Influence of excited state decay and dephasing on phonon quantum state preparation
IInfluence of excited state decay and dephasing on phonon quantum state preparation
Thilo Hahn, Daniel Groll, Tilmann Kuhn, and Daniel Wigger ∗ Institute of Solid State Theory, University of Münster, 48149 Münster, Germany (Dated: July 18, 2019)The coupling between single-photon emitters and phonons opens many possibilities to store andtransmit quantum properties. In this paper we apply the independent boson model to describethe coupling between an optically driven two-level system and a discrete phonon mode. Tailoredoptical driving allows not only to generate coherent phonon states, but also to generate coherentsuperpositions in the form of Schrödinger cat states in the phonon system. We analyze the influenceof decay and dephasing of the two-level system on these phonon preparation protocols. We findthat the decay transforms the coherent phonon state into a circular distribution in phase space.Although the dephasing between two exciting laser pulses leads to a reduction of the interferenceability in the phonon system, the decay conserves it during the transition into the ground state.This allows to store the phonon quantum state properties in the ground state of the single-photonemitter.
I. INTRODUCTION
The link between nanophotonics and phononics has re-cently been gaining more and more attention. The reasonis that many promising studies have shown that phonons– besides giving rise to dephasing of optical transitionsin nanosystems [1, 2] – offer a robust and controllableway to act on nanosystems. It has been demonstratedthat externally excited surface acoustic waves [3–6] orcoherent bulk acoustic waves [7] can be used to tune thetransition energies of single-photon emitters, such as ex-citons in quantum dots or defect centers in diamond [8].For example, based on this modulation of the transitionenergy, the output of a semiconductor microcavity lasercan be either enhanced or completely switched off [9, 10].The coupling between electrons and phonons can alsobe utilized in the opposite way. As a consequence ofrapid optical excitations of such single-photon emitters,phonons are generated [11]. Whereas the excitation ofacoustic phonons leads to the emission of phonon wavepackets [12–14], longitudinal optical (LO) phonons arecharacterized by a negligible dispersion and, therefore,cannot leave the region of creation. In Ref. [15], it wasshown that a single ultrafast optical excitation may leadto the generation of coherent LO phonon states whereastwo-pulse sequences can be used to generate Schrödingercat states. Considering the LO phonon energies, whichare in the range of tens of meV for typical III-V semicon-ductors [16] or even at a few hundreds of meV for hexago-nal boron nitride [17], the phonon-related time scales aretypically much faster than characteristic radiative decayor dephasing times of the corresponding single-photonemitters. Therefore excited state decay and dephasingare usually neglected in this context.However, LO phonons are not the only discrete vibra-tion mode that can be coupled to single-photon emit-ters. In the field of optomechanics, the eigenmodes of ∗ [email protected] micrometer-sized solid-state resonators are investigated[18, 19]. Compared to LO phonon modes where theatomic masses determine the characteristic frequencies,the large resonators have huge masses resulting in verysmall frequencies [20–23]. The coupling between suchresonators and single-photon emitters has already beenstudied theoretically [24, 25] and experimentally [23, 26].Because of the low oscillator frequencies, here the decaytime scales of the single emitters become comparable withthe phonon-related time scales. Hence, when consider-ing phonon state preparation protocols by optical excita-tion of quantum dot excitons or defect centers interactingwith large resonator systems, spontaneous decay, e.g., byphoton emission, and dephasing have to be taken intoaccount. In this paper, we systematically discuss this in-fluence. We will show that, although excited state decayand dephasing only act on the electronic system, depend-ing on the excitation conditions, they may strongly in-fluence the quantum state of the coupled phonons. Thisinfluence also opens up new possibilities to generate spe-cific phonon quantum states. II. THEORYA. Model
We consider a two-level system (TLS), representing thesingle-photon emitter, coupled to a single discrete phononmode. Since the energy of the phonon mode is typicallymuch lower than the transition energy of the TLS andphonons, therefore, do not lead to transitions between theground and the excited states, we describe the opticallydriven coupled emitter-phonon system by the standardsingle-mode independent boson Hamiltonian, H = (cid:126) Ω | x (cid:105) (cid:104) x | − (cid:2) M · E ( t ) | x (cid:105) (cid:104) g | + M ∗ · E ∗ ( t ) | g (cid:105) (cid:104) x | (cid:3) + (cid:126) ω ph b † b + (cid:126) g (cid:0) b + b † (cid:1) | x (cid:105) (cid:104) x | , (1)where | g (cid:105) and | x (cid:105) denote the ground and excited statesof the TLS, respectively, and (cid:126) Ω is the transition energy. a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l The bosonic operators b and b † act on a single-phononmode with energy (cid:126) ω ph and corresponding oscillation pe-riod t ph = 2 π/ω ph . The optical transitions, treated interms of the usual rotating-wave and dipole approxima-tions, are mediated by the dipole matrix element M andthe optical field E ( t ) . The phonon coupling to the TLSis treated via the pure dephasing mechanism with thecoupling strength g , taken to be real.In addition we include excited state decay (xd), e.g.,by means of radiative decay due to the coupling to thephoton vacuum, and pure dephasing (pd), e.g., due to theinfluence of a fluctuating environment, which are both as-sumed to be Markovian processes described by two phe-nomenological Lindblad dissipators [27], D i ( ρ ) = η i (cid:20)(cid:16) A i ρA † i (cid:17) − (cid:110) A † i A i , ρ (cid:111)(cid:21) (2a)with A xd = | g (cid:105) (cid:104) x | , η xd = Γ , (2b) A pd = | x (cid:105) (cid:104) x | , η pd = 2 ˜ β . (2c)Here, Γ is the excited state decay rate labeled as xd, and ˜ β is the pure dephasing rate labeled as pd. B. Equations of motion
Based on the Hamiltonian (1) and the Lindblad dissi-pators (2), the equation of motion for the density matrixof the coupled electron-phonon system reads dd t ρ = 1 i (cid:126) [ H, ρ ] + D xd ( ρ ) + D pd ( ρ ) . (3)As has been discussed in Ref. [28] for the single-mode caseand in Ref. [29] for the multi-mode case, the quantumstate of the coupled electron-phonon system can be com-pletely described in terms of three generating functionsfor phonon-assisted density matrices defined according to Y ( − α ∗ , α, t ) = (cid:10) | g (cid:105) (cid:104) x | exp( − α ∗ b † ) exp( αb ) (cid:11) , (4a) C ( − α ∗ , α, t ) = (cid:10) | x (cid:105) (cid:104) x | exp( − α ∗ b † ) exp( αb ) (cid:11) , (4b) F ( − α ∗ , α, t ) = (cid:10) exp( − α ∗ b † ) exp( αb ) (cid:11) . (4c)Based on the general equation of motion (3), the equa-tions of motion for the generating functions can be de-rived leading to i (cid:126) ˙ Y = (cid:126) (cid:2) Ω + ω ph ( α∂ α − α ∗ ∂ α ∗ )+ g ( α + ∂ α − ∂ α ∗ ) (cid:3) Y + M · E (2 C − F ) − i (cid:126) βY , (5a) i (cid:126) ˙ C = (cid:126) [ ω ph ( α∂ α − α ∗ ∂ α ∗ ) + g ( α + α ∗ )] C + M ∗ · E ∗ Y − M · E Y T − i (cid:126) Γ C , (5b) i (cid:126) ˙ F = (cid:126) ω ph ( α∂ α − α ∗ ∂ α ∗ ) F + (cid:126) g ( α + α ∗ ) C , (5c) where β = ˜ β + Γ / and Y T ( − α ∗ , α, t ) = Y ∗ ( α ∗ , − α, t ) .Following Ref. [28], by introducing Y ( α, t ) = Y ( − α ∗ , α, t ) (and corresponding definitions for C and F ), the dimensionless phonon coupling strength γ = g/ω ph and the polaron-shifted transition energy Ω = Ω − γ ω ph and using the transformations, Y ( α, t ) = exp (cid:0) i Ω t + γαe iω ph t (cid:1) Y (cid:0) αe iω ph t − γ, t (cid:1) , (6a) C ( α, t ) = exp (cid:2) i γ Im ( αe iω ph t ) (cid:3) C (cid:0) αe iω ph t , t (cid:1) , (6b) F ( α, t ) = F (cid:0) αe iω ph t , t (cid:1) , (6c) E ( t ) = 1 (cid:126) M · E ( t ) e i Ω t , (6d)the partial derivatives with respect to α and α ∗ can beeliminated. The resulting equations of motion for thecase without excited state decay and dephasing can befound in Refs. [28, 30]. Including decay and dephasingis straightforward. For arbitrary shapes of the drivingelectric field, they can be solved numerically.In Ref. [29], it was shown that in the limit of excita-tion by an arbitrary sequence of ultrafast optical pulsesthe equations of motion for Y , C , and F can be solvedanalytically. From a physical point of view, the ultrafastlimit is reached if the pulse duration is much shorter thanany phonon-related time scale, here, in particular, muchshorter than the phonon oscillation period t ph . In thiscase, the phonon influence on the dynamics during theexcitation is negligible, and the pulse sequence can bemathematically well approximated by a series of δ pulsesat times t j , E ( t ) = (cid:88) j θ j e i Ω t j + iϕ j δ ( t − t j ) = (cid:88) j θ j e iφ j δ ( t − t j ) , (7)with θ j being the pulse area, ϕ j being the carrier-envelope phase, and φ j is the total phase of the j th pulsearriving at t = t j .Following the derivation in Ref. [29] but additionallyincluding the decay rate Γ for the excited state occupa-tion and the total dephasing rate β = Γ / β for theinterband coherence, we can split the time evolution intorecursion relations that: (i) link the transformed gener-ating functions at t − j immediately before pulse j withthe ones at t + j directly after this pulse and (ii) connectthese functions at t + j with the ones immediately beforethe subsequent pulse at t − j +1 .Between the pulses: (i) the coherence function Y de-cays due to the dephasing constant β , (ii) the excitedstate occupation function C decays due to the decayconstant Γ , and (iii) the phonon function F is modifieddue to the excited state occupation. Denoting, e.g., thecoherence function just before and after the pulse j by Y ± j = Y ( t ± j ) and correspondingly for the other functions,the connection between the functions just after pulse j FIG. 1. Schematics of the system and its temporal evolution.(a) Mutual influence of the generating functions caused by alaser pulse with phase φ [Eq. (16)] and during the evolutionbetween the pulses [Eq. (8)]. The labels at the arrows indi-cate the phase that is transferred in the respective coupling.(b) Illustration of the phonon potentials associated with theground state | g (cid:105) (blue) and the excited state | x (cid:105) (red) of theTLS. The phonon state is depicted by the green dots. Laserexcitation and decay induce transitions between the electronicstates. and immediately before pulse j + 1 read Y − j +1 = e − β ( t j +1 − t j ) Y + j , (8a) C − j +1 = e − Γ( t j +1 − t j ) C + j , (8b) F − j +1 = F + j + C + j (8c) × t j +1 (cid:90) t j e − Γ( t − t j ) ∂∂t exp (cid:0) − i γ Im (cid:2) αe iω ph t (cid:3)(cid:1) d t . Due to the ultrashort pulse limit, the recursion rela-tions across the pulses are not influenced by decay anddephasing. Therefore, they agree with those derived inRef. [29], reduced to the single-mode case. For complete-ness, these relations, in the notation of the present paper,are summarized in the Appendix in Eq. (16). Note that,in particular, the phonon function F is not changed dur-ing the pulses.Figure 1(a) schematically shows the interplay amongthe three generating functions Y , C , and F as describedby Eqs. (8) and (16). The arrows reflect the influence ofa laser pulse with phase φ and the couplings induced bythe phonons. The labels at the arrows indicate the orderof φ entering the coupling. We see, for example, that thephonon function F is only driven by the excited stateoccupation C , whereas it influences both, the occupation C and the coherence Y . C. Wigner representation
We are primarily interested in the properties of thephonon system. As its central quantities, we define thedisplacement u and momentum p via the quadratures, u = ( b + b † ) , p = 1 i (cid:0) b − b † (cid:1) , (9) with their eigenfunctions u | U (cid:105) = U | U (cid:105) and p | Π (cid:105) =Π | Π (cid:105) . In Fig. 1(b), it is schematically shown that thephonon system is split into two subsystems, one associ-ated with each state of the TLS. The phonons attributedto the excited state | x (cid:105) have an equilibrium displacement,which is shifted with respect to the ground state | g (cid:105) by γ [15]. The laser pulse excitation mediates transitions be-tween the two systems, whereas the decay of the excitedstate acts only in one way from | x (cid:105) to | g (cid:105) .A useful tool to study the quantum state of the phononsystem is the Wigner function, which is defined via thephonon density matrix as W ( U, Π) = 14 π (cid:90) (cid:28) U + X (cid:12)(cid:12)(cid:12)(cid:12) ρ ph (cid:12)(cid:12)(cid:12)(cid:12) U − X (cid:29) e − iX Π / d X , (10)with ρ ph = Tr TLS ( ρ ) , where we take the trace with re-spect to the TLS. It is especially handy as it can di-rectly be calculated from the generating function F viathe Fourier transform, W ( U, Π) = 14 π (cid:90) e −| α | / F ( α ) e i [Re( α )Π+Im( α ) U ] d α . (11)The Wigner function is a quasiprobability distributionin the phase space spanned by U and Π . Although itcan take negative values the calculation of expectationvalues, of all operators which can be expressed as (sym-metrized) functions of u and p is performed, as in the caseof common probability density functions. For example,it is (cid:104) u (cid:105) = (cid:90) (cid:90) U W ( U, Π) d U dΠ or (12a) (cid:10) u (cid:11) = (cid:90) (cid:90) U W ( U, Π) d U dΠ . (12b)By performing the same transform as in Eq. (11) butwith C ( α ) instead of F ( α ) , we isolate the part of thephonon state that is associated with the excited state W x ( U, Π) . With this, we can also identify the part of theground state W g from W = W g + W x . (13) III. RESULTS
In the following sections, we will discuss the influenceof excited state decay and dephasing on the propertiesof the generated phonon state. Before addressing spe-cific excitation scenarios, let us briefly come back to theschematic in Fig. 1(a). There it is seen that the phononfunction F , while influencing both other functions C and Y , is only influenced by the occupation function C . Thistells us that, in the case of excitation by a single ultra-short pulse, the phonon dynamics cannot be influencedby pure dephasing processes. In contrast, in the caseof excitation by multiple pulses, pure dephasing occur-ring between the pulses influences the light-induced TLS − − Π W x (a) t/t ph = 0 + / / / (b) / − − U − − Π W g − − U − − U − − U − − U FIG. 2. Partition of the phonon Wigner function into the(a) excited state and (b) ground state subspace. The corre-sponding subspace Wigner functions are shown at five differ-ent times after excitation by a π pulse. The system parame-ters are γ = 2 , Γ = 0 . ω ph . dynamics of later pulses and, thus, indirectly also thephonon properties.To clearly identify the role of excited state decay andpure dephasing processes on the dynamical phonon state,we start by considering a single laser pulse excitation andafterwards move to two-pulse excitations where we willfirst investigate the influence of each process separatelyand finally look at their combined effect. A. Single-pulse excitation
As it has been discussed in Ref. [15] without consid-ering excited state decay and dephasing, a single-pulseexcitation of the TLS with a pulse area of θ = π in-stantaneously shifts the equilibrium position of the en-tire phonon system. The initial phonon vacuum state,thus, becomes a coherent phonon state that oscillatesaround a displaced equilibrium position in phase spaceat ( U, Π) = (2 γ, . Considering an excitation at t = 0 with pulse area θ but taking into account a nonvanish-ing decay rate Γ , the characteristic phonon function F ( t ) after the pulse, i.e., for t ≥ , reads F ( α, t ) = 1 − i γω ph sin (cid:18) θ (cid:19) t (cid:90) e − Γ t (cid:48) Re (cid:104) αe iω ph ( t (cid:48) − t ) (cid:105) × exp (cid:110) i γ Im (cid:104) αe − iω ph t (1 − e iω ph t (cid:48) ) (cid:105)(cid:111) d t (cid:48) . (14)Inserting this in Eq. (11) leads to the correspondingWigner function of the phonon system.As schematically shown in Fig. 1(b), the phonon stateis partially attributed to the ground state | g (cid:105) of the TLSand the rest to the excited state | x (cid:105) . The pulse area θ in Eq. (14) determines to which amount the systemis brought into the excited state and, therefore, whichfraction of the phonon state switches into the shifted po-tential. A nonvanishing decay rate of the excited statemakes the phonons go back to the unshifted potential as-sociated with the ground state of the TLS as depicted in Fig. 1(b).Let us start by considering the excitation by a pulsewith θ = π in a system with an electron-phonon cou-pling constant γ = 2 and an excited state decay constant Γ = 0 . ω ph . Snapshots of the Wigner functions describ-ing the dynamics of the phonons corresponding to the ex-cited state W x and to the ground state W g are plotted forfive different times in Figs. 2(a) and 2(b), respectively. A π pulse completely inverts the system, therefore, immedi-ately after the pulse ( t = 0 + ), the phonon system is com-pletely attributed to the excited state. The initial stateis a coherent state in this subspace, which rotates aroundthe shifted equilibrium position (black circle), much likein the case without decay. The excited state decay leadsto a gradual damping of the phonon state in the excitedstate subspace. The ground state contribution W g inFig. 2(b) is more involved. Starting with an empty phasespace at t = 0 + , the Wigner distribution builds up dur-ing the dynamics. The excited state successively decaysinto the ground state, which adds contributions to W g atthe point in phase space where the coherent state in theexcited state in Fig. 2(a) is located. Subsequently, thesecontributions continue to rotate, but since the equilib-rium position for W g is the origin ( U, Π) = (0 , , thisrotation occurs on circles around the origin. Interest-ingly, at the last considered time t/t ph = 1 / , they all lieon the lower half of the circle reflecting the trajectory of W x . Indeed, it is clearly seen that at this time we ob-tain a Wigner function W g distributed along the circularshape of this trajectory.The total phonon Wigner function is the sum of theground and the excited state contributions [Eq. (13)].Snapshots of this function are shown at four differenttimes for the same case as discussed above in Fig. 3(a).For short times after the laser pulse excitation ( t/t ph =1 / ), a large fraction of the distribution moves on themarked circle and keeps its Gaussian form. However, theinfluence of the decay already manifests in a curved tailconnecting the main Gaussian part with the origin of thephase space. The reason for this shape is simply the sumof the two contributions W g and W x discussed before.After half a phonon period ( t/t ph = 1 / ), the fractionof the Gaussian on the marked circle has become weakerwhereas the curved tail has gained weight. This processgoes on until the excited state has entirely decayed intothe ground state. After a full phonon period at t = t ph inFig. 3(a), the Wigner function is distributed along a cir-cle of the same size as the marked one, but shifted in theopposite U direction. At this time, the excited state occu-pation is (cid:104)| x (cid:105) (cid:104) x |(cid:105) ≈ . meaning that a large fraction ofthe phonon state already oscillates around (0 , . Whenthe TLS has entirely decayed into the ground state, theWigner function will rotate stable in shape around theorigin of the phase space.When we increase the decay rate to Γ = 2 ω ph ,the phonon state’s dynamics expressed in terms of theWigner function is depicted in Fig. 3(b). Without anydecay, the Wigner function would move stable in shape − − Π t/t ph = 1 / / / − − U − − Π / − − U / − − U / − − U − . − . . . . − . − . . . . FIG. 3. Phonon Wigner function at four different times afterexcitation by a single π pulse for an electron-phonon coupling γ = 2 and decay constants (a) Γ = 0 . ω ph and (b) Γ = 2 ω ph . on the black circle around the shifted origin at ( U, Π) =(2 γ,
0) = (4 , . However, we find that, already in thefirst shown time step ( t/t ph = 1 / ), the largest part ofthe distribution has left the marked circle of the coherentstate. The reason is that the TLS has already decayedby almost into the ground state. The overall shapeof the Wigner function is similar to the one in Fig. 3(a)but with a shorter tail, i.e., it has a larger weight nearthe origin of the phase space.Figure 4(a) summarizes the influence of the excitedstate decay on the phonon state by showing the Wignerfunction for different values of the decay constant at t/t ph = 10 when the TLS has completely decayed. TheWigner function is located on a circle which rotatesaround the origin. At integer values t/t ph = n as shownhere, the circle is a mirror image of the trajectory ofthe coherent state without decay (black circle), whereasat half-integer values t/t ph = n + , it coincides withthis trajectory. For decay constants much smaller thanthe phonon oscillation period, the Wigner function isessentially homogeneously distributed along the circle,whereas for increasing decay constants, it becomes moreand more concentrated on a part of the circle. Finally, ifthe decay is much faster than the oscillation period, theexcited state occupation has already decayed before thephonons could react on the excitation. Therefore, in thiscase the phonons essentially remain in the initial vacuumstate.The Wigner function reflects the full quantum state ofthe phonon system. The resulting temporal evolution ofthe mean displacement is depicted in Fig. 4(b) for thesame parameters as in part (a) of the figure. The dashedline displays the dynamics of the mean displacement forvanishing excited state decay. As discussed above, thephonons form a coherent state with a displacement oscil-lating around γ , i.e., in the present case around (cid:104) u (cid:105) = 4 .Including excited state decay, the equilibrium position ofthe phonons returns to zero. Therefore, at long times,all solid curves in Fig. 4(b) oscillate around zero. Forweak damping (dark lines), we clearly see this transi-tion between an oscillation around almost four initiallyand around zero finally, whereas the peak-to-peak am- t/t ph − h u i (b) Γ = 0 . ω ph . ω ph . ω ph . ω ph . ω ph − U − − Π (a) Γ = 0 . ω ph − U . ω ph − U . ω ph − U . ω ph − U . ω ph FIG. 4. (a) Phonon Wigner function after excitation by a sin-gle π pulse at the time t/t ph = 10 , i.e., after the excited stateoccupation has completely decayed for an electron-phononcoupling γ = 2 and five different values of the decay con-stant Γ . (b) Expectation value of the displacement (cid:104) u (cid:105) as afunction of time after the laser pulse for the same values of Γ . The dashed line shows the expectation value for the casewithout decay (i.e., Γ = 0 ). plitude of the oscillation remains essentially the same.With increasing excited state decay, the oscillation am-plitude decreases because the excited state occupationdecays before a complete oscillation around the shiftedequilibrium is finished. Once the TLS has returned to itsground state, there is no more coupling to the phonons[see Eq. (1)]. Thus, in the present model, the oscilla-tions in Fig. 4(b) remain undamped. They would onlybe damped by taking into account phonon-phonon inter-actions, i.e., anharmonicities, in the Hamiltonian. B. Two-pulse excitation
1. Pure dephasing
From Eqs. (8) and (16), we know that the phonon stateis only driven by the excited state occupation. Therefore,in a single-pulse excitation the dephasing has no influenceon the phonon state. The same holds for phase φ of thelaser pulse, which does not enter in Eq. (14) after a sin-gle pulse. However, in Ref. [15], it was shown that byan excitation with two pulses the phonon system can bebrought into a statistical mixture of two Schrödinger catstates. Cat states are superpositions of coherent states.The interference between these coherent phonon statescan be controlled by the relative phase of the laser pulses ∆ φ = φ − φ . To study the influence of a pure dephas-ing contribution, i.e., a nonvanishing β , whereas keeping Γ = 0 , we consider the same two-pulse excitation as inRef. [15] but with β = ˜ β (cid:54) = 0 . Having a look at theschematic picture of the generating functions in Fig. 1(a),we find that the coherence function Y has no direct in-fluence on the phonon state F . Therefore, once a phonon − − − − Π (a) γ = 2 ˜ β = 0 ω ph − − γ = 2 ˜ β = 0 . ω ph − − γ = 2 ˜ β = 0 . ω ph − . − . . . . − − U − − Π (b) γ = 0 . − − Uγ = 0 . − − Uγ = 0 . − . − . . . . FIG. 5. Phonon state after a two-pulse excitation. The pulseareas are θ i = π/ , the delay is t − t = t ph / , and the de-picted time is t/t ph = 3 / after the second pulse. (a) Wignerfunction for the coupling strengths (a) γ = 2 and (b) γ = 0 . .The pure dephasing rate increases from ˜ β = 0 (left) via . ω ph (center) to . ω ph (right). state is prepared by a pulse sequence and the populationfunction C is not affected by any decay, the phonon statebehaves periodically with the phonon frequency ω ph , andthe dephasing has no impact. But, in the case of a two-pulse excitation, the coherence function Y between thetwo pulses can act on the phonon state indirectly via C .To visualize the influence of a nonvanishing pure dephas-ing rate on the generation of phonon cat states, in Fig. 5,we plot the Wigner function after a pulse sequence withpulse areas θ = θ = π/ , delay t − t = π/ω ph = t ph / ,and relative phase ∆ φ = π/ . The phonon state is de-picted at t/t ph = 3 / after the second laser pulse excita-tion. In Fig. 5(a), we choose a large coupling strength of γ = 2 and increase the pure dephasing rate from ˜ β = 0 (left) via . ω ph (center) to . ω ph (right). The typi-cal structure of the Schrödinger cat states consisting oftwo Gaussians with a striped structure between them isvisible. The striped pattern is oriented along the con-nection line between the two Gaussians; it takes positiveand negative values and is a direct consequence of thequantum-mechanical interference between the two coher-ent states in the phonon system. We observe two suchcat states, one in each electronic subspace (ground andexcited states). When increasing the dephasing rate, wefind that the interference features faint, which nicely il-lustrates the loss of coherence during the time intervalbetween the laser pulses. It should be noted that, once aquantum coherence is generated in the phononic system,it is unaffected by the pure dephasing processes actingon the excited state of the TLS. The reason is that thephonon state is only driven by the occupation function C , which is not affected by the dephasing. Additionally,the pure dephasing does not alter the coherent phononstates, which still appear as symmetric Gaussians.In Ref. [15], it was also shown that these mixtures of . . . . . . t/t ph − E I F S Π S U ˜ β = 0˜ β = 0 . ω ph ˜ β = 0 . ω ph FIG. 6. EIFs S U (blue) and S Π (red) as functions of time afterthe second pulse. The phonon coupling strength is γ = 0 . ,and the pure dephasing rate increases from dark to brightlines. The dashed line is the limiting case of a statisticalmixture, i.e., ˜ β (cid:29) ω ph . cat states in the phonon system can exhibit squeezing,i.e., the quantum fluctuations of displacement or momen-tum can fall below the respective vacuum value. This ispossible when the coupling strength between the phononand the TLS is small enough, e.g., γ = 0 . . This effectstrongly depends on the relative phase between the twolaser pulses ∆ φ and is a consequence of the quantum in-terferences. Under the right conditions, the different con-tributions of the Wigner function overlap in phase spacein such a way that the distribution becomes narrowerthan the vacuum state. Of course, the Heisenberg uncer-tainty relation has to be fulfilled, therefore, this squeez-ing of the Wigner function is always accompanied by abroadening in the perpendicular direction. This effectis visible in Fig. 5(b) where we show the same Wignerfunctions as in Fig. 5(a) but for γ = 0 . . In the case ofno pure dephasing ˜ β = 0 (left), the two slightly nega-tive contributions lead to an additional narrowing of thedistribution. When increasing the dephasing rate, thesenegative parts vanish, and the Wigner function becomesentirely positive. It directly follows that the squeezingvanishes because without any interferences, the phononstate is a statistical mixture of coherent states and allquantum effects that could lead to a reduction of thefluctuations disappear.To obtain an overview of the influence of pure dephas-ing on the quantum fluctuations of the phonon states,we consider the excitation-induced fluctuations (EIFs) ofthe displacement and the momentum defined by [15, 31] S U = (∆ u ) − (∆ u vac ) γ , (15a) S Π = (∆ π ) − (∆ π vac ) γ . (15b)In Fig. 6, these quantities are plotted as functions oftime t after the second pulse. The blue curves show S U ,the red ones show S Π , and the pure dephasing rate in-creases from dark to bright colors from ˜ β = 0 via . ω ph to . ω ph . As already discussed in the context of theWigner functions, we find that the squeezing shrinks, i.e.,the negative values reduce. At the same time, also themaximal positive values of the fluctuations get smallerwhen increasing the dephasing. This shows that the in-fluence of the quantum interference not only leads to thereduction, but also causes increased fluctuations. Thisis in direct correspondence with the Heisenberg uncer-tainty principle. For the largest considered dephasingrate of ˜ β = 0 . ω ph the incoherent limit of a statisticalmixture of all four coherent states (dashed lines) is al-most reached. The minimum of the momentum’s EIF is S Π = 0 whereas the displacement’s EIF remains alwayspositive as is expected for the statistical mixture of fourcoherent states oscillating around equilibrium positionsshifted in the U direction [32].
2. Influence of excited state decay on cat state dynamics
The considered generation mechanism of phonon catstates, in general, leads to a state which is partly at-tributed to the ground state | g (cid:105) and partly to the excitedstate | x (cid:105) . The phonon state attributed to the groundstate oscillates stable in shape because it is not affectedby decay and dephasing. Therefore, from the mixture ofcat states, the one in the ground state will not changeonce it is generated. But the state attributed to the ex-cited state will be affected by the decay into the groundstate. Although in the considered system it is not pos-sible to generate an ideal cat state only in the excitedstate, in the following, we will study the evolution ofsuch a state to understand the influence of the excitedstate decay. In the next section, we will then analyze thefull dynamics including the generation of the cat states.Mathematically, we can initialize such an ideal catstate by applying two laser pulses with delay t − t = t ph / , pulse areas θ = θ = π/ , relative phase ∆ φ = π/ , and disregarding decay and dephasing during thephonon state preparation. As already discussed, this ex-citation scheme will generate one cat state in each elec-tronic subspace. In order to isolate the one that belongsto the excited state, we set F ( α, t = 0) = C ( α, t = 0) as initialization after the second pulse. This removes thephonon state associated with the ground state because C ( α ) carries the entire phonon state attributed to theexcited state [30]. Note that, after this procedure, thephonon state is not normalized any longer. For the dy-namics following this initialization, we consider again anonvanishing decay rate to investigate the transition intothe ground state. In this case, the dephasing has againno influence on the phonon dynamics.Figure 7 shows snapshots of the Wigner function’s dy-namics for an initial cat state in the excited state sub- − − Π t/t ph = 0 + / / − − Π / / − − U − − Π / − − U / − − U / . . − . . . − . . . − . FIG. 7. Excited state decay induced decay of a cat state.Snapshots of the Wigner function’s dynamics for a cat stateprepared in the excited state | x (cid:105) . The decay rate is Γ =0 . ω ph , and the coupling strength is γ = 2 . space at t = 0 and a decay rate of Γ = 0 . ω ph . Thenatural propagation of the phonon state in the excitedstate is a rotation around the shifted equilibrium positionat ( U, Π) = (4 , , which is the center of the interferenceterm. The rotating dynamics of the Wigner function isaccompanied by a continuous flow into the ground state,i.e., into the unshifted phase space. Because the decay israther slow compared to the phonon period, the Wignerfunction evolves into a double-ring structure in the shapeof an 8 rotating around its center. For the last depictedtime t/t ph = 5 / , the system has not yet completely de-cayed, and we find four interference terms, two at (0 , ± ,one at the original position of the cat state’s interference (4 , , and a very weak one at ( − , . Already, afterthe second time step at t/t ph = 1 / , it can be seen thatthe interference term of the cat state is transferred to-gether with the coherent contributions into the groundstate subsystem as it clearly appears as striped structurearound (0 , − . This and the interference on the oppositeside of the origin survive the decay process and remaineven after the full decay.Figure 8 shows the Wigner function at the time t/t ph =9 . , i.e., at a time when the excited state occupation hascompletely decayed, for different values of the decay con-stant Γ . Such as in the case of the single coherent state[see Fig. 4], we observe a transition from a ringlike struc-ture for a decay time much longer than the oscillationperiod to a more localized structure for very short de-cay times. Here, the ringlike structure consists of twocircles attached to each other. For small values of thedecay constant, we observe interference patterns in theregions around ( U, Π) = (0 , ± . When increasing thedecay constant, the distribution of the Wigner function − − U − − Π Γ = 0 . ω ph − − U . ω ph − − U . ω ph − − U . ω ph − − U . ω ph FIG. 8. Phonon Wigner function resulting from the excitedstate-induced decay of a cat state at the time t/t ph = 9 . , i.e.,after the excited state occupation has completely decayed foran electron-phonon coupling γ = 2 and five different values ofthe decay constant Γ . along the circles becomes more and more inhomogeneous.Inside the left circle, an interference pattern builds upwhereas the patterns between the circles fade away, firstin the region of positive Π and then also for negative Π .For the very strong decay corresponding to Γ = 5 ω ph ,the initial cat state in the excited state subspace is al-most instantaneously transferred to the ground state sub-space by keeping the entire interference between the twopeaks. Whereas the initial cat state rotated along thesolid circle around the shifted equilibrium position of theexcited state subspace, the final cat states rotate aroundthe ground state equilibrium position at the origin.The previous discussion clearly explains why, in Fig. 8,in the limit of very strong decay, there is an interferencepattern around ( U, Π) = ( − , , but we still have to un-derstand why this pattern vanishes for weak decay andother patterns at ( U, Π) = (0 , ± show up. This behav-ior results from the continuous transfer of the cat statefrom the excited state subspace to the ground state sub-space. So we can imagine the distribution as the sum ofmany cat states transferred to the ground state at dif-ferent times. At half-integer times t/t ph = n + , thetwo Gaussians of each of these cat states are aligned hor-izontally, and they are all separated by ∆ U = 8 , i.e., thediameter of the two circles. Thus, we obtain a superpo-sition of interference patterns which are all aligned hori-zontally but shifted vertically. This is indicated in Fig. 9where the position at times t/t ph = n + ( n ≥ ) of thecat states resulting from decay processes at four differ-ent times t/t ph = 0 , , , are plotted schematically to-gether with their respective interference patterns. Sincethe decay processes occur continuously, the interferencepatterns are continuously distributed along the dottedcircle. Due to the continuous superposition of these ver-tically shifted interference patterns, inside the solid anddashed circles there is a destructive interference, whichremoves the interference patterns in these regions. Incontrast, at the top and the bottom of the dotted cir-cle, there is a stationary phase of the interference pat-tern due to the horizontal slope of the circles resulting inconstructive interference. This explains why, in the caseof weak decay, when the distribution of the Wigner func-tion along the circles is almost homogeneous, interferencepatterns remain around ( U, Π) = (0 , ± , whereas they U Π – – ¼0 ¼ ½0,½¾ ¾ – FIG. 9. Schematic of the Wigner function in the ground statesubspace at times t/t ph = n + ( n ≥ ). The four horizontallyaligned cat states are examples for cat states resulting fromdecay processes at different times. The numbers indicate thetime of the decay (in units of t/t ph ). For a continuous decay,the interference patterns are continuously distributed alongthe dotted circle. vanish around ( U, Π) = ( ± , as it was produced ata later time. With increasing the decay constant, thedecay at later times is continuously reduced, which firstleads to the vanishing of the interference pattern around ( U, Π) = (0 , and finally around ( U, Π) = (0 , − ,whereas the pattern around ( U, Π) = ( − , remains be-cause of the negligible vertical shift of the contributingcat states.
3. Cat state generation with excited state decay anddephasing
Finally we go a step further and study the influence ofa nonvanishing decay rate Γ also during the generationprocess of the phononic Schrödinger cat states. We donot consider an additional pure dephasing as the decayalready has a dephasing influence on the coherence func-tion with β = Γ / . With what we have learned so far, weexpect a more complex structure of the Wigner functionsof the phonon states after the second laser pulse becausethe single contributions will not be coherent any moreand the characteristic tail structures from Sec. III A willappear.For the dynamics of the Wigner function shown inFig. 10, we consider a decay rate of Γ = 0 . ω ph . Wesee that, after the second laser pulse at t = 0 , the Gaus-sians are accompanied by a half circle distribution as aresult of the decay between the two laser pulses. De-spite the nonvanishing decay rate, still a major part ofthe phonon state is coherent, and interferences betweenthese states build up in the following dynamics. As dis-cussed in Sec. III B 2, the phonon part attributed to theground state is not affected by the decay, whereas thepart in the excited state decays as previously explained, − − Π t/t ph = 0 + / / − − U − − Π / − − U − − U / . . − . . . − . FIG. 10. Cat state generation with excited state decay anddephasing. Snapshots of the Wigner function’s dynamicsduring the first phonon period after the second laser pulse( t = 0 → t = t ph ) and at the time t/t ph = 9 / (bottom right).The decay rate is Γ = 0 . ω ph , and the coupling strength is γ = 2 . i.e., it evolves into the eight-shaped structure. Comparedto the shape-invariant cat state in the ground state sub-space, the overall amplitude of this part of the Wignerfunction is rather small because the distribution covers alarger area in phase space. So the main contribution ofthe phonon state after the full decay of the excited stateis a single cat state as depicted in Fig. 10 at t/t ph = 9 / .It is now possible to bring this state into the excitedstate potential by applying a third laser pulse with pulsearea of θ = π , which entirely inverts the TLS. Whenthis is performed at full phonon periods after the secondpulse, i.e., at t/t ph = n ( n > ), it leads to approximatelythe situation artificially prepared in Fig. 7 at t = 0 . Af-ter the following decay into the ground state, the entirephonon state will have evolved into the eight-shaped dis-tribution depicted in Fig. 8 at the end of the relaxation.Any other choice of the time or pulse area for the thirdexciting laser pulse will lead to a more complex combina-tion for the phonon cat state in the excited state, meaningthat only one of the two coherent states will move on thecircle depicted in the figures. The other one will move ona larger circle around the same center. IV. CONCLUSIONS
We have studied an optically driven single-photonemitter coupled to a single discrete phonon mode. Underthe assumption that the decay and dephasing time scaleof the TLS is in the range of the phonon period, we havesystematically analyzed the preparation of phonon quan-tum states by ultrafast optical excitations of the TLS.After a single-pulse excitation, only the decay rate hasan impact on the phonon state. During the decay of theexcited state, the Wigner function of the phonon statetransforms from a coherent Gaussian into a ring-shapedstructure, which reduces the amplitude of the lattice dis-placement. For a two-pulse excitation, also the dephas-ing between the two pulses acts on the phonon state. Forstrong dephasing rates, the phonon system loses its coher-ence, and the interferences representing the Schrödingercat states vanish. This also removes the possibility ofthe phonon state to exhibit squeezing. When taking intoaccount decay and dephasing during a two-pulse exci-tation, more complex Wigner functions emerge, and wehave shown that interferences in the phonon system canbe transferred from the excited state subspace of theTLS to its ground state subspace. Because of destruc-tive interference of cat states transferred to the groundstate subspace at different times, the interference pat-terns may be blurred in some regions of phase space. Atthe same time they survive in other regions. Our pa-per shows that there is a subtle interplay amng differenttime scales, in particular, the phonon oscillation period,the decay and dephasing times, and the delay of the ex-citing laser pulses, which can be used to tailor the finalphonon quantum state.
APPENDIX
The recursion relations connecting the values of thecharacteristic functions Y , C , and F immediately beforeand after the pulse j , i.e., from t − j to t + j , read0 Y + j ( α ) = 12 [1 + cos( θ j )] Y − j ( α )+ sin (cid:18) θ j (cid:19) Y −∗ j (cid:0) γe − iω ph t j − α (cid:1) exp (cid:8) i φ j + 2 γ Re (cid:2)(cid:0) αe iω ph t − γ (cid:1)(cid:3)(cid:9) + (cid:104) F − j (cid:0) α − γe − iω ph t j (cid:1) − C − j (cid:0) α − γe − iω ph t j (cid:1) e − i γ Im ( αe iω ph tj ) (cid:105) i θ j ) exp (cid:0) iφ j + γαe iω ph t j (cid:1) (16a) C + j ( α ) = C − j ( α )+ exp (cid:2) i γ Im (cid:0) αe iω ph t j (cid:1)(cid:3) (cid:32) sin (cid:18) θ j (cid:19) (cid:110) F − j ( α ) − C − j ( α ) exp (cid:2) − i γ Im (cid:0) αe iω ph t j (cid:1)(cid:3)(cid:111) − i θ j ) (cid:26) Y − j (cid:0) α + γe − iω ph t j (cid:1) exp (cid:2) − iφ j − (cid:0) γ + γαe iω ph t j (cid:1)(cid:3) − Y −∗ j (cid:0) − α + γe − iω ph t j (cid:1) exp (cid:2) iφ j − (cid:0) γ − γα ∗ e − iω ph t j (cid:1)(cid:3) (cid:27)(cid:33) (16b) F + j ( α ) = F − j ( α ) . 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