation

\n \n

Circuit-QED Platform

\n •\n 3-D niobium cavity: $\\\\omega \\\\approx 6$ GHz, $Q \\\\approx 10^8$ serves as harmonic oscillator\n
\n •\n Three transmon qubits: Provide finite-dimensional spin register\n
\n •\n Two-mode squeezing: $r \\\\approx 0.5$ prepares $|\\\\Psi_{\\\\text{TFC}}\\\\rangle$ in 400 ns\n
\n •\n Fast homodyne detection: Selects clock phases for conditioning\n

Measurement Protocol

\n State Preparation:\n 400 ns\n

\n Squeezing Parameter:\n $r = 0.5$\n

\n Dephasing Error:\n < 5%\n

\n Coherence Time:\n ~100 μs\n

\n \n

Energy Variance & Clock Entropy

\n $$\\\\sigma_E^2 = (\\\\hbar\\\\omega)^2 \\\\frac{e^{-\\\\beta\\\\hbar\\\\omega}}{(1-e^{-\\\\beta\\\\hbar\\\\omega})^2}$$\n

\n $$S_C = \\\\frac{\\\\beta\\\\hbar\\\\omega e^{-\\\\beta\\\\hbar\\\\omega}}{1-e^{-\\\\beta\\\\hbar\\\\omega}} - \\\\ln(1-e^{-\\\\beta\\\\hbar\\\\omega})$$\n

\n \n

\n All required coherence and control capabilities are already available in current circuit-QED setups, making this an immediately testable theoretical framework.\n

\n \n

\n\n \n

Broader Implications

\n \n

Foundational Physics

\n →\n Unifies Page-Wootters conditional dynamics with thermal-time hypothesis\n
\n →\n Energy variance becomes the progenitor of dynamics, not just a symptom\n
\n →\n Provides experimental handle on irreversibility in quantum systems\n

\n \n

Future Directions

\n →\n Finite-dimensional clock variants for continuous time studies\n
\n →\n Relativistic extensions on quantum networks\n
\n →\n Interplay with quantum thermodynamic resource theories\n

\n \n

Key Insight

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