An exact mapping between the states of arbitrary N-level quantum systems and the positions of classical coupled oscillators
aa r X i v : . [ qu a n t - ph ] F e b An exact mapping between the states of arbitrary N-level quantum systemsand the positions of classical coupled oscillators
Thomas E. Skinner ∗ Physics Department, Wright State University, Dayton, OH 45435 (Dated: September 30, 2018)The dynamics of states representing arbitrary N-level quantum systems, including dissipativesystems, can be modeled exactly by the dynamics of classical coupled oscillators. There is a directone-to-one correspondence between the quantum states and the positions of the oscillators. Quantumcoherence, expectation values, and measurement probabilities for system observables can thereforebe realized from the corresponding classical states. The time evolution of an N-level system isrepresented as the rotation of a real state vector in hyperspace, as previously known for densitymatrix states but generalized here to Schr¨odinger states. A single rotor in n -dimensions is thenmapped directly to n oscillators in one physical dimension. The number of oscillators needed torepresent N-level systems scales linearly with N for Schr¨odinger states, in contrast to N for thedensity matrix formalism. Although the well-known equivalence (SU(2), SO(3) homomorphism) of2-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitraryN-level systems, representing quantum dynamics by a system of coupled harmonic oscillators inone physical dimension is general for any N. Values for the classical coupling constants are readilyobtained from the system Hamiltonian, allowing construction of classical mechanical systems thatcan provide visual insight into the dynamics of abstract quantum systems as well as a metric forcharacterizing the interface between quantum and classical mechanics. I. INTRODUCTION.
The density matrix formalism [1–3] provides a straight-forward procedure for predicting quantum dynamics. Atany given time, the density matrix provides a completestatistical characterization of the system in terms of themean values of measurable states. It includes both thequantum uncertainty in predicting the results of singlemeasurements on pure states (even though such statesrepresent maximal possible information for the system)and the classical uncertainty in measurements on mixedstates of less than maximal information.Although the theory needs no supporting visual modelfor its application, the Liouville-von Neumann equationgoverning the time evolution of the density matrix pro-vides little physical insight into system dynamics. Therehas therefore been considerable effort towards represent-ing, where possible, quantum systems using more intu-itive classical models. Of particular influence and im-portance is the classical representation for quantum two-level systems [4], sometimes referred to as the Feynman–Vernon–Hellwarth (FVH) Theorem. The behavior of anyquantum mechanical two-level system can be modeled byclassical torque equations, providing a one-to-one corre-spondence between the time evolution of the system andthe dynamics of a spinning top in a constant gravitationalfield or a magnetic moment in a magnetic field.Work by Fano [3] published concurrently with the FVHresult also provides geometrical interpretation of spin dy-namics for more complex quantum systems. The densitymatrix for an N-level system is represented as an expan- ∗ Electronic address: [email protected] sion in Hermitian operators, resulting in a vector withreal components. The time development of this gener-alized Bloch vector is a real rotation in a hyperspace of( N −
1) dimensions. Constants of the motion can be de-rived [5, 6] that constrain the system’s dynamics and pro-vide physical insight. However, the states of the systemas given by the components of this vector (also referredto as a coherence vector [5] and, more recently, a Stoke’stensor [7]), do not evolve in a physical space amenable tovisualization, with its attendant advantages, except forthe case N = 2.Thus, no completely general mapping has been realizedthat yields a one-to-one correspondence, similar to theFVH result, between the states of a quantum-mechanicalN-level system and classical dynamical variables, provid-ing the possibility for direct mechanical insight into thedynamics of abstract quantum systems. Analogies be-tween quantum and classical systems have been noted[8–20] almost from the beginning. But exact equivalencebetween the quantum and classical equations of motionhas been obtained only for certain limiting conditions[21–28] such as weak perturbations of the system (weakcoupling limit) and the aforementioned 2-level systems.Recently, the possibility of exact representations of N-level quantum systems in terms of classical coupled oscil-lators, with no restriction to weak coupling, was demon-strated [29]. However, the formalism is limited to real,invertible Hamiltonians applied only to pure states. Itis insufficiently general to represent statistical mixturesand density matrix evolution classically, as well as open,dissipative systems.In the present work, a very simple approach is pre-sented for mapping an arbitrary N-level quantum systemto a system of coupled harmonic oscillators, available forover a decade. It both complements and augments theresults presented in [29] and is made more relevant bythat study. The salient features representing the dynam-ics of N-level systems as rotations in Liouville space arereviewed first. The desired one-to-one correspondencebetween the states of the quantum system, representedas a density matrix, and classical dynamical variablesis provided by a mapping representing harmonic oscilla-tors. The quantum states, either pure or mixed, are rep-resented exactly by the time-dependent displacements ofclassical coupled oscillators.Hilbert space rotations are reviewed next to general-ize this option for representing quantum spin dynamicsclassically. Results inspired by [29] are presented withno restriction to real, invertible Hamiltonians. An exactmapping of Schr¨odinger states to the physical displace-ments of coupled oscillators is provided, in contrast tothe displacement and velocity originally required. Thisapproach to representing spin dynamics is then extendedto mixed states. Whereas N − N oscillators.Open (dissipative) N-level systems are considered next,showing they also can be exactly and simply modeled asclassical coupled oscillators. The present treatment re-veals the necessity for negative couplings in closed sys-tems as well as antisymmetric couplings in open systems.The paper closes with illustrative examples of thequantum-classical mapping. II. TIME EVOLUTION OF QUANTUMN-LEVEL SYSTEMS
A brief synopsis of formalisms for representing the dy-namics of N-level systems is presented. The standard Li-ouville space and Hilbert space representations are con-sidered first, enabling simple generalizations that lendthemselves to a classical interpretation. In all cases, thetime evolution of the system can be reduced to the formΦ( t ) = U ( t ) Φ(0) . (1)The representation chosen determines the particularforms for Φ and the propagator U ( t ). For notational con-venience and interchangeability of energy and frequencyunits, ~ is set equal to 1 in what follows. A. Liouville equation
The Liouville-von Neumann equation for the time evo-lution of a density matrix ρ governed by system Hamil-tonian H is ˙ ρ = − i [ H, ρ ] , (2)with formal solution ρ ( t ) = e − i Ht ρ (0) e i Ht = U ρ (0) U † , (3) which defines U ( t ) = e − i H t .The time evolution can be related to a rotation by firstexpanding ρ in terms of a complete set of basis operators[3]. Orthogonal bases are particularly convenient and aretypically normalized for further convenience. Denotingthe basis elements as ˆ e i for state i and requiring onlythat the basis be orthonormal gives h ˆ e i | ˆ e j i = Tr ( ˆ e † i ˆ e j ) = δ ij , (4)where the inner product for the vector space comprisedof matrices is given by the operator Tr, which returns thetrace (sum of diagonal elements) of its argument, and † denotes the operation of Hermitian conjugation. In lieuof explicitly normalizing the ˆ e i , the inner product canbe defined with the appropriate factor multiplying Tr.Summing over repeated indices gives ρ as ρ = r j ˆ e j (5)where the coefficients in the expansion are the projectiononto the basis states. Each r j in Eq. (5), r j = h ˆ e j | ρ i = Tr ( ˆ e † j ρ ) , (6)is thus the expectation value of the quantum state ˆ e j .Then ˙ r i = Tr( ˆ e † i ˙ ρ ) = − i Tr( ˆ e † i [ H, ρ ])= − i Tr( ˆ e † i [ H, ˆ e j ]) r j = Ω ij r j . (7)Expanding the commutator, using Tr( AB ) = Tr( BA ) =[Tr( AB ) † ] ∗ and [ ˆ e † i , ˆ e j ] † = [ ˆ e † j , ˆ e i ] givesΩ ij = − i Tr( ˆ e † i [ H, ˆ e j ])= i Tr( [ ˆ e † i , ˆ e j ] H )= −{ i Tr( [ ˆ e † j , ˆ e i ] H ) } ∗ = − Ω ∗ ji (8)in terms of its complex conjugate elements, denoted by *.Thus, Ω = − Ω † is antihermitian and can be diagonalized.The evolution of the density matrix is given by ˙ r = Ω r , (9)with solution r ( t ) = e Ω t r (0) . (10)The propagator U ( t ) = e − Ω t , and therefore U † = U − is unitary, since Ω † = − Ω. Thus, Eq. (9) represents arotation, albeit still most generally in complex space. Rotation in real space
In the case where the orthonormal basis states are Her-mitian operators, the components of the density matrix(coherence vector or Stokes tensor, in this case) are real,and Ω is also a real antisymmetric matrix. The rotationof Eq. (9) is then a rotation in real, multidimensionalCartesian space, which is the generalization [3, 5] to N-level systems of the result obtained in [4]. The quan-tum dynamics are thus fully classical in the additionaldimensions exceeding 3D physical space. However, clas-sical rotations in more than three dimensions are onlymarginally less abstract than rotations in Hilbert spaceand Liouville space. More accessible insight can be ob-tained by mapping the real-valued quantum states tophysical space as follows. Exact mapping to classical coupled oscillators
A textbook exercise for deriving the Larmor precessionof a spin-1/2 in a static magnetic field B differentiatesthe first-order derivative in Ehrenfest’s theorem to obtaina harmonic oscillator equation for the time evolution ofexpectation values of the spin components transverse to B . Similarly, differentiate Eq. (9) again to obtain ¨ r = Ω r . (11)Since Ω is real, symmetric, and diagonalizable, the so-lution is readily written in terms of the normal-mode so-lutions obtained by diagonalizing Ω . The eigenvalues ofantihermitian Ω are pure imaginary, resulting in negativeeigenvalues − ω a ( a = 1 , , , . . . , n ) for n × n matrix Ω .The n distinct eigenvectors | ω a i constitute a basis setsatisfying the completeness relation P a | ω a ih ω a | = r a = h ω a | r i becomes simply¨ r a = − ω a r a (12)with standard harmonic oscillator solution r a ( t ) = r a (0) cos ω a t + ˙ r a (0) ω a sin ω a t (13)and ˙ r dependent on r according to Eq. (9), giving | r ( t ) i = n X a =1 | ω a ih ω a | r ( t ) i = n X a =1 | ω a ih ω a | (cid:20) cos ω a t + Ω sin ω a tω a (cid:21) | r (0) i = U ( t ) | r (0) i . (14)Representing the propagator in the eigenbasis whereΩ is diagonal gives Eq. (12). Using the representationfor non-diagonal Ω and its eigenvectors to calculate U ( t )gives the physical displacements r i ( t ) for each of the n oscillators. This solution for U ( t ) must be identical to thepropagator given in Eq. (10). It is included primarily forconsistency in the presentation, but also to emphasize thefundamental differential equation under consideration is first order and only requires specification of r (0) (or r evaluated at any other fixed time).To complete the explicit identification of Eq. (11) withmechanical oscillators, consider equal masses, m , on africtionless surface, with mass m i connected by spring ofstiffness k ij = k ji to mass m j ( i, j = 1 , , , . . . , n ), as inFig. 1 for an illustrative case n = 3. The classical matrix FIG. 1: Schematic of three masses at equilibrium positions r i = 0 coupled with springs of stiffness k ij . Ω Cl relating the displacement from equilibrium of the i th mass to its acceleration, as in Eq. (11), is(Ω Cl ) ij = 1 m k ij i = j − n P l =1 | k il | i = j (15)The above expression assumes reciprocal couplings k ij = k ji and employs the absolute value in the sumto accommodate negative couplings. A system of pendu-lums consisting of masses attached to rigid rods can becoupled negatively by attaching a spring to rod i belowthe fulcrum of oscillation and to rod j above the fulcrum.Displacing mass m i to the right exerts a force on m j tothe left, ie, the coupling k ij <
0. A pendulum can beinverted with its mass above the fulcrum to implement k ii <
0. Inverting transformers can be used to implementnegative couplings in LC circuits.Setting Ω Cl = Ω gives the spring constants k ij m = (Ω ) ij i = j − ( Ω ii + n P l = i | Ω | il ) i = j (16)in terms of the matrix Ω (squared) representing the quan-tum system, as derived from Eq. (8). There is thus a one-to-one mapping of the quantum states to the oscillatordisplacements embodied in both systems in r i ( t ). Giventhe initial states r i (0) of the system, the necessary ˙ r i (0)follow from Eq. (9).This mapping is very general. It is not limited toparticular values of the spin, numbers of interactingspins, specific forms of the commutation relations, or rel-ative fractions of mixed and pure states comprising ρ .An N × N density matrix generates N components inEq. (5), which requires N oscillators. The static com-ponent of the identity element can be eliminated, andthe structure of the Hamiltonian may generate evolutiononly in a smaller subspace of states, further reducing thenumber of required oscillators.For pure states, the time-dependent elements c i ( t )comprising the state vector can readily be obtained, ifdesired, from ρ reconstructed in matrix form using the r i ( t ) and Eq. (5). Each resulting element ρ ij is equal to c i c ∗ j . Assigning any one of the c i to the square root of ρ ii sets the arbitrary global phase of the pure-state ele-ments. In terms of this real c i , the remaining c j are equalto ρ ji /c i . B. Schr¨odinger equation
The solution | Ψ( t ) i = e − i Ht | Ψ(0) i . (17)to the time-dependent Schr¨odinger equation i | ˙Ψ( t ) i = H | Ψ( t ) i (18)represents a rotation of | Ψ(0) i in Hilbert space, since H is Hermitian and the propagator U ( t ) = e − i H t is uni-tary. Most generally, H and the components c i of | Ψ i in a chosen basis are complex, so there is no classicalinterpretation for the time evolution of the state.In [29], the authors derive a methodology for repre-senting the complex c i in terms of the displacement andvelocity of classical coupled oscillators for the special caseof real, invertible H . These restrictions can be removed,as will be shown in what follows. Rotation in real space
The results of the preceding sections can be extendedto the Schr¨odinger equation using only the quantumHamiltonian, in contrast to the approaches in [8, 9, 29]which start with the real, classical Hamiltonian H andrelate it to the quantum Hamiltonian H . The expressionfor H in terms of H then represents classical coupledharmonic oscillators only for real H [29].Instead, start with Eq. (18) and represent the compo-nents of | Ψ i as vector c , giving ˙ c ( t ) = − iH c ( t ) . (19)Write complex c = q + i p and complex H = Q + iP . Forreal H , q and p are conjugate variables [29] but are notmost generally so. Equating the real and imaginary partsafter performing the multiplications in Eq. (19) recaststhe Schr¨odinger equation as a real equation (cid:18) ˙ q ˙ p (cid:19) = (cid:18) P Q − Q P (cid:19) (cid:18) qp (cid:19) = Ω (cid:18) qp (cid:19) (20) in the form of Eq. (9) with Ω † = − Ω again antiherme-tian, since Hermitian H requires Q † = Q and P † = − P .The results and implications for real rotations then followfrom § II A 1 and the discussion following Eq. (9). Exact mapping to classical coupled oscillators
Similarly, differentiating Eq. (20) gives (cid:18) ¨ q ¨ p (cid:19) = (cid:18) P Q − Q P (cid:19) (cid:18) qp (cid:19) = (cid:18) P − Q P Q + QP − ( P Q + QP ) P − Q (cid:19) (cid:18) qp (cid:19) = (cid:18) −ℜ ( H ) ℑ ( H ) −ℑ ( H ) −ℜ ( H ) (cid:19) (cid:18) qp (cid:19) = Ω (cid:18) qp (cid:19) (21)in the form of Eq. (11) for real symmetric (Hermitian)matrix Ω constructed from the real and imaginary partsof H . The mapping of q and p to mechanical oscillatorsthen follows from § II A 2.For complex N × N Hamiltonian, there are thus mostgenerally 2 N mutually coupled oscillators. There can befewer oscillators and no mutual coupling between spe-cific oscillators, depending on the structure of H . Thedisplacements q and p provide an exact one-to-one map-ping to the real and imaginary components, respectively,of the quantum state | Ψ i . The state c (0) = q (0) + i p (0)uniquely determines the initial displacements, with theinitial velocities then given by Eq. (20).The present treatment produces the results in [29] inthe case of real H = Q , P = 0. Note that q and p then evolve independently according to the same prop-agator, with no mechanical coupling between q and p oscillators. The initial conditions are the only differencein the solutions. Calculating p = H − ˙ q separately as in[29] imposes an additional unnecessary restriction that H , already constrained in [29] to be real, must be invert-ible (ie, no eigenvalues equal to zero). Extension to mixed states
A statistical mixture can not be represented in terms ofa state | Ψ i , but is written in terms of the probability p k for being in each of the possible states | Ψ k i as a densitymatrix ρ ( t ) = X k p k | Ψ k ( t ) ih Ψ k ( t ) | (22)which evolves according to Eq. (2). The results in [29]and extensions in the previous section are limited to purestates evolving according to the Schr¨odinger equation.The methodology would appear to be inapplicable tomixed states. However, the density matrix representinga given system is not unique. It is an average over the N constituents comprising a macroscopic system, whichcan be astronomically large, precluding an exact deter-mination of the exact state of each of the N constituents.But the identical density matrix can also be con-structed from a completely specified set of N ≪ N non-interacting pure states, with the N elements of ρ deter-mined from measurable macroscopic (average) propertiesof the system, such as energy or polarization. In thatcase, both the weights p k and corresponding states areknown exactly, so each | Ψ k i can be used independentlyto construct a set of coupled oscillators representing thecomponents c ( k ) i ( t ) of | Ψ k ( t ) i . In lieu of density matrixevolution ρ ( t ) = U ρ (0) U † , the simpler and more efficientSchrodinger evolution | Ψ( t ) i = U | Ψ(0) i can be appliedto each pure state | Ψ k i comprising ρ in Eq. (22). Sub-sequently, the weights p k can be used to calculate expec-tation values, measurement probabilities, or reconstructthe density matrix at later times t if desired.In addition, as shown in [30], at least one of the | Ψ k i comprising the initial density matrix is redundant andcan be removed from the calculation, since it providesa relatively uninteresting constant contribution to thesystem dynamics. Choose one of the weights, for ex-ample, p . The density matrix can be rewritten as p times the identity element plus a “pseudo” density matrixconstructed from the | Ψ k i with weights ( p k − p ). Theterm that is proportional to the identity element doesn’tevolve in time under unitary transformations and can beignored.Thus, the state | Ψ i has been removed from the den-sity matrix, along with any other | Ψ k i that had originalweights p k = p . In the general case of m ≥ p k , only m < N of the | Ψ k i arerequired. The number of oscillators is correspondingly re-duced. Choosing the weight with the largest degeneracyprovides the maximum reduction. The explicit contribu-tion of each pure state to the system dynamics is readilyapparent in utilizing this approach. The density matrixat any given time is easily reconstructed as described in[30]. III. DISSIPATIVE SYSTEMS
The modifications necessary to model open systemsas a set of damped oscillators can be found very gener-ally, with minimum detail concerning the relaxation for-malism. The Wangsness-Bloch equation expressing theevolution of the density operator in the presence of re-laxation adds a relaxation operator term to Eq. (2) thatoperates on the density matrix [3, 31]. Expanding ρ ina basis of orthonormal operators as in Eq. (5) gives thereal equation ˙ r = Ω r + R r + F ( r eq ) , (23) which can be transformed to a homogeneous equation asin [3]. The relaxation matrix R must be symmetric for re-laxation elements that act symmetrically between statesof the system, with diagonal elements providing auto-relaxation rates and off-diagonal elements giving cross-relaxation. The function F adds a constant term incorpo-rating the asymptotic decay of the system to the steadystate in terms of the equilibrium state r eq . Without thisterm, the solution decays to zero.Differentiating again gives ¨ r = (Ω + R ) ˙ r = Ω [ (Ω + R ) r + F ] + R ˙ r , (24)ie, a set of coupled oscillators with a velocity-dependentfriction term and a constant applied force Ω F . A con-stant force in the harmonic oscillator equation merelyshifts the origin of the coordinates. However, the ma-trix multiplying r , which determines the mechanical cou-plings as given in Eq. (16), is no longer symmetric due tothe sum of antisymmetric Ω and symmetric R , resultingin non-reciprocal off-diagonal couplings.The precise role of non-reciprocal couplings in a clas-sical model for quantum dissipative systems can be clar-ified by eliminating ˙ r to obtain ¨ r = (Ω + R ) r + (Ω + R ) F = Γ r + Γ F, (25)a set of ideal (frictionless) coupled oscillators subjectedto a constant applied force. In this case, however, thematrix Γ is the sum of symmetric Ω + R and anti-symmetric Ω R + R Ω. The former term corresponds toa set of undamped oscillators with symmetric couplings k ij = k ji ( § II A 2), modified in comparison to no relax-ation by inclusion of R . The normal-mode frequenciesare also modified accordingly.Damping is provided by the antisymmetric part of Γ ,which gives antisymmetric couplings γ ij = − γ ji and totalcoupling K ij = k ij + γ ij . The γ ij therefore representcouplings connected in parallel with the symmetric k ij and can be implemented, in principle, using magneticmaterials and magnetic fields. For a given positive γ ij ,a positive displacement of mass m j results in a positiveforce on m i . The resulting positive displacement of m i provides a negative force on m j due to γ ji < m j and damps themotion. Stated differently, energy transferred from m j to m i is not reciprocally transferred back from m i to m j ,and the motion is quenched.The inhomogeneous term Γ F in Eq. (25) can, equiv-alently, be included in an augmented matrix ˜Γ formedby appending a column Γ F to the right of Γ and thenadding a correspondingly expanded row of zeros at thebottom. The vector r is then augmented by including alast element equal to one to obtain the equivalent homo-geneous equation d dt ˜ r = ˜Γ ˜ r . (26)The asymmetry of ˜Γ generates unphysical couplings thatare not a problem theoretically but preclude a real,physical model. However, ˜Γ is readily written as thesum of symmetric ˜Γ S = (˜Γ + ˜Γ † ) / A = (˜Γ − ˜Γ † ) /
2, which determine the symmetric cou-plings k ij and antisymmetric couplings γ ij , respectively,as above.In comparison, the Schr¨odinger equation can only in-clude relaxation in certain special cases amenable to com-plex energies in the Hamiltonian. A typical applicationis the coupling between stable and unstable states andthe resulting lifetimes of the states. An example relat-ing velocity-dependent damping of classical oscillators toa Schr¨odinger equation treatment was provided in [29]in the weak-coupling limit. However, neither this ap-proximation nor the required complex energies can beapplied more generally. Even a simple two-level systemwith relaxation dynamics described by the Bloch equa-tion cannot be addressed by the Schr¨odinger equationand requires the density matrix approach. IV. ILLUSTRATIVE EXAMPLES
Simple two-level systems are used as a prototype forimplementing the quantum-classical mapping. Althoughthey are already known to be representable by classicalrotations in three-dimensional physical space, they pro-vide sufficient detail to clarify the connection between (i)real rotations of N-level quantum states in N − A. Quantum solution
In terms of real ∆ , ∆ and complex V = ω − iω , theHamiltonian for a general 2-level system can be writtenin terms of the Pauli matrices σ i ( i = 1 , ,
3) and σ = H = (cid:18) ∆ VV ∗ ∆ (cid:19) = X α =0 ω α σ α , (27)with ω = 1 / + ∆ ) and ω = 1 / − ∆ ). The σ term commutes with the other terms, so the propagator U ( t ) = e − iHt giving the Schr¨odinger equation solutionas in Eq. (17) is readily obtained in terms of ω i σ i ( i =1 , , e − i ω · σ t using unitvector ˆ ω = ω /ω gives U ( t ) = e − iω t e − i ω · σ = e − iω t [ cos ωt − i ˆ ω · σ sin ωt ]= e − iω t (cid:18) a b − b ∗ a ∗ (cid:19) . (28)The parameters a, b obtained from expanding ˆ ω · σ andusing the matrix forms for the σ i are a = cos ωt − i ˆ ω sin ωtb = − (ˆ ω + i ˆ ω ) sin ωt, (29)recognizable from classical mechanics as the Cayley-Kleinparameters for a rotation by angle 2 ωt about ˆ ω .Evolution of Schr¨odinger state | Ψ i ↔ ( c , c ) proceedsaccording to Eq. (17), with the corresponding densitymatrix states ρ ij = c i c ∗ j evolving according to Eq. (3).The equivalent classical evolution is considered next. B. Classical representation (Liouville equation)
Using the σ α as the basis and inner product h σ α | σ β i = 1 / σ α σ β ) = δ αβ gives Ω α = 0 = Ω α according to Eq. (8). The remaining 3 × σ i , σ j ] = 2 iǫ ijk σ k writ-ten in terms of the usual Levi-Civita tensor ǫ ijk (equalto ± j, k, l = 1 , , ij = h [ σ i , σ j ] | H i = − ǫ ijk h σ k | ω · σ i = − ω l ǫ ijk h σ k | σ l i = − ω k ǫ ijk Ω = 2 − ω ω ω − ω − ω ω . (30)The resulting equation of motion ˙ r = Ω r = 2 ω × r (31)represents a rotation of r about ω at angular frequency2 ω , as expected from the FVH Theorem for arbitrarytwo-level systems. The equivalence of quantum dynam-ics, in the case of 2-level systems, to a rotation in real,physical space cannot be generalized to arbitrary N-levelsystems. Representing quantum dynamics by a systemof coupled harmonic oscillators in one physical dimension is general for any value of N .The coupling matrix isΩ = 4 − ( ω + ω ) ω ω ω ω ω ω − ( ω + ω ) ω ω ω ω ω ω − ( ω + ω ) , (32)giving three mutually coupled oscillators as in Fig. 1. Thecouplings obtained from Eq. (16) are k ij / ω i ω j i = jk ii / ω j + ω k − ω i ω j − ω i ω k i = j = k = ω j ( ω j − ω i ) + ω k ( ω k − ω i ) . (33)For any possible ordering of the relative magnitudes ofnonzero ω i , at least one of the k ii has to be negative. C. Classical Representation (Schr¨odinger equation)
The matrix Ω leading to a solution for ( q , p ) as a rota-tion e − Ω t of the initial state ( q , p ) is comprised of the real and imaginary parts of H as in Eq. (18), givingΩ = − ω ∆ ω ω ω ∆ − ∆ − ω − ω − ω − ∆ ω (34)and coupling matrixΩ = − ∆ − ( ω + ω ) − ω (∆ + ∆ ) 0 − ω (∆ + ∆ ) − ω (∆ + ∆ ) − ∆ − ( ω + ω ) ω (∆ + ∆ ) 00 ω (∆ + ∆ ) − ∆ − ( ω + ω ) − ω (∆ + ∆ ) − ω (∆ + ∆ ) 0 − ω (∆ + ∆ ) − ∆ − ( ω + ω ) . (35)Four coupled oscillators are needed to represent( q , q , p , p ) ≡ ( r , r , r , r ). The mutual couplings k ij ( i = j ) given by Eq. (16) are the corresponding elementsof Ω . The self-couplings for i = 1 , k ii = ∆ i + ω + ω − ( ω + ω )(∆ + ∆ ) , (36)with k = k and k = k . Negative couplings arerequired in general, except for the special case ω = 0and ω < r i - r j associated with the nonzero Ω ij . Thenonzero mutual couplings represent the noncommutingrotations in Ω. One easily shows (see Appendix) thatnoncommuting rotations share a common coordinate axisin their respective rotation planes, such as r - r and r - r . Then Ω Ω = (Ω ) gives a nonzero mutualcoupling k . A rotation in the r - r plane does com-mute with a rotation in the r - r plane, so one expectsthe mapping from rotations to oscillators to generate atleast one mutual coupling equal to zero. For the par-ticular example here, the structure of Ω is such that(Ω ) = 0 = (Ω ), giving zero for k and k . Mass1 is not coupled to mass 3, and mass 2 is not coupled tomass 4.Thus, the rotation of a single vector or rotor in real4-dimensional space can be viewed equivalently as 4 os-cillators evolving in one dimension. Similarly, for N-levelsystems, the evolution of the associated rotor in N − N − D. Quantum dimer
The quantum dimer example provided in [29] is givenby real V = ω , ω = 0, and ∆ = ∆ = ω , giving ω = 0.
1. Liouville approach
The only nonzero elements of Ω in Eq. (30) are thenΩ = 2 V = − Ω , leading to diagonal entries (Ω ) =(Ω ) = − V as the only nonzero elements of Ω inEq. (33). Thus, two uncoupled oscillators, each with nat-ural frequency 2 V , represent this particular quantum sys-tem, with the initial conditions determining the specificdetails of the time evolotion.For Ψ( t ) = [ c ( t ) , c ( t )] and Ψ(0) = (1 , r (0) = (0 , , /
2) using r i = 1 / σ i ρ ), result-ing in ˙ r (0) = [0 , − V, r ( t ) = 12 − sin 2 V t cos 2
V t , (37)which is the expected rotation about axis ˆ ω = ˆ ω atangular frequency 2 ω t = 2 V t given by Eq. (29). Sincethe two oscillators are out of phase by 90 ◦ , the systemcan actually be represented by a single oscillator—theposition of one oscillator automatically gives the positionof the other from a simple phasor diagram.
2. Schr¨odinger approach
Referring to the 2 × in Eq. (35),one finds off-diagonal blocks equal to zero, since theydepend on the imaginary part of V . The two remain-ing nonzero blocks on the diagonal generate independentevolution of q and p . The q -block gives two coupledoscillators with mutual coupling k = − ω V and self-couplings k ii = ( ω − V ) from Eq. (36). The p -blockgives identical couplings. One can instead switch to apositive value for the mutual coupling, as in [29], sincethe normal-mode eigenvalues − ( ω ± V ) of Ω are onlyinterchanged by changing the sign of V . This changesthe sense of rotation generated by H in Hilbert spaceand hence, by Ω in the real 4-dimensional space. Us-ing a positive coupling in this way captures the essentialelements of the problem, but does not, strictly speak-ing, faithfully map the quantum system to the oscillatorsystem. More importantly, negative couplings cannot beavoided, most generally.With the definitions in § II B 1, the initial condition c (0) = (1 ,
0) corresponds to ( q , p ) = (1 , , , ˙ q , ˙ p ) = (0 , , − ω , − V ). Thefour oscillators must be set in motion with these initialpositions and velocities for their displacments in a me-chanical implementation to correspond to the evolutionof | Ψ( t ) i = [ c ( t ) , c ( t )].However, the propagator U ( t ) is readily obtained fromEq. (14) in terms of the eigenvectors (1 ,
1) and (1 , −
1) foreach 2 × U ( t )to reproduce the solution given in [29]. The Schr¨odingerequation requires four coupled oscillators for this partic-ular example, in contrast to two uncoupled oscillators forthe Liouville representation (equivalent to a single oscil-lator, since they are always 90 ◦ out of phase). E. Symmetric unperturbed levels
Consider ∆ = − ∆ = ω , which arises in representingtwo unequal energy levels relative to the mean energy ofthe levels.
1. Liouville approach
There are no nonzero elements of Ω derived fromEq. (32). The system is fully coupled, as illustrated inFig. 1, and represents the most general result for this ap-proach. The Schr¨odinger approach, discussed next, pro-vides a simpler representation in this case.
2. Schr¨odinger approach
The matrix Ω of Eq. (35) is now diagonal for anygeneral complex perturbation V . Four uncoupled oscil-lators, each with natural frequency ( P i ω i ) / , representthe system. Specifying | Ψ(0) i determines the initial con-ditions as discussed previously. This is a very simple sys-tem, with each oscillator evolving independently.The Liouville approach, by contrast, results in a rela-tively more complex system, albeit with one less oscilla-tor. Yet, for the dimer example, the Liouville implemen-tation is much simpler than the Schr¨odinger implementa-tion. Which approach gives the simpler set of oscillatorsand couplings depends on the specific problem. F. Bloch equation with relaxation
The solution of the Bloch equation for the time depen-dence of nuclear magnetization in a magnetic field is rel-atively simple for a field along the z -axis [32]. As is well-known, the transverse magnetization precesses about thefield at the Larmor frequency while decaying exponen-tially at a transverse relaxation rate 1 /T . The longi-tudinal magnetization relaxes to the equilibrium mage-tization at a rate 1 /T . The mapping of this motion toa system of damped oscillators illustrates the proceduredescribed in § III, as well as the role of non-reciprocal cou-plings in the model. Physical insight that may have beenoverlooked in the past is readily apparent from classicaltextbook treatments of damped oscillations.The inhomogeneous term F in Eq. (23) is (0 , , M /T ,where M is the equilibrium magnetization. Vector r represents the nuclear magnetization. Denoting ω asthe Larmor frequency, the matrix Γ = Ω + R isΓ = − T − ω ω − T
00 0 − T . (38)As described earlier, appending a column Γ F to the rightof Γ followed by a row of zeros at the bottom givesEq. (26) for the oscillator equation, with˜Γ = T − ω
23 2 ω T − ω T T − ω T − M T (39)and ˜ r = 1 augmenting r to represent a static componentthat incorporates the inhomogeneous term Γ F . The nec-essary couplings are easily read from symmetric Γ S andantisymmetric Γ A that sum to give ˜Γ:˜Γ S = T − ω T − ω T − M T − M T ˜Γ A = ω T − ω T − M T M T . (40)A symmetric coupling k = − M / (2 T ) connectedin parallel with antisymmetric (nonreciprocal) coupling γ = − M / (2 T ) provides the contribution to the finalsteady state magnetization ˜ r through coupling to ˜ r .The vanishing of k + γ ensures there is no couplingfrom ˜ r to change the static component ˜ r . Althoughthere is no friction term in Eq. (26), the mechanism thatdamps ˜ r is fairly transparent. Since k = 0 = k ,˜ r is only coupled to static ˜ r , which effectively shifts˜ r to z = ˜ r − M , giving the equivalent equation ¨ z = z/T . The self-coupling k is the source of the imaginarynatural frequency i/T . resulting in the standard dampedsolution z ( t ) = z (0) e − t/T .The mechanism for transverse relaxation is perhapsmore interesting, given that the diagonal elements ˜Γ ii ( i = 1 ,
2) cannot be the source of the damping for thecase ω = 1 /T . More generally, since ˜Γ = ˜Γ for anyvalue of ω , the eigenvalues are ˜Γ plus the eigenvaluesfor the antisymmetric block, which are ± iω /T . Thenormal mode frequencies, given by the square root of theeigenvalues, are ω ± i/T . The asymmetric coupling isthe sole source of the imaginary frequency producing therequired e − t/T decay of the transverse magnetization.The displacements of ˜ r and ˜ r exert opposing forces oneach other compared to the usual symmetric couplings,which produce oscillations, and the motion is damped. V. CONCLUSION
General N-level quantum systems can be representedas an assembly of classical coupled oscillators. The methodology provides the possibility for visual, mechan-ical insight into abstract quantum systems, as well as ametric for characterizing the interface between quantumand classical mechanics. There is a one-to-one correspon-dence between oscillator positions at time t and quantumstates of the system. The formalism presented includesboth open (dissipative) and closed systems. For systemsrepresented by a density matrix, the known evolution ofstates as rotations of a single coherence vector in a real(but unphysical) hyperspace of N − N − N dimensions, which can be mapped to 2 N oscillatorsin one-dimensional physical space. The scaling of quan-tum systems to classical systems is therefore linear forSchr¨odinger states rather than quadratic for density ma-trix representations.The results are applicable to time-independent Hamil-tonians, which is sufficiently general for a great manycases of practical interest. Time-dependent Hamiltoni-ans (ie, driven systems) can be approximated to a chosenlevel of accuracy by a sequence of constant Hamiltoni-ans over sufficiently short time steps. Modeling a time-dependent quantum Hamiltonian as classical requires achange in the spring constants of the mechanical systemat each time step, together with a reinitialization of thevelocities derived from the new positions of the oscillatorsaccording to Eq. (9). By contrast, velocities in the nat-ural dynamics of a system of coupled oscillators wouldnot change instantaneously and discontinuously with achange in spring constants. Acknowledgments
The author acknowledges support from the Na-tional Science Foundation under grant CHE-1214006 andthanks N. Gershenzon for assitance with the figure. [1] R. C. Tolman,
The Principles of Quantum Statistical Me-chanics (Clarendon Press, Oxford, 1938).[2] J. von Neumann,
Mathematical Foundations of Quan-tum Mechanics (Princeton University Press, Princeton,1955).[3] U. Fano, Rev. Mod. Phys. , 74 (1957).[4] R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwarth,J. Appl. Phys. , 49 (1957).[5] F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. , 838(1981). [6] J. N. Elgin, Phys. Lett. , 140 (1980).[7] G. Jaeger, M. Teodorescu-Frumosu, A. Sergienko, B. E.A. Saleh, and M. C. Teich, Phys. Rev. A , 032307(2003).[8] P. Dirac, Proc. R. Soc. London , 243 (1927).[9] F. Strocchi, Rev. Mod. Phys. , 36 (1966).[10] J. Frenkel, Zeitschrift f¨ur Physik A
198 (1930).[11] P. W. Anderson,
Lectures on the Many-Body Problem ,edited by E. R. Caianello (Academic Press, New York,1964), vol. 2, p. 113. [12] A. C. Scott, Amer. J. Phys. , 52 (1969).[13] D. B. Sullivan and J. E. Zimmerman, Amer. J. Phys. ,1504 (1971).[14] R. J. C. Spreeuw, N. J. van Druten, M. W. Beijersbergen,E. R. Eliel, and J. P. Woerdman, Phys. Rev. Lett ,2642 (1990).[15] W. Frank and P. von Brentano, Am. J. Phys. , 706(1994).[16] C. F. Jolk, A. Klingshirn, and R. V. Baltz, Ultra-fast Dynamics of Quantum Systems: Physical Processesand Spectroscopic Techniques (Plenum Press, New York,1998), p. 397.[17] C. L. Garrido Alzar, M. A. G. Martinez, and P. Nussen-zveig, Am. J. Phys. , 37 (2002).[18] L. Novotny, Am. J. Phys. , 1199 (2010).[19] A. Kovaleva, L. I. Manevitch, and Y. A. Kosevich,Phys. Rev. E , 026602 (2011).[20] H.-T. Elze, Phys. Rev. A , 1114 (1988). [23] R.Hemmer, M.G.Prentiss, J. Opt. Soc. Am. B , 1613(1988).[24] R. Marx and S. J. Glaser, J. Magn. Reson. , 338(2003).[25] V. Leroy, J.-C. Bacri, T. Hocquet, and M. Devaud,Eur. J. Phys. , 1363 (2006).[26] B. W. Shore, M. V. Gromovyy, L. P. Yatsenko, and V. I.Romanenko, Am. J. Phys. , 1183 (2009).[27] J. S. Briggs and A. Eisfeld, Phys. Rev. E , 051911(2011).[28] A. Eisfeld and J. S. Briggs, Phys. Rev. E , 046118(2012).[29] J. S. Briggs and A. Eisfeld, Phys. Rev. A , 052111(2012).[30] T. E. Skinner and S. J. Glaser, Phys. Rev. A , 032112(2002).[31] R. Wangsness and F. Bloch, Phys. Rev. , 728 (1953).[32] F. Bloch, Phys. Rev.70