Entanglement generation between a charge qubit and its bosonic environment during pure dephasing - dependence on environment size
aa r X i v : . [ qu a n t - ph ] S e p Entanglement generation between a charge qubit and its bosonic environment duringpure dephasing - dependence on environment size
Tymoteusz Salamon
Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,Wroc law University of Science and Technology, 50-370 Wroc law, Poland
Katarzyna Roszak
Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,Wroc law University of Science and Technology, 50-370 Wroc law, Poland (Dated: September 24, 2018)We study entanglement generated between a charge qubit and a bosonic bath due to their jointevolution which leads to pure dephasing of the qubit. We tune the parameters of the interaction,so that the decoherence is quantitatively independent of the number of bosonic modes taken intoaccount and investigate, how the entanglement generated depends on the size of the environment.A second parameter of interest is the mixedness of the initial state of the environment which iscontrolled by temperature. We show analytically that for a pure initial state of the environment,entanglement does not depend on environment size. For mixed initial states of the environment, thegenerated entanglement decreases with the increase of environment size. This effect is stronger forlarger temperatures, when the environment is initially more mixed, but in the limit of an infinitelylarge environment, no entanglement is created at any finite temperature.
I. INTRODUCTION
Decoherence due to the interaction of a qubit with itsenvironment can often be modeled by classical noise witha good deal of accuracy. This is especially true with re-spect to pure dephasing, i. e. when decoherence does notdisturb the occupations of the qubit, but affects only itscoherence (the off-diagonal elements of the density ma-trix which are responsible for quantum behavior). Suchloss of qubit coherence can always be mapped by a ran-dom unitary channel acting on the system, where thechannel describes the interaction with a fictitious classi-cal environment [1] (see Ref. [2] for examples of construc-tions of such fictitious classical environments for differ-ent types of open quantum systems). The existence ofsuch a mapping does not invalidate the importance ofqubit-environment entanglement in pure-dephasing sce-narios, since the presence of entanglement in the sys-tem will influence the evolution of the system in gen-eral, e. g. it changes the state of the environment pre-and post-measurement [3]. Furthermore, if the study ofa quantum system is not limited to its preparation, al-lowing it to evolve freely, and then measurement, butalso involves some manipulation of the system, such asperforming gates [4–6], coherence maximizing schemes[7–9], etc. then the presence of system-environment en-tanglement will influence the end result and can be higlyrelevant.Hence, it is at least in principle possible to have dif-ferent qubit-environment setups, in which the pure de-phasing of the qubit is qualitatively and quantitativelythe same, but the origin of the dephasing is different,since it can, but does not have to be the result of en-tanglement generation. If the whole system is always ina pure state (for an evolution described by a Hamiltio-nian this is equivalent to the initial states of the qubit and environment being pure) pure dephasing is unam-biguously related to entanglement [10, 11]. In the case ofmixed states, the relation between qubit coherence andqubit-environment entanglement is much more ambigu-ous and although entanglement not accompanied by de-phasing is not possible, dephasing without entanglementis [12]. In fact, the latter situation is often realized inreal systems, especially in the case of large environments,high temperatures, or noise resulting from e. g. fluctiat-ing semi-classical fields [13–16]. The distinction betweenentangling and non-entangling evolutions is not trivial initself, since the non-entangling case is not limited to ran-dom unitary evolutions [1, 17–21], and a straightforwardcriterion for the generation of qubit-environment entan-glement during pure dephasing has only recently beenfound [3].We study the amount of qubit-environment entangle-ment generated during the joint evolution of a chargequbit interacting with a bath of phonons as an ex-ample of a realistic system in which the qubit under-goes strictly pure dephasing which is always accompa-nied by the creation of entanglement (with the excep-tion of only the infinite-temperature situation) [3]. Thesystem is particularly convenient, because not only canthe decoherence-curves be reproduced using an arbi-trary number of phonon modes (for short enough timesand large enough temperatures), but the results can beobtained in a semi-analytical fashion, which simplifieschanging the number of phonon modes and later inter-pretation of the results. Furthermore, we control theinitial level of mixedness of the environment by settingthe temperature of the phonon bath.The correlation build-up between a system and itsbosonic environment have thus far been studied in thecontext of its relation towards non-Markovian dynam-ics [22], decoherence (especially for mixed initial qubitstates) [23], and two qubit correlation decay [24, 25]. Fur-thermore, studies of system-environment entanglementfor a boson system and a bosonic environment have alsobeen reported [26, 27]. In these studies, the focus wason the relation between the appearance of quantum cor-relations with a large environment with other quantumfeatures of open system dynamics. In this paper, thequantity of highest importance is the size of the envi-ronment, which is vital for the amount of entanglementgenerated in the whole system.We find that the dependence of generated entangle-ment on the temperature shows monotonously decreas-ing behavior which is steep above a physically well mo-tivated threshold temperature. The temperature depen-dence is non-trivial even above this temperature, and forreasonably low temperatures displays exponential decay,while for high temperatures the dependence becomes pro-portional to 1 /T . On the other hand, the dependenceon the number of phonon modes is much more complex.For pure states, the amount of entanglement generateddoes not depend on the number of phonon modes at all.Yet for any mixed environmental state (finite tempera-ture) there is a pronounced dependence on the numberof phonon modes (on the size of the environment) of 1 /n character (where n is the number of modes). The max-imum entanglement generated throughout the evolutionalways decreases when the environment becomes larger,even though the decoherence curve is unaffected in thestudied scenario for high enough temperatures. Further-more, this decrease is steeper when the temperature ishigher (the initial state is more mixed), but when thenumber of phonon modes approaches infinity (the contin-uous case), all finite-temperature entanglement vanishes.The article is organized as follows. In Sec. II we intro-duce the system under study, the Hamiltonian describingthis system, and the full qubit-environment evolution re-sulting from this Hamiltonian. We furthermore describe,how the respective strengths of the bosonic modes aredetermined, so that the decoherence at short times andlarge enough temperatures is qualitatively and quantita-tively the same independently of the size of the environ-ment. Sec. III contains a brief description of Negativity,which is the entanglement measure which is later usedto quantify qubit-environment entanglement. In Sec. IVthe dependence of the purity of the whole system (whichis constant throughout the evolution) on the initial stateof the environment is determined, especially on the tem-perature and size of the environment. Sec. V containsthe results pertaining to the dependence of the gener-ated entanglement on temperature and consequently, onthe degree of mixedness of the initial state of the envi-ronment, as well as the results concerning the effect ofenvironment size on entanglement, when the characteris-tics of the resulting qubit decoherence remain unchanged.The conclusions are given in Sec. VI. II. THE SYSTEM AND THE EVOLUTION
The system under study consists of a charge qubit in-teracting with phonons. The Hamiltonian of this systemis H = ǫ | ih | + X k ~ ω k b † k b k + | ih | X k ( f ∗ k b k + f k b † k ) , (1)where the first term describes the energy of the qubit( ǫ is the energy difference between the qubit states | i and | i in the absence of phonons), the second term isthe Hamiltonian of the free phonon subsystem and thethird term describes their interaction. Here, ω k is thefrequency of the phonon mode with the wave vector k and b † k , b k are phonon creation and annihilation operatorscorresponding to mode k and f k are coupling constants.The Hamiltonian (1) can be diagonalized exactly usingthe Weyl operator method (see Ref. 28 for details; thesame results can be obtained using a different approach[29, 30]). For a product initial state of the system andthe environment, σ (0) = | ψ ih ψ | ⊗ R (0), where the qubitstate is pure, | ψ i = α | i + β | i , and the environment isat thermal equilibrium, R (0) = e − kBT P k ~ ω k b † k b k Tr h e − kBT P k ~ ω k b † k b k i , (2)where k B is the Bolzmann constant and T is the temper-ature, the joint qubit-environment density matrix evolvesaccording toˆ σ ( t ) = (cid:18) | α | ˆ R (0) αβ ∗ e − iǫt/ ~ ˆ R (0)ˆ u † ( t ) α ∗ βe iǫt/ ~ ˆ u ( t ) ˆ R (0) | β | ˆ u ( t ) ˆ R (0)ˆ u † ( t ) (cid:19) . (3)Here, the matrix is written in the basis of the qubit states | i and | i , while the degrees of freedom of the environ-ment are contained in the density matrix ˆ R (0) and time-evolution operators acting only on the environmnet, ˆ u ( t ).The evolution operators can be found following Ref. [28],and are given byˆ u ( t ) = exp "X k (cid:18) f k ~ ω k (1 − e − iω k t ) b † k − f ∗ k ~ ω k (1 − e iω k t ) b k (cid:19) × exp " i X k | f k | ( ~ ω k ) sin ω k t . (4)In order to obtain the density matrix of the qubit alone,a trace over the degrees of freedom of the environmentneeds to be performed, ˆ ρ ( t ) = Tr E ˆ σ . This yields a den-sity matrix with time-independent occupations and co-herences which undergo decay governed by the function |h ˆ u ( t ) i| = exp − X k (cid:12)(cid:12)(cid:12)(cid:12) f k ~ ω k (cid:12)(cid:12)(cid:12)(cid:12) (1 − cos ω k t )(2 n k + 1) ! , (5)where n k = 1 / ( e ~ ω k /k B T −
1) is the Bose-Einstein distri-bution.If, as in our case, the quantity of interest is qubit-environment entanglement and not just the coherence ofthe qubit, the time-evolution of the full system densitymatrix (3) is needed. This can be found by acting withthe evolution operator given by eq. (4) on the initial den-sity matrix of the environment. The density matrix ofthe whole system ˆ σ can be divided into four parts withrespect to the way that the evolution operator acts onthe density matrix of the environment, which correspondto the | i qubit state occupation (for which the environ-ment remains unaffected ˆ R ( t ) = ˆ R (0)), the | i qubitstate occupation (for which ˆ R ( t ) = ˆ u ( t ) ˆ R (0)ˆ u † ( t )), andthe two qubit coherences (with ˆ R ( t ) = ˆ R (0)ˆ u † ( t ) whenthe qubit density matrix element corresponds to | ih | and ˆ R ( t ) = ˆ u ( t ) ˆ R (0) when the qubit density matrixelement corresponds to | ih | ).Since the initial density matrix of the environmentis a product of density matrices for each boson modeˆ R (0) = N k ˆ R k (0) and so is the evolution operator at alltimes ˆ u ( t ) = N k ˆ u k ( t ), each matrix ˆ R ij ( t ) ( i, j = 0 , R k ij ( t ) ma-trix is in principle of infinite dimension (the number ofphonons in each mode can be arbitrarily large; the ac-tual distribution of states for a single phonon mode isgoverned by the temperature and the qubit-phonon cou-pling) and a reasonable cut-off needs to be implementedto keep the density matrix σ manageable without the lossof physical meaning. Note, that although the ˆ R ii ( t ) ma-trices corresponding to the diagonal elements of the qubitdensity matrix are density matrices themselves, this isnot always true for the ˆ R ij ( t ) matrices with i = j (whichis a first indicator of qubit-environment entanglement).It can be shown that the evolution of any state of m phonons in mode k is given by | m ( t ) i k = ˆ u k ( t ) | m i k (6)= ∞ X p = − m (cid:18) f k ~ ω k (cid:19) p s m !( m + p )! × L ( p ) m (cid:12)(cid:12)(cid:12)(cid:12) f k ~ ω k (cid:12)(cid:12)(cid:12)(cid:12) ! | m + p i k , where L ( p ) m ( x ) is a generalized Laguerre polynomial.Given the initial state of the environment, eq. (6) is suf-ficient to find the time evolution of the whole system-environment density matrix ˆ σ , sinceˆ R ( t ) = ˆ u ( t ) ˆ R (0)ˆ u † ( t ) = O k ∞ X m k =0 c m k | m ( t ) i k k h m ( t ) | ! , ˆ R ( t ) = ˆ u ( t ) ˆ R (0) = O k ∞ X m k =0 c m k | m ( t ) i k k h m | ! , ˆ R ( t ) = ˆ R (0)ˆ u † ( t ) = ˆ R † ( t ) . Here, the initial occupations of each state | m i k are foundfor a given temperature using eq. (2), c m k = e − ~ ω k kBT m k (1 − e − ~ ω k kBT ) . (7) A. Excitonic quantum dot qubits
The exciton-phonon interaction constants used in thecalculations correspond to excitonic qubits confined inquantum dots [28, 31–33], where qubit state | i corre-sponds to an empty dot, while state | i denotes an ex-citon in its ground state confined in the dot. They aregiven by f k = ( σ e − σ h ) s ~ k ̺V N c Z ∞−∞ d r ψ ∗ ( r ) e − i k · r ψ ( r ) , (8)describing the deformation potential coupling, which isthe dominating decoherence mechanism for excitons [31].Hence, ω k = ck , where c is the speed of longitudinalsound and the phonon-bath is super-Ohmic. Here ̺ isthe crystal density, V N unit cell volume, and σ e , h aredeformation potential constants for electrons and holesrespectively. The exciton wave function ψ ( r ) is modeledas a product of two identical single-particle wave func-tions ψ ( r e ) and ψ ( r h ), corresponding to the electron andhole, respectively.The parameters used in the calculations correspondto small self-assembled InAs/GaAs quantum dots, whichare additionally assumed to be isotropic (for the sake ofsimplicity when limiting the number of phonon modesand with little loss of realism, when the evolution of co-herence is found). The single particle wave functions ψ ( r ) are modeled by Gaussians with 3 nm width inall directions. The deformation potential difference is σ e − σ h = 9 . ̺ = 5300 kg/m ,and the speed of longitudinal sound is c = 5150 m/s. Theunit cell volume for GaAs is V N = 0 .
18 nm (note, thatthis volume does not enter into the decoherence function,but is relevant, when individual elements of the system-environment density matrix needed to evaluate entangle-ment are found). B. Discretization
Typically for realistic systems, the number of phononmodes is very large and the summation over k can besubstituted by integration, which in spherical coordinatesyields X k → V (2 π ) Z π dφ Z π sin θdθ Z ∞ k dk. (9)If the studied system has spherical symmetry, as do thequantum dots, the parameters of which are used in thecalculations, the integration over the angles can be per-formed analytically. In the following, when we studyqubit-environment entanglement and qubit decoherencedue to the interaction with an environment which sup-ports only a limited number of phonon modes, we donot differentiate the modes with respect to their direc-tion, only with respect to the length of the wave vector.This means that we consider a simplified scenario, wherephonon modes are averaged over all directions. | < u ( t ) > | t [ps] FIG. 1. Evolution of the degree of qubit coherence at T = 6K for different numbers of phonon modes: n = 3 - dashedblue line, n = 5 - dotted red line, n = 7 - dashed-dotted greenline, n = 100 - solid black line, The actual discretization of the phonon modes is doneonly on the level of the length of the wave vector k . Aminimum and maximum wave vector length is, somewhatarbitrarily, chosen, so that a large enough range of k isconsidered to account for different phonon modes withvalues of the function | f k / ~ ω k | which are large enoughto be relevant for both pure dephasing and entangle-ment generation. In the following they are always setto k min = 0 .
001 nm − and k max = 0 . − . For a givennumber of phonon modes n , the range [ k min , k max + k min ]is evenly divided, and only wave vectors of lengths k i = ( i − k + k min , with ∆ k = k max / ( n −
1) and i = 1 , , ..., n , are taken into account. The slight off-setby of the range of k by k min allows for the decoherence ofthe qubit in the continuous case (when an infinite num-ber of phonon modes is taken into account) to be wellapproximated by only a few phonon modes for a widerange of temperatures as seen below.The decay of qubit coherence for a quantum dot inter-acting with an environment for which only a few discretelengths of phonon wave vectors are allowed (in which onlya few phonon modes are present) is plotted in Fig. (1) for T = 6 K (note that the coherence on the plot is limitedfrom below by 0 . n , the longer-time fea-tures of decoherence are also reproduced for a finite time,and the refocusing of the qubit (which is the result of thewhole qubit-environment density matrix returning to itsinitial state in the course of its unitary evolution) is fur-ther and further delayed in time with increasing numberof phonon modes.In fact, it is easy to reproduce continuous short-timebehavior of the coherence even with small phonon modenumbers as long as the initial temperature is above somethreshold, which depends on the phonon energy spec-trum, and is here around 3 K (this also obviously de-pends on the number of phonon modes and the thresholdis higher for higher mode numbers). Below this temper-ature, the approximation gets progressively worse, if thechoice of phonon modes remains unchanged and spansthe whole energy range where the spectral density is rea-sonably large. This is because the temperature entersinto the calculation only through the initial state of theenvironment (which is at thermal equilibrium) and, ifonly a few phonon modes with well separated energiesare allowed, then below some temperature practically nophonons will be initially excited. This can be remediedby a redistribution of the phonon modes to lower ener-gies, but since we wish to compare qubit-environmententanglement evolutions, when environments with differ-ent numbers of boson modes lead to the same dynamicsof the qubit alone, but with a set system under study fora given number of modes, we simply restrict ourselves tohigher temperatures. This means that for a set number ophonon modes the studied system is always qualitativelythe same (and even though the initial state of the systemdepends on temperature, the types of phonons which canbe excited remain unchanged). III. NEGATIVITY
The measure of entanglement which is most convenient(easiest to compute) in the context of quantifying en-tanglement between a qubit (a small quantum system)and its environment (a large quantum system) is Negavi-tity [34, 35] (or equivalently logarithmic Negativity [36]).The measure is based on the PPT criterion of separabil-ity [37, 38], which does not detect bound entanglement[39, 40]. Fortunately, in the case of the evolution of aqubit initially in a pure state interacting with an arbi-trary environment due to an interaction which can onlylead to pure dephasing of the qubit, bound entanglementis never formed [3, 41], so non-zero Negativity is a goodcriterion for the presence of entanglement in the systemand the value of Negativity unambiguously indicates theamount of said entanglement.Negativity can be defined as the absolute sum of thenegative eigenvalues of the density matrix of the wholesystem after a partial transposition with respect to one ofthe two potentially entangled subsystems has been per-formed (it does not depend on which of the subsystemsis chosen for the partial transposition), N (ˆ σ ) = X i | λ i | − λ i , (10)where λ i are the eigenvalues of ˆ σ Γ A , and Γ A denotes par-tial transposition with respect to system A = Q, E (qubitor environment). In the case of the studied system, it isparticularly simple to perform partial transposition withrespect to the qubit, as it is sufficient to exchange theoff-diagonal terms in eq. (3) to get the desired partiallytransposed state.
IV. PURITY
An important factor for the amount of entanglementgenerated between the qubit and the environment is theinitial purity of the state of the whole system. Note,that since the qubit-environment evolution is unitary, thepurity does not change with time, since P (ˆ σ ( t )) = Tr ˆ σ ( t ) = Tr h ˆ U ( t )ˆ σ (0) ˆ U † ( t ) ˆ U ( t )ˆ σ (0) ˆ U † ( t ) i = Tr ˆ σ (0) = P (ˆ σ (0)) . Taking into account that the initial state of the stud-ied system is a product state and the state of the qubitsubsystem is pure, we have P (ˆ σ (0)) = P (ˆ ρ (0)) P ( ˆ R (0)) = P ( ˆ R (0)) , (11)so the purity of the system only depends on the initialpurity of the density matrix of the environment. Fur-thermore, this initial density matrix is a product of thethermal-equilibrium density matrices for each phononmode, so the purity is a product of the purities of thestate of each mode, P ( ˆ R (0)) = Q k P ( ˆ R k (0)). The pu-rity of the initial state of mode k is easily found fromeq. (2) and is given by P ( ˆ R k (0)) = (cid:18) − e − ~ ω k kBT (cid:19) − e − ~ ω k kBT . (12)Since e − ~ ω k kBT tends to one with growing temperature moreslowly than e − ~ ω k kBT , the numerator in eq. (12) tends tozero much faster than the denominator, and the purity ofthe ininial state of a given phonon mode is a decreasingfunction of temperature (which reaches zero for infinitetemperature, since the dimension of the Hilbert space ofeach phonon mode is infinite). Consequently, the purityof the whole environment for a set choice and number ofphonon modes is always a decreasing function of temper-ature as well. Less obviously, for the system under study the pu-rity is also a decreasing function of the number ofphonon modes. For a given number of modes n , n wave vectors are taken into account which are evenly dis-tributed throughout a set wave vector lengths k wherethe coupling constants are most relevant as explained inSec. II B, and the purity of their initial state is a productof the corresponding single-phonon-mode purities (12).The energy of each phonon mode is proportional to itswave vector length, ω k = ck , so the single-phonon-modepurity is an increasing function of k for any finite tem-perature. If the number of phonon modes is increasedby one, each phonon mode is substituted by one withsmaller wave vector length k (with the exception of thetwo phonon modes limiting the rangle of k ), and hence,of lesser purity, and an additional phonon mode with alonger wave vector is taken into account (since now weare dealing with n + 1 phonon modes evenly distributedover the same range). Consequently, the product of thenew n + 1 purities, which yields the purity of the ini-tial state of the environment for an increased number ofphonon modes, must be smaller for any finite tempera-ture than the purity for n modes. The exception is thezero-temperature case, for which the purity of the ini-tial environment is always equal to one, since the initialstate is pure, and the infinite-temperature case, when thepurity is always equal to zero. Hence, although for ev-ery number of phonon modes the purity is a decreasingfunction of temperature ranging from one to zero, thedecrease is faster, if n is larger. V. RESULTS: QUBIT-ENVIRONMENTENTANGLEMENT GENERATIONA. Temperature dependence
The time-evolution of entanglement for the initialqubit state with α = β = 1 / √ n = 10boson modes at three different temperatures (the temper-ature increases from top to bottom of the figure). Therelatively large number of boson modes guarantees thatthe evolution of the qubit state due to the exciton-phononinteraction not only reproduces the fast initial decay ofqubit coherence which occurs in the first two picosec-onds after the creation of the excitonic superposition fora continuous environment, but also gives a reasonableapproximation of the coherence plateau for the next fourpicoseconds (up to slight oscillations which are absentwhen an infinite number of phonon modes is taken intoaccount). The temperatures in the plot start at 6 K, wellabove the threshold temperature, so phonon modes whichare evenly distributed over the range of relevant couplingconstants f k reproduce the dynamics of decoherence forshort times well. The plots in Fig. (2) capture a sin-gle cycle of qubit-environment evolution, so at the rightend of the plots, the density matrix of the whole systemreturns to its initial state, and then the evolution is re-peated (this is an unavoidable feature of systems withdiscrete spectra). N N N t [ps] FIG. 2. Entanglement evolution for n = 10 boson modes at T = 6 K (upper panel), T = 9 K (middle panel), and T = 12K (lower panel). N m a x T [K]
FIG. 3. Maximum Negativity as a function of temperaturefor n = 2 (solid blue line), n = 4 (dashed red line), and n = 6(dashed greeen line) wave vetors. As can be seen, changing the temperature does notchange the qualitative features of the evolution of Neg-ativity, but entanglement (at any given time) is a de-creasing function of temperature. Contrarily, decoher-ence increases with temperature, so for higher tempera-tures the effect of the environment on the qubit is larger(leading to stronger pure dephasing), but this is due to abuildup of classical qubit-environment correlations, sincethe amount of entanglement generated between the twosubsystems decreases with decreasing purity of the state.In Fig. (3) the dependence of the maximum Negativityreached during the pure dephasing evolution (Negativityreached at the first maximum which corresponds to theinitial strong loss of coherence) is plotted as a function of temperature for a choice of three different numbers ofboson modes. Below around T = 2 K, the maximumNegativity stabilizes at an almost fixed value. This is be-cause the initial density matrix of the environment belowthis temperature becomes a very weak function of tem-perature, since only phonon modes with high energiescompared to k B T are taken into account (if there areonly a few phonon modes allowed in the system). Thedensity matrix of the environment is then almost in thepure state ˆ R (0) ≈ | ih | , and the resulting qubit deco-herence is no longer a good approximation of the con-tinuous case. Note that in such situations, the plateauin Negativity is strictly related to the discrete natureof the phonon energy spectrum. An agreement betweencontinuous and few-phonon-mode decoherence could bereached also for low temperatures, but this would requirechanging k max and would qualitatively change the systemunder study (which we want to avoid). If the necessaryredistribution of the phonon modes taken into account(to account for decoherence well) were made, the plateauin low-temperature negativity would not be observed.At higher temperatures, maximum Negativity de-creases strongly with temperature, regardless of the num-ber of phonon modes taken into account, although theactual amount of entanglement in the system dependsstrongly on n . The shapes of the Negativity curves plot-ted in Fig. (3) roughly resemble the temperature depen-dence of the purity, which is found using eq. (12) withappropriate values of wave vectors k , meaning that thedependence in the shown temperature range is predom-inantly exponential decay. At high temperatures (fornanostructures, meaning far outside the 20 K range ofFig. (3)), the decay is dominated by terms proportionalto 1 /T . The fitted dependence of Negativity on tem-perature is presented at the end of Sec. (V C), since thedependence on temperature is convoluted with the de-pendence on environment size and cannot be consideredseparately.Note that the trade-off temperature behavior, whichis characteristic for the build up of correlations in thestudied system [42] and which results from the decreaseof purity with temperature accompanied by an increaseof the overall effect of the environment on the qubit, isnot present here, as only the purity of the system stateis relevant for the generation of entanglement, as long asthe system-environment interaction is capable of entan-gling the two subsystems. Contrarily, this type of trade-off behavior has been reported for boson-boson system-environment ensembles [27]. B. Dependence on environment size - pure initialstate
In the case of a pure initial state of the environment(at zero temperature in the case of the studied system, sothere are initially no phonons), the joint evolution of thesystem and the environment remains pure and entangle-ment at any time can be evaluated in a straightforwardmanner using the von Neumann entropy of one of theentangled subsystems (such von Neumann entropy is theunique entanglement measure for pure states). The mea-sure is defined as E ( | ψ ( t ) i ) = − ρ ( t ) ln ρ ( t )) , (13)where | ψ ( t ) i is the pure system-environment state and ρ ( t ) = Tr E | ψ ( t ) ih ψ ( t ) | is the density matrix of the qubitat time t (obtained by tracing out the environment). Theentanglement measure in eq. (13) is normalized to yieldunity for maximally entangled states. The same resultwould be obtained when tracing out the qubit degrees offreedom instead of the environmental degrees of freedom,but the small dimensionality of the qubit makes this waymuch more convenient.Let us denote the pure initial state of the environmentas | R i . Then qubit-environment state at time t is givenby | ψ ( t ) i = α | i ⊗ | R i + βe iǫt/ ~ | i ⊗ ˆ u ( t ) | R i , (14)and ˆ u ( t ) is given by eq. (4). The density matrix of thequbit is now of the form ρ ( t ) = (cid:18) | α | αβ ∗ e − iǫt/ ~ u ∗ ( t ) α ∗ βe iǫt/ ~ u ( t ) | β | (cid:19) , (15)where u ( t ) = h R | ˆ u ( t ) | R i and the absolute value of thefunction u ( t ) constitutes the degree of coherence retainedin the qubit system at a given time (it is given by eq. (5)with T = 0).The entanglement measure of eq. (13) can be calcu-lated using eq. (15) which yields E ( | ψ ( t ) i ) = − " p ∆( t )2 ln 1 + p ∆( t )2 (16)+ 1 − p ∆( t )2 ln 1 − p ∆( t )2 , with ∆( t ) = 1 − | α | | β | + | α | | β | | u ( t ) | . Note that thevon Neumann entropy during pure dephasing dependsonly on the degree of coherence | u ( t ) | . This means that,if two qubits lose the same amount of coherence, theymust be entangled with the environment to the samedegree, regardless of the numer of boson modes whichconstitute the environment. Hence, for pure initial envi-ronmental states, the amount of entanglement generatedduring evolution does not depend on the size of the en-vironment, as long as the degree of coherence at a giventime does not depend on its size.A similar analysis can be performed using Negativ-ity as the measure of pure state entanglement (Nega-tivity does not converge to von Neumann entropy forpure states contrarily to most entanglement measures).It is then fairly straightforward to show that entangle-ment does not depend on the size of the environment, but only on the degree of coherence u ( t ), taking into ac-count the fact that Negativity in the studied system doesnot depend on the phase relations between different com-ponents of the density matrix ˆ σ . What is not straight-forward is obtaining the explicit relation between Nega-tivity and decoherence, and therefore the von Neumannentropy was used in the analysis above. C. Dependence on environment size N t [ps] FIG. 4. Entanglement evolution for three different numbersof boson modes ( n = 6 - blue solid line, n = 8 - red dashedline, n = 10 - green dotted line) at temperature T = 6 K.Vertical line indicates the first time at which entanglement ismaximized. N m a x n 6 K9 K12 K FIG. 5. Maximal entanglement as a function of the numberof boson modes for different temperatures. The points cor-respond to numerical data (blue dots - 6 K, red triangles -9 K, green squares - 12 K), while the lines depict the fittingfunction and are color-coded in the same way.
At finite temperatures (for non-pure initial environ-mental states) the number of boson modes taken intoaccount becomes very important. In Fig. (4) the evolu-tion of Negativity for the equal superposition initial stateof the qubit ( α = β = 1 / √
2) is plotted at T = 6 K fordifferent numbers of boson modes n . Note that the tem-perature is high enough, so that the initial drop of qubitcoherence is always the same as in the continuous case,while the platou is reproduced for some short time afterthe drop up to small oscillations (and this time is longerfor larger n ). This means that the three curves in Fig. (4)correspond to the same qubit decoherence curves at shorttimes (up to roughly 5 ps here). Obviously, the amountof entanglement generated during these decoherence pro-cesses is not the same, as both the maximum values andthe values at the plateau decrease with increasing numberof boson modes. As the temperature dependence, this isrelated to the purity of the whole system during the evo-lution, but the temperature also affects the decoherencecurves, while the number of phonon modes (when theyare chosen as outlined in Sec. II) does not.The dependence of maximum Negativity as a functionof the number of boson modes taken into account is plot-ted in Fig. (5) with points for three different tempera-tures (well above the threshold value). The decrease ofNegativity with growing n is rather steep for the tem-peratures shown and this steepness increases when thetemperature grows. This corresponds to the fact thatat zero temperature, entanglement does not depend onthe size of the environment, while at infinite temperatureno entanglement between the qubit and the environmentis generated at all [3]. Furthermore, for any finite tem-perature entanglement approaches zero with growing n according to a function proportional to 1 /n , and for acontinuous environment, no entanglement is generated inthe system. Although separability is reached more slowlyat lower temperatures it is reached nonetheless for largeenough values of n (technically, zero-Negativity is onlyobtained for n = ∞ , but for high enough n the values ofNegativity will be so small that such entanglement willno longer be detectable and will have practically no effecton the properties of the system).Fitting of the curves displayed in Fig. (3) for tem-peratures above the threshold temperature and pointsdisplayed in Fig. (5) allows to find the dependence ofmaximum Negativity on temperature and environmentsize. The dependence on size exhibits good ∼ n behav-ior. The temperature dependence, on the other hand,shows strong exponential decay for low temperaturesand, while increasing temperature, a 1 /T dependencebecomes dominant. Furthermore, the temperature andsize dependencies are convoluted, so they cannot be rep-resented as a simple product of temperature-dependentand size-dependent functions. A reasonable fit is ob- tained using the function N max ( n, T ) ≈ e − αT AT ( n − BT + CT + D ) , (17)where the fitting parameters are given by α = 0 . A = 3 . B = 0 . C = 0 . D = 2 . n -dependence for differ-ent temperatures, especially for higher numbers of bosonmodes ( n ≥ VI. CONCLUSION
We have studied the generation of entanglement quan-tified by Negativity between a charge qubit and itsbosonic environment during evolution which leads topure dephasing of the qubit. In particular, we studiedand excitonic qubit confined in a quantum dot in thepresence of a super-Ohmic phonon bath, but the resultscould be easily extended to other charge qubits undergo-ing similar decoherence processes. The quantity of inter-est was the dependence of the amount of generated en-tanglement on the size of the environment (the number ofboson modes taken into account) in the situation, whenthe evolution of the qubit alone does not depend on en-vironment size (for short enough times and high enoughtemperatures, such a situation is easily obtained). Wehave found that although for pure states entanglementdoes not depend on the system size (and the amountof generated entanglement for pure dephasing is an ex-plicit function of the degree of qubit coherence), for finitetemperatures Negativity is a decreasing function of envi-ronment size proportional to 1 /n and there is no entan-glement generated for a continous bosonic environmentregardless of the temperature (as long as T = 0).The temperature, which governs the initial mixednessof the environment, and consequently the mixedness ofthe whole system throughout its unitary evolution, sim-ilarly governs entanglement generated between the sys-tem and environment. This means that for higher tem-peratures, the state of the whole system is more mixed,so less entanglement is generated. 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