entropy Article
Comparing the Models of Steepest Entropy Ascent Quantum Thermodynamics, Master Equation and the Difference Equation for a Simple Quantum System Interacting with Reservoirs Charles E. Smith Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24060, USA;
[email protected]; Tel.: +1-540-577-1390 Academic Editors: Umberto Lucia and Giuseppe Grazzini Received: 5 February 2016; Accepted: 4 May 2016; Published: 12 May 2016
Abstract: There is increasing interest concerning the details about how quantum systems interact with their surroundings. A number of methodologies have been used to describe these interactions, including Master Equations (ME) based on a system-plus-reservoir (S + R) approach, and more recently, Steepest Entropy Ascent Quantum Thermodynamics (SEAQT) which asserts that entropy is a fundamental physical property and that isolated quantum systems that are not at stable equilibrium may spontaneously relax without environmental influences. In this paper, the ME, SEAQT approaches, and a simple linear difference equation (DE) model are compared with each other and experimental data in order to study the behavior of a single trapped ion as it interacts with one or more external heat reservoirs. The comparisons of the models present opportunities for additional study to verify the validity and limitations of these approaches. Keywords: quantum thermodynamics; quantum relaxation; quantum entropy; SEAQT
1. Introduction There is ongoing interest in modeling how quantum systems interact with their environment. The understanding of the relationship between relaxation processes, entropy and quantum mechanics is increasingly considered to be of fundamental importance [1,2]. External and internal influences directly affect many, if not all quantum phenomena, such as quantum computing [3], quantum chemistry [4,5], and entanglement. Methodologies that have been developed to describe quantum dissipative systems are extensive and include stochastic quantum mechanics [6] and nonlinear versions of the Schrödinger Equation [7,8]. Much of the theoretical work describing quantum interactions with the environment has been done by using system-plus-reservoir approaches (S + R), i.e., Master Equation (ME) techniques [6,9,10]. An alternate formalism, Steepest Entropy Ascent Quantum Thermodynamics (SEAQT), developed by Hatsopoulos, Gyftopoulos, and Beretta [11–16], seeks to unify the Second Law of Thermodynamics with Quantum Mechanics. The theory asserts that it is possible for quantum systems as small as one particle to possess entropy, and claims that quantum systems that are not at stable thermodynamic equilibrium may spontaneously relax. SEAQT has been used in previous efforts to model quantum steady state conditions [17]. Single trapped atoms have proved to be an important research tool, and are described extensively in Leibfried et al. [18]. Experimental data relating to the quantum relaxation of single ions include groundbreaking work by Turchette et al. [19] and Gring et al. [20]. The main objective of what follows is to directly compare the ME and SEAQT models as well a simple linear difference equation (DE) to data for interactions of a trapped ion with the environment as presented in [19]. The intent of the comparisons is to reveal similarities, but also differences Entropy 2016, 18, 176; doi:10.3390/e18050176
www.mdpi.com/journal/entropy
Entropy 2016, 18, 176
2 of 16
in predictions that can be used to test the validity of the models against the essential but limited experimental data. It is hoped that the results will encourage discussion and help promote further experimentation that further reveals how entropy interrelates with Quantum Mechanics. 1.1. Model Descriptions For this study, the quantum states for the single trapped ion under study are limited to Fock states where off-diagonal elements are equal to zero. Fock states were chosen because they are simpler and easier to handle computationally for the SEAQT cases. Nevertheless, they represent legitimate quantum states of interest for which experimental data is available. Fock states do not, however, exhibit behavior such as interference effects and quantum entanglement. Important theoretical work using the traditional ME (S + R) approach has been done by [21,22] for trapped ions interacting with a heat bath. The ME model is said to be valid for systems that are near or far from stable equilibrium [21]. The ME model assumes that the system-reservoir coupling is relatively weak so that the system maintains eigenlevels that are unaltered by the reservoir. Though the ME model is based on the reversible Schrödinger Equation, irreversibility is achieved by various approximations [6,9,10]. A straightforward presentation is found in [19] for density matrices with or without non-zero off-diagonal elements. The Master Equation for a single particle interacting with a reservoir is expressed as ˘ γ ` ` ˘ dρ γ “ pn ` 1q 2aˆ ρ aˆ ` ´ aˆ ` aˆ ρ ´ ρ aˆ ` aˆ ` n 2aˆ ` ρ aˆ ´ aˆ aˆ ` ρ ´ ρ aˆ aˆ ` dt 2 2
(1)
where ρ is the state matrix, aˆ is the lowering operator, and γ is the system decay rate or reciprocal time constant. The dynamics of the reservoir are not contained explicitly in Equation (1), other than the quantity n, which represents the average energy of the reservoir as it interacts with the system. The energy is in terms of the number of system quanta }ω0 and is related to the reservoir temperature by: ¸ ˜ e ´} ω 0 { k B T . (2) n“ 1 ´ e ´} ω 0 { k B T The solution to the Master Equation for Fock state evolution is shown in [19,21] and takes the form ρnn ptq “
1 1` N p t q
n ´ ř j “0
N ptq 1` N p t q
¯j ´
e´γt{2 1` N p t q
¯2n´2j ˆ
8 ´ ř j “0
1´
e´γt 1` N p t q
¯j
˜
n`1´j n´j
¸
˜ ˆ
n j
¸ ρn`l ´ j,n`l ´ j p0q (3)
where N ptq is the average number of quanta in the system over time for a system initially in the ground state: ` ˘ N ptq “ n 1 ´ e´γt . (4) The index n stands for the nth energy eigenlevel above the groundstate. The difference equation DE is a simple dynamic relaxation model that can be constructed based on the difference between the current state of the system and the final mutually stable equilibrium state with the reservoir [23,24]. It is understood that the region of greatest accuracy occurs in the “linear” region close to stable equilibrium, which may include steady state behavior. The model has the advantage of being simple enough to readily approximate several interacting reservoirs. The governing equation is, 1 dρnn ptq (5) “ ´ pρnn ptq ´ ρres q dt τ where τ is a time constant. The state of the reservoir is assumed to be constant in time. The SEAQT formalism is based on the idea that entropy is a fundamental property of matter that is present for all systems, even those at quantum scales. The theory asserts that the kinematical framework for entropy as a fundamental quantity is established by the state matrix ρ of quantum
Entropy 2016, 18, 176
3 of 16
mechanics, which is fully able to describe entropy for a quantum system as well as other quantities, such as energy and momentum. The formalism asserts that any system that is not at stable thermodynamic equilibrium, even a system of one particle, is by default unstable and will tend to spontaneously relax toward the equilibrium state [25]. SEAQT postulates that the evolution takes place along a path of steepest entropy increase [26]. The change occurs due to the ability of the internal degrees of freedom; i.e., the energy eigenlevels of the system, to exchange energy with one another [27]. Relaxation may however be inhibited by the presence of local metastable states, such as those robust enough to describe chemistry and the existence of atomic elements. To enable entropy changes to occur, SEAQT adds a nonlinear operator to the time-dependent Schrödinger Equation, which produces an expression analogous to a master equation [24]. The SEAQT evolution equation claims to describe the evolution of all systems whether they are near or far from stable equilibrium. Recently, SEAQT has been extended and applied to quantum chemistry [28], fuel cells [17], and quantum decoherence [29]. The SEAQT formalism also exhibits a time lag effect that is not present with the other two approaches. When a system is near a pure state close to zero entropy or a “partial equilibrium state” [27], the energy is contained in only a few of the available energy eigenlevels. Relaxation proceeds slowly until the energy becomes more evenly distributed. The dynamics of the SEAQT model are given by the equation developed by Beretta [15]: i 1 dρ “ ´ rH, ρs ´ Dpρptqq. dt } τ
(6)
The term in brackets is the time-dependent Schrödinger term, along with D, a nonlinear irreversibility operator. Equation (6) describes the time evolution of the state of an isolated, non-equilibrium quantum system as it relaxes spontaneously to a stable equilibrium end state along a path of steepest entropy increase. If the system is in a Fock state, Equation (6) simplifies to dρ 1 “ ´ D pρptqq . dt τ
(7)
The irreversibility operator for the n-th energy eigenlevel can be represented by a ratio of Gram determinants [25]: ˇ ˇ ˇ ρ lnρ ρn en ρn ˇˇ ˇ řn n ř ˇ ˇ 1 eρˇ ˇ ρi lnρi ř ř 2i i ˇ ˇř ˇ ei ρi lnρi ei ρ i e ρi ˇ ˇ i . ˇ (8) Dn “ ř ˇ 1 ei ρi ˇˇ ˇř ř 2 ˇ ˇ ˇ ei ρ i ei ρ i ˇ 1.2. Equations for Describing System Interactions New expressions were developed herein to describe the interactions between the elements of composite systems when employing the SEAQT and DE approaches. Several cases are illustrated in Figure 1. In Figure 1a, two systems relax spontaneously and independently without mutual interaction. A relation that combines the two separate equations of motion for uncorrelated systems ρ A and ρ B into a single equation for a composite system is shown in [24]. The separate equations for systems A and B are dρ A ´i 1 rH, ρ A s ´ (9) “ D pρ A q dt } τA and
´i 1 dρ B rH, ρ B s ´ D pρ B q . “ dt } τB
(10)
Entropy 2016, 18, 176
4 of 17
d ρ B −i 1 = [H , ρB ] − D ( ρB ) . τB dt
Entropy 2016, 18, 176
4 of 16
(10)
The state matrices of the systems combined usingananouter outerproduct. product. The The equivalent equivalent The state matrices of the twotwo systems cancan be be combined using expression for the composite system becomes expression for the composite system becomes d pρd A( ρbA ρ⊗Bρq B ) ´i−i 11 pρ q b ρ ´1 1 D pρ q b ρ , “ = rH, H ,ρρAA b ⊗ ρρBBs] ´ − DD( ρ ) ⊗ ρ B B− τD ( ρ B )B⊗ ρ A , A AA dt dt } [ ττAA τB B
(11) (11)
form that that is is reminiscent reminiscent of of the the product product rule rule for for differentiation. differentiation. Equation Equation (11) (11) can can be be expressed expressed in in aa form words as “the change with time of the composite of A and B is equal to the Schrödinger term for the words as “the change with time of the composite of A and B is equal to the Schrödinger term for the composite, minus minus the the spontaneous spontaneous relaxation relaxation of of A A without without the the influence influence of of B, B, minus minus the thespontaneous spontaneous composite, relaxation of of BB without without the the influence influence of of A.” A.” The The speeds speeds of of relaxation relaxation for for A A and and BB are are governed governed by by the the relaxation time constants τ and τ . Though expressed as a composite, the two systems remain uncorrelated and time constants τAA and Bτ B . Though expressed as a composite, the two systems remain uncorrelated theirtheir relaxations taketake placeplace independently. and relaxations independently.
Figure Figure 1. 1. Representations Representations of of composite composite system system interactions. interactions. (a) (a)Two Two systems systems evolve evolve spontaneously spontaneously and independently; (b) Two systems evolve spontaneously as they also interact; and independently; (b) Two systems evolve spontaneously as they also interact; (c) (c) Three Three systems systems that interact. Each region represents a separate spontaneous evolution with its own time that interact. Each region represents a separate spontaneous evolution with its own timeconstant. constant. Each or intersection intersectionof ofinterfaces interfacesrepresents representsa mutual a mutual interaction of two or more systems Each interface interface or interaction of two or more systems that that hasown its own constant; (d) Interactions of system with reservoirs B and C, may which may also also has its time time constant; (d) Interactions of system A withAreservoirs B and C, which include include steady state behavior. steady state behavior.
To enable the interaction between systems A and B, an extension is introduced to allow the To enable the interaction between systems A and B, an extension is introduced to allow the composite ρ A ⊗ ρ B to evolve under the influence of the irreversibility operator. The resulting term, composite ρ A b ρ B to evolve under the influence of the irreversibility operator. The resulting term, 11 ´ to to Equation (11),(11), where τAB isτaABseparate time constant the interaction. added Equation where is a separate time for constant for the − τAB D (pρ ρ AA⊗bρρBB)q,, isisadded τ ForABconvenience, D pρ b ρ q is rewritten as D pρ q and can be expressed as “the relaxation of the A
B
AB
interaction. For be expressed “the ( ρ interaction ) and composite of A convenience, and B due to D the of A withasB”.D (Ifρ AB the statecan matrices ρ A and ρas A ⊗ ρ B ) is rewritten B have off-diagonal terms equal to of zero, the Schrödinger can beof dropped. TheIfequation case (b)ρin relaxation of the composite A and B due to the term interaction A with B”. the statefor matrices A Figure 1 is then and ρ B have off-diagonal terms equal to zero, the Schrödinger term can be dropped. The equation d pρ AB q 1 1 1 (12) D pρ AB q . for case (b) in Figure 1 is then“ ´ D pρ A q b ρ B ´ D pρ B q b ρ A ´ dt τA τB τAB
d (B ρ ABis) a reservoir 1 1 at equilibrium, 1 off-diagonal terms are zero), Given that system (already D ( ρ AB ) . = − D (ρ A ) ⊗ ρB − D (ρB ) ⊗ ρA − (12) τA τB τ AB Equation (12) reduces to dt
Given that system B is a reservoir (already terms are zero), Equation (12) q d pρ AB 1at equilibrium, off-diagonal 1 “ ´ D pρ A q b ρ B ´ D pρ AB q . (13) reduces to dt τA τAB d ( ρofABA) is slow, 1 the first term1on the right hand side of the equation can If the spontaneous relaxation D ( ρ AB ) . = − D (ρ A ) ⊗ ρB − (13) dt τA τ AB also be dropped.
If the spontaneous relaxation of A is slow, the first term on the right hand side of the equation can also be dropped.
Entropy 2016, 18, 176
5 of 16
For the case of three systems A, B, C, that interact with one another as shown in case (c), the above formalism for a Fock state is written as: dpρ ABC q “ ´ τ1 D pρ A q b ρ B b ρC ´ τ1B D pρ B q b ρ A b ρC ´ τ1 D pρC q b ρ A b ρ B dt A C ´ τ 1 D pρ AB q b ρC ´ τ 1 D pρ AC q b ρ B ´ τ1 D pρ BC q b ρ A ´ τ 1 D pρ ABC q . AB AC BC ABC
(14)
Each spontaneous relaxation and relaxation due to interaction is distinct and has its own time constant. For case (d), where a small system A in a Fock state interacts with two reservoirs B and C, Equation (14) becomes, d pρ ABC q 1 1 1 “ ´ D pρ A q b ρ B b ρC ´ D pρ AB q b ρC ´ D pρ AC q b ρ B . dt τA τAB τAC
(15)
The two reservoirs interact only with system A and not with each other. Again, the self-relaxation term for system A can be ignored if it is deemed slow enough. The proposed interaction term, ´ τ 1 D pρ AB q is simplistic and may not fully describe all interaction AB scenarios that physically occur. The term represents a composite of two or more uncorrelated systems, where the irreversibility operator has joint and unimpeded influence on the degrees of freedom of each as the evolution takes place. Though simplistic, the interaction term fits into the dynamic framework of Equation (6) as developed by Beretta: The combined system is itself an isolated system that spontaneously evolves to a maximal entropy state along a steepest entropy path. The interaction term therefore represents a valid case within the SEAQT theory. The extended formalism represented by Equations (12)–(15) allows for the separate relaxation of each subsystem to be included as well. 1.3. Additional Modeling Details The model system under investigation is a single trapped ion held in a quadratic potential that is able to move along one dimension, similar to experiments involving a Paul trap [18]. The system behaves as a quantum harmonic oscillator with equally-spaced energy eigenlevels. The reservoir with which the system interacts is modeled as a bath of 3-dimensional quantum oscillators having a degeneracy of pn ` 1q pn ` 2q g“ , (16) 2 where n is the n-th eigenenergy level of the oscillator. Initially, all interacting systems are regarded as independent and uncorrelated. An initial state for each system (i.e., ρp0q) is chosen. The systems are closed, with no mass flow across their boundaries, but they may exchange energy and entropy via heat interactions. For the SEAQT case, the system and reservoir are combined and the evolution for the composite is calculated. Afterward, the evolution of the system of interest is extracted from the composite results using a partial trace operation. 1.4. Treatment of Reservoirs The ME and DE have been formulated to describe the evolution of the system of interest without including explicit details of the reservoir. The system of interest moves to a final stable equilibrium state that has the same temperature as the reservoir. The ME interaction scenarios were determined directly from Equation (3). For DE, Equation (5) was used. A solution for interactions with more than one reservoir using the ME approach has yet to be developed, but is being pursued by [30]. The SEAQT technique was implemented with a reservoir that was explicitly modeled as a bath of a number of harmonic oscillators. The bath remains in a stable equilibrium state relative to itself; i.e., in a thermal state, but is free to exchange energy and entropy with the system of interest. Though finite in size, the bath of oscillators was large enough to approximate the effects of a system approaching that of a reservoir for the lowest energy eigenlevels.
conditions for the three cases are identical. Spontaneous self-relaxation for the DE and SEAQT cases are excluded, except in Figure 5 for SEAQT. Given the lack of data for this interaction, the respective time constants were adjusted to produce energy vs. time curves having the same shape in order to facilitate comparisons between the models. Entropy 2016, 18, 176 6 of 16 In Figure 2, the E–S trajectory of the SEAQT model closely resembles that for the linear DE, and was found to become more coincident as the size of model reservoir for SEAQT is increased. Both 2. Results and Discussion paths markedly diverge from the predictions of ME. The differences may be due to the simplistic Figures 2–5 show comparisons between the ME, DE, andHowever, SEAQT models for the case where by the nature of the interaction model used for the SEAQT case. near-linear behavior the single-particle system near a pure state interacts with a “cold” reservoir at lower energy. SEAQT model has been seen previously. A straightforward examination of the results for isolated Initial conditions for the three cases are identical. Spontaneous self-relaxation for the DE and SEAQT systems in [31] shows trajectories that are almost indistinguishable from simple decaying cases are excluded, except in Figure 5 for SEAQT. Given the lack of data for this interaction, the exponentials of the DE model.were In Figure other figures, thecurves units having of entropy areshape (eigen) respective time constants adjusted2toand produce energy vs. time the same in energy units/K with value comparisons of the Boltzmann set to unity. order the to facilitate betweenconstant the models.
Figure 2.Figure Comparison of Energy vs.vs. Entropy using master equation (ME), difference 2. Comparison of Energy Entropytrajectories trajectories using thethe master equation (ME), difference (DE),steepest and steepest entropy ascentquantum quantum thermodynamics (SEAQT) models for a single equationequation (DE), and entropy ascent thermodynamics (SEAQT) models for a single particle system as it interacts with a “cold” reservoir. Relaxation occurs only via interaction with particle system as it interacts with a “cold” reservoir. Relaxation occurs only via interaction with the the reservoir. Spontaneous relaxation effects are excluded. The initial state is near eigenenergy of 2. reservoir. Spontaneous relaxation effects are excluded. The initial state is near eigenenergy of 2. The The state is at stable equilibrium where the slope corresponds to the temperature of the reservoir. 7 of 17 Entropy 2016,final 18, 176 final state is at stable equilibrium where the slope corresponds to the temperature of the reservoir.
(a)
(b)
Figure 3. Comparisons of time trajectories for the ME, DE, and SEAQT models for the case in Figure 2. (a) For theofenergy vs. time comparison, the and paths for themodels ME and Figure 3. Comparisons time trajectories for the ME, DE, SEAQT forDE theare caseessentially in Figure 2. identical. The trajectory for the SEAQT model is also similar, except for the time lag, which is due to the (a) For the energy vs. time comparison, the paths for the ME and DE are essentially identical. The initial slowness of energy spreading between eigenlevels; (b) For the entropy vs. time trajectories, the trajectory for the SEAQT model is also similar, except for the time lag, which is due to the initial ME model shows aspreading markedly between greater change in entropy. Thethe DEentropy and SEAQT curves indicate similar slowness of energy eigenlevels; (b) For vs. time trajectories, the ME dynamic behavior.
model shows a markedly greater change in entropy. The DE and SEAQT curves indicate similar dynamic behavior.
Figure 3. Comparisons of time trajectories for the ME, DE, and SEAQT models for the case in Figure 2. (a) For the energy vs. time comparison, the paths for the ME and DE are essentially identical. The trajectory for the SEAQT model is also similar, except for the time lag, which is due to the initial slowness of energy spreading between eigenlevels; (b) For the entropy vs. time trajectories, the ME model shows a markedly greater change in entropy. The DE and SEAQT curves indicate similar Entropy 2016, 18, 176 7 of 16 dynamic behavior.
Figure state matrix ρptqρfor five energy eigenlevels for the system Figure 4. 4. Evolution Evolutionvs. vs.time timeofofthe the state matrix forlowest the lowest five energy eigenlevels for the (t )the interaction with the “cold” reservoir of Figure 2. The various colors represent the different energy system interaction with the “cold” reservoir of Figure 2. The various colors represent the different eigenlevels of the system. The dotted lines represent the results for thefor DEthe model, the dashed lines energy eigenlevels of the system. The dotted lines represent the results DE model, the dashed represent the ME model, and the solid lines are for the SEAQT model. The occupation probabilities lines represent the ME model, and the solid lines are for the SEAQT model. The occupation for P1 and P4for change theP4 fastest andthe over the widest ranges theranges ME model. An example of the probabilities P1 and change fastest and over the with widest with the ME model. An characteristic time delay for the SEAQT case is seen. example the characteristic time delay for the SEAQT case is seen. Entropy 2016, 18,of 176 8 of 17
Figure 5.5.Energy Energyvs.vs.Entropy Entropy curves of SEAQT as described by Equation the of system of Figure curves of SEAQT as described by Equation (13) for(13) the for system interest interest A (shown in red) which includes spontaneous relaxation effects of various magnitudes as A (shown in red) which includes spontaneous relaxation effects of various magnitudes as well as well as interaction the reservoir. The evolution of reservoir B is inand green, the composite interaction with thewith reservoir. The evolution of reservoir B is in green, the and composite system system Ab B ⊗blue. B is Magenta in blue. Magenta asterisks mark the states. initial states. Ancurve ME curve is shown for comparison. isAin asterisks mark the initial An ME is shown for comparison.
The energy vs. time diagram in 3a shows identicalthat behavior ME DE, and and DE. In Figure 2, the E–S trajectory of Figure the SEAQT modelessentially closely resembles for thefor linear The found SEAQT displays a timeas lag, similarly tofor MESEAQT and DEisat later times. There is was totrajectory become more coincident thebut sizebehaves of model reservoir increased. Both paths limited literature the time lag that The is predicted by may SEAQT. Thetoresults found by [32] for markedly diverge illustrating from the predictions of ME. differences be due the simplistic nature of the model used atoms for thehints SEAQT behavior by the model the interaction relaxation of rubidium at acase. time However, delay withnear-linear a sigmoidal profile and canSEAQT be contrasted has been previously. examination ofdisplays the results forlooks isolated in [31] to that of seen [33,34]. The data A of straightforward Figure 15 in Turchette et al. [19] what likesystems time delays for shows trajectories that are almost indistinguishable from simple decaying exponentials of the DE at least two of the cases studied. model. In Figure 2 andthat other theincreases units of entropy areto(eigen) energy units/K value of Figure 3b shows thefigures, entropy faster and a greater extent for thewith ME the model than the Boltzmann constant set to unity. either DE or SEAQT. Figure 4 illustrates how the occupation probabilities for the elements of the energy diagram Figurewith 3a shows stateThe matrices forvs. thetime three modelsin change time. essentially identical behavior for ME and DE. The SEAQT trajectory displays butdiscrepancies behaves similarly to ME DE atand laterME times. There is The E–S curves in Figure a5 time showlag, how between theand SEAQT are reduced when spontaneous relaxation effects are included according to Equation (13) for system A, the system of interest. However, the curvature for the SEAQT model appears less rounded than that for the ME model over a range of time constants. The SEAQT formalism does however allow for τ to be a functional that varies with the system state [24], so that
Entropy 2016, 18, 176
8 of 16
limited literature illustrating the time lag that is predicted by SEAQT. The results found by [32] for the relaxation of rubidium atoms hints at a time delay with a sigmoidal profile and can be contrasted to that of [33,34]. The data of Figure 15 in Turchette et al. [19] displays what looks like time delays for at least two of the cases studied. Figure 3b shows that the entropy increases faster and to a greater extent for the ME model than either DE or SEAQT. Figure 4 illustrates how the occupation probabilities for the elements of the state matrices for the three models change with time. The E–S curves in Figure 5 show how discrepancies between the SEAQT and ME are reduced when spontaneous relaxation effects are included according to Equation (13) for system A, the system of interest. However, the curvature for the SEAQT model appears less rounded than that for the ME model over a range of time constants. The SEAQT formalism does however allow for τ to be a functional that varies with the system state [24], so that (17) τ “ τ pρq Changing the value of τAB alone changes the rate of the system evolution but not the shape of the E–S curve. By appropriately varying τ pρq for both spontaneous and reservoir interaction effects, the E–S curve for SEAQT or DE could be made to conform to the ME trajectory. However, at present, there is not yet a rationale for choosing how time factors should vary, other than to avoid conflict with the Heisenberg uncertainty [24]. Figure 5 also shows the E–S curves for the composite system as well as the finite reservoir. The curves illustrate how energy and entropy are transferred between system A and the reservoir. As expected, energy is conserved for the composite system and entropy increases overall due to the irreversibilities of the relaxation. Current experimental evidence indicates that spontaneous relaxation effects for single particle systems, if they exist, are negligible. Atoms that are sufficiently isolated have been found to maintain their prepared states without degradation for long periods; in the case of rubidium atoms, for 9more Entropy 2016, 18, 176 of 17 than one minute [34], and for trapped ions, more than ten minutes [35]. In Figures Figures 66 and and 7, 7 the comparative behavior for system interactions interactions with with aa “hot” “hot” reservoir reservoir are are In shown. The same general differences between ME, DE, and SEAQT are seen. A closer match with ME shown. The general differences between ME, DE, and SEAQT are seen. A closer match with can can be obtained if spontaneous relaxation effects are included for the model. ME be obtained if spontaneous relaxation effects are included for SEAQT the SEAQT model.
Figure Figure 6. 6. Comparison Comparisonof ofEnergy Energy vs. vs. Entropy Entropytrajectories trajectoriesusing usingthe the ME, ME, DE, DE, and and SEAQT SEAQT models models for for aa single single particle particle system system interacting interactingwith withaa“hot” “hot”reservoir. reservoir. Spontaneous Spontaneousrelaxation relaxation effects effects are are excluded excluded in (a), but included for the SEAQT case in (b). A magenta asterisk marks the initial state. in (a); but included for the SEAQT case in (b). A magenta asterisk marks the initial state.
(a)
(b)
Figure 6. Comparison of Energy vs. Entropy trajectories using the ME, DE, and SEAQT models for a Entropy 2016, 18, 176 single particle system interacting with a “hot” reservoir. Spontaneous relaxation effects are excluded
9 of 16
in (a), but included for the SEAQT case in (b). A magenta asterisk marks the initial state.
(a)
(b)
Figure 7. Energy and entropy time trajectories for the ME, DE, and SEAQT models for a system
Figure 7. Energy and entropy time trajectories for the ME, DE, and SEAQT models for a system interacting with a “hot” reservoir as in Figure 6a, without spontaneous relaxation by the system. As interacting with a “hot” reservoir in Figure without spontaneous relaxation bySEAQT the system. in Figure 2, the DE and SEAQTas paths tend to6a, more closely coincide as the size of the modelAs in Figure 2, the DE and SEAQT paths tend to more closely coincide as the size of the SEAQT model reservoir is increased. In (a), energy changes vs. time are shown; In (b), the ME model predicts reservoir is increased. In (a), energy changes vs. time are shown; In (b), the ME model predicts markedly faster changes in entropy with time. markedly faster changes in entropy with time. 2.1. Comparison of the Models with Experimental Data
2.1. Comparison thefew Models with Datathat are available for the relaxation of a quantum One ofof the sources of Experimental experimental data single particle system is found in the work of Turchette et al. [19]. The data for the relaxation of Fock
One of the few sources of experimental data that are available for the relaxation of a quantum states as they interact with a surrogate reservoir are shown therein in Figure 15 for the lowest five single particle system is found in the work of Turchette et al. [19]. The data for the relaxation of energy levels. Four different cases are presented, representing initial states that are prepared near Fock the states as they interact with second a surrogate shown therein in Figure 15 for theThe lowest ground (|0>), first (|1>), (|2>), reservoir and thirdare (|3>) energy eigenstates, respectively. five energy levels. Four different cases are presented, representing initial states that are prepared thermal reservoir was created by introducing a Gaussian noise spectrum to the electromagnetic near fields the ground (|0>), firstThe (|1>), second andstate thirdover (|3>) eigenstates, respectively. that trap the atom. evolution of (|2>), the system timeenergy was achieved by varying the The thermal was over created by introducing a Gaussian noise to the electromagnetic power ofreservoir the reservoir a constant time interval rather than byspectrum keeping the reservoir power and different Maintaining a constant time the fieldsconstant that trap themaking atom. measurements The evolutionover of the systemtime stateintervals. over time was achieved by varying interval has the advantage of limiting extraneous environmental effects not represented by the power of the reservoir over a constant time interval rather than by keeping the reservoir power constant reservoir. It is not clear whether the higher power levels of the reservoir might cause unintended and making measurements over different time intervals. Maintaining a constant time interval has the advantage of limiting extraneous environmental effects not represented by the reservoir. It is not clear whether the higher power levels of the reservoir might cause unintended changes to the eigenstructure of the system, however. Each data point is an average of several thousand experimental runs. Figure A1 in the Appendix shows the application of Equation (3) of the ME model to the data. The claim in [19] is that the variation of only one parameter, the inverse time constant γ, is needed to fit the theoretical curves. In practice, the values for the data near the initial state are open to interpretation and may be modified slightly to adjust the curve fit. Moreover, the increase of the mean energy chosen for the reservoir can cause the curves to group more tightly together as equilibrium is approached. The energy for the reservoir in [19] is not known, but the mean reservoir energy for the present study was chosen to be 25 system eigenenergy units. The overall ME data match appears to be reasonably accurate. The results of the correlation with the evolution for state |1> ostensibly shows the worst fit, with traces P0 , P1 , and P2 consistently separated from the data points. Small but noticeable differences are also seen when the current ME results are compared to Figure 15 of [19]. The probabilities for the lowest five theoretical traces sum to greater than unity in the areas near the vertical axis, for example. Other slight shifts of the theoretical curves are seen to imply time delays in the evolution of certain energy levels. It was found in the present study, however, that Equation (3) by itself is not able to produce time lags, regardless of the initial state of the system. Time lags, if they exist, are noteworthy, and would support the SEAQT point of view. In Figure 8, the SEAQT model utilizing Equation (13) with spontaneous effects is fitted to the data from [19] for the lowest four of the initial states. For the horizontal scale, time is equivalenced to reservoir power as was done in [19]. Reasonable correspondence to data can be made even though the results differ somewhat from the ME result. For the individual cases illustrated, both τA and τAB were held constant. Unlike the ME and DE models, SEAQT is readily able to produce time lags as seen in
Entropy 2016, 18, 176
10 of 16
the case of |1> of Figure 8. Reducing the initial probability for a given energy eigenlevel produces a longer time delay in its corresponding probability curve. Entropy 2016, 18, 176 11 of 17
(a)
(b)
(c)
(d) Figure 8. Comparison of the SEAQT theoretical curves with spontaneous relaxation effects to the
Figure 8. Comparison of the SEAQT theoretical curves with spontaneous relaxation effects to the experimental data of [19] for initial states approximating |0> (a), |1> (b), |2> (c), and |3> (d) based on experimental data of [19] for initial states approximating |0> (a), |1> (b), |2> (c), and |3> (d) based on Equation (13). The dashed tan line for P5 of |1> (b) illustrates a time lag that can be introduced using Equation (13). The dashed tan line for P5 of |1> (b) illustrates a time lag that can be introduced using the SEAQT formalism by making the initial probability small, in this case 1 × 10−5. For case |2> (c), the SEAQT formalism by making the initial probability small, in this case 1 ˆmuch 10´5faster . For case curve rises than |2> the (c), there appears to be a time delay in the data for P0. The theoretical there appears to be a time delay in the data for P . The theoretical curve rises much faster than the 0 data, however. data, however.
Entropy 2016, 18, 176
11 of 16
Unexpectedly, even when spontaneous effects are neglected for the SEAQT model, there is still reasonable correspondence to the data, as seen in Figure 9. The ME model looks to be slightly if not conclusively better at matching the two higher energy data sets |2> and |3>. Figure 10 is an example showing that the DE model without spontaneous effects is not able to adequately capture the inflections Entropy 2016, of 18,the 176 experimental data. 12 of 17
(a)
(b)
(c)
(d) Figure 9. SEAQT theoretical curves without spontaneous relaxation effects, (a), (b), (c), (d) for initial Figure 9. SEAQT theoretical curves without spontaneous relaxation effects, (a), (b), (c), (d) for initial states |0>, |1>, |2>, |3> as fitted to the data from [19]. The correlation of the curves with the data states |0>, |1>, |2>, |3> as fitted to the data from [19]. The correlation of the curves with the data appears comparable to ME as well as SEAQT with relaxation effects. appears comparable to ME as well as SEAQT with relaxation effects.
Entropy 2016, 18, 176
12 of 16
Entropy 2016, 18, 176
13 of 17
Figure 10. 10. A A result result for for the the DE DE model model without without spontaneous spontaneous relaxation relaxation effects effects as as compared compared to to the the data data Figure for state |1>. Though accurate in many respects, the DE model does not capture local maxima for state |1>. Though accurate in many respects, the DE model does not capture local maxima (e.g.,(e.g. P3 ) P3)the in the data. in data.
To best fit the data, it was found necessary to use different time constants for each of the initial To best fit the data, it was found necessary to use different time constants for each of the initial states for both the ME and SEAQT models. The implication is that τ and γ are not constant but vary states for both the ME and SEAQT models. The implication is that τ and γ are not constant but vary as as the system state changes. The values of τ that were used for each initial state are shown in Table 1. the system state changes. The values of τ that were used for each initial state are shown in Table 1. The ME model evolved more quickly the greater the difference in energy between the initial and The ME model evolved more quickly the greater the difference in energy between the initial and final final (reservoir) equilibrium state was. Since the reservoir energy was held constant, perhaps the (reservoir) equilibrium state was. Since the reservoir energy was held constant, perhaps the smaller smaller energy differences explain the larger time constants for the higher energy |2> and |3> initial energy differences explain the larger time constants for the higher energy |2> and |3> initial states. states. For the SEAQT cases with spontaneous effects, the same general trend for τ AB was seen. For the SEAQT cases with spontaneous effects, the same general trend for τAB was seen. Curiously, the Curiously, the opposite tendency was found for the SEAQT cases without spontaneous effects. opposite tendency was found for the SEAQT cases without spontaneous effects. Table 1. A table showing how the time constants values were varied in order to fit the ME and Table 1. A table showing how the time constants values were varied in order to fit the ME and SEAQT models to the experimental data. The second column are τ values for ME, the third and SEAQT models to the experimental data. The second column are τ values for ME, the third and fourth fourth are for with SEAQT with spontaneous and the last column without for SEAQT without are for SEAQT spontaneous relaxation,relaxation, and the last column for SEAQT spontaneous spontaneous relaxation effects. relaxation effects. Initial State τ ME Initial State τ (= 1/γ) |0> ME |0> 2.1 |1> |1> 3.0 |2> |2> 3.1 |3> |3> 3.7
( = 1/γ) τ AB τAB 2.1 0.40 0.400.50 3.0 0.500.55 3.1 0.550.65 3.7 0.65
τA
τ AB (No Spontaneous)
τA
0.40 0.40 0.40 0.45 0.40 0.45 0.45 0.45
τAB (No Spontaneous) 0.60 0.60 0.32 0.32 0.25 0.25 0.26 0.26
2.2. Quantum Steady States 2.2. Quantum Steady States Figure 11 shows an evolution of a single-atom system that interacts with two reservoirs at Figure 11 shows an evolution of are a single-atom system that interacts with(15). twoThe reservoirs different temperatures. The dynamics based on a DE model using Equation system at is different temperatures. The dynamics are based on a DE model using Equation (15). The system seen to move to a steady state condition that is not at stable equilibrium. The starting points are a is seenstate to move a steady entropy state condition that is not at stable starting points are pure with to near-zero and a representative stableequilibrium. equilibriumThe state. The results are aconsidered pure stateaccurate with near-zero entropy and a representative stable equilibrium state. The results are only in the region near stable equilibrium. The model demonstrates at least considered in the The on model demonstrates at least qualitativelyaccurate how theonly system canregion evolvenear to a stable steadyequilibrium. state. Each point the steady state curve is a qualitatively how the system can evolve to a steady state. Each point on the steady state curve a linear combination of the two reservoir states. The location of a steady state point is determinedisby linear combination of the reservoir states. of agiven steadybystate is determined the the relative influence oftwo each reservoir on The the location system as thepoint magnitudes of thebytime relative influence of each reservoir on the system as given by the magnitudes of the time constants for constants for the system-reservoir interactions: the system-reservoir interactions: ρ ssA = α ρ eqA ( TB ) + (1 − α ) ρ eqA (TC ) ; 0 < α < 1 (18) ρssA “ αρeqA pTB q ` p1 ´ αq ρeqA pTC q ; 0 ă α ă 1 (18) where where τ AC α α“= τAC (19) (19) ττAB ` τAC τ + AB AC
Entropy 2016, 18, 176 Entropy 2016, 18, 176
13 of 16 14 of 17
τAB 1´α “ . (20) τ AB τAC τ 1 − α = AB ` . (20) τ + τ AC Ordinarily, a state with a given energy and AB entropy that is not at stable equilibrium can be represented by an unlimited of different distinctthat state However, sincecan thebe stable Ordinarily, a state with anumber given energy and entropy is matrices. not at stable equilibrium represented by are an unlimited numberitofcan different distinctthat statethe matrices. stablefor a equilibrium states unique [36,37], be inferred steadyHowever, states aresince alsothe unique states are unique [36,37], it can be inferred that the steady states are also unique for a givenequilibrium value of the α parameter. given value of the α parameter.
Figure 11. Evolution to steady state predicted by DE of a single atom system as it interacts with two
Figure 11. Evolution to steady state predicted by DE of a single atom system as it interacts with reservoirs at different temperatures. One of the starting states is near a pure state with energy of 2 two reservoirs at different temperatures. One of the starting states is near a pure state with energy of (red trajectories), while the other is at a stable equilibrium state (green trajectories). The reservoir 2 (redtemperatures trajectories), while thetoother is at of a stable equilibrium (green The reservoir correspond the slopes the stable equilibriumstate curve at the trajectories). indicated locations. temperatures correspond to the slopes of the stable equilibrium curve at the indicated locations.
3. Conclusions
3. Conclusions Comparisons between the ME, DE, and SEAQT methods for modeling the quantum evolution of a single trapped particle a reservoir have presented. A simple, novel Comparisons between the that ME,interacts DE, andwith SEAQT methods forbeen modeling the quantum evolution formalism for describing interactions between two or more systems has also been andnovel of a single trapped particle that interacts with a reservoir have been presented.presented A simple, applied to the DE and SEAQT approaches. A straightforward adaptation of the DE model was used formalism for describing interactions between two or more systems has also been presented and to demonstrate steady state dynamics for a single particle system that interacts with two reservoirs applied to the DE and SEAQT approaches. A straightforward adaptation of the DE model was used to at different temperatures. demonstrate state for a single that interacts Bothsteady the ME anddynamics SEAQT approaches areparticle shown system to reasonably coincidewith withtwo the reservoirs existing at different temperatures. experimental data. The SEAQT model matches the data, even without the addition of spontaneous Both ME and SEAQT are shown to reasonably coincide the and existing effectsthe to interactions with theapproaches reservoir. However, the available data is too limitedwith in extent accuracy to conclusively show which of the models is the most physically correct. There are experimental data. The SEAQT model matches the data, even without the addition of spontaneous indications that thewith time the factors used in the models are constantdata but vary the state of the and effects to interactions reservoir. However, the not available is toowith limited in extent system. It is interesting to note thatof asthe applied, both the ME and SEAQT approaches, along the accuracy to conclusively show which models is the most physically correct. There arewith indications existing data, presume that entropy as described by the density matrix ρ exists and can be that the time factors used in the models are not constant but vary with the state of the system. It is measured for a single particle. The data Figure Appendix further supports notion. interesting to note that as applied, bothinthe ME A2 andin SEAQT approaches, alongthis with the The existing SEAQT point of view additionally claims that quantum systems that are not at stable thermal data, presume that entropy as described by the density matrix ρ exists and can be measured for a equilibrium can spontaneously increase in entropy, a claim that remains to be conclusively shown. single particle. The data in Figure A2 in Appendix further supports this notion. The SEAQT point SEAQT is also able to describe time lags that may exist, which the ME and DE models are unable to of view claims quantum systems that are models not at stable thermalprovide equilibrium do. additionally The discrepancies and that similarities between the theoretical will hopefully added can spontaneously increase entropy, a claim thatnew remains to be conclusively shown. SEAQT is also able rationale for further in investigation, including experiments to show the link between the entropy to describe time lags that exist, Mechanics which the more ME and DE models are unable to do. The discrepancies of the Second Law andmay Quantum clearly. and similarities between the theoretical models will hopefully provide added rationale for further Acknowledgments: The author gratefully acknowledges Clinton L. Dancey for helpful discussions concerning investigation, this paper. including new experiments to show the link between the entropy of the Second Law and Quantum Mechanics more clearly. Conflict of interest: The author claims no conflict of interest. Acknowledgments: The author gratefully acknowledges Clinton L. Dancey for helpful discussions concerning this paper.
Entropy 2016, 18, 176
14 of 16
Conflicts of Interest: The author claims no conflict of interest. Entropy 2016, 18, 176
Appendix: Comparison of the ME Model to Experimental Data
15 of 17
Appendix: Comparison of the ME Model to Experimental Data
(a)
(b)
(c)
(d) Figure A1. Comparison of the ME theoretical relaxation curves to the experimental data of [19] for
Figure A1. Comparison of the ME theoretical relaxation curves to the experimental data of [19] initial states approximating |0> in (a), |1> (b), |2> (c), and |3> (d) for the lowest five energy for initial statesThe approximating |0> in (b), |2> (c), and |3> (d) forlag theislowest five eigenlevels. results are similar but(a), not|1> identical to [19]. A possible time seen for theenergy P0 eigenlevels. The are similar but not identical to [19]. A possible time lag is seen for the P0 data data of the |2>results (c) case. of the |2> (c) case.
Entropy 2016, 18, 176 Entropy 2016, 18, 176
15 of 16 16 of 17
Figure Figure A2. A2. Energy Energy vs. vs. entropy entropyevolution evolutioncurve curvefor forexperimental experimentaldata datafrom from [19] [19] for for an an initial initial state state approximating the |0> eigenstate. The data follow but do not go to the right of the predicted approximating the |0> eigenstate. The data follow but do not go to the right of the predicted limit limit curve equilibrium states—physical states—physicalevidence evidencethat thatsingle-particle single-particlequantum quantumsystems systems can exist curve of of stable stable equilibrium can exist in in states with entropy greater than zero while obeying the limitations imposed by the Second Law. states with entropy greater than zero while obeying the limitations imposed by the Second Law.
References 1.
Rauer, B.; Schweigler, T.; Langen, T.; Schmiedmayer, J. Does an isolated quantum system relax? 2015, References
1. 2. 2.
3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 10. 9. 11. 10. 11. 12. 12. 13. 13. 14. 14. 15. 15.
16. 16. 17.
18.
arXiv:1504.04288. Rauer, B.; T.; Langen, T.; Schmiedmayer, J. Does Available an isolatedonline: quantum system relax? 2015. Stuart, C. Schweigler, The Quantum Thermodynamics Revolution. http://fqxi.org/community/ Stuart, C. The Quantum Thermodynamics Revolution. Available online: http://fqxi.org/community/ articles/display/202 (accessed on 6 May 2016). articles/display/202 onR.; 6 May 2016). Y.; Monroe, C.; O’Brien, J.L. Quantum computers. Nature Ladd, T.D.; Jeleszk, F.;(accessed Laflamme, Nakamura, Ladd,464, T.D.; Jeleszk, F.; Laflamme, R.; Nakamura, Y.; Monroe, C.; O’Brien, J.L. Quantum computers. Nature 2010, 45–53. 2010, 464, 45–53. [CrossRef] P.; [PubMed] Cohen, A.J.; Mori-Sanchez, Yang, W. Challenges for density functional theory. Chem. Rev. 2012, 112, Cohen, A.J.; Mori-Sanchez, P.; Yang, W. Challenges for density functional theory. Chem. Rev. 2012, 112, 289–320. 289–320. [CrossRef] [PubMed] theory: Its origins, rise to prominence, and future. Rev. Mod. Phys. 2015, 87, Jones, R.O. Density functional Jones, R.O. Density functional theory: Its origins, rise to prominence, and future. Rev. Mod. Phys. 2015, 87, 897–923. 897–923. Weiss, U. [CrossRef] Quantum Dissipative Systems; World Scientific: Singapore, Singapore, 2008. Weiss, U.AQuantum Dissipativedissipative Systems; World Scientific: Singapore, Singapore, Gisin, N. simple nonlinear quatum evolution equation. J. Phys. A 2008. 1980, 14, 2259–2267. Gisin, N. A simple nonlinear dissipative quatum evolution equation. J. Phys. A 1980, 14, 2259–2267. Schuch, D. Nonunitary connection between explicitly time-dependent and nonlinear approaches for the [CrossRef] description of dissipative quantum systems. Phys. Rev. A 1997, 55, 935, doi:10.1103/PhysRevA.55.935. Schuch, Nonunitary connection between explicitly time-dependent and nonlinear for the Blum, K. D. Density Matrix Theory and Applications; Springer: Berlin/Heidelberg, Germany, approaches 2012. description of dissipative quantum systems. Phys. Rev. A 1997, 55, 935. [CrossRef] Carmichael, H. An Open Systems Approach to Quantum Optics; Springer: Berlin/Heidelberg, Germany, 1991. Blum, K. Density Matrix Theory andE.P. Applications; Berlin/Heidelberg, Germany, 2012. Hatsopoulos, G.N.; Gyftopoulos, A UnifiedSpringer: Quantum Theory of Mechanics and Thermodynamics. Carmichael, H. An Open Systems Approach to Quantum Optics; Springer: Berlin/Heidelberg, Germany, 1991. Part I. Postulates. Found. Phys. 1976, 6, 15–31. Hatsopoulos, G.N.; AA Unified Quantum Theory of Mechanics and Thermodynamics. Part I. Hatsopoulos, G.N.; Gyftopoulos, Gyftopoulos,E.P. E.P. Unified Quantum Theory of Mechanics and Thermodynamics. Postulates. Found. Phys. 1976, 6, 15–31. [CrossRef] Part IIa. Available Energy. Found. Phys. 1976, 6, 127–141. Hatsopoulos, G.N.; G.N.; Gyftopoulos, Gyftopoulos,E.P. E.P. A A Unified Unified Quantum Quantum Theory Theory of of Mechanics Mechanics and and Thermodynamics. Thermodynamics. Hatsopoulos, Part IIa. Available Energy. Found. Phys. 1976, 6, 127–141. [CrossRef] Part IIb. Stable Equilibrium States. Found. Phys. 1976, 6, 439–455. Hatsopoulos, G.N.; G.N.; Gyftopoulos, Gyftopoulos,E.P. E.P. A A Unified Unified Quantum Quantum Theory Theory of of Mechanics Mechanics and and Thermodynamics. Thermodynamics. Hatsopoulos, Part IIb. Stable Equilibrium States. Found. Phys. 1976, 6, 439–455. [CrossRef] Part III. Irreducible Quantal Dispersions. Found. Phys. 1976, 6, 561–570. Hatsopoulos, G.N.; E.P. A of Unified Quantum Theory of Mechanics and Thermodynamics. Beretta, G.P. On theGyftopoulos, General Equation Motion of Quantum Thermodynamics and the Distinction Part III. Irreducible Quantal Dispersions. Found. Phys. 1976, 6, 561–570. [CrossRef] between Quantal and Nonquantal Uncertainties. Sc.D. Thesis, Massachusetts Institute of Technology, Beretta, G.P. MA, On the General Cambridge, USA, 1981.Equation of Motion of Quantum Thermodynamics and the Distinction between QuantalG.P. andWhat Nonquantal Uncertainties. Sc.D. Thesis, Massachusetts Institute of Technology, Cambridge, Beretta, is Quantum Thermodynamics. Available online: http://quantum-thermodynamics.unibs.it/ MA, USA, 1981. WebSite1.pdf (accessed on 6 May 2016). Beretta, G.P. What is Quantum Thermodynamics. Available online: http://quantum-thermodynamics.unibs. Younis, A.M. Modeling the Non-Equilibrium Phenomenon of Diffusion in Closed and Open Systems at an it/WebSite1.pdf (accessed on 6 May 2016). Atomistic Level Using Steepest-Entropy-Ascent Quantum Thermodynamics, Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA, 2015. Leibfried, D.; Blatt, R.; Monroe, C.; Wineland, D. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 2003, 75, 281–324.
Entropy 2016, 18, 176
17.
18. 19.
20.
21. 22. 23. 24. 25. 26.
27. 28.
29.
30. 31.
32. 33. 34. 35. 36. 37.
16 of 16
Younis, A.M. Modeling the Non-Equilibrium Phenomenon of Diffusion in Closed and Open Systems at an Atomistic Level Using Steepest-Entropy-Ascent Quantum Thermodynamics. Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA, 2015. Leibfried, D.; Blatt, R.; Monroe, C.; Wineland, D. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 2003, 75, 281–324. [CrossRef] Turchette, Q.A.; Myatt, C.J.; King, B.E.; Sackett, C.A.; Kielpinski, D.; Itano, W.M.; Monroe, C.; Wineland, D.J. Decoherence and Decay of Motional Quantum States of a Trapped Atom Coupled to Engineered Reservoirs. Phys. Rev. A 2000, 62, 053807. [CrossRef] Gring, M.; Kuhner, T.M.; Langen, T.; Kitagawa, T.; Rauer, B.; Schreitl, M.; Maxets, I.; Smith, D.A.; Demler, E.; Schmiedmayer, J. Relaxation and prethermalization in an isolated quantum system. Science 2012, 337, 1318–1322. [CrossRef] [PubMed] Pefinova, V.; Luks, A. Exact quantum statistics of a nonlinear dissipative oscillator evolving from an arbitrary state. Phys. Rev. A 1990, 41, 414. [CrossRef] Kim, M.S.; Buzek, V. Schrödinger-cat states at finite temperature: Influence of a finite-temperature heat bath on quantum interferences. Phys. Rev. A 1992, 46, 4239. [CrossRef] [PubMed] Kreuzer, H.J. Nonequilibrium Thermodynamics and Its Statistical Foundations; Oxford University Press: Oxford, UK, 1981. Beretta, G.P. Maximum Entropy Production Rate in Quantum Thermodynamics. J. Phys. Conf. Ser. 2010, 237, 012022. [CrossRef] Beretta, G.P. A theorem on Lyapunov stability for dynamical systems and a conjecture on a property of entropy. J. Math. Phys. 1986, 27, 305. [CrossRef] Beretta, G.P. Steepest-Entropy-Ascent Irreversible Relaxation Towards Thermodynamic Equilibrium: The Dynamical Ansatz that Completes the Gyftopoulos-Hatsopoulos Unified Theory with a General Quantal Law of Causal Evolution. Int. J. Thermodyn. 2006, 9, 117–128. Beretta, G.P. Nonlinear Model Dynamics for Closed-System, Constrained, Maximal-Entropy-Generation Relaxation by Energy Redistribution. Phys. Rev. E 2006, 73, 026113. [CrossRef] [PubMed] Li, G.; Al-Abbasi, O.; von Spakovsky, M.R. Atomistic-level non-equilibrium model for chemically reactive systems based on steepest entropy-ascent quantum thermodynamics. J. Phys. Conf. Ser. 2014, 538, 012013. [CrossRef] Cano-Andrade, S.; Beretta, G.P.; von Spakovsky, M.R. Steepest-entropy-ascent quantum thermodynamic modeling of decoherence in two different microscopic composite systems. Phys. Rev. A 2015, 91, 013848. [CrossRef] Hsiang, J.T.; Hu, B.L. Nonequilibrium steady state in open quantum systems: influence action, stochastic equation and power balance. 2014. Smith, C.E. Intrinsic Quantum Thermodynamics: Application to Hydrogen Storage on a Carbon Nanotube and Theoretical Consideration of Non-Work Interactions. Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA, 2012. Nagel, M.; Haworth, F.E. Advanced Laboratory Experiments on Optical Pumping of Rubidium Atoms—Part II: Free Precession. Am. J. Phys. 1966, 34, 559. [CrossRef] Kukolich, S.G. Time Dependence of Quantum-State Amplitudes Demonstrated by Free Precession of Spins. Am. J. Phys. 1968, 36, 420. [CrossRef] Balabas, M.V.; Karaulanov, T.; Ledbetter, M.P.; Budker, D. Polarized Alkali Vapor with Minute-Long Transverse Spin-Relaxation Time. Phys. Rev. Lett. 2010, 105, 070801. [CrossRef] [PubMed] Bollinger, J.J.; Heinzen, D.J.; Itano, W.M.; Gilbert, S.L.; Wineland, D.J. A 303-MHz frequency standard based on trapped Be/sup +/ ions. IEEE Trans. Instrum. Meas. 1991, 40, 126–128. [CrossRef] Hatsopoulos, G.N.; Keenan, J.H. Principles of General Thermodynamics; Wiley: New York, NY, USA, 1965. Gyftopoulos, E.P.; Beretta, G.P. Thermodynamics, Foundations and Applications; Dover: Mineola, NY, USA, 1991. © 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
