Report Bachelor Project Physics and Astronomy
Quench action approach for an
interaction quench in the
Lieb-Liniger Bose gas
by
Mnˆeme Stapel 10914781
June 18, 2020
15 EC
conducted between 30-03-2020 and 26-06-2020
Supervisor:
Prof. dr. Jean-S´
ebastien Caux
Second examiner:
Dr. Vladimir Gritsev
Institute for Theoretical Physics Amsterdam
Faculty of Science, University of Amsterdam
Abstract
The behaviour of an isolated quantum system in an out-of-equilibrium state is strongly dependent on the integrability of the system. While non-integrable sys-tems are expected to thermalize, integrable syssys-tems do not relax towards a thermal equilibrium. The quench from the non-interacting Bose-Einstein condensate to the Lieb-Liniger model with finite repulsive interaction strength is particularly inter-esting. For this interaction quench, the generalized Gibbs ensemble fails due to infinities in the expectation values of the conserved charges. However, another method, the quench action, can be used to find the post-quench time evolution of the system. Indeed, the post-quench steady state shows to be different from a thermal state.
Samenvatting
De afgelopen decennia zijn er grote stappen gemaakt in de ontwikkeling van experimenten met zeer koude atomen. Deze ontwikkelingen hebben de theoretische interesse voor gesloten quantumsystemen die niet in hun evenwichtstoestand zijn aangewakkerd, omdat deze kunnen worden gemaakt met ultrakoude atomen. Een gesloten systeem kan uit evenwicht worden gebracht met een zogenaamde ‘quan-tum quench’. Tijdens een quan‘quan-tum quench wordt een interne parameter plotseling aangepast. Dit kan bijvoorbeeld de sterkte van de interactie tussen de deeltjes zijn. De vraag is dan hoe het systeem reageert op deze quench; belandt het na een tijdje weer in een evenwichtstoestand? Zo ja, kan worden berekend wat de eigenschappen van deze evenwichtstoestand zullen zijn?
Gesloten quantumsystemen zijn onder te verdelen in twee categorie¨en: integreer-bare en niet-integreerintegreer-bare. De twee groepen vertonen zeer verschillend gedrag na een quench. Niet-integreerbare modellen bevinden zich na enige tijd inderdaad weer in een evenwichtstoestand en die toestand is ook het thermisch evenwicht van het systeem. Een integreerbaar systeem heeft echter een heleboel behouden grootheden die niet-integreerbare modellen niet hebben. Energie en impuls zijn natuurlijk in alle systemen behouden, maar integreerbare systemen hebben daarnaast ook andere behouden grootheden. Doordat de waarden van deze grootheden gelijk moeten blij-ven na de quench, kan een integreerbaar systeem niet terugvallen naar een thermisch evenwicht. Een integreerbaar model ontwikkelt zich wel naar een andere evenwichts-toestand.
Een specifiek geval van een quantum quench is een interactie quench in een Lieb-Liniger Bose gas. In deze scriptie wordt de quench van een Bose-Einsteincondensaat, waarin geen interactie is tussen de deeltjes, naar een Lieb-Liniger model, waarin de deeltjes elkaar afstoten, beschreven. Het Lieb-Liniger gas is een integreerbaar mo-del, het zou na de quench dus in een niet-thermische toestand moeten belanden. Deze toestand vinden bleek lastiger dan voor sommige andere integreerbare syste-men, omdat de standaard methode niet kon worden gebruikt. Om toch het gedrag van het Lieb-Liniger model na de quench te kunnen beschrijven wordt de ‘quench action’ methode gebruikt. De quench action maakt gebruik van de overlappingen van de golffuncties van de modellen voor en na de quench. In dit geval zijn dat dus die van het Bose-Einsteincondensaat en het Lieb-Liniger gas. Met de quench action wordt inderdaad gevonden dat het gas na de quench niet in een thermisch evenwicht belandt, maar wel in een andere evenwichtstoestand. Deze toestand is stabiel, het systeem zal niet verder evolueren zolang de integreerbaarheid niet gebroken wordt.
Contents
1 Introduction 1
2 Fundamental concepts 1
2.1 Relaxation . . . 1
2.2 Integrability . . . 2
2.3 Lieb-Liniger model and Bose-Einstein condensate . . . 2
2.4 Quench . . . 2
2.5 Generalized Gibbs ensemble . . . 3
3 Interaction quench 3 3.1 The GGE for the BEC to Lieb-Liniger quench . . . 3
3.2 Thermodynamic Bethe Ansatz . . . 4
3.3 The quench action . . . 5
4 Saddle point equation 7 4.1 Solving the saddle point equation . . . 7
4.2 Saddle point density distribution . . . 9
5 Discussion 11
6 Conclusion 11
7 Acknowledgements 12
1
Introduction
The many developments in the field of ultracold atom experiments in the past few decades have given impetus to the theoretical study of the relaxation behaviour of quantum sys-tems [1]. The evolution of integrable one-dimensional many body syssys-tems was considered to be of purely theoretical value, but with the possibility of performing experiments with ultracold gases of bosons confined to one dimension in an optical trap, the relaxation of these systems can also be experimentally investigated. A way to bring a system in an out-of-equilibrium state is to perform a quantum quench.
Post-quench behaviour turns out to be vastly different for a non-integrable system than for an integrable system. A non-integrable system relaxes towards a thermal equilibrium, an integrable system, on the other hand, does not evolve towards a thermal state. In [2] the lack of thermalization in an integrable system is shown in an experiment with a one-dimensional Bose gas. An interesting question therefore is whether an out-of-equilibrium integrable quantum system does evolve towards a steady state at all, and if so, how that steady state can be found.
An example of such an integrable system out of equilibrium is the repulsive Lieb-Liniger model after a quench from a Bose-Einstein condensate. In this case the standard method of using the generalized Gibbs ensemble to find the post-quench steady state does not work. Instead, another method, the quench action, can be applied. The quench action utilizes the Bethe Ansatz solutions of the Lieb-Liniger model. In this thesis it is shown that the quench action formalism is indeed successful in describing the evolution of the Lieb-Liniger Bose gas after an interaction quench. In the next chapter the concepts needed for the description with the quench action method are explained. Chapter 3 contains the actual description of the application of the quench action method, and the properties of the post-quench steady state are calculated in chapter 4.
2
Fundamental concepts
Before tackling the question of post-quench evolution, some of the important background concepts need some explanation. In this chapter the fundaments for the application of the quench action method are laid down.
2.1
Relaxation
Relaxation of a quantum system is the evolution of that system from an out-of-equilibrium state towards a steady state. Once the system has reached a steady state it can stay there for long periods of time. When this steady state is a thermal equilibrium, the evolution of the system is called thermalization. A thermal steady state is a state that has a distribution that corresponds to the distribution of one of the statistical ensembles in equilibrium. However, in some systems this thermal state is not reached, or at least not directly, but only via some other metastable state. Whether or not a system can thermalize, depends on its integrability [3]. The notion of integrability thus plays a key role in understanding the relaxation behaviour of quantum systems.
2.2
Integrability
Although integrability is a well defined concept for classical systems, for quantum systems finding a suitable definition is not straightforward [4]. Sometimes a system is said to be quantum integrable if the system is exactly solvable, for example by making use of the Bethe Ansatz [5]. Other definitions rely on the existence of a complete set of independent commuting operators. For our purpose it is relevant to note that when discussing an integrable quantum system we should not only take trivial conservation laws into account, such as conservation of momentum and energy, but also the many other conserved charges the system has. A non-integrable system would uniformly fill the available Hilbert space, but the extra, nontrivial conserved quantities an integrable quantum system has, prevent the system from exploring the entire Hilbert space [6].
2.3
Lieb-Liniger model and Bose-Einstein
condensate
An example of such an integrable model is the Lieb-Liniger model. This model describes a one-dimensional gas of bosons interacting via a delta function potential. The Hamiltonian is given by HLL = − N X j=1 ∂2 ∂x2 j + 2c X j1<j2 δ(xj1 − xj2), (1)
where 2c indicates the character and the strength of the delta function [7]. When c > 0 the interaction is repulsive, for c < 0 the interaction is attractive. The Schr¨odinger equa-tion for the Lieb-Liniger model is exactly solvable with the Bethe Ansatz.
A non-interacting gas of bosons in one dimension is described by the Bose-Einstein con-densate (BEC) state
|Ψ0i = 1 √ LNN !(ψ † k=0) N|0i , (2)
where ψ†k=0 is the creation operator for a zero-momentum particle working on the Fock vacuum |0i [8].
2.4
Quench
In order to study the relaxation process of a system, the system needs to be in an out-of-equilibrium state. This can be achieved by performing a quantum quench. Before the quench, the system should be in an eigenstate of Hi, the initial Hamiltonian. This means
that the state is described by the eigenfunction ψn of the Hamiltonian, that satisfies the
equation
Hi|ψni = En|ψni , (3)
where En is the corresponding eigenvalue of the Hamiltonian. The quantum quench
then takes place at t = 0, when the Hamiltonian Hi is suddenly replaced by another
Hamiltonian Hf. For t > 0 the system evolves unitarily according to the new Hamiltonian.
by Hf and will therefore not necessarily relax to its ground state [9]. The question if, and
towards which steady state, the system will relax after a quantum quench is not easily answered.
2.5
Generalized Gibbs ensemble
In some cases, the quenched system evolves towards a state that can be described by the generalized Gibbs ensemble (GGE). The density matrix of the GGE is given by
ρGGE =
e−PmβmQbm
Z , (4)
where the Lagrange multipliers {βm} are fixed by the expectation values h bQmi, { bQm} is
the full set of conserved charges, and the partition function is given by Z = Tr[e−PmβmQbm]
[10].
Since the Lieb-Liniger model is solvable by the Bethe Ansatz, the eigenstates can be described by a set of rapidities λj [11]. The eigenvalues of all conserved charges Qm are
given by the power sums over the rapidities Qm ≡
X
j
λmj . (5)
Once the expectation values hQmi are known, the density of the final state ρLL can than
be found, using the relation [12]
hQmi = L
Z
dλρLL(λ)λm. (6)
3
Interaction quench
In section 2.5 the generalized Gibbs ensemble was explained. The following section shows that the GGE cannot be used for the interaction quench in the Lieb-Liniger Bose gas. The Bethe equations for the Lieb-Liniger model are given in section 3.2. The thermodynamic Bethe Ansatz is used because the evolution of the system after the quench is described in the thermodynamic limit. This means that the number of particles N → ∞ and the sys-tem size L → ∞ while keeping NL constant. The advantage of taking the thermodynamic limit is that internal fluctuations do not influence the thermodynamic quantities of the system.
3.1
The GGE for the BEC to Lieb-Liniger quench
In order to apply the GGE to the quench from a BEC to the Lieb-Liniger model, the expectation values of the conserved charges from equation 5 need to be calculated on the BEC state as well. Calculating these expectation values fixes the Lagrange multipliers {βm} using the expression
hBEC| Qm|BECi = T r[e−PmβmQmQ m] T r[e−P mβmQm] . (7)
The expectation values of Qm can be calculated using the expression for the charges in real space (8) Qm ∼ N X j=1 ∂m ∂xm j + αm(1)c N X j=1 ( ∂ m ∂xm j )m−2 X 1≥i>j>≥N δ(xi− xj) + . . . + α(m) c X 1≥i>j>≥N δ(xi− xj) !n/2 ,
where c is the coupling constant of the Hamiltonian [13]. Because the wave functions of the Lieb-Liniger model have cusps when the position of two particles is the same, the powers of the delta-functions get cancelled. With these cancelled powers, equation 8 gives the same eigenvalues as those found in equation 5. For a Hamiltonian with a different coupling constant c0 the powers of the delta-functions do not get cancelled, and a divergence occurs. This divergence makes using the GGE for this interaction quench at best very complicated. The post-quench relaxation can however be described using the quench action method that is explained in the last section of this chapter.
3.2
Thermodynamic Bethe Ansatz
The eigenstates of a Bethe Ansatz-solvable model are labeled by a set of quantum numbers {Ij}, with j = 1, ..., N [8]. The quantum numbers are integers if N is odd, and half-odd
integers for N even. These quantum numbers are mapped to a set of rapidities {λj} by the
Bethe equations [14]. In the thermodynamic limit a continuum version of the rapidities, the function λ(x), can be defined. The values of λ(x) on the set {xj =
Ij
L} are fixed to
λj. Using this new function λ(x) the continuum version of the Bethe equations can be
written as
λ(x) + Z ∞
−∞
dyφ(λ(x) − λ(y))ρ(y) = 2πx, (9)
with φ(λ) = 2 atanλc, the scattering phase shift, and ρ(x) = L1 PN
j=1δ(x − Ij
L), the density
distribution. Equation 9 provides a mapping between position and rapidity space, so the density distribution can be rewritten as functions of λ. Using this density distribution in rapidity space, the continuum version of the Bethe equation can be written as
λ + Z ∞
−∞
dλ0φ(λ − λ0)ρ(λ0) = 2πx(λ), (10) where ρ(λ) is the particle density. A relation between de particle density and the hole density ρh(λ) is found by differentiating equation 10 with respect to λ. Because the
derivative of x(λ) is given by dx(λ)dλ = ρ(λ) + ρh(λ), doing so gives the relation
1 + 2π Z ∞
−∞
dλ0C(λ − λ0)ρ(λ0) = 2π(ρ(λ) + ρh(λ)), (11)
with the Cauchy kernel C(λ) = 2π1 dφ(λ)dλ = π1c2+λc 2. When the distribution ρ(λ) is known,
the densities of the conserved quantities are easily calculated. The momentum and energy densities are p = P L = Z ∞ −∞ dλλρ(λ) and e = E L = Z ∞ −∞ dλλ2ρ(λ). (12)
Figure 1: The quantum numbers are divided in boxes with width ∆xi, the filled in circles
represent occupied quantum numbers, the empty ones are holes.
The density of the higher conserved charges of the system is given by qn = Qn L = Z ∞ −∞ dλλnρ(λ), n ∈ N. (13)
3.3
The quench action
The particle and hole density of the TBA are used to find the quench action. With the action the post-quench evolution of the system can be determined. For this approach, the overlaps between the initial and the final state h{I}|ψ(t = 0)i have to be calculated [8]. At t = 0 the wavefunction can be written as a sum over the eigenstates of the post-quench Hamiltonian
|ψ(t = 0)i =X
{I}
e−S{I}|{I}i , (14)
with the overlap coefficients
S{I} = − log h{I}|ψ(t = 0)i. (15)
In the eigenbasis of the post-quench Hamiltonian, the Schr¨odinger equation is easily solved with the eigenvalues ω{I}. Using these eigenvalues and the overlap coefficients, the
time-dependent wavefunction can be written as |ψ(t)i =X
{I}
e−S{I}−iω{I}t|{I}i . (16)
Now that the exact time-dependent wavefunction is known, the expectation value of a physical operator O can be determined. Calculating the expectation value of O
hψ(t)| O |ψ(t)i = X
{I}
X
{I0}
e(−S{I})∗−S{I0}+i(ω{I}−ω{I0})th{I}| O |{I0}i (17)
gives a double summation. However, equation 17 can be simplified by going to the thermo-dynamic limit [8]. Before doing so, the set of quantum numbers {I} should be rewritten by dividing the real line of the quantum number coordinates into boxes Bi. ∆xi denotes
the size of a box, the integer i labels the boxes. In figure 3.3 a sketch of the boxes dividing the quantum numbers can be found. Within each box, there are Lρt(xi)∆xi
allowed quantum numbers. Of these allowed places Lρ(xi)∆xi are filled and Lρh(xi)∆xi
are holes. To describe the set of quantum numbers {I} completely, the configuration of the holes and particles in the boxes should be known. However, in the thermodynamic
limit, differences between in-box configurations do not change the expectation value to leading order [14]. Only the total number of ways to distribute the Lρ(xi)∆xi particles
over the allowed places is used to calculate the expectation value. This number is given by Lρt(xi)∆xi Lρ(xi)∆xi = [L(ρ(xi) + ρh(xi))∆xi]! [Lρ(xi)∆xi]! [Lρh(xi)∆xi]! , (18)
where the constraint ρt(x) = ρ(x) + ρh(x) has been used. Stirling’s approximation allows
equation 18 to be rewritten as eSiL∆xi, with
Si = (ρ(xi) + ρh(xi)) log (ρ(xi) + ρh(xi)) − ρ(xi) log ρ(xi) − ρh(xi) log ρh(xi). (19)
Taking the thermodynamic limit, the double summation in equation 17 can be simplified using the expression from 19 in the thermodynamic limit. The summation over the quantum numbers can be replaced by a functional integral
limth X {I} (. . .) = Z D[ρ]eSY Y[ρ](. . .), (20) using SY Y[ρ(x)] = L Z ∞ −∞
dx[(ρ(x) + ρh(x)) log (ρ(x) + ρh(x)) − ρ(x) log ρ(x) − ρh(x) log ρh(x)],
(21) the thermodynamic limit of equation 19. SY Y[ρ(x)], the Yang-Yang entropy, is given in
quantum number space, so in view of the Bethe equations from section 3.2 it is convenient to express equation 21 in rapidity space. After the transformation, the Yang-Yang entropy can be written as
SY Y[ρ(λ)] = L Z ∞
−∞
dλ[(ρ(λ) + ρh(λ)) log (ρ(λ) + ρh(λ)) − ρ(λ) log ρ(λ) − ρh(λ) log ρh(λ)].
(22) With the thermodynamic limit of the overlaps S = −limthlog h{I}|ψ(t = 0)i the quench
action can be defined as the difference between the extensive part of the overlaps and the Yang-Yang entropy [15]:
SQ[ρ] ≡ 2S[ρ] − SY Y[ρ]. (23)
The quench action can be seen as the counterpart of the Gibbs free energy for out-of-equilibrium systems [8]. Equation 23 can be evaluated with the saddle-point approxima-tion of the quench acapproxima-tion. The condiapproxima-tion that needs to be satisfied for this approximaapproxima-tion is given by δSQ[ρ] δρ ρsp = δ(2S[ρ] − S Y Y[ρ]) δρ ρsp = 0, (24)
with ρsp the saddle point distribution [15]. In the infinite time limit, the saddle point
distribution can be used to calculate the expectation value of operator O lim
t→∞limthhψ0| O(t) |ψ0i = hρsp| O |ρspi . (25)
This provides a possibility to calculate numerous physical properties of the post-quench steady state once the saddle point distribution is known. However, not any expectation
value can be calculated using this method, the operator O does need to be weak, smooth and thermodynamically finite [8]. These conditions are usually met by local operators.
4
Saddle point equation
In order to calculate the post-quench time evolution of the system, a saddle point ap-proximation is used. The saddle point describes the steady state after the quench to an interacting Lieb-Liniger model. Since this model is integrable, the system has an infinite amount of conserved charges. Because of these infinities the system will not relax further after it has reached the saddle point state. In the next sections the saddle point equation 24 is solved, and the distribution ρsp is shown to be different from a thermal density
function.
4.1
Solving the saddle point equation
The overlaps between the initial state and the Bethe states of the Lieb-Liniger model play an essential role in the application of the quench action method. These overlaps have been calculated in [15]:
hλ, −λ|0i = s (cL)−NN ! detNj,k=1Gjk detN/2j,k=1GQjk N/2 Y j=1 λj c s λ2 j c2 + 1 4 . (26)
The states |λ, −λi represent parity invariant Bethe states, these are the only sates that give a nonzero overlap with the initial states. To calculate the quench action 23, the thermodynamic limit of the overlaps needs to be determined. This is also done in [15]. The extensive part of the overlaps is
(27) S[ρ] = −limth(log hλ, −λ|0i)
= Ln 2 (log γ + 1) + L 2 Z ∞ 0 dλρ(λ) log[λ 2 c2( λ2 c2 + 1 4)] + O(L 0),
with γ = nc. As can be seen in equation 24, only the ρ-dependent part of the overlaps is used to evaluate the quench action. Rewriting the quench action with the ρ-dependent part of the overlaps S[ρ] and the Yang-Yang entropy gives
SQ[ρ]/L = Z ∞ 0 dλ[ρ(λ) log λ 2 c2( 1 4 + λ2 c2)
−ρt(λ) log ρt(λ)+ρ(λ) log(ρ(λ))+ρh(λ) log ρh(λ)].
(28) The functional that is used to find the saddle point equation
(29) F = Z ∞ 0 dλ[ρ(λ) log λ 2 c2( 1 4 + λ2 c2) − ρt(λ) log ρt(λ) + ρ(λ) log(ρ(λ)) + ρh(λ) log ρh(λ)] + Lh 2 (n − Z ∞ −∞ dλρ(λ))
consist of the quench action with an added Lagrange multiplier, to fix the average density to the value in accordance with the normalisation conditions on ρ. In order to find the saddle point equation, the functional derivative of F with respect to ρ is taken. This gives
(30) δF L = Z ∞ 0 dλ[δρ(λ) log λ 2 c2( 1 4+ λ2 c2) − δρt(λ) log ρt(λ) − ρt(λ) 1 ρt(λ)δρ t (λ) + δρ(λ) log(ρ(λ)) + ρ(λ) 1 ρ(λ)δρ(λ) + δρh(λ) log ρh(λ) + ρh(λ) 1 ρh(λ)δρ h (λ)] −h 2 Z ∞ −∞ dλδρ(λ) .
The only states that are left after calculating the overlaps are the parity-invariant Bethe states, since the overlap with the BEC state is zero for all other states. This parity-invariance allows the second integral in equation 30 to be rewritten as −hR∞
0 dλδρ(λ).
Using ρt(λ) = ρ(λ) + ρh(λ) and the new form of the integral, equation 30 can be rewritten
as (31) δF L = Z ∞ 0 dλ[δρ(λ) log λ 2 c2( 1 4+ λ2 c2) − δρ(λ) log 1 + ρ h(λ) ρ(λ) − δρh(λ) log 1 + ρ(λ) ρh(λ) − hδρ(λ)].
The thermodynamic form of the Bethe equations relates the particle density to the hole density. The relation between δρ(λ) and δρh(λ) is found after taking the derivative of
equation 11. The relation is given by δρ(λ) + δρh(λ) =
Z ∞
−∞
dλ0C(λ − λ0)δρ(λ0), (32) where C(λ) is the Cauchy kernel. Equation 31 can now be rewritten using the relation between δρ(λ) and δρh(λ). This gives
(33) δF L = Z ∞ 0 dλ[δρ(λ) log λ 2 c2( 1 4+ λ2 c2) − δρ(λ) log ρ h(λ) ρ(λ) − hδρ(λ) − log 1 + ρ(λ) ρh(λ) Z ∞ −∞ dλ0C(λ − λ0)δρ(λ0)]. Every term in this equation contains δρ(λ), so equation 33 can be simplified to
(34) δF L = Z ∞ 0 dλδρ(λ) log λ 2 c2( 1 4+ λ2 c2) − log ρ h(λ) ρ(λ) − h − log 1 + ρ(λ) ρh(λ) Z ∞ −∞ dλ0C(λ − λ0) log 1 + ρ(λ) ρh(λ) .
To make this expression more clear, a new, dimensionless variable x = λc can be intro-duced, together with K(x) = 2
x2+1. Using the function a(x) =
ρ(λ)
ρh(λ) will prove to be useful.
of equation 34 should be changed to λ0 = cy. With these new variables and functions, equation 34 becomes δF L = Z ∞ 0 dλδρ(λ) log x2(1 4+ x 2 ) + log(a(x)) − h − Z ∞ −∞ dy 2πK(x − y) log(1 + a(y)) . (35) The form of δFL in equation 35 can be used to apply the saddle point condition δF = 0. Imposing the saddle-point condition is the same as equating the integrand to zero. Since the integrand has to be zero for all variations δρ(λ), applying the saddle point condition boils down to log x2(1 4+ x 2) + log(a(x)) − h − Z ∞ −∞ dy 2πK(x − y) log(1 + a(y)) = 0. (36) Using a different variable τ = eh2 instead of just the Lagrange multiplier h, gives the final
version of the saddle point condition log(a(x)) = log τ2 − log
x2(1 4 + x 2) + Z ∞ −∞ dy 2πK(x − y) log(1 + a(y)). (37) This saddle point equation has an analytical solution that can be found in [15]. With the relation between ρ(λ) and ρh(λ) given by the Bethe equation 11, an expression for ρ(x) in
terms of a(x) can be found by taking the partial derivative with respect to τ of equation 37 2πρ(x) = τ 2∂τa(x) 1 + a(x) = a(x) 1 + a(x) τ ∂τ 2 log a(x). (38)
Because the variable x is defined as x = λc, the saddle point distribution is given by ρsp(λ) = ρ(λ/c).
4.2
Saddle point density distribution
Now that the density distribution for the saddle point state has been calculated, it can be compared to the thermal steady state. In figure 4.2 the density distribution of the saddle point state ρsp and the density distribution of the thermalized state ρth are shown [15].
From these plots it is evident that the thermalized density distribution and the saddle point density distribution are different. Therefore, the post-quench steady state cannot be a thermal state.
The difference between the saddle point state and the thermal equilibrium can also be seen by calculating the expectation value of a suitable operator using equation 25. Such operators are for example the static density moments
gK = hρsp| : (ˆρ(0)/n)K : |ρspi , (39)
where ˆρ(x) = Ψ†(x)Ψ(x) and the commutation relation [Ψ(x), Ψ†(x)] = δ(x − x0). In figure 4.2 the difference between g2 and g3 evaluated on the thermal state and evaluated
on the saddle point state is shown [15]. From these static density moments it is also clear that the system does not thermalize, even at times long after the quench.
Figure 2: Adapted from [15]. For different values of the interaction parameter γ the den-sity distribution is shown, the solid line represents the saddle point state and the dashed line the thermal state. The difference between the density function of the thermalized state and that of the saddle point state is clearly visible.
Figure 3: Reprinted from [15]. The static density moments g2 and g3 are shown as a
function of γ. Like in figure 4.2 the solid lines represent the density moments evaluated on the saddle point state and the dashed lines show g2 and g3 evaluated on the thermal
state. The red curve is g2, the green one g3and the black dashed curves give the asymptotic
behaviours for γ → 0 and γ → ∞. The insets are the same plots but for different values of γ.
5
Discussion
In chapter 4 the saddle point density distribution was calculated. The post-quench steady state found with the quench action method is indeed different from a thermal equilibrium, as would be expected because of the integrability of the Lieb-Liniger model. The density distribution found in section 4.1 corresponds to the distribution found in [15].
While this thesis has mainly focused on the post-quench steady state, finding the full time evolution is, in principle, possible with the quench action method. The time evolution is ‘reverse engineered’ from the saddle point state with excitations around that point. In [15] the full time evolution of the density-density correlation is calculated for the limiting case of the Tonks-Girardeau gas.
The quench action has already been proven to be a successful approach to several other quench problems, like the N´eel to XXZ spin chain quench [16, 17], and the quench from a BEC to an attractive Lieb-Liniger gas [18]. One of the main challenges for applying the quench action method is finding exact overlaps between the initial state and the eigenstates of the final Hamiltonian. However, when not interested in the full post-quench time evolution but only in the saddle point state, only the extensive part of the overlaps needs to be known [8]. By calculating the post-quench steady state in this way, the full time evolution of operators should even be recovered by using the quench action method on the results found with the non-exact overlaps.
Experimentally, performing a quantum quench protocol with integrable systems is possible by using ultracold bosons. By trapping the system in such a way that the gas can only move in one dimension, a Lieb-Liniger Bose gas can be achieved. The two other spatial dimensions are still present and can influence the behaviour of the trapped gas [3]. So while the system discussed in this thesis is not expected to relax further after it has reached the saddle point, experimentally the integrability can be broken. When integrability is broken, the system is expected to thermalize, although it happens at much larger time scales. The link between integrability and thermalization can be investigated in experiments by controlling the influence of the two remaining spatial dimensions.
6
Conclusion
In this thesis the evolution of an out-of-equilibrium Lieb-Liniger gas after a quench from a Bose Einstein condensate is examined. Because of the many conserved charges of the Lieb-Liniger model, the system does not thermalize. However, relaxation towards a steady state, the so-called saddle point state, does occur. Even though the generalized Gibbs ensemble proved to be inapplicable to this interaction quench, the post-quench steady state can be found using the quench action method.
In the future, the quench action method could be applied to other models, such as spin chains. Even when the exact overlaps cannot be calculated, but the extensive part is known, the quench action might be used to recover the full time evolution of the system. The ability to perform experiments with ultracold bosons has opened up the possibility of studying these quantum quenches not only theoretically. With experiments, the rela-tion between integrability breaking and relaxarela-tion towards a thermal equilibrium can be explored.
7
Acknowledgements
I would like to thank prof. dr. Jean-S´ebastien Caux for taking the time to supervise my thesis despite his busy schedule. In this era of video conferencing, the meetings were inspiring, and provided me with the right amount of science park-feeling to avoid pre-COVID-19 nostalgia. Dr. Vladimir Gritsev, thank you for your prompt, affirmative answer when I asked you to be the second examiner for my thesis.
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