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Communications in Theoretical Physics

Study on Quantum Entanglement Between Mesoscopic Circuit and Environment at Coherent State

Yinghua Ji

Published Oct 1, 2008• 2 citations
Publisher

Abstract

Taking into account the interaction between electrons and phonons, in the case without-rotating-wave approximation, we study the entangling property between the mesoscopic circuit and environment at coherent state or equilibrium state. The result indicates that, in long time limit t → ∞, the averages of charge and current in the circuit only depend on the average of the system at the initial state when the environment is initially at thermal equilibrium. However, when the environment is initially at coherent state, the average of charge and current in the circuit is determined by the specific coherent state ensemble. Generally speaking, the entanglement between the circuit and environment will lead to the quantum state purity declining of the circuit, then the circuit emerges decoherent phenomenon, and so a mixed state appears. Purity changes are related to the initial quantum state of environment and circuit. With the further evolution of time, coherence will be gradually restored, but cannot return to 1.

DOI
10.1088/0253-6102/50/4/21

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Taking into account the interaction between electrons and phonons, in the case without-rotating-wave approximation, we study the entangling property between the mesoscopic circuit and environment at coherent state or equilibrium state. The result indicates that, in long time limit t → ∞, the averages of charge and current in the circuit only depend on the average of the system at the initial state when the environment is initially at thermal equilibrium. However, when the environment is initially at coherent state, the average of charge and current in the circuit is determined by the specific coherent state ensemble. Generally speaking, the entanglement between the circuit and environment will lead to the quantum state purity declining of the circuit, then the circuit emerges decoherent phenomenon, and so a mixed state appears. Purity changes are related to the initial quantum state of environment and circuit. With the further evolution of time, coherence will be gradually restored, but cannot return to 1.
Taking into account the interaction between electrons and phonons, we study the dynamical behavior of a mesoscopic dissipative circuit for two classes of pure initial states (grand canonical ensemble and number state) of phonon bath modes. In the long time limit t→∞: when the environment is initially at thermal equilibrium, the average values of current and charge in the circuit system will only depend on the average values of the circuit in initial quantum state, and they are not related with the environment; when the environment is initially in number state, the time evolution of the average values of current and charge are just the same as that in thermal equilibrium, which indicates that the average values of current and charge are related to certain set of occupied numbers. The quantum fluctuations of current and charge in the circuit not only depend on the initial state of the system but also depend on the quantum state and temperature of the environment. Generally, the entanglement between the circuit system and environment will lead to the reduction of the purity of the quantum state. The mesoscopic circuit will remain in a pure quantum state only when the environment is at very low temperature and at thermal equilibrium.
Based on the information theory, we present a density matrix to discuss coherent and squeezed states in a mesoscopic LC circuit with time-dependent frequency. With the relevant operators included in the density matrix, a connection between the appearance of coherent and squeezed states is established, i.e., the quantum state evolution of the system is closely related to the initial state. Generally speaking, due to the effect of environment temperature, the mesoscopic LC circuit will evolve to a squeezed state when it initially lies in an excited state. In particular, at a low temperature, step changes of circuit parameters will result in a squeezed minimum uncertainty state if the resonance frequency remains the same after the change.
Starting from the classical movement equation of a non-dissipative inductance coupled mesoscopic circuit,the circuit is quantized by using linear transformation method,and the quantum fluctuations of currents and charges at excited coherent states are calculated.The results show that the average and square average of current and charge of each loop are not zero without power,i.e.,there exist quantum fluctuations. And the fluctuations are not only determined by the loop parameters,but also related to another loop's parameters and coupled inductance parameters.The quantum fluctuations in two circuits are relevant,and obviously depend on the state parameters of circuits,i.e.,parameters of particle count states and coherent states.
The coupling of a mesoscopic system with its environment usually causes total decoherence: at long times the reduced density matrix of the system evolves in time to a limit which is independent of its initial value, losing all the quantum information stored in its initial state. Under special circumstances, a subspace of the system's Hilbert space remains coherent, or "decoherence free", and the reduced density matrix approaches a non-trivial limit which contains information on its initial quantum state, despite the coupling to the environment. This situation is called "partial decoherence". Here we find the conditions for partial decoherence for a mesoscopic system (with NNN quantum states) which is coupled to an environment. When the Hamiltonian of the system commutes with the total Hamiltonian, one has "adiabatic decoherence", which yields N-1 time-independent combinations of the reduced density matrix elements. In the presence of a magnetic flux, one can measure circulating currents around loops in the system even at long times, and use them to retrieve information on the initial state. For N=2, we demonstrate that partial decoherence can happen only under adiabatic decoherence conditions. However, for N>2N>2N>2 we find partial decoherence even when the Hamiltonian of the system does not commute with the total Hamiltonian, and we obtain the general conditions for such non-adiabatic partial decoherence. For an electron moving on a ring, with N>2N>2N>2 single-level quantum dots, non-adiabatic partial decoherence can arise only when the total flux through the ring vanishes (or equals an integer number of flux quanta), and therefore there is no asymptotic circulating current.
Taking into consideration the interactions between electrons and phonons,we have studied the temporal evolution of the average charge and current in a dissipative mesoscopic RLC circuit.Our results show that a mesoscopic RLC circuit can be treated as an interactive system between an electromagnetic harmonic oscillator and many lattice harmonic oscillators;this is called the bathing of the harmonic oscillators.The results also show that the quantum equation of motion of the linear mesoscopic RLC circuit is identical in form to its classical equation of motion,the only difference between them being their respective meanings.In order to thoroughly study the quantum properties of a dissipative mesoscopic circuit,we have to consider not only the electromagnetic energy of the circuit,but also the crystal lattice vibration energy and the interactive energy between electrons and phonons.
Taking into consideration the interactions between electrons and phonons, we have studied the temporal evolution of the average charge and current in a dissipative mesoscopic RLC circuit. Our results show that a mesoscopic RLC circuit can be treated as an interactive system between an electromagnetic harmonic oscillator and many lattice harmonic oscillators; this is called the bathing of the harmonic oscillators. The results also show that the quantum equation of motion of the linear mesoscopic RLC circuit is identical in form to its classical equation of motion, the only difference between them being their respective meanings. In order to thoroughly study the quantum properties of a dissipative mesoscopic circuit, we have to consider not only the electromagnetic energy of the circuit, but also the crystal lattice vibration energy and the interactive energy between electrons and phonons.