Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians
aa r X i v : . [ m a t h - ph ] J u l Complexified coherent states and quantumevolution with non-Hermitian Hamiltonians
Eva-Maria Graefe and Roman Schubert Department of Mathematics, Imperial College London, London SW7 2AZ, UK School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK
Abstract.
The complex geometry underlying the Schr¨odinger dynamics of coherentstates for non-Hermitian Hamiltonians is investigated. In particular two seeminglycontradictory approaches are compared: (i) a complex WKB formalism, for which thecentres of coherent states naturally evolve along complex trajectories, which leads to aclass of complexified coherent states; (ii) the investigation of the dynamical equationsfor the real expectation values of position and momentum, for which an Ehrenfesttheorem has been derived in a previous paper, yielding real but non-Hamiltonianclassical dynamics on phase space for the real centres of coherent states. Bothapproaches become exact for quadratic Hamiltonians. The apparent contradiction isresolved building on an observation by Huber, Heller and Littlejohn, that complexifiedcoherent states are equivalent if their centres lie on a specific complex Lagrangianmanifold. A rich underlying complex symplectic geometry is unravelled. In particulara natural complex structure is identified that defines a projection from complex to realphase space, mapping complexified coherent states to their real equivalents.PACS numbers: 03.65Sq, 02.40Tt
1. Introduction
We analyse the geometric structure related to complexified coherent states, that isGaussian states with a formal complex centre. These states appear naturally insituations where the classical Hamiltonian function is complex valued, or in classicallyforbidden regions in the description of tunneling processes. Here we focus in particularon the quantum counterpart of complex Hamiltonians, that is the quantum timedependence generated by the Schr¨odinger equation with a non-Hermitian Hamiltonoperator. Such operators are of interest in many areas in science, in particular in physicsand chemistry. They appear, e.g., in the description of decay processes in quantummechanics, from early models in nuclear physics to the use of complex scaling in thecomputation of resonances [1]. In optics they naturally appear in the study of absorbingor optical active materials [2, 3], and in chemistry absorbing complex potentials arefrequently used for numerical simulations [4]. From a more mathematical perspectivethe spectral theory of non-Hermitian operators has received renewed interest recently, omplexified Coherent States
2. Coherent states and complex structures
It is well known that a manifold of coherent states can be interpreted as the phase spaceof the corresponding classical system [11, 23, 24], and how the symplectic structure ofclassical mechanics naturally arises from the geometry of coherent states. What isperhaps less appreciated, is that the coherent state manifold is further equipped with ametric and a complex structure, which is as well inherited to the classical system. As themetric structure does not appear in classical Hamiltonian equations of motion, it can beeasily overlooked. This is different in the context of dissipative classical systems, wherein addition to the symplectic flow of Hamiltonian dynamics, a metric gradient flow oftenappears. These types of dynamics are sometimes referred to as metriplectic flows [25].It has recently been pointed out, how similar structures arise in the semiclassical limitof non-Hermitian quantum theories, where the metric of the classical phase space isprovided by the metric on the space of coherent states [21, 26, 27].Let us now recall how certain classes of Gaussian coherent states endow classical omplexified Coherent States ψ BZ ( x ) = (det Im B ) / ( π ~ ) n/ e i ~ [ P · ( x − Q )+ ( x − Q ) · B ( x − Q )] , (1)with Z = ( P, Q ) ∈ R n × R n , and B ∈ M n ( C ), where B is symmetric and has positiveimaginary part, Im B >
0. This last condition ensures that the state is in L ( R n ) andthe prefactor is chosen such that the state is normalised to one.This coherent state manifold can in the semiclassical limit be identified with theclassical phase space via the centre Z , and the matrix B defines a metric and a complexstructure on phase space. The metric emerges in a natural way in a phase spaceformulation of quantum mechanics, using for example the Wigner function, see [12]for the following. The Wigner function of the state (1) is a Gaussian centred around Z = z , and localised on the order of ~ : W ( z ′ ) = 1( π ~ ) d e − ~ ( z ′ − Z ) · G ( z ′ − Z ) , (2)where z ′ = ( p ′ , q ′ ) denotes the coordinate and momentum variables, and the positivesymmetric matrix G is related to B via G = I − Re B I ! [Im B ] −
00 Im B ! I − Re B I ! . (3)Hence the matrix G defines a metric on phase space. This metric G has the additionalproperty that it is symplectic, i.e., it satisfies G Ω G = Ω, where Ω denotes theantisymmetric matrix Ω = − I n I n ! . (4)Since ΩΩ = − I this implies that − Ω G Ω = G − and using this it is easy to see that J := − Ω G , (5)defines a complex structure on phase space, i.e., it satisfies J = − I . Recall that ageneral Ω-compatible complex structure on phase space is a symplectic matrix J suchthat J = − I , and the matrix Ω J is positive definite. For later use we note that by (3)the complex structure J can be expressed in terms of B as J = − Re B [Im B ] − Im B + Re B [Im B ] − Re B − [Im B ] − [Im B ] − Re B ! = − Re B I − I ! [Im B ] −
00 Im B ! I − Re B I ! . (6) omplexified Coherent States Z and the metric G become apparent whenconsidering expectation values and variances of physical observables ˆ A . Let ˆ A be theWeyl quantisation of a smooth classical observable A , then the expectation value andvariance of ˆ A in the state (1) are given by h ψ, ˆ Aψ i|| ψ || = A ( Z ) + O ( ~ ) , (∆ ˆ A ) ψ = ~ ∇ A ( Z ) · G − ∇ A ( Z ) + O ( ~ ) , (7)i.e., Z is the centre of the phase space distribution of ψ and G determines its variance.Thus, in the limit of ~ → Z , andthe matrix G encodes a local metric at this point.Let us now extend the previous considerations to the case that the coherent stateis formally centred at a complex phase space point, i.e., we consider Gaussian coherentstates on R n similar to (1), but with a complex centre z = ( p, q ) ∈ C n × C n , ψ Bz ( x ) = (det Im B ) / ( π ~ ) n/ e i ~ [ p · ( x − q )+ ( x − q ) · B ( x − q )] , (8)and B ∈ M n ( C ) is again a symmetric n × n matrix with Im B >
0. Note that while thecentre is formally chosen complex, the wave function can still be viewed as a function ofa real coordinate x ∈ R n , and the condition Im B > L ( R n ).Similar states were considered previously by Huber, Heller and Littejohn, [13,22], and itwas noted that different choices of the complex centre z can lead to the same quantumstate. In particular, it was found that two centres, z and z ′ , define the same quantumstate if z − z ′ ∈ L B := { ( Bq, q ) ; q ∈ C n } , (9)where L B is a natural complex Lagrangian space associated with the state (8) which wewill analyze in more detail below. We will show here that this result is closely relatedto the complex structure J induced by B and can be reformulated in terms of a naturalprojection from complex phase space to real phase space defined by P J ( z ) := Re z + J Im z , (10)i.e., P J (Re z + i Im z ) = Re z + J Im z , where z ∈ C n × C n and the real and imaginaryparts are taken component-wise.The main result of this section can now be formulated as follows. Theorem 2.1.
Let ψ Bz be the coherent state (8) and P J be the projection (10) definedin terms of the complex structure (6) , then ψ Bz = e i ~ σ ( z,P J ( z )) ψ BP J ( z ) , (11) where with z = ( p, q ) and P J ( z ) = ( P, Q ) we have σ ( z, P J ( z )) = 12 ( P + p ) · ( Q − q ) . (12) omplexified Coherent States Furthermore the Wigner function of this state is W ( z ′ ) = e − σ ( z,P J ( z )) / ~ ( π ~ ) n e − ~ ( z ′ − P J ( z )) · G ( z ′ − P J ( z )) . (13)In other words, the complex ”centre” z = Re z + i Im z of the state (8) is projectedto the real centre Z = Re z + J Im z . Hence the coherent state centred at z is physicallyequivalent to the one centred at Z = P J ( z ). Proof.
We can write a coherent state (8) in the form ψ ( x ) = C e i ~ S ( x ) with S ( x ) = p · ( x − q ) + 12 ( x − q ) · B ( x − q ) , (14)and some constant C . The crucial step is to note that this state is concentratedaround the point where the imaginary part of S ( x ) is minimal, but since the parameter z = ( p, q ) ∈ C n × C n can be complex the minimum need not be located at x = q . Letus introduce Z = ( P, Q ) by the conditions ∇ Im S ( Q ) = 0 and P = ∇ Re S ( Q ) = ∇ S ( Q ) (15)then Q is the minimum of the imaginary part of S and by expanding S ( x ) up to secondorder around x = Q we can rewrite S ( x ) as S ( x ) = S ( Q ) + P · ( x − Q ) + 12 ( x − Q ) · B ( x − Q ) . (16)The complex structure will now appear if we express Z in terms of z . We find ∇ Im S ( x ) = Im[ p + B ( x − q )] = Im p + Im B ( x − Re q ) − Re B Im q and thus thecondition ∇ Im S ( Q ) = 0 gives Q = Re q + [Im B ] − Re B Im q − [Im B ] − Im p. (17)Since ∇ Re S ( x ) = Re[ p + B ( x − q )] = Re p + Re B ( x − Re q ) + Im B Im q we obtainfurther P = Re p − Re B [Im B ] − Im p + (Re B [Im B ] − Re B + Im B ) Im q . (18)These two equations yield Z = ( P, Q ) = P J ( z ), with J given by (6), and hence with(16) we find ψ Bz = e i S ( Q ) / ~ ψ BP J ( z ) . (19)It remains to compute S ( Q ) = p · ( Q − q ) + ( Q − q ) · B ( Q − q ). From (17) and (18) wefind B ( Q − q ) = P − p and hence S ( Q ) = 12 ( P + p ) · ( Q − q ) = σ ( z, P J ( z )) . (20)The form of the Wigner function (13) follows from (11) and (2). omplexified Coherent States B in the definition of acoherent state (11), the complex structure J (6), and the Lagrangian submanifold L B (9). Obviously J and L B are both defined in terms of B . We can further show thatthere are one-to-one relationships between all three of them.Let us recall that a linear subspace L ⊂ C n × C n is called Lagrangian if Ω | L = 0and dim L = n , and positive Lagrangian if in addition the quadratic form h ( z, z ′ ) := i2 z · Ω¯ z ′ (21)is positive on L , i.e., h ( z, z ) > z ∈ L . It is a well known result [28] that anypositive Lagrangian subspace can be written in the form (9): Lemma 2.2.
The subspace L B = { ( Bq, q ); q ∈ C n } defined in (9) is Lagrangian if B issymmetric, and positive Lagrangian if Im B > . On the other hand, if L ⊂ C n × C n isa positive Lagrangian subspace then there exists a symmetric B ∈ M n ( C ) with Im B > such that L = { ( Bx, x ) ; x ∈ C d } .Proof. It is clear from the definition that dim L = n . To check that Ω | L = 0 we choose z = ( Bx, x ) ∈ L and z ′ = ( Bx ′ , x ′ ) ∈ L and find z · Ω z ′ = − Bx · x ′ + x · Bx ′ = x · ( B T − B ) x ′ = 0, since B is symmetric. To check positivity we consider h L ( z, z ) = i z · Ω¯ z/ z = ( Bx, x ) which gives h L ( z, z ) = i2 [ − ( Bx ) · ¯ x + x · ¯ B ¯ x ] = i2 x · [ ¯ B − B T ]¯ x = x · Im B ¯ x ≥ . (22)Now assume L to be a positive Lagrangian subspace and consider the projection π : L → C n defined by π ( p, q ) = q . Then ker π = { } because if z = ( p, q ) ∈ ker π ,then q = 0 and hence i z · Ω¯ z/ L implies z = 0. Therefore themap π is invertible and since it leaves the q component invariant the inverse must be ofthe form π − ( q ) = ( Bq, q ) for some matrix B , i.e., L = { ( Bq, q ) ; q ∈ C n } . That B issymmetric and has positive imaginary part follows now as before from the fact that L is positive and Lagrangian.This establishes the one-to-one correspondence between Lagrangian subspaces andcomplex symmetric matrices with positive imaginary part. Let us now relate complexstructures and positive Lagrangian subspaces. By (10) and (11), the complex centres z and z ′ define the same state if P J ( z − z ′ ) = 0. Hence, the set of equivalent complexcentres is given by L := ker P J = { z ∈ C n × C n : P J ( z ) = 0 } . (23)According to the work of Heller et. al. [13, 22] we expect that L = L B . Let us, however,first show that L is actually a positive Lagrangian manifold, and furthermore, that theset of Ω-compatible complex structures is isomorphic to the set of positive Lagrangiansubspaces of C n × C n . omplexified Coherent States Lemma 2.3.
Let J be a Ω -compatible complex structure on R n × R n (see the definitionafter equation (5) ) and define P J ( z ) := Re z + J Im z , (24) then L := ker P J = { z ∈ V C ; Re z + J Im z = 0 } (25) is a positive Lagrangian subspace. Conversely, for every positive Lagrangian subspace L there exists a compatible complex structure J L such that L = ker P J L , i.e., z ∈ L ⇔ Re z + J L Im z = 0 . (26) Proof.
Note that since J = − I the relation Re z + J Im z = 0 can be rewritten asIm z = J Re z , (27)i.e., z ∈ L means z = ( I + i J ) Re z . Since J is non-degenerate we clearly havedim C L = n , and for z, z ′ ∈ L we get z · Ω z ′ = Re z · ( I + i J T )Ω( I + i J ) Re z ′ = Re z · (Ω − J T Ω J ) Re z ′ + i Re z · ( J T Ω + Ω J ) Re z ′ (28)and if J is symplectic and Ω J = G symmetric we get that z · Ω z ′ = 0, and thus L isLagrangian. Furthermore we find for z = ( I + i J ) Re z ∈ L i2 z · Ω¯ z = Re z · G Re z , (29)hence L is positive.On the other hand, assume L ⊂ C n × C n to be a positive Lagrangian subspaceand consider the map Re L : L → R n × R n , defined by Re L ( z ) = Re z . We claimthat this map is invertible. To see this assume z ∈ ker Re L , i.e, Re z = 0, theni z · Ω¯ z = i Im z · Ω Im z = 0, hence z = 0 by the positivity of L , so ker Re L = { } and Re L is invertible as claimed. The inverse must be of the form Re − L ( v ) = v + i J v for a linear map J : R n × R n → R n × R n . Then (28) with Re z = v shows that if L is Lagrangian J must be symplectic and G := Ω J symmetric, and (29) shows that G must be positive. Then J = Ω G Ω G = ΩΩ = − I , therefore J is a compatible complexstructure.In summary, we have shown that the set of complex symmetric matrices withpositive imaginary part, the set of positive Lagrangian subspaces, and the set of Ω-compatible complex structures are all isomorphic to each other. What we have notshown yet is that L B is actually mapped to the complex structure (6), i.e., thatker P J = L B . (30) omplexified Coherent States P J = dim L B it is enough to show that L B ⊂ ker P J , i.e., that for any z ∈ L B we have Re z + J Im z = 0. Now any element in L B is of the form z = ( Bq, q )for some q ∈ C n and a short calculation gives z = (cid:20) Re B − Im BI ! + i Im B Re B I ! (cid:21) Re q Im q ! , (31)and hence z ∈ ker P J for all z ∈ L B means Re B − Im BI ! + J Im B Re B I ! = 0 . (32)Solving this equation for J then gives the expression (6) which we have alreadyencountered. Hence (30) holds.For completeness we finally note that the metric G = Ω J defines a K¨ahler structureon complex phase space which turns P J into an orthogonal projection: Lemma 2.4.
Let h ( z, z ′ ) := z · G ¯ z ′ − i z · Ω¯ z ′ be the hermitian inner product on C n × C n defined by G = Ω J , then P J is the unique projection onto R n × R n which is hermitianwith respect to h , i.e., h ( P J z, z ′ ) = h ( z, P J z ′ ) .Proof. P J is a projection, so it is hermitian with respect to h ( z, z ′ ) if the kernel andimage are orthogonal to each other. Then it is as well uniquely determined by itsimage. Since by (27) any z ∈ L = ker P J is of the form z = ( I + i J ) x for some x ∈ R n × R n , we get for z = ( I + i J ) ∈ L and z ′ = x ′ ∈ R n × R n = Im P J that h ( z, z ′ ) = x · ( I + i J ) t ( G − iΩ) x ′ = x · [ G + J t Ω + i( J t G − Ω)] x ′ . But since G issymmetric G = Ω J implies G = − J t Ω and from GJ = − Ω we obtain J t G = Ω,therefore h ( z, z ′ ) = 0 for all z ∈ ker P J and z ′ ∈ Im P J .We have shown that coherent states with a complex centre are organised alongLagrangian submanifolds of physically equivalent coherent states one of which has areal centre. In what follows we shall investigate the time dependence of these structuresunder the evolution with non-Hermitian Hamiltonians. In particular, we will focuson the analytically solvable case of quadratic Hamiltonians, which lies at the heart ofsemiclassical considerations for more general systems.
3. Schr¨odinger dynamics with complex quadratic Hamiltonians
Here we will investigate the Schr¨odinger dynamics generated by complex quadraticHamiltonians that are given as Weyl quantisations of complex quadratic forms onphase space. For these Hamiltonians semiclassical approximations are exact, and werestrict ourselves to these purely quadratic Hamiltonians to understand the essence ofthe dynamics in detail. It is straightforward to include also linear terms; here, however,we want to keep the discussion concise. omplexified Coherent States z = ( p, q ) ∈ R n × R n points in phase space and set H ( z ) = 12 z · Hz , (33)where H ∈ M n ( C ) is a complex symmetric 2 n × n matrix and the quantumHamiltonians we will consider are given by the Weyl quantisation of quadratic functionsof the form H ,ˆ H = − ~ ∇ x · H pp ∇ x + ~ i x · H qp ∇ x + 12 x · H qq x − i ~ H qp , (34)where H = H pp H pq H qp H qq ! . We will in general allow the matrix H to be time dependentwithout explicitly indicating this in the notation. Our aim is to study the solutions tothe time dependent Schr¨odinger equationi ~ ∂ t ψ = ˆ H ψ , (35)for initial states given by coherent states. Since our Hamilton operator is in generalnot self-adjoint the question of whether this equation has solutions in suitable functionspaces is not trivial. To illustrate the issue, consider the following simple example: Ifthe Hamiltonian is given by H ( z ) = i q / U ( t ) = e t ~ x and taking for instance an initial state of the form ψ ( x ) = e − b ~ x it followsthat ψ ( t, x ) = e t − b ~ x (36)and hence ψ ( t, x ) / ∈ L ( R ) for t ≥ b .Problems of this kind are avoided if the imaginary part of H is chosen to be non-positive. For Im H ≤ H isoften of interest, in particular in the context of PT-symmetric quantum systems. Thus,we allow for general complex H here, but we only consider special initial conditions forwhich explicit solutions can be computed, at least for short times.We will investigate the dynamical behaviour of initially Gaussian coherent statesthat is generated by a Hamiltonian operator of the form (34). Similar to the real valuedcase, the class of Gaussian coherent states, now with a complex centre, is invariant underthis time evolution, as we shall see in the following. For this purpose we consider timedependent Gaussian coherent states of the form ψ ( t, x ) = e i α ( t ) (det Im B ( t )) / ( π ~ ) n/ e i ~ [ p ( t ) · ( x − q ( t ))+ ( x − q ( t )) · B ( t )( x − q ( t ))] = e i α ( t ) ψ B ( t ) z ( t ) ( x ) , (37) omplexified Coherent States z ( t ) = ( p ( t ) , q ( t )) ∈ C n × C n , B ( t ) ∈ M n ( C ) is symmetric and has positiveimaginary part, Im B ( t ) >
0, and α ( t ) ∈ C . Inserting the state (37) as an ansatz intothe Schr¨odinger equation (35) and separating terms with different powers of ( x − q )yields the following set of differential equations for ( p ( t ) , q ( t )), B ( t ) and α ( t ): − ˙ p + B ˙ q = H ′ q + B H ′ p (38) − ˙ B = H ′′ qq + H ′′ pq B + B H ′′ qp + B H ′′ pp B (39) − ˙ α + i4 tr( ˙ BB − ) = − ~ [ p · ˙ q − H ] − i2 [tr H ′′ pq + tr( H ′′ pp B )] , (40)where H ′ p , H ′′ pq , ... denote derivatives of H ( z ) with respect to p , and p and q , etc.. Ifwe choose p and q to be solutions to Hamiltons equations, i.e., ˙ p = −H ′ q and ˙ q = H ′ p ,then the first equation is satisfied, and furthermore using the second equation we cansimplify the third, thus arriving at the simplified system˙ z = Ω Hz (41)˙ B = − H qq − H pq B − BH qp − BH pp B (42)˙ α = 1 ~ [ p · ˙ q − H ( z )] + i4 tr[ H pp B − H qq B − ] . (43)Here the first equation is Hamilton’s equation with a complex Hamilton function andthe third equation can be integrated once the first and the second are solved. Thesolutions to the second equation can be obtained most easily using symplectic geometrywhich will be reviewed in what follows. This set of equations is a complex extension ofthe classical approach to coherent state propagation of Hepp, [9], and Heller [10], whichis used and developed further in many areas (see, e.g., the review [12] or [31] for anoverview of more recent mathematical developments).For complex H equation (41) leads to complex solutions z ( t ), even if the initialcondition is chosen to be real, and thus we will obtain coherent states with complexcentres. As discussed in the previous section a complex centre has no direct physicalmeaning, but using a complex structure it can be projected to a physically meaningfulreal centre. We will now apply the complex symplectic geometry we developed in thelast section to understand the relation between the dynamics of the complex centre andits projection to real space.In a previous paper [21] we concentrated on the dynamics of the Wigner functionwhich directly yields the expectation values and hence the real centre of a state. Thisconsiderations led to a non-Hermitian version of Ehrenfest’s theorem with a new type ofclassical dynamics emerging in the semiclassical limit. We derived an evolution equationfor the Wigner function, which in the case of a quadratic Hamiltonian reduces to ~ ∂ t W ( t, z ) = − (cid:18) − ~ Im H − ~ z · Re H Ω ∇ − z · Im Hz (cid:19) W ( t, z ) , (44)where all derivatives are with respect to z , and∆ Im H := −∇ · Ω T Im H Ω ∇ . (45) omplexified Coherent States ψ is of the type (37) the Wigner function is of the form W ( t, z ) = e − β ( t ) ( π ~ ) n e − ~ ( z − Z ( t )) · G ( t )( z − Z ( t )) (46)with Z ( t ) ∈ R n × R n , a symmetric G ( t ) ∈ M n ( R ), and β ( t ) ∈ R . Inserting the ansatz(46) into equation (44), and separating different powers of ( z − Z ), leads to the followingset of equations ˙ Z = Ω Re HZ + G − Im HZ (47)˙ G = Re H Ω G − G Ω Re H − Im H + G Ω T Im H Ω G (48)˙ β = − ~ Z · Im HZ −
12 tr[Im H Ω G Ω T ] (49)It can be verified, that this set of equations is also compatible with the dynamicalequations (38), (39), and (40) obtained from the coherent state ansatz in the Schr¨odingerequation, if we demand p and q to be real. Thus, equations (41), (42), and (43) are notthe unique dynamical equations for the propagation of coherent states for non-HermitianHamiltonians.The two different sets of equations that we have obtained, (41), (42), and (47),(48), are supposed to describe the dynamics of the same physical state. In what followswe will discuss how they can be related using complex structure associated with thecoherent states. To solve the evolution equations obtained above, in particular the nonlinear matrixRicatti equations (42) and (48), we have to understand how the geometric structuresdiscussed in the previous section evolve in time under the action of complex Hamiltoniandynamics. For this purpose, we first investigate the action of a linear symplectic mapon a positive Lagrangian subspace L , i.e., we change L to SL with S ∈ Sp ( n, C ).Here Sp ( n, R ) and Sp ( n, C ) denote the set of real or complex 2 n × n matrices S with S T Ω S = Ω, i.e., the real and complex linear symplectic groups. Since any z ∈ SL is ofthe form z = Sz for some z ∈ L we get z · Ω z ′ = z · S T Ω Sz ′ = z · Ω z ′ = 0, since L isLagrangian, and thus SL is, too. Furthermorei2 z ′ · Ω¯ z ′ = i2 z · S T Ω S ¯ z , (50)thus, if S = S , i.e., S ∈ Sp ( n, R ), then SL is positive, too. If S is complex, SL doesnot have to be positive any more.We will mainly consider situations in which S is the solution to Hamilton’s equation,i.e, S ( t ) satisfies ˙ S = Ω HS , with S ( t = 0) = I , (51)where H ∈ M n ( C ) is symmetric. omplexified Coherent States Lemma 3.1.
Assume L to be a positive Lagrangian subspace and S ( t ) a solution of (51) . Then there exists a T H,L such that for all t ∈ [0 , T H,L ) S ( t ) L is again a positiveLagrangian subspace. If Im H ≤ we can take T H,L = ∞ .Proof. Since S ( t ) is close to the identity for small t , S ( t ) L will be positive by continuityfor sufficiently small t . If Im H ≤ Sz ∈ SL for z ∈ L wehave to consider i( Sz ) · Ω Sz/ z · S T Ω ¯ S ¯ z/ z ∈ L . From (51) we finddd t (cid:18) i2 S T Ω ¯ S (cid:19) = i2 S T [ − H ΩΩ + ΩΩ ¯ H ] ¯ S = − S T Im H ¯ S. (52)Thus, if Im H ≤ dd t i2 z · S T Ω ¯ S ¯ z ≥ S ( t ) L is therefore positive for all t ≥ B and the complex structure transform if we apply asymplectic map to L . Proposition 3.2.
Let L be a positive Lagrangian subspace and S ∈ Sp ( n, C ) such that SL is still positive. Then(i) B SL = S ∗ B L (53) where the action of S = S pp S pq S qp S qq ! on B L is defined by S ∗ B L := ( S pp B L + S pq )( S qp B L + S qq ) − , (54) (ii) and J SL = (Re S − Im SJ L ) J L (Re S − Im SJ L ) − (55) G SL = Ω(Re S − Im SJ L )Ω T G L (Re S − Im SJ L ) − . (56) Proof.
Let z ∈ L , then there exists a q ∈ C n such that z = ( B L q, q ), by Lemma2.2, and since Sz ∈ SL there exists a q ′ ∈ C n such that Sz = ( B SL q ′ , q ′ ). Now Sz = S ( B L q, q ) = ( S pp B L q + S pq q, S qp B L q + S qq q ) and hence we obtain the two equations( S pp B L + S pq ) q = B SL q ′ , ( S qp B L + S qq ) q = q ′ . (57)From the second equation we get q = ( S qp B L + S qq ) − q ′ and inserting this into the firstgives ( S pp B L + S pq )( S qp B L + S qq ) − q ′ = B SL q ′ , which is the first result.To derive the second result we note that z ∈ L means z = Re z + i J L Re z , by (27),and similarly Sz ∈ SL means Sz = Re( Sz ) + i J SL Re( Sz ) and thus we arrive at theexpressions Im[ Sz ] = J SL Re[ Sz ] = J SL (Re S − Im SJ L ) Re z (58)Im[ Sz ] = Im[ S (Re z + i J L Re z )] = (Im S + Re SJ L ) Re z . (59)Comparing these two expressions for Im( Sz ) gives J SL = (Im S + Re SJ L )(Re S − Im SJ L ) − and with J L = − S + Re SJ L ) = (Re S − Im SJ L ) J L . The result for G SL then follows from G SL = − Ω J SL . omplexified Coherent States J SL = (Re SJ L + Im S )( − Im SJ L + Re S ) − (60) G SL = (Ω Re S Ω T G L + Ω Im S )( − Im S Ω T G L + Re S ) − (61)then J SL = ˜Φ ∗ J L and G SL = Φ ∗ G L , (62)with ˜Φ = Re S Im S − Im S Re S ! and Φ = Ω Re S Ω T Ω Im S − Im S Ω T Re S ! . (63)If the symplectic matrix S is a solution of the differential equation (51) then thisinduces corresponding differential equations for the evolution of the matrices B SL and J SL which we shall now derive. Theorem 3.3.
Let S ( t ) be a solution to (51) with H = H pp H pq H qp H qq ! , and L a positiveLagrangian subspace, then there exists a T H,L > such that S ( t ) L is positive for t ∈ [0 , T H,L ] and we have(i) ˙ B SL = − H qp B SL − B SL H pq − H qq − B SL H pp B SL (64) (ii) and ˙ J SL = Ω Re HJ SL − J SL Ω Re H + Ω Im H + J SL Ω Im HJ SL (65)˙ G SL = Re H Ω G SL − G SL Ω Re H − Im H + G SL Ω T Im H Ω G SL (66) Furthermore if Im H ≤ we can take T H,L = ∞ .Proof. Since S (0) = I it is clear that for small t the space S ( t ) L will still be positive,hence there exists a T H,L such that SL is positive for t ∈ [0 , T H,L ]. Now from (51) weget ˙ S pp ˙ S pq ˙ S qp ˙ S qq ! = − H qp S pp − H qq S qp − H qp S pq − H qq S qq H pp S pp + H pq S qp H pp S pq + H pq S qq ! (67)then differentiating the relation (53) and using (67) gives˙ B SL = ( ˙ S pp B L + ˙ S pq )( S qp B L + S qq ) − − B SL ( ˙ S qp B L + ˙ S qq )( S qp B L + S qq ) − = − H qp ( S pp B L + S pq )( S qp B L + S qq ) − − H qq ( S qp B L + S qq )( S qp B L + S qq ) − − B SL H pp ( S pp B L + S pq )( S qp B L + S qq ) − − B SL H pq ( S qp B L + S qq )( S qp B L + S qq ) − = − H qp B SL − H qq − B SL H pp B SL − B SL H pq (68) omplexified Coherent States J SL = AJ L A − with A = Re S − Im SJ L , then˙ J SL = ˙ AJ L A − − AJ L A − ˙ AA − = ˙ AA − J SL − J SL ˙ AA − . (69)Then from (51) we find Re ˙ S = Ω Re H Re S − Ω Im H Im S and Im ˙ S = Ω Im H Re S +Ω Re H Im S and using these relations we find˙ A = Re ˙ S − Im ˙ SJ L = [Ω Re H − Ω Im HJ SL ] A (70)and this leads to˙ J SL = Ω Re HJ SL + Ω Im H − J SL Ω Re H + J SL Ω Im HJ SL . (71)The result for G SL then follows using the relation G SL = Ω J SL The formal similarity of the equations (66) and (64) suggests to define a Hamiltonian K ( ζ , z ) on the doubled phase space by K ( ζ , z ) = 12 ( ζ , z ) − Ω T Im H Ω Ω Re H − Re H Ω Im H ! ζz ! (72)then the matrix Φ( t ) from (63) satisfies˙Φ = − II ! − Ω T Im H Ω Ω Re H − Re H Ω Im H ! Φ . (73)And so by solving (73) with Φ( t = 0) = I we find a matrix such that G ( t ) = Φ( t ) ∗ G (74)is a solution to (48) with G ( t = 0) = G . The results from Theorem 3.3 allow us to solve the non-linear Riccati equations (42)and (48) in terms of solutions to linear Hamiltonian equations, which we will exploit inwhat follwos.We first consider the Schr¨odinger equation for a coherent state in positionrepresentation, (37). Let S ( t ) ∈ Sp ( n, C ) be the solutions to˙ S = Ω HS , with S (0) = I (75)then z ( t ) = S ( t ) z is a solution to (41) and by Theorem 3.3, part (i), and Proposition3.2, part (i), S ∗ B is a solution to (42). Hence we conclude omplexified Coherent States Theorem 3.4.
Let L = L B be a positive Lagrangian subspace, then there exists a T H,L > such that for t ∈ [0 , T H,L ) the solution to the Schr¨odinger equation with ψ ( t = 0) = ψ Bz is given by ψ ( t ) = e i α ( t ) ψ S ( t ) ∗ BS ( t ) z (76) where α ∈ C is the solution to (43) with α (0) = 0 . The phase factor is related to the action along S ( t ) z and also contains Maslov-phasetype contributions.The matrix S ( t ) ∗ B defines a time dependent complex structure J ( t ) via (6) whichprojects the complex centre S ( t ) z to the real centre Z ( t ) = P J ( t ) ( S ( t ) z ) (77)and we can use Theorem 2.1 to express the Wigner function in terms of projections fromthe complex dynamics S ( t ).Alternatively we can solve the purely real set of equations (48) and (47) to directlyobtain the motion of the real centre. Let Φ( t ) be the solution to (73) and Z ( t ) a solutionto (47) with G ( t ) = Φ( t ) ∗ G then we have Theorem 3.5.
Let G be a symplectic positive definite symmetric matrix. Then thereexists a T H,G > such that for t ∈ [0 , T H,G ) the unique solution to the Wigner vonNeuman equation (44) with initial condition W ( z ) = π ~ ) n e − ~ ( z − Z ) · G ( z − Z ) is given by W ( t, z ) = e − β ( t ) ( π ~ ) n e − ~ ( z − Z ( t )) · [Φ( t ) ∗ G ]( z − Z ( t )) (78) where β ( t ) ∈ R is a solution to (49) with β (0) = 0 . One of the characteristic features of the dynamical equation (47) for the real centre Z ( t ) is that it is in general not autonomous, the coefficients of this equation will dependon t via the metric G ( t ). However, in many cases there are special solutions for whichthe metric is time independent, corresponding to fixed points of the evolution equation(48). To analyse the possible time independent complex structures we have to set theexpression for the time derivative of the metric G ( t ) in (48), or equivalently the timederivative of the matrix B ( t ) in (42), to zero. Thus we obtain quadratic matrix equationsfor the fixed points G and B , respectively. Let us illustrate this observation with afew examples.(1) Assume the Hamiltonian is anti-Hermitian, i.e., Re H = 0, then (48) with ˙ G = 0becomes Im H = G Ω T Im H Ω G , and if we assume furthermore that for some γ > H = − γS , where S is symplectic, symmetric and positive, then we findthat G = S (here we used that Ω T S Ω = S − ). The assumptions on Im H hold forinstance if n = 1 and Im H is negative definite (with γ = det Im H ). Thus in thiscase the metric and the associated complex structure are constant and the equationof motion for the centre simplifies to˙ Z = − S − γSZ = − γZ . (79) omplexified Coherent States G the solution to (48) is given by G ( t ) = ( G + tanh( γt ) S )(tanh( γt ) G + S ) − S = S + O (e − γt ) , (80)hence the stationary solution G = S we found above is a global attractor, to whichany other solution converges exponentially fast to. Note as well that G ( t ) can beextended to some negative t but will eventually become singular.(2) We previously discussed the example H ( z ) = i q / B = − i and hence B ( t ) = B − i tI .Since Im B ( t ) = Im B − tI we see that the condition Im B ( t ) > B = 0, then˙ P = 0 and the position reaches infinity in finite time Q ( t ) = B B − t Q . (81)(3) We now have a look at a harmonic oscillator with damping induced by a momentumdependent imaginary part. We choose H ( p, q ) = ¯ δ p + ω q , (82)where the parameter δ ∈ C is assumed to satisfy | δ | = 1 and Re δ, Im δ >
0, henceIm H ( p, q ) = − Re δ Im δ p ≤
0. Therefore δ parametrizes the strength of thedamping relative to the kinetic energy. Note that choosing | δ | 6 = 1 just amounts torescaling of ω → | δ | ω and t → | δ | t . Using (42) we find that B = i ωδ is a constantsolution with Im B >
0. We can then determine the corresponding metric G andthe equations of motions for the centre which read˙ p = − ω q − ω Im δ p , ˙ q = p . (83)For comparison with the classical damped harmonic oscillator we transform this setof first order equations into a second order equation for q ,¨ q + 2 ω Im δ ˙ q + ω q = 0 . (84)We see that due to the metric this describes an underdamped oscillator, sinceIm δ ≤ | δ | = 1, irrespective of the choice for δ .(4) It is instructive to include an example with a linear term in z = ( p, q ) ∈ R , H γ ( z ) = 12 z · z + i γ · Ω z (85) omplexified Coherent States γ ∈ R . The inclusion of Ω in the linear term is convenient, it implies thatif z is to the right of γ the term is negative and we have damping, and if z is to theleft of γ the term is positive and we have enhancement. This is a PT symmetricsystem. It can be brought to the more familiar form H ( p, q ) = p + V ( q ) with V ( − q ) = ¯ V ( q ) via a canonical rotation of the phase space variables. Since Re H = I and Im H = 0 we find that G = I is a solution for all times and with this initialchoice the equation of motion for the centre Z , see [21], becomes ˙ Z = Ω( Z − γ ).Hence Z ( t ) = γ + O ( t )( Z − γ ) with O ( t ) ∈ SO (2) denoting a rotation by t . Wecan as well solve the equation for β and find β ( t ) = − γ · ( Z ( t ) − Z ). Thus, thecentre of the Wigner function evolves along circles as for the harmonic oscillator,but the circles are shifted due to damping and enhancement in different parts ofphase space. The relation to a real harmonic oscillator H can be directly seen inthe following way. Introducting the complex translation T ( γ ) := e − ~ γ · ˆ z we have T ( γ ) − ˆ H T ( γ ) = H γ − | γ | /
2, and thus the operator is conjugated to the harmonicoscillator by a non-unitary operator, and thus the spectrum is purely real. Thenorm of the state stays bounded over time although it oscillates, which reflects thefact that the eigenvalues of the Hamiltonian are real, but the eigenfunctions arenot orthogonal for γ = 0.
4. Summary
Coherent states are a useful tool for the investigation of semiclassical limits of quantumtheories. The investigations presented here can be viewed as part of a programme tounderstand the classical dynamics emerging from the semiclassical limit of general non-Hermitian operators. We recently formulated an Ehrenfest Theorem for non-Hermitianoperators [21], in which the classical dynamics is given by a combination of a symplecticand a metric gradient field, which are generated by the real and imaginary part of theHamilton function, respectively. This is a very different type of dynamics compared towhat one would expect from extending standard WKB theory to complex Hamiltonians,which results in a Hamiltonian flow on complexified phase space. The main result here,is the proof that these two approaches are physically equivalent and are related by aprojection from complexified phase space to real phase space,i J , (86)where J is a complex structure on phase space which is determined by the physicalstates and becomes a dynamical variable in our theory.We restricted ourselves to quadratic Hamiltonians and Gausssian coherent stateshere, because both semiclassical approaches become exact in this case, and we couldfocus on the complex symplectic geometry relating them. This will form the basis forextension to more general systems following [21]. It is well known that semiclassicalmethods based on dynamics in complexified phase space often run into difficultiesrelated to analytic extensions, e.g., complex trajectories often develop singularities, and omplexified Coherent States Acknowledgments
EMG acknowledges support from the Imperial College JRF scheme. [1] Moiseyev N 2011
Non-Hermitian Quantum Mechanics (Cambridge: Cambridge University Press)[2] El-Ganainy R, Makris K G, Christodoulides D N and Musslimani Z H 2007
Opt. Lett. Laser and Photon. Rev. Phys. Rep.
Spectra and pseudospectra :the behavior of nonnormal matricesand operators (Princeton: Princeton University Press)[6] Bender C M, Boettcher S and Meisinger P N 1999
J. Math. Phys. Phys. Rev. Lett. Coherent States, Wavelets and Their Generalizations (NewYork: Springer)[9] Hepp K 1974
Comm. Math. Phys. J. Chem. Phys. Rev. Mod. Phys. Phys. Rep.
J. Chem. Phys. Ann. Phys.
Comm. Math. Phys.
Phys. Lett. A
J. Phys. A F793[18] Curtright T and Mezincescu L 2007
J. Math. Phys. Phys. Scr. J. Phys. A Phys. Rev. A (6) 060101[22] Huber D and Heller E J 1987 J. Chem. Phys. Rev. Mod. Phys. J. Phys. A J. Math. Anal. Appl.
J. Phys. A Phys. Rev. A The analysis of linear partial differential operators. III (Berlin: Springer)[29] H¨ormander L 1995
Math. Z.