Wave packet evolution in non-Hermitian quantum systems
WWave packet evolution in non-Hermitian quantum systems
Eva-Maria Graefe and Roman Schubert Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom Department of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom
The quantum evolution of the Wigner function for Gaussian wave packets generated by a non-Hermitian Hamiltonian is investigated. In the semiclassical limit (cid:126) → Effective non-Hermitian Hamiltonians have long beenused for the description of a wide range of open quan-tum systems [1]. Their applications range from chemicalreactions to ultra cold atoms and laser physics. Com-plex potentials for matter waves can be tailored exper-imentally using standing light waves [2], and the com-plex Schr¨odinger equation appears in optics using ma-terials with complex refractive index. The latter anal-ogy was used in the recently reported first experimentalrealizations of PT-symmetric non-Hermitian Hamiltoni-ans [3, 4], which boosted the interest in the field further.While much attention has been paid to the theoreticalstudy of example systems [5, 6], the generic dynamicalfeatures remain hitherto mostly unexplored. The presentpaper aims to fill this gap by investigating wave packetdynamics for general non-Hermitian systems.The time evolution of wave packets is a powerful toolfor the understanding of both dynamical and stationaryproperties of Hermitian quantum systems [7, 8]. Further-more, it is a convenient way to investigate the semiclas-sical limit and thus forms the basis of many semiclassicalmethods. In what follows we shall generalize the funda-mental ideas of wave packet dynamics to non-Hermitiansystems. From this we derive classical equations of mo-tion in the spirit of a generalized Ehrenfest theorem. Thiswill pave the way for the development of a semiclassicalframework for non-Hermitian quantum systems, or moregenerally, absorbing wave equations.The semiclassical properties of non-Hermitian dynam-ics have recently been approached from various direc-tions. Examples include the study of ray dynamics ofabsorbing wave equations for weak non-Hermiticities [9];the mean-field approximation for a non-Hermitian many-particle system [6]; the quantum classical correspondencefor open quantum maps in the chaotic regime [10]; andcomplex extensions of the quantum probability distribu-tion [11]. In [12] a coherent state approximation hasbeen applied to non-Hermitian systems, and a general-ized canonical structure involving a metric gradient flowhas been identified. Here we go beyond this study by in-vestigating general Gaussian states that are allowed tochange their shapes during time evolution, which canbe interpreted as a time-dependent metric on the cor- responding classical phase space. To derive the classicalevolution equations in the spirit of the Ehrenfest theo-rem, we study the quantum evolution equation for theWigner function and take the semiclassical limit. Thisresults in a new type of classical phase space dynam-ics in which the evolution equations for the phase spacecoordinates depend on the local metric, and vice versa.The complex structure of the classical phase space, whichis rarely considered in Hermitian systems, thus becomeshighly relevant in the presence of non-Hermiticity. Themain dynamical effect of the anti-Hermitian part of thequantum Hamiltonian in this semiclassical approxima-tion is a damping of the motion. This strengthens theoften speculated connection to classical dissipation wherenon-Hermitian Hamiltonians are a recurrent theme in thesearch for quantum counterparts (see e.g., [13] and refer-ences therein).
Gaussian coherent states.
For a general n -dimensionalquantum system we study the evolution of initial wavepackets of the form ψ ( q ) = (det Im B ) / ( π (cid:126) ) n/ e i (cid:126) [ P · ( q − Q )+ ( q − Q ) · B ( q − Q )] , (1)where P, Q ∈ R n , and B is a complex symmetric ma-trix with positive definite imaginary part so that ψ islocalized around q = Q and normalized to unity. heGaussian states (1) with fixed B form a submanifoldof Hilberts space wich has a natural complex structureassociated with B . In the semiclassical limit this sub-manifold can be identified with the classical phase space,which thus inherits not only the symplectic but also ametric structure. These relations become most trans-parent using a phase space representation [14]. In thefollowing we will focus on the evolution of the Wignerfunction W of ψ . The Wigner function alllows a directcomputation of expectation values via phase space inte-grals (cid:104) ψ, ˆ Aψ (cid:105) / (cid:104) ψ, ψ (cid:105) = (cid:104) A (cid:105) W := (cid:82) W A d z/ (cid:82) W d z , whereˆ A is the Weyl quantization of A ( z ) and z = ( p, q ) arecanonical phase space coordinates. The Wigner functionof a Gaussian state (1) is Gaussian: W ( z ) = ( π (cid:126) ) − n e − (cid:126) ( z − Z ) · G ( z − Z ) . (2)Here Z = ( P, Q ) ∈ R n × R n and the matrix G is related a r X i v : . [ qu a n t - ph ] M a r to B by G = (cid:18) I − Re B I (cid:19) (cid:18) (Im B ) −
00 Im B (cid:19) (cid:18) I − Re B I (cid:19) . The matrix G is nondegenerate, positive, and symmet-ric, and thus acts as a metric on phase space. It is alsosymplectic, i.e. it satisfies G Ω G = Ω, whereΩ = (cid:18) − II (cid:19) (3)is the symplectic, or canonical, structure on phase space.With such a metric we can associate a compatible com-plex structure J with J = − I and Ω J = G [15]. Thestructure defined by Ω, G , and J turns the phase spaceinto a K¨ahler manifold.The Wigner function (2) is localized of order √ (cid:126) around the maximum Z = ( P, Q ). In the semiclassi-cal limit (cid:126) → Z .Hence the expectation value of an observable ˆ A satisfies (cid:104) ˆ A (cid:105) W = A ( Z ) + O ( (cid:126) ) (4)for A ( z ) smooth. The metric G describes the shape andorientation of W in phase space, and therefore determinesthe variance of observables:(∆ ˆ A ) ψ = (cid:126) ∇ A ( Z ) · G − ∇ A ( Z ) + O ( (cid:126) ) . (5)In the Hermitian case an initially Gaussian state staysapproximately Gaussian during the time evolution up tothe Ehrenfest time [7]. The center moves according tothe classical canonical equations of motion ˙ Z = Ω ∇ H ,and the evolution of the metric is governed by the lin-earized Hamiltonian flow around the classical trajectory.In the framework of the time-dependent variational prin-ciple [16] it has been shown that this dynamics can alsobe described by Hamiltonian equations of motion. Al-though it plays a central role in semiclassical methodsinvolving families of coherent states [16, 17], the met-ric is often little investigated, as it does not enter thedynamical equations for the phase space variables. Wewill see shortly that this is fundamentally changed in thepresence of non-Hermiticity. Non-Hermitian Wigner-von Neumann equation.
De-composing the Hamiltonian in its Hermitian and anti-Hermitian part ˆ H − iˆΓ, where we assume ˆ H and ˆΓ tobe given as the Weyl quantizations of sufficiently well-behaved classical observables H ( z ) and Γ( z ), the evolu-tion equation for a density operator ˆ W follows from theSchr¨odinger equation asi (cid:126) ∂ t ˆ W = [ ˆ H, ˆ W ] − i[ˆΓ , ˆ W ] + , (6)where [ · , · ] + denotes the anti commutator. Thus, the evo-lution equation of a general Wigner function is given byi (cid:126) ∂ t W = ( H(cid:93)W − W (cid:93)H ) − i(Γ (cid:93)W + W (cid:93) Γ) , (7) where ( A(cid:93)B )( z ) denotes the Weyl product [14] for twophase space functions A ( z ) and B ( z ): A(cid:93)B = A e i (cid:126) ←−∇ z · Ω −→∇ z B ∼ ∞ (cid:88) k =0 k ! (cid:18) i (cid:126) (cid:19) k A (cid:0) ←−∇ z · Ω −→∇ z (cid:1) k B .
Here the arrows over the differential operators indicatewhether they act on the function to the right or to theleft. We will now evaluate the leading order terms in (cid:126) of (7). The Hermitian part of the evolution equation(7) is the well known Moyal bracket with an asymptoticexpansion in odd powers of (cid:126) whose leading term givesthe Poisson bracket
H(cid:93)W − W (cid:93)H = i (cid:126) ∇ H · Ω ∇ W + O ( (cid:126) ) . (8)The anti-Hermitian part has an expansion in even powersof (cid:126) , with the first two terms given byΓ (cid:93)W + W (cid:93)
Γ = 2Γ W − (cid:126) Γ W + O ( (cid:126) ) , (9)where we introduced a second order differential operatordefined by Γ as ∆ Γ W := Γ (cid:0) ←−∇ z · Ω −→∇ z (cid:1) W . Denoting thematrix of second derivatives of Γ( z ) at z by Γ (cid:48)(cid:48) ( z ) theoperator ∆ Γ can be written in the form ∆ Γ = ∇ · Γ (cid:48)(cid:48) Ω ∇ ,with Γ (cid:48)(cid:48) Ω ( z ) := Ω t Γ (cid:48)(cid:48) ( z )Ω. It follows that ∆ Γ is Hermitian.Furthermore, if Γ (cid:48)(cid:48) ( z ) is symplectic, then Γ (cid:48)(cid:48) Ω = Γ (cid:48)(cid:48)− , and∆ Γ is the Laplace-Beltrami operator defined by Γ (cid:48)(cid:48) .Summarizing, in leading order of (cid:126) the dynamical equa-tion for the Wigner function reads (cid:126) ∂ t W = − (cid:18) − (cid:126) Γ − (cid:126) ∇ H · Ω ∇ + 2Γ (cid:19) W . (10)For vanishing Γ we recover the classical Liouville equa-tion for the transport of phase space densities. For non-vanishing and positive Γ, on the other hand, the Γ termdefines a diffusion equation.The higher order terms are of order (cid:126) | ∂ W | ( | ∂ H | + | ∂ Γ | ), i.e. they are small if the derivatives of H ( z ),Γ( z ) and W ( z ) do not grow too fast as (cid:126) →
0. If thederivatives of W are bounded, the first term on the rightside of (10) is of lower order than the other terms. Inthis case the solution W ( t, z ) is obtained by transportingthe initial W ( z ) along the Hamiltonian flow generated by H , multiplied by a damping factor, which is determinedby the integral of Γ along the Hamiltonian trajectoriesof H . This behavior is well known from damped waveequations. For a Gaussian initial state (2), on the otherhand, the term (cid:126) ∆ Γ W = O ( (cid:126) ) is of the same order asthe Hamiltonian term in (10), and the dynamics differdrastically from the Hermitian case. Gaussian evolution.
In what follows we investigate thesolution of (10) for an initial Gaussian Wigner function(2). Inserting a Gaussian ansatz for the time evolvedWigner function W ( t, z ) = α ( t )( (cid:126) π ) n e − (cid:126) δz · G ( t ) δz , with δz := z − Z ( t )into (10) yields (cid:20) (cid:126) ˙ αα + 2 ˙ Z · Gδz − δz · ˙ Gδz (cid:21) W ( z )= (cid:20) δz · G Γ (cid:48)(cid:48) Ω δz − ∇ H · Ω Gδz − (cid:126) (cid:0) Γ (cid:48)(cid:48) Ω G (cid:1) − (cid:21) W ( z ) . (11)Following the well established method of Heller and Hepp[7], we expand Γ( z ) and H ( z ) up to second order around z = Z : Γ( z ) ≈ Γ( Z ) + ∇ Γ( Z ) · δz + δz · Γ (cid:48)(cid:48) ( Z ) δz and ∇ H ( z ) ≈ ∇ H ( Z ) + H (cid:48)(cid:48) ( Z ) δz . Since W ( z ) is localizedaround z = Z with a width proportional to √ (cid:126) the re-mainder terms are of order (cid:126) / . Separating differentpowers of δz = z − Z in (11) then yields the followingthree equations of motion for Z ( t ), G ( t ) and α ( t ):˙ Z = Ω ∇ H ( Z ) − G − ∇ Γ( Z ) (12)˙ G = H (cid:48)(cid:48) ( Z )Ω G − G Ω H (cid:48)(cid:48) ( Z ) + Γ (cid:48)(cid:48) ( Z ) − G Γ (cid:48)(cid:48) Ω ( Z ) G (13)˙ αα = − (cid:126) Γ( Z ) −
12 tr (cid:2) Γ (cid:48)(cid:48) Ω ( Z ) G (cid:3) . (14)To obtain (13) the symmetry enforcing convention G =( G t + G ) / W depends only on the sym-metric part of G any anti-symmetric part is unobservable.The time evolution of expectation values and vari-ances of arbitrary observables in Gaussian coherent statesfor small (cid:126) is determined by Z ( t ) and G ( t ) accordingto (4) and (5). Equations (12)-(14) can be interpretedas the hitherto unidentified semiclassical limit of non-Hermitian quantum dynamics. This result goes beyondprevious studies of the non-Hermitian Ehrenfest theo-rem [12, 13, 18] for two reasons. First, previous stud-ies usually focussed on unnormalised expectation values,which prevented the identification of a classical structure,and second, disregarded the role of the metric, related tothe widths of the quantum wave packet. The dynamics(12) emerging as the classical limit is no longer Hamil-tonian, but has a Hamiltonian part and a gradient part,determined by the Hermitian and anti-Hermitian partsof ˆ H − iˆΓ, respectively. The main dynamical effect of theanti-Hermitian part is to drive the motion towards theminima of Γ. In addition, this gradient part is coupledto an evolution equation (13) for the metric G which inturn depends on (12). In this context it is importantto note that (13) preserves the symplectic nature of G and hence describes an evolution of the complex struc-ture on phase space. Further, the anti-Hermitian partleads to a change of the overall probability according toequation (14), which can be interpreted as absorption oramplification. The first term gives the contribution fromthe center and the second term captures the influence ofthe width of the Wigner function. Note, however, that af-ter renormalization the non-Hermitian Schr¨odinger equa-tion is equivalent to norm-conserving nonlinear modelsfor quantum dissipation [19]. p q − − − − − p q − − − − − p q − − − − − p q − − − − − p q − − − − − p q − − − − − − t ! t ! m a x FIG. 1. Time evolution of the exact Wigner function (leftcolumn) and the semiclassical approximation (right column)for an initial state at ( p, q ) = (5 ,
0) at different times ( t =1 , . ,
4) for the anharmonic oscillator. The white line showsthe motion of the center. The left panel on the bottomshows the norm of the exact quantum state (black dashedline) and the semiclassical approximation (blue line), and theright panel shows the largest eigenvalue of G ( t ) (blue line) incomparison with the Hermitian case γ = 0 (pink line). The quadratic approximation around z = Z ( t ) to H ( z )and Γ( z ) is expected to remain accurate so long as W ( t, z )stays strongly localized around z = Z ( t ). Since for asymplectic G we have (cid:107) G − (cid:107) = (cid:107) G (cid:107) , a suitable criterionfor this is (cid:126) (cid:107) G ( t ) (cid:107) (cid:28) . (15)The wave packet becomes delocalized at the Ehrenfesttime T E defined by (cid:126) (cid:107) G ( T E ) (cid:107) = 1 and the semiclassi-cal approximation based on the central trajectory Z ( t )breaks down. The nonlinear term in the equation for G ( t ) that is induced by Γ can have a stabilizing effecton the long-time evolution of G ( t ). Therefore, the non-Hermitian part can increase the Ehrenfest time, i.e. thetime scale for which the semiclassical approximation isvalid as compared to the Hermitian case. Examples.
To illustrate our results we consider twoexamples. The first example is a non-Hermitian an-harmonic oscillator with ˆ H = ω (ˆ p + ˆ q ) + β ˆ q and − − − − − q ! FIG. 2. Quantum evolution (black dashed line) versus semi-classical approximation (blue line) of a PT-symmetric waveg-uide for an initial state at ( p, q ) = (0 , ˆΓ = γ (ˆ p + ˆ q ) respectively, with ω = 1, γ = 0 .
2, and β = 0 .
5. This can be interpreted as a quantum analogueof a damped anharmonic oscillator (see, e.g., [13] andreferences therein). The simple structure of this modelmakes it an ideal testing ground for the semiclassicalapproximation proposed here. We set (cid:126) = 1, which isequivalent to a rescaling upon which β plays the role ofan effective (cid:126) . For β = 0 the semiclassical approxima-tion becomes exact. While the Hermitian part tries topropagate a state along closed curves of constant ener-gies around the origin, the anti-Hermitian part drives ittowards the origin, thus acting as a damping. Figure 1shows the exact numerical propagation and the semiclas-sical approximation for an initial Wigner function with G = I , which are in good agreement. Also the totalmass of the exact state, a measure for the absorption,due to the anti-Hermitian part is well described by thesemiclassical approximation α ( t ), as illustrated in the leftpanel on the bottom. The right panel shows the time-dependence of the larger eigenvalue of G as a measure for (cid:107) G ( t ) (cid:107) in comparison with a Hermitian case γ = 0. Theresult indicates that the Ehrenfest time, see (15), is in-creased in the non-Hermitian case, i.e. the semiclassicalapproximation is accurate over a longer time scale thanin a comparable Hermitian case.Second, we consider a simple model system for a PT-symmetric optical waveguide [3]: A single waveguide withharmonic confinement described by H = (ˆ p + ˆ q ), andan anti-Hermitian part Γ = 5 tanh(0 . q ) that models ab-sorption on one side and equally strong amplification onthe other side with a smooth transition in between. Al-though the Hamiltonian is complex, due to its specialsymmetry, it has real eigenvalues which can lead to apseudo-closed behavior. This phenomenon is captured byour classical approximation. Figure 2 shows an exampleof the full quantum evolution and its classical counter-part. Both the phase space evolution and the dynamicsof the norm are well approximated by the classical de-scription. In particular, despite the anti-Hermitian partin the Hamiltonian, no sink of the dynamics is observed.Note that there is a stable fixed point at ( p, q ) = (0 , Conclusion.
The results presented here for the evolu-tion of a Gaussian coherent state generated by a non-Hermitian Hamiltonian provide the basis for a more pro-found understanding of non-Hermitian time evolution, atopic of considerable interest in a wide range of subjects.In particular, the application to realistic examples of typ-ically non local non-Hermitian Hamiltonians appearing inresonance physics is an interesting task for future studies.The authors thank D. C. Brody and H. J. Korsch foruseful comments. EMG is supported by an Imperial Col-lege Junior Research Fellowship. [1] Moiseyev, N. 2011
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