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arXiv:0809.2338 · Journal of Physics Conference Series

Purity sieve for models with factorizable interactions

Oleg Valentinovich Lychkovskiy

Published Jun 1, 2009• 2 citations
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Abstract

Evolution of purity in case of factorizable interaction between an open system and an environment is investigated. We derive a simple expression for purity decrease at the first instants of evolution (when purity is close to unity), which appears to be quadratic in time. According to this expression, purity at very small times is maximal when the initial state of an open system coincide with one of the eigenstates of the interaction operator, no matter how strong the interaction is. However, it is widely known that eigenstates of the interaction Hamiltonian are not pointer states when, for example, interaction is weak (compared to self-Hamiltonian). Therefore the procedure of selecting pointer states by purity maximization (known as "purity sieve") should not rely on short-time purity behavior. We propose a modification of the purity sieve criterion which takes into account purity evolution at longer times. As an example of its applicability we recover known results for pointer states of a particle undergoing quantum Brownian motion; we point out that the criterion is not applicable for some other models, however. It is argued that the proposed modified purity sieve may be used for selecting pointer states of a particle undergoing decoherence through colliions.

DOI
10.1088/1742-6596/174/1/012030

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Evolution of purity in case of factorizable interaction between an open system and an environment is investigated. We derive a simple expression for purity decrease at the first instants of evolution (when purity is close to unity), which appears to be quadratic in time. According to this expression, purity at very small times is maximal when the initial state of an open system coincide with one of the eigenstates of the interaction operator, no matter how strong the interaction is. However, it is widely known that eigenstates of the interaction Hamiltonian are not pointer states when, for example, interaction is weak (compared to self-Hamiltonian). Therefore the procedure of selecting pointer states by purity maximization (known as "purity sieve") should not rely on short-time purity behavior. We propose a modification of the purity sieve criterion which takes into account purity evolution at longer times. As an example of its applicability we recover known results for pointer states of a particle undergoing quantum Brownian motion; we point out that the criterion is not applicable for some other models, however. It is argued that the proposed modified purity sieve may be used for selecting pointer states of a particle undergoing decoherence through colliions.
We study various measures of classicality of the states of open quantum systems subject to decoherence. Classical states are expected to be stable in spite of decoherence, and are thought to leave conspicuous imprints on the environment. Here these expected features of environment-induced superselection are quantified using four different criteria: predictability sieve (which selects states that produce least entropy), purification time (which looks for states that are the easiest to find out from the imprint they leave on the environment), efficiency threshold (which finds states that can be deduced from measurements on a smallest fraction of the environment), and purity loss time (that looks for states for which it takes the longest to lose a set fraction of their initial purity). We show that when pointer states---the most predictable states of an open quantum system selected by the predictability sieve---are well defined, all four criteria agree that they are indeed the most classical states. We illustrate this with two examples: an underdamped harmonic oscillator, for which coherent states are unanimously chosen by all criteria, and a free particle undergoing quantum Brownian motion, for which most criteria select almost identical Gaussian states (although, in this case, the predictability sieve does not select well defined pointer states).
We study various measures of classicality of the states of open quantum systems subject to decoherence and continuous quantum measurement. Classical states are expected to be stable in spite of decoherence, and are thought to leave conspicuous imprints on the environment. Here these expected features of environment-induced superselection (einselection) are quantified using four different criteria: predictability sieve (which selects states that produce least entropy), purification time (which looks for states that are the easiest to find out from the imprint they leave on the environment), efficiency threshold (which finds states that can be deduced from measurements on a smallest fraction of the environment), and purity loss time (that looks for states for which it takes the longest to lose a set fraction of their initial purity). We show that when pointer states, the most predictable states of an open quantum system selected by the predictability sieve, are well defined, all four criteria agree that they are indeed the most classical states. We illustrate this with two examples: an underdamped harmonic oscillator, for which coherent states are unanimously chosen by all criteria, and a free particle undergoing quantum Brownian motion, for which most criteria select almost identical Gaussian states (although, in this case, predictability sieve does not select well defined pointer states.)
We investigate the purity dynamics of a quantum system coupled to a continuous-variable pointer through a von Neumann-type interaction Hamiltonian of the form H^int=g H^⊗p^\hat H_{\mathrm{int}}=g\,\hat H\otimes\hat pH^int​=gH^⊗p^​. For an initially Gaussian pointer state, the interaction generates energy-dependent conditional translations whose mutual overlaps are determined explicitly by the populated spectral separations of the system Hamiltonian. After tracing out the pointer degrees of freedom, we obtain the reduced density operator and derive an exact analytical expression for the time-dependent purity. Using this expression, we establish two-sided purity bounds governed by the minimum and maximum nonzero energy gaps on the populated spectral support. These bounds provide a state-dependent spectral characterization of the loss of purity and become exact for two-level systems. We further show that the short-time decrease of purity is controlled by the Hamiltonian variance of the initial state, with P′′(0)=−(g2/σ2)Var⁡ψS(H^)\mathcal P''(0)=-(g^2/σ^2) \operatorname{Var}_{ψ_S}(\hat H)P′′(0)=−(g2/σ2)VarψS​​(H^). In addition, an explicit sufficient timescale is derived for the purity to approach its asymptotic value within a prescribed tolerance, revealing the scaling tε∝σ/(∣g∣Δmin⁡)t_{\varepsilon}\proptoσ/(|g|Δ_{\min})tε​∝σ/(∣g∣Δmin​). Finally, the general results are illustrated for an equally weighted NNN-level system with an equally spaced spectrum, for which the asymptotic purity is 1/N1/N1/N. The analysis clarifies the distinct roles of spectral separation, energy variance, coupling strength, and pointer width in Hamiltonian-conditioned purity loss.
We investigate the time evolution of the reduced density matrix (RDM) and its purity in the dynamics of a two-level system coupled to a dissipative harmonic bath, when the system is initially placed in one of its eigenstates. We point out that the symmetry of the initial condition confines the motion of the RDM elements to a one-dimensional subspace and show that the purity always goes through its maximally mixed value at some time during relaxation, but subsequently recovers and (under low-temperature, weakly dissipative conditions) can rise to values that approach unity. These behaviors are quantified through accurate path integral calculations. Under low-temperature, weakly dissipative conditions, we observe unusual, nonmonotonic population dynamics when the two-level system is initially placed in its ground state. We also analyze the origin of the system-bath interactions responsible for the nonmonotonic behavior of purity during relaxation. Our results show that classical dephasing processes arising from site level fluctuations lead to a monotonic decay of purity, and that the quantum mechanical decoherence events associated with spontaneous phonon emission are responsible for the subsequent recovery of purity. Last, we show that coupling with a low-temperature bath can purify a mixed two-level system. In the case of the maximally mixed initial RDM, the purity increases monotonically even during short time.