SSuper Bloch Oscillation in a PT symmetric system Z. Turker † , C. Yuce (cid:63) † Engineering Faculty, Near East University, North Cyprus, (cid:63)
Physics Department, Anadolu University, Eskisehir, Turkey
Abstract
Wannier-Stark ladder in a PT symmetric system is generally complex that leads to amplified / damped Bloch oscil-lation. We show that a non-amplified wave packet oscillation with very large amplitude can be realized in a non-Hermitian tight binding lattice if certain conditions are satisfied. We show that pseudo PT symmetry guarantees thereality of the quasi energy spectrum in our system.
1. Introduction
A quantum particle in a periodical potential drivenby a constant force performs an oscillatory motion in-stead of uniformly accelerated motion. This oscillationis known as Bloch oscillation [1]. Long after its theo-retical prediction, it was observed in some physical sys-tems such as superlattice [2], ultracold [3] and opticalsystems [4]. If the system is subjected to an oscillat-ing force instead of the constant force, then localiza-tion occurs at some certain oscillation frequencies. Thiskind of localization is known as dynamical localiza-tion and its physical origin is di ff erent from Andersonlocalization [5]. The combined presence of both con-stant and oscillating forces leads to the photon-assistedtunneling. In this case, wave packet delocalization oc-curs for an initially localized wave packet when mod-ulation frequency is at multiple integers of the Blochfrequency [6]. Another interesting e ff ect is the superBloch oscillations, which arises when an integer multi-ple of the oscillation frequency is only slightly detunedfrom the Bloch frequency associated with the constantforce [7, 8, 9, 10, 11, 12]. Note that the amplitude of su-per Bloch oscillation is much larger than that of Blochoscillation. It was experimentally realized in an ultra-cold system, where a giant matter-wave oscillation (su-per Bloch oscillation) that extend over hundreds of lat-tice sites was observed [8]. A generalization of superBloch oscillation to correlated particles was consideredin [11]. We refer the reader to [12] for the theoretical Email address: (cid:63) [email protected] (Z. Turker † , C.Yuce (cid:63) ) analysis of the Hermitian case.The experimental realizations of non-Hermitian PT symmetric optical systems [13, 14, 15] have triggeredinterests in the investigation of oscillatory dynamics in PT symmetric systems. Bloch oscilation in a PT sym-metric complex crystal was first addressed theoreticallyin [16]. It was shown that either amplified or dampedBloch oscillation occurs depending on the sign of theconstant force. This is simply due to broken PT sym-metry by the external force that leads to the appear-ance of complex Wannier Stark (WS) ladders. Blochoscillations were also shown to display unusual fea-tures such as non-reciprocal cycles related to a viola-tion of Friedel’s law of Bragg scattering and cascadeof wave packet splittings. Bloch oscillations for a non-Hermitian infinitely-extended, ring and truncated tight-binding lattices with unidirectional hopping were inves-tigated [17]. It was shown that the oscillatory motion isnot secularly damped nor amplified for that system as aresult of the equally-spaced real Wannier-Stark ladder.It is well known that the acceleration theorem explainsBloch oscillation semi-classically for Hermitian lattice.An extension of the the acceleration theorem to non-Hermitian lattices was discussed and Bloch oscillationfor the generalized non-Hermitian Hatano and Nelsonwas specifically studied [18]. Another theoretical pa-per investigated Bloch-Zener oscillation for locally PT symmetric system [19]. PT symmetric Bloch oscilla-tion is not only theoretical interests. It was also exper-imentally realized in [20, 21, 22]. Although the non-Hermitian Hamiltonian considered in [20] is PT sym-metric, the total power was observed to be changed intime. In the experiment [21], pseudo-Hermitian wave Preprint submitted to Physics Letters A September 10, 2018 a r X i v : . [ phy s i c s . op ti c s ] M a y acket propagation with a vanishing net increase inpower was observed by controlling the period of theBloch oscillation in a global and local PT symmetricmesh lattice. A resonant restoration of PT symme-try and secondary emissions, which occurs each timethe wave packet passes through the exceptional pointwas also observed in that experiment. Another interest-ing dynamical e ff ect in solid state and optical systemsis the dynamic localization e ff ect. The dynamic local-ization for non-Hermitian PT symmetric systems havebeen investigated by various authors [23, 24, 25]. It wasshown that the reality of the quasi-energy spectrum ispreserved by the PT symmetry and Bragg scatteringin the crystal becomes highly nonreciprocal at the PT symmetry-breaking point [24].In this paper, we study another interesting dynam-ical e ff ect, super Bloch oscillation, for a complexlattice. Bloch oscillation in non-Hermitian latticesis generally either amplified or damped because ofcomplex Wanner-Stark ladders. Here, we show thatundamped / non-amplified large amplitude oscillationoccurs in a non-Hermitian lattice if certain conditionsare satisfied.
2. Model
Consider an array of tight-binding one dimensionallattice with alternating gain and loss. The system is alsosubject to a time-dependent force. The non-HermitianHamiltonian for our system is given by H = − (cid:88) n T ( | n >< n + | + | n + >< n | ) + (cid:88) n ( F ( t ) a n + i ( − n γ ) | n >< n | (1)where the constant T is tunneling amplitude throughwhich particles are transferred between neighboringsites, F ( t ) is a real valued time-dependent force, a isthe distance between the centers of adjacent sites andthe real constant γ is non-Hermitian degree describingthe strength of the gain / loss materials. Note that thesame Hamiltonian can also describe an array of opti-cal waveguides if the time parameter t is replaced withthe propagation distance z .Let us first study the reality of the spectrum of this non-Hermitian Hamiltonian. As a result of the balancedgain and loss, the non-Hermitian Hamiltonian becomes PT symmetric if no force exists F =
0. Thereforethe corresponding spectrum is real unless γ exceeds acritical value γ PT , which corresponds to the transitionfrom unbroken to broken PT symmetry. In the pres-ence of force, the reality of the spectrum depends on the form of the force. For example, a time-independentforce, F ( t ) = const . , breaks PT symmetry and conse-quently complex-conjugate eigenvalues appear. In otherwords, Wannier-Stark ladders become complex-valuedand then a localized wave packet is amplified / dampedduring Bloch oscillation. The physics changes signifi-cantly if the force changes periodically in time. In thiscase, the energy spectrum is replaced by a quasi-energyspectrum, which is real for a wide range of parameters[24]. At some certain parameters of the Hamiltonian,dynamic localization was shown to occur. In this paper,we suppose that the force is composed of both a con-stant and an oscillating terms F ( t ) = ω ( l + κ cos( ω t + φ ) ) (2)where ω is the modulation frequency, l (cid:44) ω κ is the strength of the oscillating termand φ is the initial phases. Apparently, the constant termin (2) breaks the PT symmetry of the Hamiltonian (1).So, one expects that the system has zero threshold for PT symmetry breaking, i.e. γ PT =
0. This means thatwave packet is either amplified or damped during oscil-lation. However, this is not the case since pseudo PT symmetry guarantees the reality of the correspondingspectrum as it was shown in [26, 27, 28]. As a result,pseudo PT symmetry allows us to study large ampli-tude oscillation of a wave packet that is neither dampednor amplified.We now show that the spectrum in our system is real.One can use high-frequency Floquet approach to con-struct a time-independent e ff ective Hamiltonian to studythe spectrum of the periodical Hamiltonian. In this ap-proach, which is valid if ω is large, the tunneling pa-rameter is replaced by an e ff ective tunneling parameter, T e f f . [9, 29, 30] H e f f . = − (cid:88) n = T e f f . | n >< n + | + T (cid:63) e f f . | n + >< n | + i γ N (cid:88) n = ( − n | n >< n | (3)where star denotes the complex conjugate and the e ff ec-tive tunneling is given by T e f f . T = (cid:90) t e i η dt (cid:48) (4)where overline denotes the average over time and η is given by η ( t ) = (cid:90) t F ( t (cid:48) ) dt (cid:48) . To evaluate theintegral (4), we use the Jacobi-Anger expansion; e i κ sin( x ) = (cid:88) m J m ( κ ) e imx , where J m is the m -th order2essel function of first kind. If we expand the oscil-latory term e i η in terms of Bessel functions and take thetime average of the integration, we get the e ff ective tun-neling expression T e f f . T = J − l ( κ ) e − il φ (5)We conclude that the force (2) modifies the tunnelingamplitude in the Hamiltonian. As it was discussedabove, this type of force breaks the PT symmetry ofthe original Hamiltonian. However, this PT symme-try breaking term is absent in the e ff ective Hamiltonian.The system is called pseudo PT symmetric, whicharises when not the original Hamiltonian but the e ff ec-tive Hamiltonian is PT symmetric [26, 27].The absolute value of the e ff ective tunneling parame-ter changes with κ and l . The Bessel function J − l ( κ ) isroughly like a decaying sine function. In the absenceof the oscillating force term, κ =
0, the Bessel func-tion is always zero, J − l (0) =
0. Therefore, the e ff ectivetunneling is suppressed and the spectrum becomes com-plex. This result is a direct consequence of the broken PT symmetry that occurs when l changes from zero tononzero. Note that vanishing T e f f . does not mean entiredestruction of tunneling because of the neglected o ff -resonant terms in the derivation of the e ff ective tunnel-ing. The tunneling is partially restored and the systementers the pseudo PT symmetric phase with the addi-tional application of oscillating force term. This is theregion where we study super Bloch oscillation. We em-phasize that the pseudo PT symmetry is spontaneouslybroken whenever κ is a root of Bessel function of order l since the e ff ective tunneling vanishes.Having derived the e ff ective Hamiltonian, we can dis-cuss the reality of the spectrum. For an infinitelyextended lattice, the energy eigenvalues of the e ff ec-tive Hamiltonian are composed by two minibands andgiven by E e f f . = ∓ (cid:113) | T e f f . | cos ( ka ) − γ . Thereforethe spectrum is complex at any γ for the periodical lat-tice. If the system, on the other hand, is truncated, thespectrum of the e ff ective system becomes real providedthat non-Hermitian degree is below than a critical num-ber. If it is beyond the critical number, spontaneous PT symmetry breaking occurs and the eigenfunctions of theHamiltonian are no longer simultaneous eigenfunctionof PT operator. The energy spectrum of the e ff ectiveHamiltonian (3) for the truncated lattice can be foundnumerically. In our system, the critical value decreaseswith increasing number of truncated lattice sites. As N → ∞ , the critical value goes to zero. A discussionand analytical formula for open and periodical bound-ary conditions can be found in [31]. The above formal- ism is valid when t >> /ω . It does not account for thedynamics of the system, either. However, it gives an in-sight to us on the reality of the spectrum. This motivatesus to study undamped / non-amplified super Bloch oscil-lation in our non-Hermitian system. Below, we studythe dynamics of the system numerically.
3. Super Bloch Oscillations
To observe super Bloch oscillation, the frequency ω should be slightly detuned from the Bloch frequency. Tosatisfy this o ff -resonance condition, we can make thefollowing replacement in the expression (2): l → l + δ ,where the parameter δ << δ makes the amplitude of oscillation much largerthan that of Bloch oscillation as can be seen below. Westress that δ acts perturbatively to the e ff ective Hamilto-nian (3) and hence we say that corresponding spectrumchanges slightly.Let us first review the super Bloch oscillation brieflyfor the Hermitian system. The semiclassical expla-nation for the super Bloch oscillation is as follows.In the absence of gain and loss, γ =
0, the en-ergy expression for the tight binding lattice is givenby E = − T cos( ka ), where the quasi-momentum k changes with the force according to dk / dt = F ( t ).Then the instantaneous group velocity for the force (2)reads v g = dE / dk = T a sin(( l + δ ) ω t + κ sin( ω t )) (Weset φ = v g over one period using the Jacobi-Anger relation.Then the mean position of the wave packet can be foundusing ¯ v g = d ¯ x g / dt , where overline denotes average¯ x g ( t ) = T a J − l ( κ ) cos( δω t ) δ ω (6)This expression clearly shows us why the oscillation iscalled super Bloch oscillation. This is because of thefact that the amplitude of the oscillation scales with theinverse of the detuning δ . The amplitude of the oscil-lation can be enhanced and oscillation with a giant am-plitude can be observed if the detuning is made smallenough. Not only the amplitude but also the period ofthe super Bloch oscillation is large. The higher the am-plitude of the oscillation, the larger the period is.The semi-classical picture for the Hermitian lattice [12]can not be directly generalized to a non-Hermitian lat-tice as discussed in [17]. Below, we perform numericalcomputation to investigate super Bloch oscillation forthe non-Hermitian truncated lattices. We take T = N =
30 from now on. Let us first write the solution3s | ψ ( t ) > = N (cid:88) n = c n ( t ) | n > , where n is the lattice site num-ber. The corresponding equation for the time-dependentcoe ffi cients c n ( t ) can be obtained by substituting this ex-pansion into the Schrodinger-like equation, H | ψ > = i ˙ ψ .Suppose first that γ =
0. The Fig. 1 depicts the evolu-tion of a wave packet (snapshot of | c n ( t ) | ) for a singlesite initial input, c n ( t ) = δ n , for the parameters l = δ = .
1. For comparison, the intensity profile forBloch ( κ =
0) and super Bloch oscillations ( κ = .
5) areshown in the figure. The presence of the constant forceforms the Wannier-Stark ladder and therefore Bloch os-cillation occurs as can be seen from the upper panel.The additional presence of the oscillating force togetherwith the detuning δ = . PT symmetric super Bloch os-cillation, γ (cid:44)
0. In the absence of the oscillatingforce, κ =
0, the energy spectrum becomes complexand consequently the total intensity grows exponentiallyin time. As discussed above, the presence of an addi-tional oscillating force makes the e ff ective Hamiltonian PT symmetric although the original Hamiltonian is not.Therefore, super Bloch oscillation without amplified in-tensity can be observed as long as γ is below than acritical value, γ PT . If it exceeds the critical value, am-plification / damping occurs. Consider first γ < γ PT . Inthis case, energy spectrum is real and we don’t expectexponential growth of the total particle number. We nu-merically compute time evolution for a single site and abroad Gaussian initial excitations. The figure-2 plots thebreathing and oscillatory super Bloch oscillation corre-sponding to an initial single-site and broad-site excita-tions of the lattice. As can be seen from the figure, largeamplitude oscillation similar to the one in the Hermitianlattice occurs. However, we find that particle numberis not conserved but changes slightly with time. Thisis because of the fact that eigenstates of non-HermitianHamiltonian are not orthogonal to each other. A similare ff ect was also observed in an experiment on Bloch os-cillation for a PT symmetric Hamiltonian [20]. We em-phasize that non-conservation of the total particle num-ber breaks the exact self-imaging property of the lattice.However, this e ff ect is weak and the oscillation is almostperiodical if γ < γ PT . Note that the amplitude of the os-cillation can be made higher by choosing the detuning δ smaller than 0 .
1. However, our lattice is truncatedand once the wave packet reaches the lattice boundary,the oscillating character of the wave packet is lost. To
Figure 1: The parameter for both figures are given by N = l = δ = . ω =
1. In the upper figure, localization occurs when κ = κ = .
5. The horizontal andvertical axes represent N and time, respectively. sum up, we show that although (small amplitude) stableBloch oscillation does not occur in our non-Hermitianlattice, large amplitude Bloch oscillation is possible inthe same system. This is the main finding of this pa-per. Suppose next that the e ff ective system is in thespontaneously broken PT symmetric phase, γ > γ PT , where pairs of complex-conjugate eigenvalues appear.In this case, wave packet is amplified during the oscilla-tion. In fact, the total number of particles grows expo-nentially at almost every lattice sites even if only singlesite is initially excited. Therefore, super Bloch oscilla-tion does not occur in this case. If, on the other hand, γ is very close to γ PT , we can still observe at least a fewcycle large amplitude oscillation. As time goes on, newparticles that appear at every lattice sites become toomuch and oscillatory motion of the wave packet is con-4 igure 2: The parameters are the same as in the figure 1, but γ = . γ PT = .
1) If only single site in initially populated, breathing superBloch oscillations occurs (a). In the case of broad Gaussian initialexcitation, the wave packet undergoes an oscillatory motion (b). Thehorizontal and vertical axes represent N and time, respectively. Notethat super Bloch oscillation is destroyed after a couple of cycles thatcorresponds to a large propagation distance. sequently destroyed. The Fig-3 shows numerical sim-ulation for this case when the initial wave function isGaussian. As can be seen, the wave packet undergoeshigh amplitude oscillations in real space and secondaryemission occurs.So far, we have only considered super Bloch oscilla-tion. For our problem, dynamic localization and photonassisted tunneling in the PT symmetric region can alsobe investigated. Let us mention them briefly. The dy-namic localization appears in the presence of the oscil-lating force, l = , κ (cid:44)
0. At some particular strength ofthe oscillating force, an initially localized wave-packetstops spreading in time. In other words, the system isin the dynamically localized regime if the e ff ective tun- Figure 3: The parameters are the same as in the figure 1, but γ = . neling vanishes, which occurs whenever κ is a root ofthe Bessel function of order l . Note that the phase φ plays no role on dynamic localization. If the e ff ectivetunneling is zero, then the critical point γ PT is zero, too.We numerically solve our equation for l = κ = . J (2 . = γ becomes dif-ferent from zero. Secondly, let us mention about pho-ton assisted tunneling. It is well known that the trans-lational symmetry of the lattice is broken and tunnel-ing is consequently suppressed by the presence of theconstant force. Thus, an initially localized wave-packetremains localized in the Hermitian lattice. However,tunneling is partially restored if an additional oscillat-ing force is also present in the system. This e ff ect isknown as photon assisted tunneling [9] since the role ofthe photons is played by the oscillation of the force atresonant frequencies. In our system, partial restorationof the tunneling increases γ PT . In the presence of onlyconstant force, γ PT = γ PT from zero value to a non-zerovalue.In conclusion, the main finding of our paper is the ex-istence of non-amplified super Bloch oscillation in a PT symmetric lattice even though Bloch oscillation isamplified in the same lattice. In a non-Hermitian lat-tice, the standard semi-classical picture does not work.Using the high-frequency Floquet analysis, we haveshown that super Bloch oscillation can occur if the non-Hermitian system is in pseudo PT symmetric region.We have numerically seen that the super Bloch oscilla-tion in our system with γ < γ PT is very similar to the5ne in the Hermitian lattice, γ = γ > γ PT . On the otherhand, we have shown that a few cycle super Bloch oscil-lation with secondary emission that has no analogue ina Hermitian system can be observed if γ is very close to γ PT . We have also discussed dynamic localization andphoton assisted tunneling in our non-Hermitian system. References [1] F. Bloch, Z. Phys. , 555 (1928).[2] C.Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz and K.Kohler, Phys. Rev. Lett. , 3319 (1993).[3] B.P. Anderson and M.A. Kasevich, Science , 1686 (1998).[4] R. Morandotti, U. Peschel, J.S. Aitchison, H.S. Eisenberg, and Y.Silberberg, Phys. Rev. Lett. , 4756 (1999).[5] D. H. Dunlap and V. M. Kenkre, Phys. Rev. B , 040404 (2008).[7] A. Alberti V. V. Ivanov G. M. Tino and G. Ferrari, Nature Physics
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