Quantum tunneling on graphs - National Taiwan · PDF fileThe Gap of the First Two Eigenvalues...

The Gap of the First Two EigenvaluesQuantum tunneling

Results and methodsExamples

Quantum tunneling on graphs

Shing-Tung Yau

Harvard University

Talk at S. R. Srinivasa Varadhan’s 70th Birthday ConferenceTaiwan University, July 11-15, 2011

1



Overview

1 The Gap of the First Two Eigenvalues of the SchrödingerOperator

2 Quantum tunneling (joint work with Yong Lin and GaborLippner)

3 Results and methods

4 ExamplesDouble wellsTriple wells

2



For a second order self-adjoint elliptic operator acting on theHilbert space of functions defined on a compact closedmanifold or a compact manifold with zero boundary data, thevariational principle shows that the first eigenfunction of theoperator does not change sign and has multiplicity equal to one.

The most important linear operator is the Schrödinger operator−1

2∆ + V where V is the potential function. This operatordescribes the behavior of particles moving in a potential fieldwith energy given by

E(x) =12

∫ T

0|x |2 dt +

∫ T

0V (x(t))dt

where x : [0,T ] → R3 is the path of the particle.

3



The least action principle shows that the paths which arecritical point of this functional describes motions of classicalparticles. When V = 0, these are geodesics of the Riemannianmetric that defines the Laplacian.

The first eigenfunction for V = 0 is given by the constantfunction and its eigenvalue is zero. However, in the geometryliterature, the first nontrivial eigenvalue is called the firsteigenvalue of the manifold and denoted by λ1. So when weestimate the first eigenvalue, we are actually estimating thedifference between the first two eigenvalues.

The gap of the first two eigenvalues is very important for bothgeometry and physics. It can be used to measure convergenceof solutions of differential equations. It can be used to measurethe tunneling effect or ionization in quantum mechanics.

4



There were works of S.Y. Cheng (1973, 1975) who gave sharpupper estimates of the first eigenvalue of the Laplacian formanifolds whose Ricci curvature has a lower bound. He provedthat for closed manifolds with non-negative Ricci curvature andthe same diameter, the sphere of the same dimension has thehighest first eigenvalue.

When the Ricci curvature is strongly positive; there were worksof Lichnerowicz (1958) and Obata (1962), who gave sharplower estimates of the first eigenvalue. They proved that forclosed manifolds with Ricci curvature ≥ c > 0, the sphere hasthe lowest first eigenvalue.

5



Peter Li (1979) then generalized the result for manifolds withnon-negative Ricci curvature. Li-Yau (1981) made use of theidea of the gradient estimate of Yau (1973) to find a lowerestimate of the eigenvalue in terms of the lower bound of theRicci curvature and a upper bound of the diameter.

Li and I conjectured in 1982 that for compact manifolds withnon-negative Ricci curvature, a circle will give the lower valuefor λ1d2 where d is the diameter. Improving the method ofLi-Yau, Zhong-Yang in 1984 proved this conjecture.

It turns out most of the arguments that Li and I used can begeneralized to the case when there is a potential.

6



An important potential is the scalar curvature of the manifold.The Schrödinger operator was then related to the secondvariation of the area of a minimal hypersurface.

In fact, in an impressive work, Barbosa-Do Carmo (1973)studied the first eigenvalue of the Schrödinger operator,−∆ + 2K , where K is the Gaussian curvature of the minimalsurface and is associated to the second variation of the area.They concluded that if the absolute total curvature of a piece ofthe minimal surface is less than 2π, then the surface is stable.This argument is based on the Faber-Krahn inequality fordomains on the sphere.

7



Note that estimates of the eigenvalue of −∆ + 2K can be linkedto estimates of the eigenvalues of −K ds2, which is the metricon the sphere.

Shortly after, Nitsche (1973) was able to use Barbosa-DoCarmo’s remarkable theorem to prove the uniqueness of theminimal disk spanning a Jordan curve whose absolute totalcurvature is less than 4π.

8



When one studies conformal deformation of metrics, there is anatural conformal invariant operator

−∆ + C(n)R

where R is the scalar curvature of the metric. Schoen-Yau(1979, 1981, 1988) studied this extensively in relation to thepositive mass conjecture and the geometry of conformally flatmanifolds with positive scalar curvature.

9



Schoen-Yau (1983) also found an upper bound of the firsteigenvalue of the operator in terms of a suitably defineddiameter of the manifold. Specifically, the first eigenvalue of−∆ + (1/6)R for a three dimensional manifold with Dirichletboundary problem is bounded from above by (3π2/2)1/d2

where d is the diameter defined by considering the width of thetube embedded in the domain.

Note that the scalar curvature comes from gravity and it wouldbe interesting to find the best constant for such an inequality tohold true. It may be achieved by domains with an SU(2)symmetry.

10



When the potential is convex, the idea was carried out in apaper by I. Singer, B. Wong, S.-T. Yau and S. S.-T. Yau (SWYY)in 1985.

An important ingredient is the log concavity result ofBrascamp-Lieb (1976). I developed a continuity argument toreprove the Brascamp-Lieb result. I found this method in 1980.Through my former student A. Treibergs, my argument wasspread to experts who found my method quite useful.

I shall use similar arguments to give a lower estimate of theHessian of − log u , where u is the ground state, withoutassuming the potential is convex. This estimate is the key togiving a good lower estimate of the gap λ2 − λ1.

11



In SWYY, we proved that the gap λ2 − λ1 for a convex potentialcan be bounded from below in terms of the diameter of thedomain.

In 2003, I strengthened this result to the case when the Hessianof the potential is greater than c > 0. Specifically, I proved that

λ2 − λ1 ≥θ2(β)

diam(Ω)2 + β√

c

where θ(β) = sin−1 1√1 + β√

2−β

and 0 < β <√

2 .

Recently, J. Ling has improved on this estimate based onsimilar gradient estimate.

12



Drawing an analogy with the manifold case, we expect that for aconvex domain Ω with (inner) diameter d and for a potentialwhose Hessian has eigenvalue greater than c ≥ 0, (λ2 − λ1)d2

is minimized when the domain becomes an interval with lengthd and the potential become a quadratic function c x2/2.

13



Let us now turn to non-convex potentials. The most well-knownresults come from the double well potential. Consider thefollowing simple one dimensional operator:

− d2

dx2 + a2 x2(x − a−1)2

x → a−1− x gives a reflection with the center ata−1

2. There are

two wells

x = 0and x = a−1.

When a is small, the behavior of the system is decoupled andbehave like a single well.

However, quantum mechanically, the two wells are coupled bytunnelling and the degeneracy of the lowest eigenvalue isrelated to this coupling.

14



Roughly speaking, if u1 and u2 are the normalized eigenvectors

with lowest eigenvalues for the operator −12∆ + V , u1 + u2

should concentrate in the well at x = 0 and u1 − u2 at the wellx = a−1.

Now,

e−itH (u1 + u2) = e−itλ1(

u1 + e−it(λ2−λ1) u2

)we see that

π

λ2 − λ1is the time needed to evolve from one

state concentrated in one well into the other well.

When λ2 is very close to λ1, we observe eigenvaluedegeneracy. This is relevant for many interesting physicalphenomena.

15



For example, C. J. Thomson and M. Kac [in “Phase transitionand eigenvalue degeneracy of a one dimensional anharmonicoscillator." Studies in Appl. Math. 48 1969 257–264] discussedsome lattice models with exponential interactions. Wereproduce their discussion in the following:

Take a one dimensional chain of N spins, µi = ±1 , withinteraction energy

E = −Jγ∑

1≤i<j≤N

exp(−γ|i − j |)µi µj .

16



The free energy per spin, ψ , in the limit N →∞ is given by

− ψ

kT= log 2 − ν γ + logλ1

where ν =J

kT, k is the Boltzmann constant, T is the absolute

temperature and λ1 is the maximum eigenvalue of the integralequation of the form∫ ∞

−∞K (x , y)ϕ(y)dy = λϕ(x) .

17



The correlation function ρ(r) is given by

ρ(r) = limN→∞

< µk µk+r >

=∑j=2

(λj

λ1

)r (∫ ∞

−∞ϕj(x)ϕ1(x) tanh (

√νγ x) dx

)2

where λj are the eigenvalues and ϕi are the eigenfunctions ofthe integral equation.

One sees thatlimr→α

ρ(r) > 0

iff λ1 is asymptotically degenerate.

18



In fact,λ2

λ1= 1−O

(exp

(−cγ

))for c > 0

as γ → 0 , and

logλ1 =γ

2− γ E0 +O(γ2)

where E0 is the smallest eigenvalue of

d2ψ

dx2 −(

14

x2 − γ−1 log [cosh(√νγ x)]

)ψ = −Eψ .

19



When 2ν < 1, the equation can be approximated by

d2ψ

dx2 − v(x)ψ = −Eψ

where v(x) = αx2 + βγx4 .When γ → 0, α < 0 .

By a change of variable to (−α)−14 x , the equation becomes

d2udx2 + (x2 − εx4)u = −λu

where

u(x) = ψ((−α)−14 x)

λ = (−α)−12 E

ε = βγ(−α)−32

20



By a heuristic argument, C. J. Thompson and M. Kac found

λ2 − λ1 ≤ A exp(− 1

16ε

).

They noted that the potential

v(x) = −x2 + εx4

has two equal minima at x = ±(2ε)−1/2 and vmin(x) = −(4ε)−1.

If one expands the potential about the point x = (ε)−1/2 theunperturbed potential is

− 14ε

+ 2(

x − 1√2ε

)2

.

This potential represents a perfectly well-behaved oscillator.21



The unperturbed principal eigenfunction then consists of twoGaussians centered at ±(2ε)−1/2

u(0)1 (x) = u(0)

(x − 1√

2ε

)+ u(0)

(x +

1√2ε

)(∗)

whereu(0)(x) = exp(−x2/

√2) .

The unperturbed eigenfunction corresponding to the nextlowest eigenvalue λ2 is the antisymmetric form of equation (∗)

u(0)2 (x) = u(0)

(x − 1√

2ε

)− u(0)

(x +

1√2ε

). (∗∗)

22



The two eigenfunctions u(0)1 and u(0)

2 give the sameunperturbed eigenvalue. But taking u(0)

1 and u(0)2 as trial

functions in the Rayleigh-Ritz principle for λ1 and λ2

respectively, a simple calculation shows that

λ2−λ1 ∼∫ ∞

−∞u(0)

(x − 1√

2ε

)u(0)

(x +

1√2ε

)dx ∼ exp(−c/ε)

with c > 0.

This heuristic result is very suggestive but it is not a proof, as itis derived from perturbation theory.

23



Consider the analogous problem for a two dimensional latticemodel. It has exponential interaction in one direction andnearest neighbor interactions in the other. For an M ×∞ lattice,the analogous differential equation is now the M-dimensionalequation

M∑k=1

∂2ψ

∂x2k−

14

M∑k=1

x2k − γ−1

M∑k=1

log cosh√νγ

2(xk + xk+1)

ψ = −λψ

The problem is to show that for sufficiently large ν (i.e.sufficiently small temperatures) and sufficiently small γ

λ2 − λ1 = O(exp(−cM/γ)) , c > 0 .

Thompson and Kac heuristic perturbative argument gives thisresult but they do not have a proof.

24



R. B. Griffiths has argued that long range order exists forsufficiently large ν for all finite γ, in the limit M →∞, so that λ1

must be asymptotically degenerate.

Also, if the predicted eigenvalue difference is correct, the limitM →∞ alone is sufficient to produce long range order. Smallγ, which was essential for long range order in one dimension isthen only a convenience and not a necessity in two dimensions.

25



In a different direction, B. Simon [in "Instantons, double wells,and large deviations," Bull. Amer. Math. Soc. 8 (1983)323-326] applied semi-classical analysis to the study ofeigenvalues of the Schrödinger operator

−12∆ + λ2V

in the limit λ→∞.

26



Assume the following for V .

1. V is C∞.

2. V (x) ≥ 0 for all x and lim|x |→∞ V (x) > 0 .

3. V vanishes at exactly two points a and b and at thesepoints ∂2V/∂xi∂xj is strictly positive definite.

Under these conditions, it can be proved that the ground stateu1 is concentrated as λ→∞ near the points a, b and that λi/λ

has a finite nonzero limit.

27



Assume also that for all ε small and for y = a and for y = b

limλ→∞

∫|x−y |≤ε

|u1(λ, x)|2 dx > 0 .

A basic geometric object

ρ(x , y) = infγ,T

(12

∫ T

0γ(s)2ds +

∫ T

0V (γ(s))ds

γ(0) = x , γ(T ) = y

).

With all of the above assumptions for V , Simon showed that

limλ→∞

−1λ

ln |λ2 − λ1|

= ρ(a,b) .

28



The semi-classical calculation of Simon requires the Planckconstant = 1/λ to tend to zero. It does not provide muchinformation if the constant remains non-zero.

I proved in 2003 that for a convex domain Ω, the gap of the firsttwo eigenvalues of −∆ + V is greater than

2 d(Ω)−2 exp(−α d(Ω)2

)where d(Ω) is the inner diameter of Ω and α is the lower boundof the Hessian of − log u1 where u1 is the first eigenfunction.

29



Estimate of λ2 − λ1

Let u1 be the first eigenfunction (positive)

(−4+ V )u1 = λ1u1

and ϕ = − log u1.

30



Let t ≥ (inf∂Ω V )−12 and infΩ(V − λ2) ≤ ε inf∂Ω V . Then

infΩt |u|supΩt

|u|≤ ε

1− ε.

33



Assume|∇V | ≤ Cα(V + α)

32 ,

|(4V )+| ≤ Cα(V + α)3.

Define

dα(x0, x1) = inf∫ 1

0

√V + α|x |dt ,

where the infimum is taken among all paths x : [0,1] → Ω

joining x0 to x1.

34



Estimate of (− log u1)ii

Suppose there is a function u1 > 0

(−4+ V )u1 = λu1

so that u1 = 0 on ∂Ω and (− log u1)ii has a lower bound.

36



Define u1 = u1W .

On ∂Ω, W can be estimated in terms of ∂u1∂ν and all the

derivatives of W can be estimated in terms of the derivatives ofu1 on ∂Ω. But

4W = (V − V − λ1 + λ1)W − 2(∇ log u1) · ∇W

Since we have estimates of W along the boundary, we canestimate W . As a result, we can give lower estimate of the− log of the first eigenfunction.

37



The case of a graph

The standard shortest-path metric on a graph is counterintuitivefor graphs coming from geometry.

Quantum tunneling on manifolds is related to the geodesicdistance of the potential wells. Hence this might be used todefine a more intuitive metric on graphs.

Discrete approximation to the physical problem: manifolds →graphs; Laplace operator → Laplace matrix.

Discrete evolution: continuous quantum walk.

Recent applications in quantum computing: examples ofquantum speed-up, where hitting times of quantum walks aremuch smaller than the corresponding classical walks. This canbe used to construct problems which can be solved faster usingquantum computers than classical computers.

38



States and potentials

Let G(X ,E) be a finite graph.

X , the set of vertices of G represent the "pure" states of aquantum particle.

A "mixed" state is given by a function ϕ : X → C.Interpretation: |ϕ(v)|2 is the probability that the particle isin v .

Potential energy is given by a function V : X → R. We areinterested in asymptotics, hence we assume V = Q ·Wwhere Q →∞. For simplicity we assume W : X → R takesonly 0’s and 1’s.

Discrete Schrödinger equation: −i ddtϕt = Hϕt

39



Three types of tunneling

DefinitionLet x , y ∈ X . Let us start a particle from the pure state x andlook at the probability T (x , y) = supt |ϕt(y)|2. Now let thepotential grow to infinity. The tunneling from x to y is

perfect if lim T (x , y) = 1,partial if 0 < lim T (x , y) < 1,none if lim T (x , y) = 0.

DefinitionThe tunneling time is the first t (as a function of the potential)such that ϕt(y) → 1.

40



The main reduction

Need to analyze eigenfunctions and eigenvalues of ann × n matrix.

First step: eigenfunctions become concentrated on thewells.

Main idea: reduction to a k × k matrix where k is thenumber of wells.

Generalization of the Dirichlet problem for harmonicfunctions =⇒ consider random walks on the graph.

The new matrix will have power-series entries, coefficientsrelated to paths between the wells.

41



Cospectrality

Neccessary for tunneling: the difference of the diagonalentries has to be small compared to the off-diagonalentries.

This leads to a notion of spectral symmetry. Let PR(x , k)

denote the probability that the simple random walk startedat x ∈ X returns to x at time k .

DefinitionGiven two vertices x , y ∈ X we say that they are m-cospectral ifthe probability of return is the same for x and y up to time m,that is PR(x , k) = PR(y , k) for every 0 ≤ k ≤ m. Thecospectrality of x and y is the maximal m for which they arem-cospectral. This will be denoted by co(x , y).

42



Results 1

TheoremIn the case of a double-well potential where the graph-distanceof the two wells x , y is denoted by d = d(x , y) the following istrue between x and y:

if the two wells are d-cospectral (co(x , y) ≥ d) then thereis perfect asymptotic tunnelingif co(x , y) = d − 1 then there is partial tunnelingif co(x , y) < d − 1 then there is no tunneling.

When there is tunneling, the tunneling time is of magnitude∼ Qd−1.

43



Results 2

Let now x , y , z ∈ X be the wells. Let their distances bed(x , y) = a ≤ d(x , z) = b ≤ d(y , z) = c. We further assumethey are at least pairwise 2a-cospectral.

TheoremIf a < b ≤ c then the system splits into (x , y) and z. The(x , y) part behaves like a double-well system.If a = b < c then a particle started from x tunnelssymmetrically to y and z at the same time. This happenson the scale of t ∼ Qa−1.A particle started from y exhibits dual behavior: on thescale of t ∼ Qa−1 it tunnels to z through x, while on thescale of t ∼ Qc−1 states y and z become coupled and thendecoupled.

44



Results 3

RemarkIn the last case it is interesting to note, that adding a third wellactually speeds up the tunneling. The time needed to get fromy to z would be Qc−1 if there were only these two wells, while itdecreases to Qa−1 in the presence of the well at x.

RemarkLet us note that in all the preceding cases the tunnelingbehavior turned out to be a local property of the graph: theradius d (respectively c in the triple well case) neighborhoodsof the wells completely determine what kind of tunneling takesplace and what is the speed of the tunneling. This is in sharpcontrast with the following result.

45



Results 4

Theorem (Instability)There exists graphs with three wells x , y , z such that thepairwise distances are all the same, and

there is partial tunneling between y and z withlim T (y , z) = 5/9.by adding a single edge to the graph arbitrarily far from thewells, the tunneling between y and z becomes perfect.Hence the value of lim T (y , z) is unstable under smallperturbations, and the tunneling phenomenon becomescompletely non-local.

46



Double wellsTriple wells

Perfect tunneling

1

2

3

45

6

7

8

9

10

11

1213

14

15

16

17

0 100 000 200 000 300 000 400 0000.0

0.2

0.4

0.6

0.8

1.0

Wells: 4,15; Cospectrality = 9; Perfect tunneling

47




Partial tunneling

12

3

4

5

6

7

8

910

11

12

13

14

15

16

17

0 50 100 1500.0

0.2

0.4

0.6

0.8

1.0

Wells: 1,15; Distance = 2; Cospectrality = 1; Partial tunneling

48




No tunneling

12

3

4

5

6

7

8

910

11

12

13

14

15

16

17

0 100 000 200 000 300 000 400 000 500 0000.0

0.2

0.4

0.6

0.8

1.0

Wells: 4,15; Distance = 5; Cospectrality = 1; No tunneling

49




Splitting

12

3

4

5

6

7

8

9 10

11

12

13

14

15

16

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

Wells: 1,4,15; Perfect tunneling between 1 and 15;

50




Distant tunneling 1

12

3

4

5

6

7

8

9 10

11

12

13

14

15

16

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

Wells: 1,3,15; The middle well tunnels symmetrically to theother two;

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Distant tunneling 2

12

3

4

5

6

7

8

9 10

11

12

13

14

15

16

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

Wells: 1,3,15; There is perfect tunneling between the distantwells (through the middle well) on the short timescale;

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Distant tunneling 3

12

3

4

5

6

7

8

9 10

11

12

13

14

15

16

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

On the long scale the distant wells become coupled, then againdecoupled

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Distant tunneling 4

12

3

4

5

6

7

8

9 10

11

12

13

14

15

16

1000 1200 1400 1600 18000.0

0.2

0.4

0.6

0.8

1.0

On the long scale the distant wells become coupled, then againdecoupled

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Further work

Detailed analysis of the missing symmetric triple well case.

Investigating whether more than 3 wells could lead to newinteresting phenomena.

Non-asymptotic analysis of tunneling, in particularbehavior for smaller, but still significant potentials.

This should hopefully lead to applications in quantumcomputing.

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Quantum tunneling on graphs - National Taiwan · PDF fileThe Gap of the First Two Eigenvalues...

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