Hartman effect from layered PT -symmetric system
HHartman effect from layered
P T -symmetric system
Mohammad Hasan , , Bhabani Prasad Mandal Space Science Programme Office, Indian Space Research Organisation,Bangalore-560094, INDIA , Department of Physics, Banaras Hindu University, Varanasi-221005, INDIA.
Abstract
The time taken by a wave packet to cross through a finite layered
P T -symmetric systemis calculated by stationary phase method. We consider the
P T - symmetric system offix spatial length L consisting of N units of the potential system ‘+ iV ’ and ‘ − iV ’ ofequal width ‘ b ’ such that L = 2 N b . In the limit of large ‘ b ’, the tunneling time isfound to be independent of L and therefore the layered P T -symmetric system displaythe Hartman effect. The interesting limit of N → ∞ such that L remains finite isinvestigated analytically. In this limit the tunneling time matches with the time taken tocross an empty space of length L . The result of this limiting case N → ∞ also shows theconsistency of phase space method of calculating the tunneling time despite the existenceof controversial Hartman effect. The reason of Hartman effect is unknown to presentday however the other definitions of tunneling time that indicate a delay which dependsupon the length of traversing region have been effectively ruled out by recent attosecondmeasurements. e-mail address: [email protected], [email protected] e-mail address: [email protected], [email protected] a r X i v : . [ qu a n t - ph ] A ug Introduction
The tunneling of a particle from a classically forbidden region is one of the earliest studiedproblems of quantum mechanics which started in the year 1928 [1, 2]. Since then thequantum mechanical tunneling has long been studied by several authors [3, 4, 5]. Howeverhow much time does a particle take to tunnel through the barrier is an open questionboth theoretically and experimentally. In the year 1962, Hartman applied the concept ofstationary phase method to calculate the time taken by a particle to cross a classicallyforbidden region. He considered the tunneling region imposed by metal-insulator-metalsandwich and shown that tunneling time is independent of the barrier thickness for largebarriers[6]. This paradox was later called as Hartman effect. This phenomena was alsofound by Fletcher in an independent study [7] in later years. The saturation of thetunneling time with barrier thickness is an obvious unexpected result as it contradictwith the principles of special relativity. This has also created doubt on the methodologyof stationary phase method of calculating the tunneling time and has prompted newdefinitions of tunneling time (see [8]). However the numerical calculation of the tunnelingtime by monitoring the time evolution of a particle wave packet have also indicated thatthe tunneling time agrees well with the one calculated by stationary phase method [9].Tunneling time have been studied for double and multi-barrier structures as well. Thestudies on double barrier structure have revealed the existence of generalized Hartmaneffect where the tunneling time is independent of the intervening gap of a double barriersystem for large thickness [10]. This also holds for multi-barrier case [11]. See [12, 13,14] for critical comments on generalized Hartman effect. For the case when particleenergy lies in the energy gap of super- lattice structure, the tunneling time throughthe super-lattice can be smaller than the free motion time [15]. Ref [15] also derivesthe close form expression of the tunneling time for super-lattice tunneling. Owing to thegeneralization of standard quantum mechanics to non-Hermitian quantum mechanics, theparticle tunneling have also been studied for complex barriers. Hartman effect doesn’texist in complex barrier tunneling [16]. However the approach presented in [16] have beenquestioned in [17] and a two channel formalism for incorporating inelasticity in barrierpotential shows that Hartman effect exist for inelastic barriers with weak absorption[17, 18]. We have also calculated tunneling time in space fractional quantum mechanics(SFQM) and it is shown analytically that Hartman effect doesn’t occur in SFQM [19].Experimental attempts have been made to test the finding of theoretical results abouttunneling time. Many authors have contributed towards this. The earlier experimentshave indicated the superluminal nature of the tunneling time [20] and are found to beindependent of the thickness of the tunneling region [21, 22, 23, 24, 25, 26]. The tunnelingtime was also studied with double barrier optical gratings [25] and double barrier photonicband gaps [27] and is found to be paradoxically short. Very recently, the attosecondmeasurement on one-electron tunneling dynamics has indicated that the tunneling is an2nstantaneous phenomena [28] within the experimental limitations of 1 . P T -symmetric quantum mechanics [29]. The non-Hermitian Hamiltonian display several new features which are originally absent in Hermi-tian Hamiltonians. The important features are exceptional points (EPs) [30, 31], spectralsingularity (SS) [32, 33],coherent perfect absorption (CPA) [34]-[38], critical coupling (CC)[39]-[42] and CPA-laser [43]. Others notable features are invisibility [44, 45, 46] and reci-procity [47]. Recently CPA and SS have also been studied in the context of non-Hermitianfractional quantum mechanics [48].In the present work, we investigate the existence of Hartman effect from a layered
P T -symmetric potential. We first take a ‘unit cell’
P T -symmetric system of width 2 b made by two complex rectangular barriers iV and − iV each of width b . We repeat thisunit cell N times without any intervening gap covering the total span L = 2 N b . It isfound that for b → ∞ , the tunneling time is independent of L which shows Hartmaneffect from this system. For a constant finite support L , each complex barriers becomeinfinitely thin in the limit N → ∞ (as b = L N ). In this limit it is expected that the effectof adjacent barriers iV and − iV will cancel each other and the tunneling time shouldequate to free passage time by the particle traversing the length L in vacuum. This isindeed the case and the proof is shown analytically. This also shows the consistency ofphase delay method of computing the tunneling time. We organize the paper as follows:In section 2 we briefly mention the phase space methodology of tunneling time. Section3 provides the detail steps of calculating the tunneling time from our locally periodic P T -symmetric system. In this section we give the close form expression of the tunnelingtime. Sub-section 3.3 and 3.4 discusses the limiting case to obtain Hartman effect andfree passage time. Finally the results are discussed in section 4.
In this section we briefly introduce the reader about the stationary phase methodologyof calculating the tunneling time [49]. In stationary phase method, the tunneling timeof a localized wave packet traversing a potential barrier is defined as the time differencebetween the incoming and the outgoing peak of the wave packet. Consider a normalizedGaussian wave packet U k ( k ) with mean momentum (cid:126) k . The time evolution of the wavepacket propagating to positive x direction would be (cid:90) U k ( k ) e i ( kx − Et (cid:126) ) dk (1)3here k = √ mE . On traversing the potential barrier of width L , the transmitted wavepacket would be (cid:90) U k ( k ) | A ( k ) | e i ( kx − Et (cid:126) + θ ( k )) dk (2)where A ( k ) = | A ( k ) | e iθ ( k ) is the transmission coefficient through the potential barrier V ( x ) ( V ( x ) = V for 0 ≤ x ≤ L and zero elsewhere). According to stationary phasemethod the tunneling time τ is given by ddk (cid:18) kL − Eτ (cid:126) + θ ( k ) (cid:19) = 0 (3)The tunneling time expression then becomes, τ = (cid:126) dθ ( E ) dE + L ( (cid:126) km ) (4)For a square barrier potential V ( x ) = V of width L , the tunneling time is τ = (cid:126) ddE tan − (cid:18) k − q kq tanh qL (cid:19) (5)Here q = (cid:112) m ( V − E ) / (cid:126) . It is seen that for L →
0, we have τ →
0. This is expected,however for L → ∞ , τ = m (cid:126) qk , This doesn’t involve any dependency on L i.e. tunnelingtime is independent of the width of the barrier L for thick barrier. This is the famousHartman effect. In the system of unit 2 m = 1, (cid:126) = 1, c = 1lim L →∞ τ = 1 qk (6) P T -symmetric system
As discussed in the previous section, the calculation of tunneling time by stationary phasemethod involves the calculation of phase of the transmission coefficient and differentiationof the same with respect to energy of the particle to obtain phase delay time. Thephase delay time when combined with the free passage time gives the net tunnelingtime of the particle. To achieve this we first calculate the transmission coefficient of thelocally periodic
P T -symmetric system by using transfer matrix method. For the sake ofcompleteness, we also describe the transfer matrix method of calculating the transmissioncoefficient. The details of the methodology and calculations are illustrated in subsequentsection.
The Hamiltonian operator in one dimension for a non-relativistic particle is (in the unit (cid:126) = 1 and 2 m = 1) H = − d dx + V ( x ) (7)4here V ( x ) ∈ C . V ( x ) → x → ±∞ . If (cid:82) U ( x ) dx , where U ( x ) = (1 + | x | ) V ( x ) isfinite over all x , then the Hamiltonian given above admits a scattering solution with thefollowing asymptotic values ψ ( k, x → + ∞ ) = A + ( k ) e ikx + B + ( k ) e − ikx (8) ψ ( k, x → −∞ ) = A − ( k ) e ikx + B − ( k ) e − ikx (9)The coefficients A ± , B ± are connected through a 2 × M , called as transfer matrixas given below, (cid:18) A + ( k ) B + ( k ) (cid:19) = M ( k ) (cid:18) A − ( k ) B − ( k ) (cid:19) (10)where, M ( k ) = (cid:18) M ( k ) M ( k ) M ( k ) M ( k ) (cid:19) (11)With the knowledge of the transfer matrix M ( k ), the transmission coefficient t ( k ) is easilyobtained as the inverse of the lower diagonal element (for wave incidence from left of thepotential) i.e. t ( k ) = 1 M ( k ) (12)The transfer matrix shows composition property. If the transfer matrix for two non-overlapping finite scattering regions V and V , where V is to the left of V , are M and M respectively, then the net transfer matrix M net of the whole system ( V and V ) is M net = M .M (13)The composition result can be generalized for arbitrary numbers of non-overlapping fi-nite scattering regions. Knowing the transfer matrix, one easily compute the scatteringcoefficients by inversing the lower diagonal element of the net transfer matrix. P T -symmetricsystem and tunneling time
Fig 1 represent our
P T -symmetric ‘unit cell’ barrier configuration. The transfer matricesfor the two barriers labeled as ‘1’ and ‘2’ are M , ( k ) = 12 (cid:18) e − ikb P , e − ikb (1+2 j ) S , − e ikb (1+2 j ) S , e ikb P , − (cid:19) (14)In the above j = 0 for barrier-1 and j = 1 for barrier-2 as labeled in Fig 1 and, P , ± = 2 cos k , b ± i (cid:18) µ , + 1 µ , (cid:19) sin k , b (15)5igure 1: A P T -symmetric ‘unit cell’ consisting a pair of complex conjugate barrier. y -axis represent the imaginary height of the potential. S , = i (cid:18) µ , − µ , (cid:19) sin k , b (16) µ , = k , k , k , = (cid:112) E − V , (17)For the present problem V = iV and V = − iV . By using the composition properties ofthe transfer matrix, the net transfer matrix of our P T -symmetric ‘unit cell’ can be easilyfound as M ( k ) = 14 (cid:18) e − ikb ( P P − S S ) e − ikb ( P S + P − S ) − e ikb ( P − S + P S ) e ikb ( P − P − − S S ) (cid:19) (18)If the transfer matrix of the ‘unit cell’ potential is known, one can obtain the transfermatrix of the corresponding locally periodic [50] as well as locally super periodic potential[51] and therefore the transmission coefficient from such a system is easily obtained. InFig 2 we show the locally periodic P T -symmetric potential obtained by periodic repetitionof our
P T -symmetric ‘unit cell’. For the system to be
P T -symmetric, there required tobe total n barriers ( combined iV and − iV barriers and not the ‘unit cell’) on either sideof the origin. Thus the total barriers are n = 2 N , N ∈ I + and the net ‘unit cells’ are N .On doing the algebraic simplification of the transfer matrix 18 for the potential V = iV , V = − iV by using eqs. 15, 16, 17 and following the method outlined in [50] we obtain theclose form expression of the transmission coefficient. Below we write the final expression6igure 2: A periodic
P T -symmetric potential made by the periodic repetition of the ‘unitcell’ potential shown in Fig 1. y -axis represent the imaginary height of the potential. of transmission coefficient from our locally periodic P T -symmetric system as t = e − ikL G ( k ) (19)where G ( k ) is given as G ( k ) = ( ξ − iχ ) U N − ( ξ ) − U N − ( ξ ) (20) U N ( ξ ) is Chebyshev polynomial of second kind and L = 2 N b is the net spatial extent ofthe periodic potential. { ξ, χ } ∈ R . The expressions for ξ and χ are given below ξ = 12 (cos 2 α + cosh 2 β ) − cos 2 φ (cosh β sin α + cos α sinh β ) (21) χ = 12 ( U + cos φ sin 2 α + U − sin φ sinh 2 β ) (22)In the above equations, U ± = kρ ± ρk , α = bρ cos φ , β = bρ sin φ . ρ and φ are themodulus and phase of k = √ k + iV = ρe iφ respectively such that ρ = ( k + V ) and φ = tan − (cid:0) Vk (cid:1) . It can be noted that k = ρe − iφ . The detail exercise of the above is leftto the interested reader. To find the tunneling time, we now separate G ( k ) in real andimaginary parts and calculate the phase θ of the transmission coefficient t (eq. 19). Theexpression of θ is calculated to be θ = tan − ( qχ ) − kL (23)7ere, q = U N − ( ξ ) T N ( ξ ) (24)Using θ in eq. 4 the final expressions for the tunneling time from our P T -symmetricsystem is found to be τ = 12 k (1 + q χ ) (cid:20) qχ (cid:48) + χ (cid:18) N ξ (cid:48) ξ − − N ξ (cid:48) q − qξξ (cid:48) ξ − (cid:19)(cid:21) (25)In the above (cid:48) denote the derivative with respect to k . The expression for ξ (cid:48) and χ (cid:48) are ξ (cid:48) = 2 β (cid:48) sin φ sinh 2 β − α (cid:48) cos φ sin 2 α + φ (cid:48) sin 2 φ (cosh 2 β − cos 2 α ) (26) χ (cid:48) = U + ( α (cid:48) cos 2 α cos φ − φ (cid:48) sin φ sin 2 α ) + U − ( β (cid:48) cosh 2 β sin φ + 12 φ (cid:48) cos φ sinh 2 β )+ 12 U (cid:48) + cos φ sin 2 α + 12 U (cid:48)− sin φ sinh 2 β (27)Other quantities with prime are defined below, U (cid:48)± = V ρ (cid:18) ∓ ρ k (cid:19) , ρ (cid:48) = (cid:18) kρ (cid:19) , φ (cid:48) = − kVρ α (cid:48) = bkρ ( V sin φ + k cos φ ) , β (cid:48) = bkρ ( − V cos φ + k sin φ ) To study the existence of Hartman effect we take the limiting case b → ∞ of the tunnelingtime τ and evaluate the dependency on b in this limiting case. To arrive at the final resultwe evaluate the limiting case of the following quantitieslim b →∞ ξ ∼ f e β (28)lim b →∞ ξ (cid:48) ∼ ( f + bf ) e β + bf (29)lim b →∞ q ∼ ξ ( b → ∞ ) (30)lim b →∞ χ ∼ U − e β sin φ (31)lim b →∞ χ (cid:48) ∼ bg + ( bg + g ) e β (32)lim b →∞ χξ = γ (33)8n eqs 28-33, we have used ‘ ∼ ’ when the right hand side (RHS) depends upon ‘ b ’. In casewhen ‘ b ’ doesn’t appear in RHS, we have used ‘=’ sign. Various quantities appearing inthe eqs. 28- 33 are defined below. f = 12 sin φ (34) f = 12 φ (cid:48) sin 2 φ (35) f = − kρ sin 2 α cos φ ( k cos φ + 12 V sin 2 φ ) (36) f = k sin φρ ( k sin φ − V sin 2 φ ) (37) g = kU + cos 2 αρ ( k cos φ + 12 V sin 2 φ ) (38) g = kU − ρ ( k sin φ − V sin 2 φ ) (39) g = 14 ( φ (cid:48) U − cos φ + U (cid:48)− sin φ ) (40) γ = 12 U − csc φ (41)The quantities defined from eqs 35- 41 are independent of ‘ b ’. Due to lim b →∞ q ∼ ξ and ξ >> b → ∞ , the first and second term of the parenthesis in the eq. 25 cancelsand we are left with lim b →∞ τ = 12 k lim b →∞ (cid:20)(cid:18)
11 + γ (cid:19) (cid:18) χ (cid:48) ξ − χξ (cid:48) ξ (cid:19)(cid:21) (42)From eq. 42 , it is evident that the tunneling time is independent of the repetitions N of the unit cell in the limit b → ∞ i.e. independent of the net spatial extent. This isHartman effect. We further illustrate that the right hand side of eq. 42 is independent ofthe width ‘ b ’ as well. Eq. 42 can be written aslim b →∞ τ = 12 k (1 + γ ) (cid:20) g − γf f + bf ( g − γf ) (cid:21) (43)where we have used eqs 28, 29, 29 and 33 in arriving at eq. 43. From the expressions of g , γ and f it can be easily shown that g − γf = 0 (44)Thus, lim b →∞ τ = 12 k (1 + γ ) (cid:20) g − γf f (cid:21) (45)Right hand side of the above equation is independent of the net span L = 2 N b of thepotential as γ , f , f and g are independent of b . This proves the Hartman effect for our P T -symmetric system. This is also shown graphically for different N values in Fig 3.9igure 3: Variation of tunneling time τ with width ‘ b ’ of the ‘unit cell’ barrier for differentvalues of N . The black line is drawn at the theoretical value of τ in the limit b → ∞ obtained in eq. 45. It is seen that all curve of different N approaches to this value withincreasing b . In the figure V = 20 and E = 1 . The free propagation limit is expected when the width of each barriers are infinitely thinand number of the barriers are infinitely large such that the net span L = 2 N b of the
P T -symmetric potential is fixed. In this limiting case the gain ‘+ iV ’ and loss ‘ − iV ’ partare expected to cancel each other and the net medium of length L is free from any net gainor loss. In this case the tunneling time of a particle will approach to the free propagationtime L/ k . To illustrate this , we take b = L N where L is fixed and study the limitingcase of N → ∞ in eq. 25. Taking the limiting case of N → ∞ , it can be shown thatlim N →∞ τ = L kρ (cid:20) ( k U + + ρ U (cid:48) + ) cos φ + ( k U − + ρ U (cid:48)− ) sin φ + 12 ( U + − U − )( kV sin 2 φ − φ (cid:48) ρ ) (cid:21) (46)Using the expressions of U + , U − and U (cid:48) + , U (cid:48)− , derived earlier this further simplifies tolim N →∞ τ = L kρ (cid:20) ρ + (cid:18) k − V k (cid:19) ρ cos 2 φ + 2 V ρ sin 2 φ (cid:21) (47)Using sin 2 φ = Vρ , cos 2 φ = k ρ in the above equation, the term in the square parenthesissimplifies to 2 ρ and therefore, lim N →∞ τ = L k , for fixed L (48)10 iiFigure 4: Variation of tunneling time τ with the repetitions ‘ N ’ of the P T -symmetricbarrier for fixed spatial extent L . The horizontal black curve is drawn at the value τ = L k .The curves correspond to different value of V shown in the figure. The Fig-i is for E = 1 and Fig-ii is for E = 4 . It is observed that with increasing N , all curves approach to thefree propagation time L k i.e. the time taken by a particle of wave vector k to travel anempty region L . k traversing thefree length L (without medium). We have demonstrated this graphically in Fig 4. In the present work, we have studied the tunneling time from a layered
P T -symmetricsystem by stationary phase method. We consider the
P T - symmetric system of fix spatiallength L consisting of N units of the potential system ‘+ iV ’ and ‘ − iV ’ of equal width ‘ b ’such that L = 2 N b . We have derived closed form expression of tunneling time for such asystem for arbitrary finite repetitions N . It is shown analytically and also demonstratedgraphically that in the limit of large b , the tunneling time is independent of b as well asthe repetitions N . Thus the Hartman effect exist for layered P T -symmetric system.We have chosen our
P T -symmetric system in such a way that for a fixed spatial extent‘ L ’ of the system, the ‘unit cell’ P T -symmetric system becomes infinitely thin when thenumber of repetitions ‘ N ’ is infinitely large. In this particular limit the whole system isexpected to resemble an empty space of length L and the tunneling time would approachthe free propagation time L k (chosen unit 2 m = 1, (cid:126) = 1, c = 1) where k is the wavevector of the particle. This is indeed the case and is proven analytically. We call thislimiting case as free propagation limit and is demonstrated graphically as well. Theanalytical proof of free propagation limit also indicates the consistency of stationary phasemethod of calculating the tunneling time. To the best of our knowledge , tunneling timeobtained by stationary phase method are the only known time which shows insensitivityof barrier thickness for large barriers and thus tunneling can be instantaneous. The recentattosecond measurement have indicated the instantaneous nature of tunneling of singleelectron system and thus have ruled out the other used definitions of tunneling timewhich are sensitive to the thickness of the tunneling region. It is an extraordinary factto see that the mathematically complicate expressions of the derived tunneling time bystationary phase method for finite layered P T -symmetric system correctly reduces to freepropagation time for length L in the limit N → ∞ for fix L . Acknowledgements :MH acknowledges supports from Director-SSPO for the encouragement of research activ-ities. BPM acknowledges the support from CAS, Department of Physics, BHU.
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