Emergence of Periodic Structure from Maximizing the Lifetime of a Bound State Coupled to Radiation
EEmergence of Periodic Structure from Maximizing the Lifetime of aBound State Coupled to Radiation
Braxton Osting and Michael I. WeinsteinNovember 21, 2018
Keywords:
Fermi’s golden rule, quality factor, constrained optimization, ionization, parametricforcing, Schr¨odinger Equation, Bragg resonance, spectral gap
Abstract
Consider a system governed by the time-dependent Schr¨odinger equation in its ground state.When subjected to weak (size (cid:15) ) parametric forcing by an “ionizing field” (time-varying), thestate decays with advancing time due to coupling of the bound state to radiation modes. Thedecay-rate of this metastable state is governed by
Fermi’s Golden Rule , Γ[ V ], which depends onthe potential V and the details of the forcing. We pose the potential design problem: find V opt which minimizes Γ[ V ] (maximizes the lifetime of the state) over an admissible class of potentialswith fixed spatial support. We formulate this problem as a constrained optimization problemand prove that an admissible optimal solution exists. Then, using quasi-Newton methods, wecompute locally optimal potentials. These have the structure of a truncated periodic poten-tial with a localized defect. In contrast to optimal structures for other spectral optimizationproblems, our optimizing potentials appear to be interior points of the constraint set and to besmooth. The multi-scale structures that emerge incorporate the physical mechanisms of energyconfinement via material contrast and interference effects.An analysis of locally optimal potentials reveals local optimality is attained via two mech-anisms: (i) decreasing the density of states near a resonant frequency in the continuum and(ii) tuning the oscillations of extended states to make Γ[ V ], an oscillatory integral, small. Ourapproach achieves lifetimes, ∼ (cid:0) (cid:15) Γ[ V ] (cid:1) − , for locally optimal potentials with Γ − ∼ O (10 ) ascompared with Γ − ∼ O (10 ) for a typical potential. Finally, we explore the performance ofoptimal potentials via simulations of the time-evolution. In many problems of fundamental and applied science a scenario arises where, due to physical lawsor engineering design, a state of the system is metastable ; the state is long-lived but of finite lifetimedue to coupling or leakage to an environment . In settings as diverse as linear and nonlinear optics,cavity QED, Bose-Einstein condensation (BEC), and quantum computation, one is interested inthe manipulation of the lifetime of such metastable states. Our goal in this paper is to explore theproblem of maximizing the lifetime of a metastable state for a class of ionization problems . Theapproach we take is applicable to a wide variety of linear and nonlinear problems. Specific exampleswhere metastable states exist include: 1 a r X i v : . [ m a t h . A P ] O c t . the excited state of an atom, e.g. hydrogen, where due to coupling to a photon field, theatom in its excited state spontaneously undergoes a transition to its ground state, after someexcited state lifetime [4],2. an approximate bound state (quasi-mode) of a quantum system, e.g. atom in a cavity orBEC in a magnetic trap, which leaks (“tunnels”) out of the cavity and whose wave functiondecays with advancing time [28],3. an approximate guided mode of an optical waveguide which, due to scattering, bends in thewaveguide, or diffraction, leaks out of the structure, resulting in attenuation of the wave-fieldwithin the waveguide with increasing propagation distance [24], and4. a “scatterer” which confines “rays” , but leaks energy to spatial infinity due to their wavenature, e.g. Helmholtz resonator, traps rays between obstacles [23].These examples are representative of a class of extended (infinite spatial domain), yet energy-preserving ( closed ) systems, where the mechanism for energy loss is scattering loss , the escape ofenergy from a compact spatial domain to spatial infinity.Such systems can often be viewed as two coupled subsystems, one with oscillator-like degrees offreedom characterized by discrete frequencies and the other a wave-field characterized by continuousspectrum. When (artificially) decoupled from the wave-field, the discrete system has infinitely long-lived time-periodic bound states. Coupling leads to energy transfer from the system with oscillatorlike degrees of freedom to the wave field. In many situations, a (typically approximate) reduceddescription, which is a closed equation for the oscillator amplitudes, can be derived. This reductioncaptures the view of the oscillator degrees of freedom as an open system with an effective (radiation)damping term. In the problems considered in this paper, the reduced equation is of the simple form: ı∂ t A (cid:15) ( t ) ∼ (cid:15) ( Λ − ı Γ ) A (cid:15) ( t ) . (1.1)Here, (cid:15) is a real-valued small parameter, measuring the degree of coupling between oscillator andfield degrees of freedom. A (cid:15) ( t ) denotes the slowly-varying complex envelope amplitude of theperturbed bound state. Λ is a real frequency and Γ > ı∂ t φ (cid:15) = H V φ (cid:15) + (cid:15) W ( t, x, | φ (cid:15) | ) φ (cid:15) , H V ≡ − ∆ + V ( x ) . (1.2)Here, V ( x ) is a real-valued time-independent potential and W ( t, x, | φ | ) is a time-dependent potential(parametric forcing), W = ˜ β ( t, x ), or nonlinear potential, e.g. W = ±| φ | . Equation (1.2) definesan evolution which is unitary in L ( R ).In this article we focus on the class of one-dimensional ionization problems, where W ( t, x ) = cos( µt ) β ( x )where β ( x ) is a spatially localized and real-valued function and µ > ı∂ t φ (cid:15) = H V φ (cid:15) + (cid:15) cos( µt ) β ( x ) φ (cid:15) . (1.3)2e focus on the case where the parameter (cid:15) is real-valued and assumed sufficiently small. Assumptions for the unperturbed problem, (cid:15) = 0 : Initially we assume that the potential V ( x ), decays sufficiently rapidly as | x | → ∞ , although we shall later restrict to potentials with afixed compact support. Furthermore, we assume that H V has exactly one eigenvalue λ V <
0, withcorresponding (bound state)eigenfunction, ψ V ( x ): H V ψ V = λ V ψ V , (cid:107) ψ V (cid:107) = 1 . (1.4)Thus, φ ( x, t ) = e − ıλ V t ψ V ( x ) is a time-periodic and spatially localized solution of the unperturbedlinear Schr¨odinger equation: ı∂ t φ = H V φ We indicate an explicit dependence of λ V and ψ V on V , since we shall be varying V . Fermi’s Golden Rule:
We cite consequences of the general theory of [33, 19, 20]. If µ , the forcingfrequency, is such that λ V + µ >
0, then for initial data, φ ( x,
0) = ψ V ( x ) (or close to ψ V ), thesolution decays to zero as t → ∞ . On a time scale of order (cid:15) − the decay is controlled by (1.1), i.e. | A ( t ) | ∼ | A (0) | e − (cid:15) Γ[ V ] t , < t < O ( (cid:15) − ) (1.5)where | A ( t ) | = |(cid:104) ψ V , φ (cid:15) ( t ) (cid:105)| and Γ[ V ] is a positive constant. Thus, we say the bound state has alifetime of order (cid:0) (cid:15) · Γ[ V ] (cid:1) − and the perturbation ionizes the bound state.The emergent damping coefficient, Γ[ V ], is often called Fermi’s Golden Rule [35], arising in thecontext of the spontaneous emission problem. However, the notion of effective radiation dampingdue to coupling of an oscillator to a field has a long history [22]. In general, Γ [ V ] is a sum ofexpressions of the form: (cid:12)(cid:12)(cid:12) (cid:104) e V ( · , k res ( λ V )) , G W ( ψ V ) (cid:105) L ( R d ) (cid:12)(cid:12)(cid:12) = | t V ( k res ) | (cid:12)(cid:12)(cid:12) (cid:104) f V ( · , k res ( λ V )) , G W ( ψ V ) (cid:105) L ( R d ) (cid:12)(cid:12)(cid:12) , (1.6)(see (3.2)) where G W ( ψ V ) depends on the coupling perturbation W in (1.2) and the unperturbedbound state, ψ V . Here, e V ( · , k res ) = t V ( k res ) f V ( · , k res ) is the distorted plane wave (continuumradiation mode) associated with the Schr¨odinger operator, H V , at a resonant frequency k res = k res ( λ V ), for which k res ∈ σ cont ( H V ). t V ( k ) denotes the transmission coefficient and f V ( x, k ) a Jost solution . In Secs. 2 and 3 we present an outline of the background theory for scattering andthe ionization problem, leading to (1.5), (1.6); see [33].We study the problem of maximizing the lifetime of a metastable state, or equivalently, mini-mizing the scattering loss of a state due to radiation by appropriate deformation of the potential, V ( x ), within some admissible class, A ( a, b, µ ):min V ∈A ( a,b,µ ) Γ [ V ] . (1.7)We refer to Eq. (1.7) as the potential design problem (PDP). Our admissible class, A ( a, b, µ ), isdefined as follows: Definition 1.1. V ∈ A ( a, b, µ ) if 3. V has support contained in the interval [ − a, a ], i.e. V ≡ | x | > a V ∈ H ( R ) and (cid:107) V (cid:107) H ≤ b H V = − ∂ x + V ( x ) has exactly one negative eigenvalue, λ V , with corresponding eigenfunction ψ V ∈ L ( R ), which satisfies Eq. (1.4) : H V ψ V = λ V ψ V , (cid:107) ψ V (cid:107) = 1.4. k res ≡ √ λ V + µ > Remark 1.1.
Based on our numerical simulations, we conjecture that the hypothesis 2., imposinga bound of V , can be dropped.The idea of controlling the lifetime of states by varying the characteristics of a backgroundpotential goes back to the work of E. Purcell [29, 30], who reasoned that the lifetime of a state canbe influenced by manipulating the set of states to which it can couple, and through which it canradiate. Remark 1.2.
We discuss the potential design problem where1. β ( x ) is a fixed function, chosen independently of V , for example, β ( x ) = [ − , ( x )2. β ( x ) = V ( x ). Remark 1.3. How does one minimize an expression of the form (1.6) ? We can think of two ways in which (1.6) can be made small:
Mechanism (A)
Find a potential in A for which the first factor in (1.6), | t V ( k res ) | is small,corresponding to low density of states near k res . N.B.
As proved in Proposition 2.6, | t V ( k ) | ≥ O ( e − Ka ) for V with support contained in [ − a, a ]. Mechanism (B)
Find a potential in A which may have significant density of states near k res (say | t V ( k res ) | ≥ /
2) but such that the oscillations of f V ( x, k res ) are tuned to make the matrix elementexpression (inner product) in (1.6) small due to cancellation in the integral.Indeed, we find that both mechanisms occur in our optimization study. Remark 1.4.
We are interested in the problem of deforming V within an admissible set in such away as to maximize the lifetime of decaying (metastable) state. Intuitively, there are two physicalmechanisms with which one can confine wave-energy in a region: via the depth of the potential(material contrast) and via interference effects. We shall see that our (locally) optimal solutions,of types (A) and (B) find the proper balance of these mechanisms.