Cooperative e ects and many-body tunnelling...Cooperative e ects and many-body tunnelling Relatore: Ch.mo Prof. Fausto Borgonovi Correlatore: Ch.mo Prof. Giuseppe Luca Celardo Candidato:

Università Cattolica del Sacro Cuore

Sede di Brescia

Facoltà di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea in Fisica

Tesi di Laurea Magistrale

Cooperative effects and many-bodytunnelling

Relatore:

Ch.mo Prof. Fausto Borgonovi

Correlatore:

Ch.mo Prof. Giuseppe Luca Celardo

Candidato:

Guido Farinacci

Matricola: 4813507

Anno Accademico 2019/2020

Contents

1 Introduction 3

2 Single-body dynamics 6

2.1 Generic double-well potential solution . . . . . . . . . . . . . . . . . 6

2.2 Symmetric double-well potential . . . . . . . . . . . . . . . . . . . . 8

2.3 Dynamics of the symmetric potential . . . . . . . . . . . . . . . . . . 11

2.4 Validity of the tight-binding approximation . . . . . . . . . . . . . . 14

2.5 Loss dynamics for asymmetric potentials . . . . . . . . . . . . . . . . 22

3 Many-body dynamics 25

3.1 Interacting N -body double-well dynamics . . . . . . . . . . . . . . . 25

3.2 Generalization to N interacting bosons . . . . . . . . . . . . . . . . . 29

3.3 Non-interacting tight-binding dynamics . . . . . . . . . . . . . . . . 30

3.4 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 Extended Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . 42

3.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Conclusions 57

Appendices

A Computational methods 58

B Numerical interaction matrices 65

Acknowledgements 73

Bibliography 74

2

Chapter 1

Introduction

It is difficult to deny that tunnelling is one of the most fascinating phenomena in

quantum physics as it seemingly defies any classical intuition. First introduced as

an explanation for the overcome of the Coulomb barrier in alpha decay, it has since

been at the core of many physical theories such as that of Josephson oscillations, the

spatial flipping of the nitrogen atom in the ammonium molecule or the dynamics of

a lattice potential, among many other examples.

The phenomenon is well understood in the context of elementary quantum mechanics

and is a direct consequence of the so-called wave-particle duality. However, its

implications on the dynamics of many-body systems are still unclear. In recent

years, it has been argued that the presence of inter-particle interactions may lead to

cooperative behaviours such as the simultaneous tunnelling of a few particles as a

single object through a potential barrier ([10], [3]): this may explain the simultaneous

double ionization that is observed for some atoms (see for example [9]), which for

the moment remains unexplained.

Previously, the issue has been faced with a more mathematical approach (see [4] and

[2]); in [3] it has been demonstrated that two electrons in a double-well potential

should exhibit cooperative effects in the presence of a strong interaction in the

means of a simultaneous two-body tunnelling contribution to the dynamics. We

shall instead employ a more physical approach by computing the exact dynamics of a

double-well potential inhabited by a few particles and check a posteriori the presence

of co-tunnelling processes in the dynamics. Even if previous works have hinted at

the presence of a co-tunnelling contribution in the numerically computed dynamics

3

4

([10], [6]), none have actually verified if this corresponds to an actual two-particle

tunnelling amplitude in the interaction matrix in the lattice site representation; in

other words, the effect has been only described qualitatively and not quantitatively,

as we shall instead do. This has the inherent benefit of not only verifying if such

process is physical, but also to gauge how physically relevant it is.

Another open question is whether many-body tunnelling processes may exist for

more than two particles: as we will show, the presence of a two-body interaction

alone suggests the absence of such N -body effect. Other mechanisms should be

probably adopted to see such an effect.

We shall begin by reviewing the theory of a double-well potential in the single-

body case: we will find the exact eigenfunctions and eigenvalues (section 2.2) and

then compute the dynamics in the symmetric potential case (section 2.3). We shall

then find some analytical predictions for the dynamics by means of a tight-binding

approximation: in section 2.4 we will discuss in detail the construction of a site

localized basis, which is a debated topic for double-well potentials. Finally, we

shall briefly review the dynamics of strongly asymmetrical double-wells (section

2.5), where one can see the transition from usual Rabi oscillations to a particle loss

regime (as observed in [10]).

Subsequently, we shall move to analysing the dynamics of the many-body system:

first, we will provide a generalization of the single-body tight-binding model to the

many-body non-interacting case (section 3.3). Then, in section 3.4 we will review

the Hubbard model ([5]), which is the most commonly used approximation in the

study of the dynamics of lattice potentials (of which the double-well represents a

special case): in the strongly-interacting regime analytical solutions to the dynamics

can be found for two particles. The full exact dynamics of the system will then be

computed (section 3.6) as a comparison in the case of a δ-style “contact” potential to

find the shortcomings of the Hubbard approximation; as a middle-ground, in section

3.5 we will propose an extension to the Hubbard model which considers the exact

contributions of the interaction to the system, but is limited to the first few states

in the many-body spectrum. Its validity can be defined in terms of the interaction

strength, both attractive and repulsive: in particular it should be weak enough to

give rise to a negligible probability of occupation of the high energy states.

Our results confirm the presence of two-particle simultaneous tunnelling processes

5

in the dynamics, but also show that their amplitude is several orders of magnitude

smaller compared to the single-body tunnelling terms, implying that the overall

effect on the dynamics is small. This is in agreement with the findings of [6] where

it has been observed that for symmetric double-wells the dynamics are dominated

by sequential single-particle tunnelling processes, which are faster due to the larger

coupling. Still, we see that the overall effect is appreciable and leads to a slightly

different behaviour when compared to the traditional Hubbard model.

Chapter 2

Single-body dynamics

2.1 Generic double-well potential solution

In this introductory section we will focus on the single-body dynamics of a one-

dimensional double-well potential: this will constitute the groundwork of our later

investigation of the many-body dynamics of such system.

By one-dimensional double-well potential we assume any potential defined as follows:

V (x) :=

∞ for x < x0,

a for x0 ≤ x ≤ x0 + l,

a+ V0 for x0 + l < x < x0 + l + b,

a for x0 + l + b ≤ x ≤ x0 + l + b+ r,

∞ for x > x0 + l + b+ r.

(2.1)

It comes natural to refer to l and r respectively as the left and right well sizes, while

b and V0 respectively are the barrier size and height. As any potential can be defined

up to an arbitrary constant, it is common to choose a = 0; one is also free to translate

the coordinate system so that x0 = 0 to further simplify calculations. It must be

admitted that this is not the only possible formulation for a double-potential well,

as one may as well choose non-square wells, or even have the two wells at different

energies; the proposed model is the simplest and the most general.

Naturally, the next step is to solve the Schrödinger equation for such a potential; to

write our results in a more streamlined way we get rid of the constants by setting

6

2.1 ∼ Generic double-well potential solution 7

Figure 2.1: example of a double-well potential with parameters l = 5, b = 2, r = 7, V0 = 5.

~ = 2m = 1, with m being the mass of the particle that lives inside our one-

dimensional world. For stationary states of the system, which form a basis for the

whole Hilbert space of the possible solutions, the equation reads:(− ∂

2

∂x2+ V (x)

)ψ(x) = Eψ(x) (2.2)

In general, such equation admits infinitely many solutions ψ(x): however, abundance

is not a particularly appreciated quality, especially in the context of numerical sim-

ulations. Therefore, we chose to discard a part of the Hilbert space (alas, infinitely

large) by setting an upper cut-off on the energy spectrum so that we only consider

the stationary states whose energy is lower than that of the potential barrier, V0:

naturally, this limits our ability to study the dynamics of the system only to those

states that have negligible projections on the states over the barrier energy V0.

Before blindly employing our calculus machinery to find ψ(x), we make some logical

assumptions based on elementary quantum mechanics: we know that any wavefunc-

tion with energy lower than V0 will have oscillatory nature inside the two wells, while

it will be exponentially suppressed inside the barrier. Of course, ψ(x) must also be

0 wherever the potential is infinite. Therefore we can rightfully set:

ψ(x) := A ∗

φ1(x) := sin(kx) for 0 ≤ x < l,

φ2(x) := (Beλx + Ce−λx) for l ≤ x < l + b,

φ3(x) := D sin(k(l + b+ r − x)) for l + b ≤ x < l + b+ r,

0 elsewhere.

(2.3)

with k :=√E and λ :=

√V0 − E. Coefficient A is to be calculated via normaliza-

tion, while B, C and D will be determined by setting continuity conditions for the

2.2 ∼ Symmetric double-well potential 8

wavefunction and its derivative with respect to the spatial coordinate x:

φ1(l) = φ2(l),

φ2(l + b) = φ3(l + b),

φ′1(l) = φ′2(l),

φ′2(l + b) = φ′3(l + b).

(2.4)

The reader will surely have spotted that we are trying to solve a system of four

equations in only three variables B, C and D, seen as functions of E (or rather k

and λ). Therefore, coefficients B(E), C(E) and D(E) can be uniquely determined

by choosing a subsystem of only three of the equations, while the fourth one will

act as a quantization condition for the spectrum E. Such considerations lead to the

solution:

B(E) =e−λl

2

(sin(kl) +

k

λcos(kl)

), (2.5)

C(E) =e+λl

2

(sin(kl)− k

λcos(kl)

), (2.6)

D(E) =1

2 sin(kr)

(eλb(sin(kl) +

k

λcos(kl)) + e−λb(sin(kl)− k

λcos(kl))

), (2.7)

while the quantization condition for the momenta k is given by the solutions of the

equation:

f(E) := e2λb −sin(kl)− kλ cos(kl)sin(kl) + kλ cos(kl)

∗sin(kr)− kλ cos(kr)sin(kr) + kλ cos(kr)

= 0. (2.8)

Finally, by imposing the normalization condition∫|ψ(x)|2dx = 1 we obtain:

A(E) =( l

2− sin(2kl)

4k+D2

(r2− sin(2kr)

4k

)+ 2bBC+

+1

2λ

(B2e2λl(e2λb − 1) + C2e−2λl(e−2λb − 1)

))− 12.

(2.9)

2.2 Symmetric double-well potential

As we have just demonstrated, the eigenvalues of the Hamiltonian in equation 2.2

can be obtained by finding the roots of equation 2.8. However, such task cannot be

accomplished analytically and one must employ some sort of numerical algorithm:

unfortunately this can be somewhat difficult in the case of a symmetric double-well

potential (l = r), as the eigenstates come in couples of quasi-degenerate states, one

spatially symmetric and one anti-symmetric with respect to the center of the barrier


(x = l + b/2), only separated by a small splitting in energy; a not so careful choice

of the precision of the algorithm may either cause the loss of some of the eigenval-

ues if the scanning is too rough, while a very meticulous scanning greatly inflates

computational times.

A safer path is to impose the spatial symmetry (or anti-symmetry) a priori alongside

the continuity conditions for ψ(x), so that this way we obtain two different quanti-

zation conditions for the momenta, one for the even states and one for the odd ones.

This is achieved by adding equation

ψ(x) = ±ψ(l + b+ r − x) (2.10)

to system 2.4, choosing + for the even states and − for the odd ones.

Naturally, the previous definitions of A(E), B(E), C(E) and D(E) still hold true

by setting l = r, while as expected we obtain two uncoupled equations to determine

the eigenvalues of the Hamiltonian:

feven(E) :=λ

ktanh(λa)− tan(kc) sin(ka) + cos(ka)

sin(ka)− tan(kc) cos(ka)= 0, (2.11)

fodd(E) :=k

λtanh(λa)− sin(ka)− tan(kc) cos(ka)

tan(kc) sin(ka) + cos(ka)= 0, (2.12)

where for the sake of readability we have set a = b2 and c = a+ l.

Figure 2.2: this plots verifies that the quantization conditions for even (2.11) and odd (2.12) states

yield the same eigenvalues as 2.8 in the case of a symmetric potential; the parameters used in this

example are l = r = 5, b = 0.5 and V0 = 5.


Figure 2.3: lowest 8 eigenstates of a symmetric double-well potential with parameters l = r = 2,

b = 0.2, V0 = 500; in total, such potential parameters allow for 28 bound eigenstates with energy

lower than the central barrier. The reader can notice the aforementioned fact that states come

in doublets of spatially symmetric (left column) and anti-symmetric (right column) wavefunctions,

with the states inside each doublet having the same number of nodes inside each well. In red:

schematic representation of the double-well potential.

2.3 ∼ Dynamics of the symmetric potential 11

2.3 Dynamics of the symmetric potential

To evaluate the behaviour of the system we calculate its evolution in time: we expect

that even in the case of many non-interacting particles we should obtain the same

results, provided analogous starting conditions.

Following the results from previous sections, given a set of potential parameters

we can calculate the single-body spectrum Ei and the corresponding eigenfunctions

ψi(x). We are free to choose any initial state Ψ0(x); it then follows that the state

at any given time t must be:

Ψ(x, t) = 〈x|Ψ(t)〉 = 〈x|e−iĤt|Ψ0〉 = 〈x|e−iĤt(∑

n

〈ψn|Ψ0〉)|ψn〉 :=

=∑n

cne−iEnt 〈x|ψn〉 =

∑n

cne−iEntψn(x).

(2.13)

Naturally, the evolved state 2.13 in itself contains all the information about the dy-

namics of the system. However, to make sense of such information we must calculate

some more elementary observables, as for example the probability of occupation of

the left and right halves of the system:

PL(t) :=

∫ l+b/20

|Ψ(x, t)|2dx; (2.14)

PR(t) :=

∫ l+b+rl+b/2

|Ψ(x, t)|2dx. (2.15)

We may start from any arbitrary initial state |Ψ0〉. However in the case of a single

particle there are only two sensible choices: we can either start with the particle

in the left or in the right well; we choose to start from the left well by convention.

It is clear that such concept, while being perfectly intuitive, makes no sense from

a mathematical point of view: in technical terms, we must choose a starting state

where PL(0) ≈ 1 � PR(0) ≈ 0.1 A very simple choice that satisfies such condition

is |Ψ0〉 = |∞L〉, with |∞L〉 being the ground state of the left potential well with

V0 →∞. The reader may easily verify that this implies:

Ψ0(x) := 〈x|∞L〉 =

√

2l sin

(πxl

)for 0 ≤ x < l,

0 elsewhere.

(2.16)

1This is unfortunately easier said than done in practical terms: as previously noted, to make it

possible to perform any numerical computation one must truncate the Hilbert space, so that we

lose its completeness and therefore we lose the ability to expand any arbitrary initial state onto a

given basis. One must be cautious to choose an initial state that has negligible projections over the

states excluded by the truncation.


As we have already underlined, in the two observable quantities 2.14 and 2.15 we

are not actually looking at the occupation of the sole two wells, but more precisely

at the occupation of the two halves of the system with respect to the center of the

barrier: this has been done for the sake of convenience as then the two probabilities

are complementary and sum up to 1. In any case, results will not be much different

than the more natural choice of only calculating the occupation of the two wells

as, first and foremost, the eigenfunctions are strongly suppressed inside the barrier;

moreover, we will usually choose the size of the barrier b to be negligible in compar-

ison to the size of the two wells l and r.

Naturally, any numerical result must be partnered to some analytical approximation

for comparison; we provide such in the form of a tight-binding model where we ne-

glect the presence of any single-particle eigenstates besides the lowest two in energy.

This hypothesis obviously discards any contribution given by higher states of the

spectrum to the dynamics, but allows us to analytically compute the left and right

well occupations in a simple and readable form. One must be sure to check that the

higher states in the spectrum do not play an important role in the dynamics before

making any comparison: in the case of a single particle, this is simply guaranteed,

following 2.13, by choosing an initial state where c0, c1 � cn, ∀n ≥ 2 because coeffi-

cients cn do not evolve in time. Further details on the validity of the tight-binding

model will be delayed until the following section.

Our tight-binding basis is restricted to only the lowest two eigenstates of the total

Hamiltonian, |ψ0〉 and |ψ1〉. Following our previous findings, | 〈x|ψ0〉 |2 is spatially

symmetric around the center of the barrier, while | 〈x|ψ1〉 |2 is antisymmetric, there-

fore for the sake of clarity in the next calculations we make this property explicit by

calling |ψ0〉 := |s〉 (with s standing for symmetric) and |ψ1〉 := |a〉 (with a standing

for antisymmetric). We can exploit such symmetry to define the localized wavefunc-

tions for each potential well in a very simple – although approximate – form:

|L〉 := 1√2

(|s〉+ |a〉

), (2.17)

|R〉 := 1√2

(|s〉 − |a〉

), (2.18)

with |L〉 being localized inside the left well and |R〉 in the right one. Unfortunately,

the contributions of |a〉 and |s〉 to |L〉 do not cancel out exactly inside the right well,


leaving a small oscillating probability amplitude; the same can be said for |R〉 in the

left well. Therefore we cannot consider |L〉 and |R〉 as an exact basis for the left and

right sites; once again, further discussion on this matter is delayed until the next

section.

It is convenient to calculate the Hamiltonian matrix terms in the {|L〉 , |R〉} basis.

As |s〉 and |a〉 are by definition eigenstates of the Hamiltonian, we have:

Ĥ |s〉 = Es |s〉 , (2.19)

Ĥ |a〉 = Ea |a〉 , (2.20)

thus:

〈L|Ĥ|L〉 = 〈R|Ĥ|R〉 = 12

(Es + Ea) := E0, (2.21)

〈L|Ĥ|R〉 = 〈R|Ĥ|L〉 = 12

(Es − Ea) := Ω0. (2.22)

To obtain analogous dynamics as obtained in the numerical calculations, we choose

our initial state to be |L〉, which guarantees that the particle at the initial time is

mostly2 in the left well. so that we can calculate the occupation probabilities for

the two wells in time by approximating them as the projections of the evolved state

onto the |L〉 and |R〉 states:

PL(t) ≈∣∣∣ 〈L|e−iĤt|L〉 ∣∣∣2 = ∣∣∣1

2

(e−iEst + e−iEat

)∣∣∣2 ==∣∣∣e−iEa+Es2 t

2

(e−i

Ea−Es2

t + e+iEa−Es

2t)∣∣∣2 =

= cos2(Ω0t),

(2.23)

PR(t) ≈∣∣∣ 〈R|e−iĤt|L〉 ∣∣∣2 = ∣∣∣1

2

(e−iEst − e−iEat

)∣∣∣2 == (...) = sin2(Ω0t).

(2.24)

2It is not confined in the left well due to the aforementioned fact that |L〉 has a small but non-zero

probability amplitude outside of the left well.

2.4 ∼ Validity of the tight-binding approximation 14

Figure 2.4: numerical results for the left and right well occupation probabilities 2.14 and 2.15 at

different times for a single particle starting from state 2.16 and potential parameters l = r = 2,

b = 0.2, V0 = 500; the black dotted lines represent the tight-binding predictions 2.23 and 2.24,

which in this case overlap the numerical results. Ω0 ≈ 0.00238 as defined in 2.22.

2.4 Validity of the tight-binding approximation

The goodness of the tight-binding approximation relies on the fact that the |L〉 and

|R〉 states, as defined 2.17 in and 2.18, approximate reasonably well the “site” basis3

for the double-well potential. Now, there is no clear-cut definition of such basis;

however, one reasonable choice would be to consider the ground state of each of the

wells with a very large barrier size, i.e. b → ∞ (which is equivalent to filling the

second well up to V0). This way, most of the contribution to the probability is given

by the region inside the considered well, because outside of it the wavefunction is

either zero or exponentially suppressed. For example, to obtain such state for the

left well we must find the solutions of the Schrödinger equation(− ∂

2

∂x2+ VL(x)

)ψL(x) = EψL(x), (2.25)

with:

VL(x) :=

∞ for x < 0,

0 for 0 ≤ x ≤ l,

V0 elsewhere.

(2.26)

3The term basis is used here quite liberally; our scope is to find a set of (two) states for which

we would be allowed to say that the particle resides inside one of the wells (almost) exclusively,

while keeping the treatment restricted to as few eigenstates as possible as to allow for analytical

predictions to be calculated in a simple way.


(a) (b)

(c)

Figure 2.5: (a): first two eigenfunctions of the double-well potential with parameters l = r = 2,

b = 0.2, V0 = 500; (b): näıve site basis as defined in 2.17 and 2.18 for the same potential parameters;

(c): zoom of plot (b) to highlight the contribution of each näıve site basis state in the opposite

potential well.

Without dwelling into details, we infer that any wavefunction with energy smaller

than the barrier height V0 must have oscillatory nature inside the well, while it must

be exponentially suppressed inside the barrier:

ψL(x) := N ∗

sin(kLx) for 0 ≤ x < l,

sin(kLl)e−λL(x−l) for x ≥ l,

0 elsewhere.

(2.27)

where kL :=√EL and λL :=

√V0 − EL, with EL to be determined numerically via

the level quantization condition given by wavefunction continuity:

tan(kLl) = −kLλL. (2.28)

Via normalization we also get:

N =( l

2− 1

2kLsin(kLl) cos(kLl) +

1

2λLsin2(kLl)

)− 12. (2.29)

Naturally, by symmetry the same holds true for the right potential well, with the

isolated state being ψR(x) := ψL(l + b+ r − x).


(a) (b)

(c) (d)

(e) (f)

Figure 2.6: left column: 〈x|L〉 and ψL(x) defined respectively as in 2.17 and 2.27 for different

sizes of the barrier b; right column:∣∣∣| 〈x|L〉 |2 − |ψL(x)|2∣∣∣ for the corresponding barrier size on the

left (b = 0.5 for (a) and (b), b = 1 for (c) and (d), b = 2.5 for (e) and (f)). The other potential

parameters are kept fixed at l = r = 5 and V0 = 10.


(a) (b)

(c) (d)

(e) (f)

Figure 2.7: left column: 〈x|L〉 and ψL(x) defined respectively as in 2.17 and 2.27 for different

heights of the barrier V0; right column:∣∣∣| 〈x|L〉 |2 − |ψL(x)|2∣∣∣ for the corresponding barrier height

on the left (V0 = 0.5 for (a) and (b), V0 = 1 for (c) and (d), V0 = 25 for (e) and (f)). The other

potential parameters are kept fixed at l = r = 5 and b = 1. Note: barrier height in the plots is not

to scale.


As the reader may appreciate from figures 2.6 and 2.7, our set of states {|L〉 , |R〉}

approximates the localized states {|ψL〉 , |ψR〉} only when the two potential wells

are separated by a barrier of sufficiently large width (b) and/or height (V0): this is

particularly highlighted by the figures on the right-hand side, which represent the

difference in the probability distributions, which are of increasingly smaller order

of magnitude as the barrier gets stronger. Another interesting feature is that for

strong barriers 〈x|L〉 and ψL(x) are well localized inside the left-well region.

Our findings can be further confirmed by comparing the Rabi oscillation frequency

obtained for our tight-binding approximation, which accordingly to 2.14 and 2.15

is Ω0, with the model proposed by Landau and Lifshitz (page 175-176 of [7]): as

we have verified by solving the Schrödinger equation in the previous sections, the

presence of a finite potential barrier splits the single-well4 energy levels into doublets

of even and odd states, separated by a small splitting energy.5 If, for example, we

take a single-well normalized wavefunction φ0(x) with corresponding energy E0, the

presence of the barrier will split such level into a doublet φ1(x) and φ2(x), with

respective energies E1 and E2. As noted, if we take our coordinate system to be

centred around the middle of the barrier, such wavefunctions can be approximated

as the symmetric and anti-symmetric combinations of φ0(x) and φ0(−x):

φ1(x) :=1√2

(φ0(x) + φ0(−x)

),

φ2(x) :=1√2

(φ0(x)− φ0(−x)

).

(2.30)

According to Schrödinger’s equation we have:

∂2

∂x2φ0(x) +

2m

~2(E0 − U(x)

)φ0(x) = 0,

∂2

∂x2φ1(x) +

2m

~2(E1 − U(x)

)φ1(x) = 0,

(2.31)

with U(x) being a generic double-well potential. By multiplying each equation in

2.30 respectively by φ1(x) and φ0(x), this implies:φ1(x)φ

′′0(x) +

2m~2

(E0 − U(x)

)φ1(x)φ0(x) = 0

φ0(x)φ′′1(x) +

2m~2

(E1 − U(x)

)φ0(x)φ1(x) = 0

(2.32)

4We are here referring to each one of the two-wells from the double-well potential taken sepa-

rately; in our treatment it is analogous to 2.26.5In general, the splitting energy is not constant along the spectrum and is heavily dependent on

the shape of the potential barrier.


where short-hand apostrophe notation has been used for the derivative over x. Com-

puting the difference of the two equations and integrating over x between 0 and ∞

one obtains: (φ1(x)φ

′0(x)

)∣∣∣∞0−(φ0(x)φ

′1(x)

)∣∣∣∞0

+

+2m

~2(E0 − E1)

∫ ∞0

φ0(x)φ1(x)dx = 0.(2.33)

Finally, if we remember that φ1(0) =√

2φ0(0), φ′1(0) = 0 and we approximate

6∫ ∞0

φ0(x)φ1(x)dx ≈1√2

∫ ∞0

φ20(x)dx =1√2, (2.34)

where the last equality holds true because φ0(x) is normalized and lives inside a

single well, we finally get:

E1 − E0 = −~2

m

(φ0(0)φ

′0(0)

). (2.35)

The same process can be repeated for φ2(x) and subtracting the two results we

finally get an expression for the splitting energy:

∆ := E2 − E1 =2~2

m

(φ0(0)φ

′0(0)

). (2.36)

The power of this simple models relies on the fact that it makes no assumptions

about the nature of the double-well potential; our treatment instead is only valid

for square double-wells.

Adapting result 2.36 to our model implies rescaling the units such that ~ = 2m = 1

and taking φ0(x) = ψL(x), according to 2.27, with a simple translation of the x

coordinate (x → x + l + b/2) to account for the fact that our coordinate system is

not centred around the middle of the barrier:

|∆| =∣∣∣4ψL(l + b

2

)ψ′L

(l +

b

2

)∣∣∣ = ∣∣∣4λLψ2L(l + b2)∣∣∣, (2.37)where the last equality directly descends from definition 2.27. In our tight-binding

model, the Rabi oscillation frequency for one particle is equal to half the splitting

energy, Ω0; therefore, according to 2.37 we should expect this value to approximate

∆/2 (see figure 2.8: the two models are in good agreement for sufficiently large

barrier height V0, while a large barrier width b seems to equally affect both models

by making the splitting energy smaller).6This is an approximation due to the fact that we are neglecting the contribution of φ0(−x) for

x ≥ 0; this can be done due to the fact that such contribution is exponentially suppressed.


(a) (b)

(c)

Figure 2.8: Rabi oscillation frequency for the tight-binding model Ω0 and according to Landau’s

approximation for varying barrier width b and different barrier heights (V0 = 0.5 for (a), 5 for (b)

and 10 for (c)); the remaining potential parameters are l = r = 5.

In our treatment we have given for granted that states |L〉 and |R〉 represent our best

effort in creating a site localized basis when restricting the double-well spectrum to

just the first two states; despite this is common practice in literature, we would like

to attempt to give a demonstration to the fact that this is indeed a sensible choice.

If we take {|ψ0〉 , |ψ1〉} as our basis, any state of the system can be expressed in the

form:

|χ〉 := cos(ϑ) |ψ0〉+ eiϕ sin(ϑ) |ψ1〉 , (2.38)

where ϕ is an arbitrary phase angle; the state is normalized by construction. If we

wish |χ〉 to be completely localized inside the left well, we expect:∫ l0| 〈x|χ〉 |2dx = 1. (2.39)

One can easily verify that:

| 〈x|χ〉 |2 = cos2(ϑ)|ψ0(x)|2 + sin2(ϑ)|ψ1(x)|2+

+ sin(2ϑ) cos(ϕ)ψ0(x)ψ1(x),(2.40)


where the last term can be written in such simple form due to the fact that ψ0(x)

and ψ1(x) are real-valued functions. Therefore, by defining:I1 :=

∫ l0 |ψ0(x)|

2dx

I2 :=∫ l

0 |ψ1(x)|2dx

I3 :=∫ l

0 ψ0(x)ψ1(x)dx

(2.41)

according to 2.39 we should have:

cos2(ϑ)I1 + sin2(ϑ)I2 + sin(2ϑ) cos(ϕ)I3 = 1. (2.42)

Now, as the eigenfunctions are normalized over [0, 2l + b], which is the width of

the symmetric double-well, and taken into account the spacial symmetry properties

of the wavefunctions, we expect that the integral of their square modulus over half

their support, [0, l+b/2], is 1/2; therefore, their integral over the sole left well should

surely be less than 1/2 as the integrand is always positive, implying I1, I2 < 1/2.

Moreover, as we have observed, inside the left well we have ψ1(x) ≈ ψ0(x) (and the

same holds true for the right well with the addition of a minus sign in front of either

wavefunction), so we may as well say I3 < 1/2. In general, one should then calculate

the three integrals in 2.41 and solve equation 2.42 to find the appropriate values for

ϑ and ϕ. If we set ξ := max(I1, I2) < 1/2, equation 2.42 reduces to:

sin(2ϑ) cos(ϕ) =1− ξI3

>1

2I3> 1, (2.43)

which one can quite easily see is not satisfied by any choice of ϑ and ϕ, therefore

shattering our hope of finding a fully localized state by only combining |ψ0〉 and

|ψ1〉.

However, if we take sufficiently large potential parameters b and/or V0, ψ0(x) and

ψ1(x) will be strongly suppressed inside the barrier, so that the contribution to I1,

I2 and I3 in the region [b, b/2] should be negligible,7 so that we have I1, I2, I3 / 1/2,

therefore we may light-heartedly assume I1 = I2 = I3 = 1/2, hence equation 2.42

reduces to:

sin(2ϑ) cos(ϕ) = 1. (2.44)

The reader may easily verify that, given ϑ, ϕ ∈ ]− π, π], this leads to the solutions:

ϑ =π

4, ϕ = 0 ∨ ϑ = −π

4, ϕ = π (2.45)

7As an example, for potential parameters l = r = 2, b = 0.2 and V0 = 500 we have that the

three integrals are just below 1/2 by order 10−5.

2.5 ∼ Loss dynamics for asymmetric potentials 22

which indeed imply either |χ〉 = |L〉 or |χ〉 = |R〉,8 confirming the assumption that

this represents our best effort in building a localized site basis only employing the

first two states in the spectrum in the context of strong enough potential barriers.

2.5 Loss dynamics for asymmetric potentials

Even though our scope in the next chapters will mostly concern symmetric double-

well potentials, we would like to give a brief overlook to the dynamics of a strongly

asymmetrical double-well at least for the single-particle case: the following para-

graph has no ambition to be regarded as a complete and self-standing discussion on

the topic.

Naturally, for l ≈ r we expect to observe similar dynamics as seen for the symmetric

potential, so the particle will oscillate between the left and the right well if initially

prepared inside either one of them. Instead, when the size of one of the two wells

is significantly larger than the other, for example r � l, the system will radically

change its behaviour and an exponential loss of occupation probability for the ini-

tial well is observed. This can be understood in the context of Wigner’s theory

of decaying systems if we consider the two wells as separate but interacting. This

becomes clear when we are faced with the expression for the density of states in a

square well; we know from elementary quantum mechanics that the energy levels for

a one-dimensional square potential well of size a, taken ~ = 2m = 1, are:

En =π2n2

a2, (2.46)

where n ∈ N is a label for each state. Therefore we can quite simply differentiate:

dE =2π2n

a2dn, (2.47)

from which the density of states ρ(E) descends directly:

=⇒ ρ(E) := dndE

=a2

2nπ2=

a

2π√E. (2.48)

We obtain that the spectral density of states is directly proportional to the size of

the quantum well a: if the size is large enough, the spectrum may therefore be taken

as a continuum. As anticipated, if we initially prepare the particle in the left well

8The fact that we also get |R〉 as a possible solution even though we only imposed conditions

for the integrals inside the left well is a by-product of the spatial symmetry of the potential.


with l� r, the problem can be reformulated as that of a single energy level from the

left well coupled to the continuum of states from the right well, justifying the usage

of Wigner’s prediction for which we expect that the projections of the evolved state

over the eigenstates of the Hamiltonian |cn(t)|2 := | 〈ψn|ψ(0)〉 |2 are distributed with

a Lorentzian shape of width γ in energy space, which leads to an exponential decay

with rate γ in the time domain. Following [6], we may approximate the energy

spectrum as homogeneous and instead look at the distribution of the projections

|cn(t)|2 in the space of state labels n, which under this approximation still follows a

Lorentzian shape (see figure 2.9 (a)), but with a different width parameter Γ, which

will be equal to γ multiplied by the supposedly constant density of states in the

spectrum:

Γ := γρ(E). (2.49)

This gives a first numerically computable expression for the decay rate:

γ0 :=2πΓ

r

√Ei, (2.50)

where Ei is the energy of the initial state |ψ(0)〉 at which the density of states is

numerically evaluated according to 2.48. Still following [6], we can make a sightly

more sophisticated assumption; first we define the participation ratio for a given

state:

PR(|ψ(0)〉

):=(∑

n

|cn|4)−1

, (2.51)

which gives a measure of how many eigenstates contribute to the initial state. If one

assumes a Lorentzian distribution for the projections and performs the summation

in 2.51, we get:

PR(|ψ(0)〉

)= πΓ. (2.52)

By analogy with equation 2.49, we obtain a more refined expression for the decay

rate which does not assume a perfectly Lorentzian distribution of the projections:

γ1 :=2PR(|ψ(0)〉)

r

√Ei. (2.53)

Finally, another expression can be found in literature for the decay rate (see [1] for

further details on its derivation):

γ2 :=8α3E

3/2i

V 20 (1 + αl2)e−2αb, (2.54)


where for readability’s sake α :=√V0 − Ei.

The three proposed expressions for the decay rate have been compared with the

numerical result for a particular set of parameters in figure 2.9 (b) and are all in

good agreement provided a sufficiently large right well size r; this is made evident

by the simulations shown in figure 2.9 (c), where one can appreciate the onset of the

particle loss regime only for sufficiently high r (r > 1000 for the particular choice of

parameters, refer to the figure caption for further details). For small enough r the

dynamics still follow the characteristic behaviour of Rabi oscillations, although with

a sightly smaller amplitude compared to the symmetric case.

(a) (b)

(c)

Figure 2.9: (a): distribution of the |cn|2 in the space of state labels for l = 51, b = 4, V0 =

0.1 and r = 4000; inset: zoom of the plot around the peak of the distribution, in red a least

squares Lorentzian fit with center ∼ 55.668 and width ∼ 0.985; (b): numerically computed left well

occupation 2.14 for the same potential parameters and analytical prediction according to decay rates

2.50, 2.53 and 2.54; (c): numerically computed left well occupation 2.14 for potential parameters

l = 51, b = 4, V0 = 0.1 and different values of r: the transition from Rabi oscillations to exponential

decay regime is observed for r > 1000.

Chapter 3

Many-body dynamics

3.1 Interacting N-body double-well dynamics

Now that we have laid the foundations for our analysis of the double-well potential

by studying the dynamics of a single particle, we may now start to tackle the more

sophisticated task of populating our system with a multitude of particles. In this

introductory section we will begin by looking at the problem from a general point

of view and, subsequently, we will move on to analyse more specific cases so that we

can provide both numerical simulations and analytical predictions to be confirmed

or, sometimes more interestingly, disproved.

First and foremost, dealing with more than one particle introduces a new and ex-

tremely radical feature in our system: the possible presence of an interaction be-

tween said particles, which can introduce novel effects in the dynamics of the system.

Mathematically, if for the moment we restrict our treatment to N distinguishable

particles, this translates into the presence of a new term in the total Hamiltonian:

Ĥ(x1, . . . , xN ) :=

N∑i=1

Ĥ(0)i (xi) +

1

2

N∑i=1

∑j 6=i

Û(xi, xj), (3.1)

where xi is the position of the i-th particle, Û is an interaction potential that acts

on any possible combination of two particles, and finally Ĥ(0) is the non-interacting

Hamiltonian that acts separately on each single particle (the dynamics of which have

been studied in the previous chapter):

Ĥ(0)i (xi) := T̂ (xi) + V̂ (xi), (3.2)

25

3.1 ∼ Interacting N -body double-well dynamics 26

with T̂ being the kinetic term and V̂ the double-well potential.

Now, the interaction potential Û may take any arbitrary form, one common example

would be a Coulomb type ∼ 1/r electrostatic interaction, with r being the distance

between the two involved particles. However, such a choice would make it very

difficult to express any analytical prediction as the interaction matrix terms could

only be numerically calculated so that they could not be easily expressed in terms of

the elementary quantities that characterise our physical system. Instead, following a

common convention in literature (for example see[10]), we employ a δ-type “contact”

interaction:

Û(xi, xj) := Uδ(xi − xj), (3.3)

where U is a tunable interaction strength parameter, which can take both a pos-

itive or negative sign to emulate respectively a repulsive or attractive interaction.

Of course, such a choice discards any long-range contributions to the interaction,

making it an extremely simplistic model. On the good side, in the limit of strong

interactions U → ∞ such a potential mimics the effect of the Pauli exclusion prin-

ciple as the particles then act as impenetrable balls, making it feasible to map the

problem of N interacting bosons to the possibly simpler one of N free fermions (see

for example [10] for an in-depth but still extremely straight-forward treatment of

the problem).

Getting onto more practical issues, we will now need to find some observables to

be calculated at different times to have a picture of the dynamics of the system:

naturally we wish to extend the concepts of left and right well occupations to the

many-body case. As in any quantum-mechanical problem, the first step is to find

a convenient basis. Following the results from the previous chapter, we know how

to calculate the single-body spectrum {Ei} and eigenbasis {|ψi〉}, where the i in-

dex runs over the states in the spectrum.1 Therefore an easy choice is to build

the distinguishable many-body basis as an Hartree tensor prouct of the single-body

eigenstates:

|Ψa1,...,aN 〉 := |ψa1〉1 ⊗ |ψa2〉2 ⊗ . . .⊗ |ψaN 〉N , (3.4)

where the ai is the quantum number for each particle, labelled by the subscript | · 〉i.

One can see that in the position representation this simply translates as the product

1Naturally we take for granted that all particles are identical (however not always undistinguish-

able) and therefore they all share the same single-body spectrum.


of the single-particle wavefunctions:

Ψa1,...,aN (x1, . . . , xN ) := 〈x1, . . . , xN |Ψa1,...,aN 〉 =N∏i=1

ψai(xi). (3.5)

If we define the cardinality of the single body spectrum {Ei} as C, we find that our

many-body Hilbert space will have dimension CN .2

Now, if for a moment we forget the interaction, we can verify that this Hartree

product basis is an eigenbasis for the total non-interacting Hamiltonian, which in

this case is just a sum of single-particle operators:

Ĥ(U=0) |Ψa1,...,aN 〉 =N∑i=1

Ĥ(0)i

(|ψa1〉1 ⊗ |ψa2〉2 ⊗ . . .⊗ |ψaN 〉N

)=

=

N∑i=1

Eai |Ψa1,...,aN 〉 .

(3.6)

This is, unfortunately, generally untrue for the interacting term which would oth-

erwise just act as a simple energy shift for the free many-body levels. Instead, the

eigenstates must be found by diagonalizing the total Hamiltonian for each consid-

ered value of the interaction strength U ; let us say that in general such interacting

eigenbasis will be indicated as {|α〉}, on the other hand the free eigenbasis 3.4 will

be indicated as {|k〉}.3

In general, we may choose a different starting state for each one of the N particles,

so that the many-body initial state can be expressed as:

|Ψ0〉 :=N⊗i=1

|ψ(0)〉i , (3.7)

with |ψ(0)〉i being the starting state for the i-th particle. The trick up our sleeve is

that we know how to express the free eigenbasis states in the position representation

in a simple analytical form, so that it is convenient for us to express everything in

this basis. Consequently we define:

|Ψ0〉 =∑k

〈k|Ψ0〉 |k〉 :=∑k

ck0 |k〉 . (3.8)

2Naturally, if we don’t take any approximation C = ∞; as before, to have any hope to perform

numerical calculations we will have to truncate the spectrum. Anyways, the number of basis states

grows exponentially with the number of particles, so that we may have to take a lower cut-off than

the barrier energy to keep the calculations manageable.3Formally it would be more correct to label each state in the basis with a subscript, but for the

sake of readability we will leave them unlabeled.


However, to apply the total time evolution operator to the initial state we must

make an intermediate step in the interacting eigenbasis:

|Ψ(t)〉 := e−iĤt |Ψ0〉 =∑α

∑k

ck0 〈α|k〉 e−iEαt |α〉 :=

=∑α

cα0 e−iEαt |α〉 =

∑k

∑α

cα0 e−iEαt 〈k|α〉 |k〉 :=

=∑k

∑α

cα0 e−iEαtckα |k〉 :=

∑k

ck(t) |k〉 ,

(3.9)

where the ckα are the {|α〉} → {|k〉} change of basis matrix terms.

Finally, we may now extend the hole occupation observables that we used in the

single particle case (2.14 and 2.15) to the many-body case; we define Pm(t) as the

probability of finding the first m particles in the left potential well at time t:4

Pm(t) :=

∫Ldx1 · · ·

∫Ldxm

∫Rdxm+1 · · ·

∫RdxN |Ψ(x1, . . . , xN , t)|2 =

=

∫Ldx1 · · ·

∫Ldxm

∫Rdxm+1 · · ·

∫RdxN 〈Ψ(t)|x〉〈x|Ψ(t)〉 =

=

∫Ldx1 · · ·

∫Ldxm

∫Rdxm+1 · · ·

· · ·∫RdxN

∑k

∑k′

c∗k′(t)ck(t) 〈k′|x1, . . . , xN 〉〈x1, . . . , xN |k〉 ,

(3.10)

where the L and R subscripts respectively denote that the integration has to be

carried over the left and the right potential wells. If, for example, we take:

|k〉 := |ψa1〉1 ⊗ . . .⊗ |ψaN 〉N (3.11)

|k′〉 := |ψb1〉1 ⊗ . . .⊗ |ψbN 〉N , (3.12)

we get an easily computable expression for Pm(t):

=⇒ Pm(t) =∑

a1,...,aN

∑b1,...,bN

c∗b1,...,bN (t)ca1,...,aN (t)

∗m∏i=1

∫Lψ∗bi(xi)ψai(xi)dxi

N∏j=m+1

∫Rψ∗bj (xj)ψaj (xj)dxj .

(3.13)

Now, calculating the probability of finding the first m particles in the left well does

not make much physical sense: this is a consequence of the fact that this observable

is inherently more suitable for indistinguishable particles, so that we would have to

consider all possible ways in which we can fill the left well with m particles, while

4As for the single-body case, we actually extend the occupation up to the centre of the potential

barrier to have complementary probabilities.

3.2 ∼ Generalization to N interacting bosons 29

we are working with distinguishable ones; we shall later move on to analyze such

case, which is surely more interesting from a physical point of view.

Another interesting observable that can be evaluated is the probability density dis-

tribution in space for each particle: as an example, we will now show how it can

be calculated for the first particle by marginalizing the total probability density

function |Ψ(x1, . . . , xN , t)|2:

ρ(x1, t) :=

∫dx2 . . . dxN |Ψ(x1, . . . , xN , t)|2 =

=∑k

∑k′

c∗k′(t)ck(t)

∫dx2 . . . dxN 〈k′|x1, . . . , xN 〉〈x1, . . . , xN |k〉 =

=∑

a1,...,aN

∑b1,...,bN

c∗b1,...,bN (t)ca1,...,aN (t)ψ∗b1(x1)ψa1(x1) ∗

∗N∏i=2

dxiψ∗bi

(xi)ψai(xi),

(3.14)

where we have recovered definitions 3.11 and 3.12.

3.2 Generalization to N interacting bosons

Studying the dynamics of many interacting bosons can be particularly enlightening

if, for example, we choose to prepare all N particles in the same state at t = 0: in

fact, this is a very simplified model of an interacting Bose-Einstein condensate in

a double-well trap and the dynamics can be studied exactly, opposed to the usual

mean-field treatment.

On paper, the problem of switching from distinguishable particles to bosons is a

simple task: we must ensure that in our formalism the particles are indistinguish-

able and that the states of the system are totally symmetric under any exchange of

two particles (see for example chapter 7 in [8] for an in-depth analysis of the issue).

Usually, this is performed by switching to the so-called second quantization formal-

ism; however, as we wish to keep as much as we can of the results we have obtained

so far5 we shall use what therefore should be called first quantization formalism and

ensure the bosonic particle exchange symmetry for the states. In general, given a N

distinguishable (but otherwise identical) particle state |Ψa1,...,aN 〉, any permutation5We shall not forget that the scope of our overview of the dynamics is to perform numerical

simulations, so we – or rather the author – wish to adapt the code developed so far for distinguishable

particles with minimum effort.

3.3 ∼ Non-interacting tight-binding dynamics 30

of the label set {a1, . . . , aN} constitutes a new and distinguished state; the number

of such states is equal ton1! . . . nN !

N !(3.15)

where ni is the number of times ai appears in {a1, . . . , aN}. Imposing bosonic

particle exchange symmetry, all the permutations of a distinguishable particle state

contribute to a single bosonic state, defined as the sum over the permutations of the

particle labels:

|Ψa1,...,aN 〉+ :=√n1! . . . nN !

N !

∑{P}

P̂ |Ψa1,...,aN 〉 , (3.16)

where the + subscript denotes bosonic symmetry (opposed to − for fermionic anti-

symmetry), and the sum is intended to be carried over the set of all permutations

of the distinguishable particle states generated by the particle permutation operator

P̂ . This last equation constitutes our Rosetta Stone for translating bosonic states

into distinguishable particle states that we already know how to treat. Naturally,

one must also take into consideration that the dimension of the Hilbert space will

in general be different if we deal with bosons of distinguishable particles, due to the

aforementioned symmetrization of the states.

3.3 Non-interacting tight-binding dynamics

In our thirst for analytical results we should start by generalizing the single-body

tight-binding model to the many-body case. Naturally, as per nature of a single-

body Hamiltonian, no interaction terms can be included in the framework of the

tight-binding model without developing a specific many-body theory, so that we are

restricted to generalizing the results for the non-interacting case; naturally, we are

also restricted to symmetric double-well potentials. We recall that in this case the

total Hamiltonian is just a sum of single particle operators:

Ĥ(x1, . . . , xn) =N∑i=1

Ĥ(0)i (xi), (3.17)

therefore also the time evolution operator can be factorized into single-particle op-

erators:

Û(t) = e−iĤt = exp(− i

N∑i=1

Ĥ(0)i t)

=N∏i=1

e−iĤ(0)i t. (3.18)


Recalling the single-body tight-binding approximation, we postulated that the sys-

tem can be fully described in the site basis {|L〉 , |R〉}, as defined in 2.17 and 2.18.

Restricting our treatment to distinguishable particles for the moment, we can gen-

eralize the tight-binding site basis to its many-body equivalent:

{|α1〉1 ⊗ |α2〉2 ⊗ . . .⊗ |αN 〉N}, |αi〉i ∈ {|L〉i , |R〉i}, (3.19)

where the i subscript indicates that the state refers to the i-th particle. We now

have all the ingredients needed to calculate site occupation probabilities as we did

for the single-particle case; naturally we now have a plethora of such probabilities,

2N to be exact, because each particle can either occupy the left or the right potential

well.

Let us suppose that, in analogy to the single-particle case, at time t = 0 we prepare

the system with all the particles located in the left well; any of such 2N probabilities

can be calculated as:

Pw1,...,wn(t) :=∣∣∣(1 〈w1| ⊗ . . .⊗ N 〈wN |) N∏

i=1

e−iĤ(0)i t(|L〉1 ⊗ . . .⊗ |L〉N

)∣∣∣2, (3.20)with wi ∈ {L,R} expresses whether particle i occupies the left or the right well. We

also used the fact that the initial state can be expressed as:

|Ψ(0)〉 :=N⊗i=1

|L〉i . (3.21)

In the single particle case we have calculated:

PL(t) := |i 〈L|e−iĤ(0)i t|L〉i |

2 = cos2(Ω0t), (3.22)

PR(t) := |i 〈R|e−iĤ(0)i t|L〉i |

2 = sin2(Ω0t), (3.23)

following definitions 2.23 and 2.24. On the basis of these results we can write:

Pw1,...,wn(t) = cos2NL(Ω0t) sin

2NR(Ω0t), (3.24)

with NL :=∑N

i=1 δwi,L being the number of particles inside the left well in the

target state, with δ being the Kronecker delta (and analogously for NR). Therefore,

following 3.13 we have:

P(U=0)NL

(t) ≈ cos2NL(Ω0t) sin2NR(Ω0t). (3.25)

Equation 3.24 could also have been obtained by simple statistics; we have demon-

strated that the total Hamiltonian acts independently on each particle, and so does


the time evolution operator. Therefore the N particles can be regarded as a set of

N disjoint events: this implies that their conjoined probability is just the product of

the probabilities of the single events, namely PL(t) and PR(t) being the probability

of finding a particle in the left or right well at time t, in accordance with 3.24.

The previous results can be extended also to the bosonic case; we may either re-

peat the calculation in 3.20 but substituting the target state with its symmetric

counterpart

|w1, . . . , wn〉+ :=√n1! . . . nN !

N !

∑{P}

P̂( N⊗i=1

|wi〉i)

(3.26)

in accordance with definition 3.16, or once again follow our statistical analogy. The

main difference with bosonic particles resides in their indistinguishability: this means

that, unlike in the distinguishable case, there is a multitude of ways in which we

can populate the left well with NL bosons and the right well with NR bosons.6

Therefore, we must multiply result 3.25 by the number of times we can group NL

bosons from a pool of N particles:

P(U=0)NL

(t) ≈(N

NL

)cos2NL(Ω0t) sin

2NR(Ω0t). (3.27)

(a) (b) (c)

Figure 3.1: numerical simulations of |Ψ(x1, x2)|2 for 2 bosons in the non-interacting case at the

start (a), at a quarter (b) and at half (c) of the oscillation cycle. The potential parameters are

l = r = 2, b = 0.2, V0 = 500 and the starting state is 2.16 for both particles; Ω0 ≈ 0.00238. The

lower left (resp. upper right) corner represents the region where both particles occupy the left (resp.

right) well; in the other two regions they are separated.

6The reader may argue that this is obviously also true for distinguishable particles; however we

have to recall that, according to definition 3.13, in 3.25 we are looking at the probability of finding

the first NL particles in the left well, which can only be done in one way.


(a) (b)

(c)

Figure 3.2: equation 3.27 for 2 (a), 3 (b) and 4 (c) bosons; the results are in very good agreement

with the analytical results from the tight-binding approximation, even though the simulations also

include the contributions of the higher states in the spectrum. The potential parameters are l = r =

2, b = 0.2, V0 = 500 and the starting state was chosen to be 2.16 for each particle; Ω0 ≈ 0.00238.

3.4 ∼ Hubbard model 34

3.4 Hubbard model

We will now review a model which is usually employed in calculating the dynamics

of many particles in a lattice-style potential: the model can be downsized to a

lattice of just two sites, corresponding to a double-well potential. The model was

first introduced by John Hubbard in 1963 (see [5]) and is extremely streamlined in

its form: we suppose to populate with a given number of particles a lattice-style

potential made of wells, or lattice sites, each containing one single bound state and

separated by potential barriers of finite height. The total Hamiltonian of the system

is composed of three simple terms: the single-body Hamiltonian for each particle,

accounting for the bound-state energy of the occupied lattice sites; a single-body

hopping term between neighbour lattice sites, characterised by a constant tunnelling

amplitude; finally, a very simple interaction term that adds a constant contribution

to the total energy whenever a lattice site is multiply occupied: we suppose that

particles cannot interact when they occupy different sites. In the so-called second

quantization formalism the total Hamiltonian can be written as:

Ĥ : =∑i,s

(E0 â

†i,sâi,s +

∑j

U

2â†i,sâi,sâ

†j,sâj,s

)+

+ Ω0∑i,s,s̄

(â†i,s̄âi,s + â

†i,sâi,s̄

),

(3.28)

where the i and j are labels for each particle, s is a label for each lattice site, E0 is the

bound-state energy of the sites, U the constant energy contribution given whenever

two particles occupy the same site (divided by two to account for double-counting

in the summation) and Ω0 is the single-body tunnelling amplitude between site s

and its neighbours s̄. Naturally, â†i,s and âi,s represent the usual ladder operators

for particle i in the lattice site s, which act as follows:

â†i,s⊗j

|n(j)0 , n(j)1 , . . . , n

(j)s , . . .〉j =

=

√n

(i)s + 1 |n(i)0 , n

(i)1 , . . . , n

(i)s + 1, . . .〉i

⊗i 6=j|n(j)0 , n

(j)1 , . . . , n

(j)s , . . .〉j

(3.29)

âi,s⊗j

|n(j)0 , n(j)1 , . . . , n

(j)s , . . .〉j =

=

√n

(i)s |n(i)0 , n

(i)1 , . . . , n

(i)s + 1, . . .〉i

⊗i 6=j|n(j)0 , n

(j)1 , . . . , n

(j)s , . . .〉j

(3.30)


with n(i)s ∈ {0, 1} being the occupation number for particle i in site s,

∑s n

(i)s = 1.7

We will now have a brief look at the dynamics of two distinguishable particles in a

double-well potential according to the Hubbard model: in equation 3.28 this means

that we should take i, j ∈ {1, 2} and s ∈ {L,R} (referring respectively to the left

and right potential wells). In the spirit of second quantization we shall build a basis

for the Hilbert space in the site occupation number representation; there are four

possible ways in which we can populate the double-well with two particles8 (two

where both occupy the same well and two – due to the fact that we are working

with distinguishable particles – where they are in separated wells):

|ΨLL〉 := â†1,Lâ†2,L |0〉 , (3.31)

|ΨLR〉 := â†1,Lâ†2,R |0〉 , (3.32)

|ΨRL〉 := â†1,Râ†2,L |0〉 , (3.33)

|ΨRR〉 := â†1,Râ†2,R |0〉 , (3.34)

where |0〉 is the vacuum state where the system is not populated by any particle.

We may verify the action of the Hamiltonian on the basis states by computing the

matrix elements in the {|ΨLL〉 , |ΨLR〉 , |ΨRL〉 , |ΨRR〉} basis:

Ĥ.=

2E0 + U Ω0 Ω0 0

Ω0 2E0 0 Ω0

Ω0 0 2E0 Ω0

0 Ω0 Ω0 2E0 + U

. (3.35)

The reader may observe from the Hamiltonian matrix representation that there is

no non-zero matrix term linking states where both particles change their site: this

means that the Hubbard Hamiltonian only allows for single-body tunnelling events;

7This cumbersome and rather ugly looking notation is a by-product of the fact that we are

trying to use second quantization not for its intended scope, that is dealing with undistinguishable

particles; therefore we have to keep track of the occupation numbers of each single particle for all

the lattice sites, instead of just labelling the states based on how many bosons/fermions occupy

each site.8Once again, this is only true in the Hubbard model approximation where each well has only

one bound state. We have verified from exact calculations that, instead, the eigenfunctions of the

double-well potential are delocalized over both wells; the difficulties of defining a bound state for

the wells have been analyzed in section 2.4.


we shall discuss the implication of this fact in a deeper fashion in the following sec-

tions.

Due to basis completeness, any state of the system at a given time t may be decom-

posed over the basis states:

|ψ(t)〉 := bLL(t) |ΨLL〉+ bLR(t) |ΨLR〉+ bRL(t) |ΨRL〉+ bRR(t) |ΨRR〉 , (3.36)

where normalization imposes:

|bLL(t)|2 + |bLR(t)|2 + |bRL(t)|2 + |bRR(t)|2 = 1. (3.37)

Therefore, the evolution of the state of the system is dictated by the Schrödinger

equation, which applied to 3.36 yields:

iḃLL(t) =(

2E0 + U)bLL(t) + Ω0

(bLR(t) + bRL(t)

)iḃLR(t) =

(2E0

)bLR(t) + Ω0

(bLL(t) + bRR(t)

)iḃRL(t) =

(2E0

)bRL(t) + Ω0

(bLL(t) + bRR(t)

)iḃRR(t) =

(2E0 + U

)bRR(t) + Ω0

(bLR(t) + bRL(t)

). (3.38)

The exact solutions of such system can be found analytically but are extremely

cumbersome. Instead, we are looking for some simple analytical result to have

a better intuitive understanding of the system’s behaviour: following [3], we may

provide an approximate solution in the strongly-interacting regime. First, we begin

by making the solution of the system more accessible by defining new variables as a

combination of the probability amplitudes of the four basis states:

x1(t) := bLL(t)− bRR(t)

x2(t) := bLR(t)− bRL(t)

y1(t) := bLL(t) + bRR(t)

y2(t) := bLR(t) + bRL(t)

. (3.39)

This way we obtain two decoupled equations that can be solved independently from

the others:

iẋ1(t) =(

2E0 + U)x1(t) =⇒ x1(t) = Ae−i(2E0+U)t, (3.40)

iẋ2(t) =(

2E0

)x2(t) =⇒ x2(t) = De−i(2E0)t. (3.41)


We are left with two coupled equations:iẏ1(t) =

(2E0 + U

)y1(t) + 2Ω0y2(t)

iẏ2(t) =(

2E0

)y2(t) + 2Ω0y1(t)

. (3.42)

To uncouple the two equations we diagonalize the system in its matrix representa-

tion. The eigenvalues can be calculated to be:

λ1,2 = 2E0 +U

2±√U2 + 16Ω20. (3.43)

In order to obtain a simple analytical form for the solutions we approximate the

eigenvalues with a second order Taylor series expansion in function of ω := 2Ω20/U :

λ1,2 ≈ 2E0 +U

2± 1

2

(U + 4ω +O(ω2)

). (3.44)

Naturally, this holds true in the strongly-interacting regime where U � Ω20. In this

approximation the eigenvectors are:z1(t) = Ω0y1(t) + ωy2(t)

z2(t) = ωy1(t)− Ω0y2(t), (3.45)

which lead to the solutions:

iż1(t) =(

2E0 + U + 2ω)z1(t) =⇒ z1(t) = Be−i(2E0+U+2ω)t, (3.46)

iż2(t) =(

2E0 − 2ω)z2(t) =⇒ z2(t) = Ce−i(2E0−2ω)t. (3.47)

We may now finally write the solution for the basis probability amplitude coefficients:

bLL(t) = Ae−i(2E0+U)t +Be−i(2E0+U+2ω)t + Ce−i(2E0−2ω)t

bLR(t) = De−i2E0t + ωΩ0Be

−i(2E0+U+2ω)t + Ω0ω Ce−i(2E0−2ω)t

bRL(t) = −De−i2E0t + ωΩ0Be−i(2E0+U+2ω)t + Ω0ω Ce

−i(2E0−2ω)t

bRR(t) = −Ae−i(2E0+U)t +Be−i(2E0+U+2ω)t + Ce−i(2E0−2ω)t

. (3.48)

To evaluate the dynamics, we must now choose some initial conditions: we decide

to place the two particles inside the left well at t = 0. This roughly corresponds to

imposing bLL(0) = 1 and bLR(0) = bRL(0) = bRR(0) = 0 which, if we take ω ≈ 0,

implies A = B = 1/2 and C = D = 0.

Finally, we can compute the occupation probabilities for each of the basis states for


the evolving system:

PLL(t) := |bLL(t)|2 =∣∣∣ e−i(2E0+U+ω)t2 (eiωt + eiωt)∣∣∣2 = cos2 (2Ω20U t)

PLR(t) := |bLR(t)|2 = (. . .) = 4Ω0U2 sin2(U2 t)

PRL(t) := |bRL(t)|2 = (. . .) = 4Ω0U2 sin2(U2 t)

PRR(t) := |bRR(t)|2 = (. . .) = sin2(

2Ω20U t)

. (3.49)

We stress once again that those results are only valid for U � Ω20; moreover, the

applicability of this approximate solutions to the exact dynamics of the double-well

must also take into account the intrinsic limitations of the Hubbard model: in re-

ality U must be small enough not to involve the higher states of the spectrum in

the dynamics, as the Hubbard model only considers a single “band” that can be

populated by the particles.

A brief look at system 3.49 reveals that the probability of finding the two particles

separated is strongly suppressed by the presence of the interaction: this is striking,

considering that the model is insensitive to the sign of U , meaning that the dynamics

of an attractive or repulsive interaction are the same. Instead, the particles tend

to oscillate together between the two wells but a much lower frequency compared

to the non-interacting Rabi oscillations as 2Ω20/U � Ω0 in the strongly interacting

regime. This behaviour can be simply explained in terms of energy conservation:

when two particles are prepared together in the same well, their total mean energy

is 2E0 + U ; instead, when they occupy different wells, their total mean energy is

only 2E0 due to the fact that in the Hubbard regime the range of the interaction is

confined inside each lattice site. This means that, as the interaction grows stronger

(may U be either positive or negative), the two mentioned states are increasingly

detuned so that the probability of the two particles jumping from being together in

the same well to being separated is suppressed, in accordance with the approximate

results.

As an addendum, we now want to provide an example on how the Hubbard model

can be generalized to N bosons; the total Hamiltonian reads:

Ĥ :=∑s

(E0 n̂s +

U

2n̂s(n̂s − 1)

)+ Ω0

∑s,s̄

(â†s̄âs + â

†sâs̄

), (3.50)


where we have defined n̂s := â†sâs as the number operator that counts the number

of bosons inside the s-th lattice site. The reader may notice that the operators

have lost the index pertaining to the particle label, in compliance with the fact that

bosons have to be treated as identical and indistinguishable particles and therefore

they all share the same ladder operators. The interaction terms has slightly changed

its form, but not its function: it counts the number of particle pairs inside each well

and appends an U contribution to the energy for each one.

As an example, we now solve the same problem of two particles inside the symmetric

double-well potential. Differently from the distinguishable particle case, we must

keep in mind that all states must obey bosonic particle exchange symmetry rules,

according to definition 3.16; therefore we can define a basis for bosonic particles by

a symmetrization of the distinguishable particle basis:

|20〉 := |ΨLL〉 , (3.51)

|11〉 := 1√2

(|ΨLR〉+ |ΨRL〉

), (3.52)

|02〉 := |ΨRR〉 , (3.53)

where we have switched to the site occupation number representation, in great sec-

ond quantization style. In this basis the Hamiltonian matrix reads:

Ĥ.=

2E0 + U

√2Ω0 0

√2Ω0 2E0

√2Ω0

0√

2Ω0 2E0 + U

. (3.54)

The state of the system may be expanded on this basis as:

|ψ(t)〉 := b20(t) |20〉+ b11(t) |11〉+ b02(t) |22〉 . (3.55)

Applying the Schrödinger equation to state |ψ(t)〉 we get:iḃ20(t) =

(2E0 + U

)b20(t) +

√2Ω0b11(t)

iḃ11(t) =(

2E0

)b11(t) +

√2Ω0

(b20(t) + b02(t)

)iḃ02(t) =

(2E0

)b02(t) +

√2Ω0b11(t).

(3.56)

We will not go into the details of the calculations, which are similar to the distin-

guishable case. The solution of the system in the strong interaction limit U � 2Ω20


gives the probability amplitudes:b20(t) = Ae

−i(2E0+U)t +Be−i(2E0+U+2ω)t + Ce−i(2E0−2ω)t

b11(t) =√

2ωΩ0

Be−i(2E0+U+2ω)t +√

2Ω0ω Ce

−i(2E0−2ω)t

b02(t) = −Ae−i(2E0+U)t +Be−i(2E0+U+2ω)t + Ce−i(2E0−2ω)t

, (3.57)

where A, B and C are constant to be determined via initial conditions. We choose

once again to prepare the system with both particles in the left well at t = 0, which

yields A = B = 1/2 and C = 0. The state occupation probabilities can therefore be

calculated: P20(t) := |b20|2 = cos2

(2Ω20U t)

= PLL(t)

P02(t) := |b02|2 = sin2(

2Ω20U t)

= PRR(t)

P11(t) := |b11|2 =8Ω20U2

sin2(U2 t)

= PLR(t) + PRL(t)

. (3.58)

The reader may see that the dynamics are very similar to the distinguishable parti-

cle case; once again the effect of the interaction is to slow down the transfer of the

particles between the two wells and suppressing the amplitude of the state where

the two bosons are separated, independently of the sign of the interaction (in the

limits of the approximation U � 2Ω20). This can be appreciated in figure 3.3: as

long as we choose U � Ω20, but still not too large to excite the upper states in the

spectrum, the approximate Hubbard model solutions mimic quite well the behaviour

of the exact dynamics.

How high we can push the interaction strength shall be estimated from the single-

body spectrum: in the exact model, the non-interacting many-body eigenenergies

are∑

n nnEn with nn being the occupation number of the n-th eigenstate with

energy En. As we shall see in section 3.5, the single-band Hubbard model approxi-

mation translates in the exact model to truncating the single-body spectrum to the

first two states, {E0, E1}, meaning that the uppermost many-body eigenvalue shall

be �a := NE1. The next highest eigenvalue, excluded from the single-band model, is

therefore �b := (N − 1)E1 +E2: therefore, the single-band approximation holds true

until the interaction is strong enough to drive the transition �a → �b, which roughly

corresponds to U ≈ (�b − �a) = E2 − E1 := ∆E. Referring to the parameters used

in figure 3.3 (which are reported in the caption), ∆E ≈ 7.


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 3.3: numerical simulations for left well occupation probability 3.13 for 2 bosons (left

column) and 1 boson (right column) in the N = 2 system for various strengths of the interaction

(see plot title): in red the repulsive case (U = +|U |) and in blue the attractive one (U = −|U |);

the dashed black line represent the approximate solution from the Hubbard model (3.58). The

potential parameters are l = r = 2, b = 0.2, V0 = 500 and the starting state was chosen to be

2.16 for each particle; Ω0 ≈ 0.00238. The gap between the first and the second doublet in the

single-body spectrum is ∆E ≈ 7.

3.5 ∼ Extended Hubbard model 42

Finally, we would like to spend some words on a comparison between the Hubbard

model and the exact dynamics for the double-well potential. In both models, the

interaction strength can be tuned by changing the parameter U ; however, the pa-

rameter is fundamentally different in the two cases: while in the exact dynamics

it is just a multiplicative constant in front of the interaction matrix elements (see

definitions 3.1 and 3.3), in the Hubbard model it is the energy shift to the state

where two particles occupy the same lattice site due to the interaction. We shall

therefore refer to the two quantities with different labels, so we choose to indicate

the Hubbard energy shift as Ueff as it is an effective contribution given by the inter-

action to the spectrum. If we want to compare the two models we have to assure

that Ueff is equal to the contribution given by the δ-type potential for a given U

whenever two-particles reside in the same well; as we have mentioned before, there

is no clear-cut definition of such localized states, so we make the arbitrary, but sen-

sible, choice of putting each particle in the ground state of a single potential well in

the limit V0 →∞. Recalling definition 2.16 we have:

Ueff =( N⊗i=1

i 〈∞L|)Û( N⊗i=1

i |∞L〉)

=

= UN∑i=1

N∑j=i

∫dxi

∫dxj | 〈xi|∞L〉 |2| 〈xj |∞L〉 |2δ(xi − xj) =

= UN(N − 1)

2

∫dxi| 〈xi|∞L〉 |4 =

=N(N − 1)

2

4U

l2

∫ l0

sin4(πxl

)dx =

N(N − 1)2

3U

2l.

(3.59)

This is consistent with the fact that the factor N(N − 1)/2 counts the number of

distinguished particle pairs inside the left well, each contributing a factor 3U/2l.

3.5 Extended Hubbard model

We wish now to sightly improve the Hubbard model by getting rid of some of the

more simplistic approximations, in particular in regard to the interaction. We will

still keep the model as a single-band theory: as we have seen, the symmetric double-

well single-body spectrum is made of doublets of symmetric and antisymmetric

states; we will therefore only consider the contribution to the dynamics given by

the first “band” (that is the first doublet of states) for each particle, in a similar

way as we have done in the tight-binding approximation. This will allow us to


define an approximate site basis as discussed in section 2.4 for a symmetric double-

well potential.

The model can be easily built from the N -body total Hamiltonian 3.1 and restricting

the spectrum of each particle to {E0, E1}, the first two eigenvalues of the single-body

double-well Hamiltonian Ĥ(0)i ; no further approximations will be taken. This implies

that the model should give accurate results, provided that the higher states in the

spectrum do not provide significant contribution to the dynamics: in comparison to

the single-body case, now this is not only guaranteed by choosing an initial state with

small projections on the upper states, but also that the interaction strength U should

be small enough not to excite the upper states, as discussed in the previous section.

We may offer a graphical justification to this fact by looking at figure 3.4, where the

probability density distribution |Ψ(x1, x2, t)|2 for two interacting bosons has been

calculated at various points of the system’s evolution. For weak interactions (first

row), the probability distribution inside each well is described by a single “lobe”;

instead, as the interaction gets stronger (second and third rows), the probability is

either enhanced or suppressed for x1 = x2 (depending on whether the interaction is

attractive or repulsive), consistently with what we may näıvely expect from a δ(x1−

x2) interaction, so that the distribution has a more peculiar shape. Remembering

definition 3.4 and the single-body wavefunctions (see figure 2.3), one may see that

this peculiar shape requires the many-body state Ψ(x1, x2, t) to have non-negligible

projections over the states that present several nodes in their spatial distribution,

which are the more energetic ones.

Distinguishable particles

In this model, the Hartree product basis can be built as:

|Ψa1,...,aN 〉 :=N⊗i=1

|ψai〉i , (3.60)

with ai ∈ {0, 1} denoting the first two single-particle eigenstates. As we have seen,

in this basis the non-interacting part of the Hamiltonian is diagonal. Instead, the


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 3.4: numerical simulations for the probability density function |Ψ(x1, x2)|2 for 2 bosons at

the start (left column), at a quarter (central column) and at half (right column) of the oscillation

cycle (see figure 3.3) for U = 0.1 ((a), (b), (c)), U = −3 ((d), (e), (f)) and U = 3 ((g), (h), (i)). The

potential parameters are l = r = 2, b = 0.2, V0 = 500 and the starting state was chosen to be 2.16 for

both particles; Ω0 ≈ 0.00238. The lower left (resp. upper right) corner represents the region where

both particles occupy the left (resp. right) potential well; in the other two regions the particles

are separated. Compared to the non-interacting case (figure 3.1), the probability of finding the

particles separated is strongly suppressed, in accordance with the Hubbard model. We can see that

for strong attractive (resp. repulsive) interaction the probability is enhanced (resp. suppressed) for

x1 = x2, consistently with what we expect from a δ(x1 − x2) interaction. The formation of “lobes”

in the probability distribution is a symptom of the involvement of higher states of the spectrum

in the dynamics, which is the breakdown point for both the traditional and the extended Hubbard

model.


interaction matrix terms can be calculated as:

〈Ψb1,...,bN |Û |Ψa1,...,aN 〉 =( N⊗j=1

j 〈ψbj |)Û( N⊗i=1

|ψai〉i)

=

=

∫dx1 · · ·

∫dxN

N∏j=1

ψ∗bj (xj)N∏i=1

ψai(xi)U

2

N∑m=1

∑n6=m

δ(xm − xn).

(3.61)

Naturally this holds for distinguishable particles; for bosons one can employ the

same results after performing state symmetrization.

To analyse the differences with the traditional Hubbard model, we will now show a

practical example by calculating the Hamiltonian matrix terms for two distinguish-

able particles. The Hartree product basis will be composed of four states:

|Ψ00〉 := |ψ0〉1 ⊗ |ψ0〉2 , (3.62)

|Ψ01〉 := |ψ0〉1 ⊗ |ψ1〉2 , (3.63)

|Ψ10〉 := |ψ1〉1 ⊗ |ψ0〉2 , (3.64)

|Ψ11〉 := |ψ1〉1 ⊗ |ψ1〉2 . (3.65)

The non-interacting part of the total Hamiltonian is, as mentioned, diagonal in this

basis and reads:

Ĥ(U=0).=

2E0 0 0 0

0 E0 + E1 0 0

0 0 E0 + E1 0

0 0 0 2E1

. (3.66)

Instead, the interaction matrix terms will have to be calculated according to defini-

tion 3.61. As an example, we explicitly calculate the first term:

〈Ψ00|Û |Ψ00〉 =

= U

∫dx1

∫dx2ψ

∗0(x1)ψ

∗0(x2)ψ0(x1)ψ0(x2)δ(x1 − x2) =

= U

∫|ψ0(x1)|4dx1.

(3.67)

For the sake of clean notation, we define the quantities:a :=

∫|ψ0(x)|4dx

b :=∫|ψ1(x)|4dx

c :=∫|ψ0(x)|2|ψ1(x)|2dx

. (3.68)


To make writing the interaction matrix easier, we define a fourth quantity:

d : =1

4

∫ (|ψ0(x)|2 − |ψ1(x)|2

)2dx =

=1

4

∫ ((ψ0(x) + ψ1(x))(ψ0(x)− ψ1(x))

)2dx =

=

∫ (〈x|L〉〈x|R〉

)2dx,

(3.69)

where we used the fact that the wavefunctions are real valued and definitions 2.17

and 2.18 for the |L〉 and |R〉 states. The reader may notice that this is the interaction

matrix term that links the state |LL〉 := |L〉1⊗|L2〉 with the state |RR〉 := |R〉1⊗|R2〉

(and vice-versa) and therefore corresponds to the amplitude for the simultaneous co-

tunnelling process of the two particles.

The previous definitions allow us to write:

c =

∫|ψ0(x)|2|ψ1(x)|2dx =

=1

2

∫ (|ψ0(x)|4 + |ψ1(x)|4 −

(|ψ0(x)|2 − |ψ1(x)|2

)2)dx =

=a+ b

2− 2d,

(3.70)

so that the interaction matrix reads:

Û.= U

a 0 0 a+b2 − 2d

0 a+b2 − 2da+b

2 − 2d 0

0 a+b2 − 2da+b

2 − 2d 0

a+b2 − 2d 0 0 b

, (3.71)

where we have used the fact that:∫ψ30(x)ψ1(x)dx =

∫ψ0(x)ψ

31(x)dx = 0 (3.72)

due to the symmetry of the integrand function.

Now, to draw any comparison with the Hubbard model we must write the Hamil-

tonian matrix in the site basis; we follow the usual {|L〉 , |R〉} approximation that

we have developed for the single-body tight-binding model. Therefore, following


definitions 2.17 and 2.18 we can build the basis:

|LL〉 : = |L〉1 ⊗ |L〉2 =1√2

(|ψ0〉1 + |ψ1〉1

)⊗ 1√

2

(|ψ0〉2 + |ψ1〉2

)=

=1

2

(|Ψ00〉+ |Ψ01〉+ |Ψ10〉+ |Ψ11〉

),

(3.73)

|LR〉 : =(. . .)

=1

2

(|Ψ00〉 − |Ψ01〉+ |Ψ10〉 − |Ψ11〉

), (3.74)

|RL〉 : =(. . .)

=1

2

(|Ψ00〉+ |Ψ01〉 − |Ψ10〉 − |Ψ11〉

), (3.75)

|RR〉 : =(. . .)

=1

2

(|Ψ00〉 − |Ψ01〉 − |Ψ10〉+ |Ψ11〉

). (3.76)

We can now calculate the {|Ψ00〉 , |Ψ01〉 , |Ψ10〉 , |Ψ11〉} → {|LL〉 , |LR〉 , |RL〉 , |RR〉}

change of basis matrix:

T̂ :.=

1

2

+1 +1 +1 +1

+1 −1 +1 −1

+1 +1 −1 −1

+1 −1 −1 +1

, (3.77)

so that for the Hamiltonian matrix terms can be rewritten in the site basis:

Ĥ.=

2EL Ω0 Ω0 0

Ω0 2EL 0 Ω0

Ω0 0 2EL Ω0

0 Ω0 Ω0 2EL

+ U

a+ b− 3d a−b4a−b

4 d

a−b4 d d

a−b4

a−b4 d d

a−b4

d a−b4a−b

4 a+ b− 3d

, (3.78)

where we have separated the free and the interaction contributions;

EL : =1

2(E0 + E1), (3.79)

Ω0 : =1

2(E0 − E1) (3.80)

in compliance with the single-body tight binding definitions. If we compare 3.78 with

3.35, first and foremost we see that by calculating the exact interaction terms we have

a tunnelling amplitude d whenever the Hamiltonian connects two states where both

particles switch their site location: as we have noted before, those terms represent

the two-body co-tunnelling amplitudes and constitute an additional physical process

that was not accounted for in the Hubbard Hamiltonian. Moreover, the interaction

matrix provides a correction to the single-body tunnelling amplitudes Ω0:

Ω0 → Ω0 + Ua− b

4. (3.81)


This contribution is sensitive to the sign U , so it is different whether the interaction

is attractive or repulsive.

Bosons

Analogously to what we have done for the traditional Hubbard model, we will now

repeat the same calculations for two bosons to highlight any difference in the dy-

namics. We must also add that for N ≥ 2 it makes more sense to study only bosonic

dynamics, as we usually prepare the state so that at the initial time all the particles

occupy the same state (and this can only be done for bosons); instead, studying

the distinguishable particle case for N = 2 makes sense if, for example, we load the

system with a doublet of opposite spin electrons.

We will only highlight the differences from the distinguishable particle case, as oth-

erwise the calculations are similar; we must only obey the state symmetrization rule

3.16. Therefore, the Hartree product basis is:

|Ψ00〉+ := |ψ0〉1 ⊗ |ψ0〉2 , (3.82)

|Ψ01〉+ :=1√2

(|ψ0〉1 ⊗ |ψ1〉2 + |ψ1〉1 ⊗ |ψ0〉2

), (3.83)

|Ψ11〉+ := |ψ1〉1 ⊗ |ψ1〉2 . (3.84)

In this basis, the Hamiltonian matrix reads:

Ĥ.=

2E0 0 0

0 E0 + E1 0

0 0 2E1

+ U

a 0 a+b2 − 2d

0 a+ b− 4d 0

a+b2 − 2d 0 b

, (3.85)

with a, b and d defined according to 3.68 and 3.69. Once again, comparisons with

the näıve Hubbard mo

Documents

Cooperative e ects and many-body tunnelling...Cooperative e ects and many-body tunnelling Relatore: Ch.mo Prof. Fausto Borgonovi Correlatore: Ch.mo Prof. Giuseppe Luca Celardo Candidato: