Post-quench time evolution of an integrable quantum system
Application of the Quench Action after the quench of a Bose-Einstein condensate to a Lieb-Liniger gas.
Melin Walet Student number 10564551
Report Bachelor Project Physics and Astronomy, size 15 EC Conducted between 01-12-2016 and 16-07-2017
Institute for Theoretical Physics Amsterdam Faculty of Science, University of Amsterdam
Supervisor: prof. dr. J.-S. Caux Second assessor: dr. J. van Wezel
Abstract
The last decade a lot of progress have been made in the understanding of the post-quench time evolution of quantum systems. When studying the behavior of isolated out-of-equilibrium systems, an important difference between integrable and non-integrable quan-tum systems can be observed. Non-integrable system evolve towards a thermal state. In the integrable case, this thermalization appears to be absent. This thesis focuses on the quench from a non-interacting Bose-Einstein condensate to an integrable Lieb-Liniger gas. Because of the infinite initial values of the conserved charges, the Generalized Gibbs En-semble cannot be used to describe the post-quench steady state. Therefore another method, the Quench Action is applied. It is shown that relaxation towards a non-thermal steady state occurs.
Contents
1 Introduction 3
2 Context and Terminology 3
2.1 Quantum Quench . . . 3
2.2 Relaxation of a quantum system . . . 4
2.3 Integrability . . . 4
2.4 Generalized Gibbs Ensemble . . . 5
2.5 Bose-Einstein condensate and Lieb-Liniger model . . . 6
2.6 Inapplicability of GGE in BEC to LL quench . . . 6
3 Quench Action Formalism 7 3.1 Thermodynamic limit . . . 7
3.2 Thermodynamic Bethe Ansatz . . . 7
3.3 Quench Action functional . . . 8
3.4 Saddle Point Approximation . . . 9
4 Derivation of post-quench time evolution 10
5 Discussion 14
6 Conclusion 14
7 Acknowledgements 15
References 16
A Full derivation of post-quench time evolution 17
1
Introduction
When studying the behavior of a quantum gas after a quench, the question arises whether or not this gas will relax towards a thermal steady state. As Fagotti and Essler described in (1), the post-quench behavior in 1D gases is rather different than in 2D or 3D gases. These interest-ing phenomena after a quantum quench, caused by the presence of oscillatory factors became an incentive to investigate the non-equilibrium dynamics of integrable quantum systems. The past decade there has been a lot of new insights in this field: while a non-integrable system is expected to relax towards a thermal equilibrium, in the case of an integrable system a non-thermal steady state will be reached. Research involving optically trapped ultra-cold atomic gases showed the absence of thermalization, as was the case in the experiment of Kinoshita, Wenger and Weiss (2). For the description of such a post-quench steady state, various methods such as a generalized Gibbs ensemble (GGE) have been used.
One specifically interesting case is the quench of a non-interacting Bose-Einstein condensate (BEC) to a δ-interacting Lieb-Liniger (LL) Bose gas. Since GGE cannot be used to describe the dynamics of this system, another new method is put to work which is based on the Thermo-dynamic Bethe Ansatz (TBA). This method is called the Quench Action (3), using the overlap between the initial state and the eigenfunctions of the Hamiltonian which describe the post-quench time evolution in a saddle point approximation. The main objective of this thesis is to explain why the Quench Action can be used to describe the post-quench steady state of an integrable quantum system. This thesis is organized as follows: In section 2 the reader can find some relevant terminology about the quantum quench and post-quench behavior. In section 3 the Quench Action formalism is explained as used in (3). In section 4 the reader will be guided through the application of the Quench Action, obtaining a result for the post-quench steady state after a BEC to LL quench.
2
Context and Terminology
Before diving into the application of the Quench Action, it is useful to obtain some knowledge about the context and relevant terminology. In this section the principle of a quantum quench will be explained, as well as some relevant terminology involving the integrability of quantum systems.
2.1
Quantum Quench
The reason our system is put out of equilibrium and hence needs a proper description of its time evolution, is the quantum quench. The energy of a quantum system can be described with a Hamiltonian.
We start off with a system which is prepared in the ground state of a certain Hamiltonian H(h0).
Here, h0 represents the relevant system parameter. At time t=0 this system parameter is
sud-denly changed from h0 to h. This can be for example the addition of interactions in the system.
This change in the Hamiltonian is called a quantum quench (4). The system is put out of its equilibrium and can be described with a new Hamiltonian, H(h). Subsequently the system will undergo unitary evolution under this new Hamiltonian H(h). The question is raised whether or not this system will relax towards an equilibrium state (5). Whilst the original Hamiltonian describes the time evolution, our new Hamiltonian can be used to describe any equilibrium the system might reach. The fact that those Hamiltonians are not equivalent, makes the quantum quench such an interesting case. How can one take the discrepancies between those Hamilto-nians into account? Before specifying which quantum quench is considered in this thesis, first the terminology of relaxation is elaborated upon in section 2.2.
2.2
Relaxation of a quantum system
The finesse within the term relaxation is rather important for the understanding of the time evolution of a system. Relaxation is the general name for the evolution of an out-of-equilibrium system towards a steady state. A steady state is a stable condition, in which a system will remain regardless of time. A specific case of relaxation is called thermalization. The system relaxes towards a thermal steady state, i.e. a state which obeys the laws of thermodynamics.
Sometimes however, a final thermal state is not yet reached, and the system displays a pseudo-equilibrium. This evolution is called prethermalization. In this event, a metastable state is reached, characterized by the conservation laws of the system (6).
Figure 1: (Pre-)thermalization (adapted from (6). The different types of post-quench time evo-lution, prethermalization and thermalization, are visualized.
2.3
Integrability
As mentioned before, integrability plays a crucial role in the ability of a system to relax. This section will reveal the underlying reason. Multiple definitions of integrability exist in quantum physics, such as the existence of an exact Bethe Ansatz solution (6). Relevant to us is the fact that integrable quantum systems have many conserved charges. These charges, IN, commute
with the Hamiltonian H.
A nonintegrable quantum system has to take into account only trivial conservation laws, such as energy, momentum and particle number conservation. Ergodicity then tells us that all config-urations in Hilbert space are equally likely to occur (7). In the integrable case however, many more higher order conserved charges are present. Since the system retains memory of the value of those conserved charges, the ergodicity of the system is broken. This means not every con-figuration is accessible and therefore a thermal steady state cannot be reached.
2.4
Generalized Gibbs Ensemble
In order to describe the steady state a system evolves towards, scientists have come up with a so called Generalized Gibbs Ensemble (GGE). The density matrix of this GGE is
ρGGE =
e−PmβmQˆm
ZGGE
. (3)
Here βm are the generalized chemical potentials, fixed by the initial expectation values h ˆQmi.
The canonical partition function is Z = Tr(e−PmβmQˆm) (8). We benefit from the fact that one can
calculate the average values of local operators using expectation values. The Eigenstate Ther-malization Hypothesis (ETH) (4) states the following: considering a small box within Hilbert space, the average state energy within this box will provide sufficient information for the entire system. This is because in the eigenstate all states within a thermal ensemble have the same physical conditions. A generalized version of the ETH can be applied in this case. Hence the asymptotic values of the local operators can be described with the expectation values of con-served charges in the eigenstate. The GGE corresponds with these expectation values and hence characterizes the new steady state a long time after a quantum quench.
In the next section some details will be revealed about the model used for the post-quench description. Specific to the Lieb-Liniger model is the expression for the eigenstates of the op-erators. They are described as a sum over the various quasimomenta λ (3):
Qn =
X
j
λnj. (4)
Describing the expectation values of operator QM gives the following equation.
hQMi = L
Z ∞
−∞
δλρLL(λ)λM. (5)
Here the ρLL(λ) is the density of the steady state we are looking for. The initial values of
the conserved charges determine the new steady state (8). However, if these initial values are infinite, the use of the GGE becomes impossible. This will be explained in section 2.6.
2.5
Bose-Einstein condensate and Lieb-Liniger model
The specific case we will focus on is the quench of a non-interacting Bose-Einstein condensate to an integrable Lieb-Liniger gas. A Bose-Einstein condensate is the phase of matter in which atoms are cooled down to a temperature close to absolute zero. A majority of the atoms occupy the lowest single body eigenstate. We consider the one-dimensional case. This Bose-Einstein condensate (BEC) is described by (3):
|Ψ0i = 1 √ LNN !(ψ † k=0) N|0i . (6)
The ψk=0† creates a zero momentum particle and the |0i indicates the Fock vacuum (3). As a result of the BEC to LL quench, interactions between the particles occur. This results in a change of the Hamiltonian to a new Hamiltonian, i.e. the Hamiltonian of a Lieb-Liniger gas. This Lieb-Liniger model describes gases that move in one dimension and obey the Bose-Einstein statistics. It is an integrable model (9).
The Hamiltonian of the Lieb-Liniger gas listens to the following expression (3):
HLL = − N X j=1 ∂2 ∂x2 j + 2c N X 1≤j1≤j2≤N δ(xj1 − xj2). (7)
Here c characterizes the interaction, where a positive (negative) value indicates a repulsive (attractive) interaction. γ = nc is the dimensionless interaction strength (with n the number den-sity). The interaction potential is zero-range (ultra-local), indicated with the 2cδ-term (10). The Bogoliubov theory implies that the lowest momentum mode is occupied (10). The Bo-goliubov theory represents our Lieb-Liniger model properly in the following sense. After the non-interacting Bose Einstein Condensate, interactions occur because of the quantum quench. For small interactions, the Bogoliubov approach is able to represent this new state.
2.6
Inapplicability of GGE in BEC to LL quench
As mentioned in 2.4 the expectation values of the conserved charges provide information about the time evolution and the steady state. However, the GGE cannot be used in the case of a BEC to LL quench (8). When using the expression below, some important observations can be made. Let’s take a look at the expression for the expectation values of operator QM.
hQMi = L
Z ∞
−∞
δλρLL(λ)λM. (8)
For values of M equaling 1 or 2, this integral converges. For a value of M equaling 4 (for higher order conserved charges), divergence is observed. Obviously this suggests that the den-sity ρLL(λ) contains a λ14 term (3). This cancels out the λ4-term, allowing the integral to be
Z ∞
−∞
δλ ∝ δ(x − 0). (9)
The above shows that infinite conserved charges create infinities for the expectation values of the conserved charges, which makes the use of GGE impossible. Thermalization is not expected to take place because of the many conserved charges that features the integrable Lieb-Liniger model. In order to describe the time evolution GGE cannot be used. Another variational method is adopted, called the Quench Action. The next section is devoted to the application of this Quench Action.
3
Quench Action Formalism
In the past several attempts have been made to find an exact solution for the post-quench behav-ior of an integrable system. The Generalized Gibbs Ensemble seems to be a fine approximation in many cases, however fails in the BEC to LL quench. By now the reader of this thesis should be well aware of the fact that thermalization will not take place after the BEC to LL quench. However relaxation towards another steady state will occur. The Quench Action is a useful framework to compute the time-dependent expectation value of an operator after a quantum quench (3).
3.1
Thermodynamic limit
We describe the evolution of the state in the thermodynamic limit (hereafter; TD-limit). In this limit the number of particles (N) as well as the length of the box (L) are set to ∞, keep-ing the number density NL constant. The reason for describing a gas in the TD-limit is that its thermodynamic quantities (such as pressure and energy) depend only on variables like temper-ature. Fluctuations of the internal energy can thus be ignored. The expectation values of local operators in the saddle point describes the (pseudo)-stable state.
ρdist(x; I) = 1 L X j δ(x − xj). (10) where xj = Ij/L.
In this case we only have the interaction parameter γ, which is either negative (attractive) or positive (repulsive). The two limiting cases are c=0 (non-interacting Bose gas) and c=∞ (Tonks Girardeau limit).
3.2
Thermodynamic Bethe Ansatz
As described in (3) the eigenstates of a model which is solvable with the Bethe Ansatz are labeled by a set of quantum numbers {I}. The expression for the normalized N-particle Bethe
Ansatz state is obtained by evaluation at the solutions of the Bethe equations. We define the density ρ(x) and hole density ρh(x):
ρ(x) = 1 L X n∈I δ(x − n L). (11) ρh(x) = 1 L X m∈ ˜I δ(x − m L). (12) The variable x is Ij L.
In order to get a relation between the density ρ and the hole density ρh(λ), we make use of the Bethe equations. The Bethe equations map a set of quantum numbers {Ij} to a set of
(pseudo-)momenta (rapidities) λj (10): ρ(λ) + ρh(λ) = 1 2π + Z ∞ −∞ dµ 2πK(λ − µ)ρ(µ). (13)
Here K stands for the Cauchy kernel.
3.3
Quench Action functional
As mentioned before, the difference between the pre- and post-quench Hamiltonian has to be taken into account. The overlaps between the original state and the post-quench eigenstate hI|ψ0i can be calculated. The minus log of this expression is called the overlap pseudo energy.
When taking the TD-limit, we obtain the extensive part of this energy.
S0 = −limthlog hI|ψ0i . (14)
The expression for the time evolution of our initial state is (summed over the sets of quantum numbers {I}):
|ψ(t)i =X
I
e−SIO−iωt|Ii , (15)
such that the expectation value for an operator A can be written as (3): hψ(t)|A|ψ(t)i =X
I
X
I0
eSIO−SOI0e(−ωI−ωI0)thI| A |I0i . (16)
We are facing a double summation in equation 16 which can luckily be simplified. Before doing so, the notation needs some explanation. We consider the quantum numbers {I} to be uniquely characterized as a combination of in-box configurations {ci} and box fillings {ρi}.
remaining summation over {ρi} can be seen as a functional integral, including the Yang-Yang
entropy SY Y (3, 10).
The Yang-Yang entropy counts the number of thermodynamically indistinguishable states. When considering the possible microstates the system can be in, the number of possible config-urations is obtained (10):
Ω = (L(ρ + ρ
h)dλ)!
(Lρdλ)!(Lρhdλ)!, (17)
with ρ and ρhrespectively the density and hole density of our system. This gives us the needed
Yang-Yang entropy (11), with the Boltzmann constant kB set to 1:
SY Y[ρ] = ln(Ω) = Z ∞
−∞
(Ldλ(ρ + ρh)ln(ρ + ρh) − Lρlnρ − Lρhlnρh). (18) To get rid of the double summation in equation 16 we use the following fact. In the ther-modynamic limit, the summation over the box fillings can be rewritten as a functional integral containing the Yang-Yang entropy (12):
limT h X {I} ... = Z ρ∈C∞ D[ρ]eSY Y(...). (19)
The Quench Action functional is defined as the difference between the pseudo-energy ob-tained from the overlaps and the real part of the Yang-Yang entropy of the state,
SQ[ρ] = 2SO[ρ] − SY Y[ρ]. (20)
The formula for the Quench Action could be seen as a quantum analogy of the Gibbs free energy G, which equals the difference between the enthalpy H and TS (a measurement for the uncertainty of the system).
Equation 16 can now be rewritten in terms of this Quench Action: hψ(t)|A|ψ(t)i =
Z ∞
0
D[ρ]eSQ[ρ]. (21)
This integral form will be used in the description of the time evolution in section 4.
3.4
Saddle Point Approximation
The Quench Action makes use of a saddle point approximation. The saddle point characterizes a pseudo-stable state for our system (4). It is called pseudo-stable because it describes the temporarily steady state (still in the phase of prethermalization). Because of the conservation laws of the integrable system however, further relaxation will not occur. By minimizing the Quench Action functional (SQ), we meet the maximum entropy condition.
We describe the expectation values of local operators in the saddle point. These operators have to meet certain properties (3): an operator has to be weak and smooth. Also it has to be
Figure 2: Visualization of the saddle point which represents the post-quench steady state. Image used from Caux, 2017 (13)
.
thermodynamically finite, meaning it does not cause any unwelcome shifts around the saddle point.
4
Derivation of post-quench time evolution
Now that the relevant terminology is described, we turn our attention to the actual calculation of the time evolution of the post-quench quantum system. In (10) this evolution, after a quench from a BEC to Lieb-Liniger model, is described. In this section the reader will be guided through this derivation, using the Quench Action formalism. For the readability of the thesis some (trivial) steps have been omitted. In appendix A the full derivation can be found.
We start off with the overlap energy SO[ρ], calculated in (12). They used the following overlaps for the calculation.
hλ, −λ|0i = s (cL)−N! detN j,k=1Gjk detN/2j,k=1GQjk QN/2 j=1 λj c q λ2 j c2 + 1 4 . (22)
SO[ρ] = −limth(log(hλ, −λ|0i) =
Ln 2 (logγ+1)+ L 2 Z ∞ 0 dλρ(λ)logλ 2 c2 λ2 c2+ 1 4 +O(L0). (23) The −λ-term in equation 22 represents the counterpart of the positive pseudo-rapidities λ, using the parity invariance of the system (which is a direct result of the conserved charges) (12).
We fill in equation 20, using only the ρ-dependent part of the overlap SO[ρ] from equation 23 and recalling equation 18 for the Yang-Yang entropy:
SQ[ρ] L = 1 L(2S O[ρ] − S Y Y[ρ]) = 2 L L 2 Z ∞ 0 δλρ(λ)logλ 2 c2 λ2 c2 + 1 4 − 1 L L Z ∞ 0 δλ(ρ + ρh)ln(ρ + ρh) − ρlnρ − ρhlnρh. (24) SQ[ρ] L = Z ∞ 0 δλ[ρ(λ)log(λ 2 c2( 1 4 + λ2 c2) − ρ
t(λ)log(ρt(λ)) + ρ(λ)log(ρ(λ)) + ρh(λ)log(ρ(λ))].
(25) Now that the expression for the Quench Action SQ[ρ] is obtained, the functional integral from equation 21 can be expressed explicitly:
Z
D[ρ]e−SQ[ρ]. (26)
We add a Lagrange multiplier h to equation 26 to obey the normalization conditions (12): Z i∞
−i∞
dh Z
D[ρ]e−SQ[ρ]e−(Lh/2)(n−R−∞∞ dλρ(λ)). (27)
After substituting our expression for the Quench Action (equation 25) we will focus on the power-term in the integral, calling this term G. To apply the saddle point approximation, we calculate the partial derivative of G:
G = SQ[ρ] + (Lh/2)(n − Z ∞ −∞ dλρ(λ)). (28) δG L = Z ∞ 0 dλ[δρ(λ)log(λ 2 c2( 1 4+ λ2 c2))−δρ(λ)log(1+ ρh(λ) ρ(λ) )−δρ h(λ)log(1+ ρ(λ) ρh(λ))]− h 2 Z ∞ −∞ δλδρ(λ). (29) The partial derivative form of the Bethe equations (12) will be used to substitute δρh(λ) in
equation 29: δρ(λ) − Z ∞ −∞ dµ 2πK(λ − µ)δρ(µ) = −δρ h(λ). (30)
δG L = Z ∞ 0 dλ δρ(λ)log λ 2 c2 1 4 + λ2 c2 − δρ(λ)log1 + ρ h(λ) ρ(λ) + δρ(λ) − Z ∞ −∞ dµ 2πK(λ − µ)δρ(µ) ! log(1 + ρ(λ) ρh(λ)) − h Z ∞ 0 dλδρ(λ). (31) δG L = Z ∞ 0 dλδρ(λ) log λ 2 c2 1 4 + λ2 c2 − log(ρ h(λ) ρ(λ) ) − Z ∞ −∞ dµ 2πK(λ − µ)log 1 + ρ(λ) ρh(λ) − h . (32) The following variables are introduced:
x = λ/c, (33)
α(x) = ρ(λ)
ρh(λ), (34)
K(x) = 2
x2+ 1. (35)
This allows us to rewrite equation 32 into: δG L = Z ∞ 0 dλδρ(λ) log x21 4 + x 2 + log(α(x)) − Z ∞ −∞ dµ 2π 2 c(x2+ 1)log(1 + α(x)) − h . (36) At this point we can finally use the saddle point method. Hence by equating the integrand to zero, we obtain the following saddle point condition:
log x21 4+ x 2 + log(α(x)) − Z ∞ −∞ dµ 2π 2 c(x2+ 1)log(1 + α(x)) − h = 0. (37)
Introducing y = µc allows us to rewrite it into: log(α(x)) = −log x21 4 + x 2 + Z ∞ −∞ dy 2π 2 (x2+ 1)log(1 + α(x)) + h. (38)
Further we introduce the variable T = eh2 such that h = log(T2). We are then left with the
equation which was calculated in (12): log(α(x)) = log(T2) − log
x21 4+ x 2 + Z ∞ −∞ dy 2π 2 (x2+ 1)log(1 + α(x)). (39)
The expression for log(α(x)) we found in equation 39 contains all the useful physical in-formation about our steady state, since it is the ratio of the density ρ(λ) and the hole density ρh(λ). Evaluating this expression gives us the following equation for the density ρ(x) of the
steady state.
2πρ(x) = α(x) 1 + α(x)T
∂T
2 log(α(x)). (40)
where saddle point distribution is given by ρsp(λ) = ρ(λc)
Now that we have worked our way through the saddle point approximation, we can plot the obtained result for the density of our steady state (equation 66) against the density for a thermal state. This is done in (12), shown in figures 3 and 4.
Figure 3: Densities for the obtained saddle point state (solid lines) and thermal state (dashed lines) clearly differ from each other (12).
In figures 3 and 4 it is clear that our obtained saddle point state for the BEC to LL quench differs from a thermal state. Hence no thermalization occurs in this situation, as we expected from an integrable system.
Figure 4: Post-quench time evolution to the obtained steady state and a thermal state.
5
Discussion
The result we obtained in this thesis corresponds to the calculations of previous works such as (12). Indeed no thermal state is reached after a BEC to Lieb-Liniger quench. The infinite initial values of the conserved charges would have caused complications when trying to ap-ply the GGE. Using the saddle point approximation within the Quench Action bypassed these complications, allowing us to obtain a result for the post-quench steady state.
Several experiments have been conducted involving a quantum quench. By working with ultracold bosonic gases, integrable quantum systems are represented. To realize a BEC to Lieb-Liniger quantum quench, one could change the amount of interaction. This could be done by increasing the amplitude of the wavefunction of the light beam. Because the bottom of the optical potential becomes more narrow, the wave functions of the various particles get more strangled and hence more interactions will occur. As described in this thesis, the time evolution of the system will end at a state of prethermalization, where the system remains for a long time. Since integrability prevents the final thermalization of a system, it could be interesting to look for tools to break this integrability. Theoretically, such tools are absent. Experimentally however, weak forces that affect (and break) the integrability could perhaps be found. On an extended period of time, those weak terms will break the integrability. In that case, a thermal steady state could eventually be reached.
6
Conclusion
In this thesis the quench from a non-interacting Bose Einstein Condensate to an integrable Lieb-Liniger model is discussed. Integrable systems contain a lot of conserved charges, causing the
inability to relax towards a thermal state. We tracked down the expression for a density matrix describing the post-quench steady state. This steady state turned out to be a non-thermal state, as we expected from the integrability of the Lieb-Liniger model. The steady state differs from a thermal state in the way it is distributed. There are more particles with higher energy than in a thermal state.
Further research could be the application of the Quench Action formalism to other quantum models such as spin chains. Experimentally there could be a search for forces that are able to break the integrability of a system, eventually resulting in thermalization a long time after a quantum quench.
7
Acknowledgements
I would like to thank prof. dr. Jean S´ebastien Caux for his excellent supervision. Without constructive suggestions and invaluable support this thesis would not have been possible.
References
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A
Full derivation of post-quench time evolution
In this appendix the various steps of the calculation of the steady state are extensively written down. Some steps might be rather trivial to the reader. In section 4 a more depleted version can be found. We start off with the overlap energy SO[ρ], calculated in (12). They used the following overlaps for the calculation.
hλ, −λ|0i = s (cL)−N! detN j,k=1Gjk ∗ det N/2 j,k=1G Q jk QN/2 j=1 λj c q λ2 j c2 + 1 4 (41)
SO[ρ] = −limth(log(hλ, −λ|0i) =
Ln 2 (logγ +1)+ L 2 Z λ 0 ρ(λ)logλ 2 c2 λ2 c2 + 1 4 +O(L0) (42) The −λ-term in equation 41 represents the counterpart of the positive pseudo-rapidities λ, using the parity invariance of the system (which is a direct result of the conserved charges) (12). We fill in equation 20, using only the ρ-dependent part of the overlap SO[ρ] and recalling equa-tion 18 for the Yang-Yang entropy:
SQ[ρ] L = 1 L(2S O [ρ] − SY Y[ρ]) = 2 L L 2 Z ∞ 0 dλρ(λ)logλ 2 c2 λ2 c2 + 1 4 − 1 L L Z ∞ 0 dλ(ρ + ρh)ln(ρ + ρh) − ρlnρ − ρhlnρh (43) SQ[ρ] L = Z ∞ 0 dλ[ρ(λ)log(λ 2 c2( 1 4 + λ2 c2) − ρ
t(λ)log(ρt(λ)) + ρ(λ)log(ρ(λ)) + ρh(λ)log(ρ(λ))]
(44) Now that we know the expression for the Quench Action, it is time to optimize this with a functional, using the saddle point equation.
We recall the expression for the functional including the Quench Action: Z
D[ρ]e−SQ[ρ] (45)
which, by adding a Lagrange multiplier, we modify to: Z i∞
−i∞
δh Z
After substituting our expression for the Quench Action (44) we calculate the partial deriva-tive of the integrand G of this functional, before equalling that to zero (saddle point equation):
G = SQ[ρ] + (Lh/2)(n − Z ∞ −∞ dλρ(λ)) (47) G = L Z ∞ 0 dλ[ρ(λ)log(λ 2 c2( 1 4+ λ2 c2)) − ρ
t(λ)log(ρt(λ)) + ρ(λ)log(ρ(λ)) + ρh(λ)log(ρ(λ))]
+ (Lh/2)(n − Z ∞ −∞ dλρ(λ)) (48) G L = Z ∞ 0 dλ[ρ(λ)log(λ 2 c2( 1 4+ λ2 c2)) − ρ
t(λ))log(ρt(λ)) + ρ(λ)log(ρ(λ)) + ρh(λ)log(ρ(λ))]
+ (h/2)(n − Z ∞ −∞ dλρ(λ)). (49) δG L = Z ∞ 0 dλ[δρ(λ)log(λ 2 c2( 1 4+ λ2 c2))−δρ t(λ)log(ρt(λ))−ρt(λ) 1 ρt(λ)δρ t(λ)+δρ(λ)log(ρ(λ)) + ρ(λ) 1 ρ(λ)δρ(λ) + δρ h(λ)log(ρ(λ)) + ρh(λ) 1 ρh(λ)δρ h(λ)] − h Z ∞ 0 dλδρ(λ). (50) Mind the change of limits in the second integral of equation 50 and the correcting factor 2. We were able to do this because of the parity invariance of ρ(λ), which states that ρ(−λ) equals ρ(λ). In the next equation we have replaced ρt(λ) for ρ(λ) + ρh(λ).
δG L = Z ∞ 0 dλ[dρ(λ)log(λ 2 c2( 1 4+ λ2 c2))−(δρ(λ)+δρ h (λ))log(ρ(λ)+ρh(λ))−(δρ(λ)+δρh(λ)) + δρ(λ)log(ρ(λ)) + δρ(λ) + δρh(λ)log(ρ(λ)) + δρh(λ)] − h Z ∞ 0 dλδρ(λ). (51) Realizing the following:
log ρ(λ) + ρh(λ) − log(ρ(λ)) = log 1 + ρ h(λ) ρ(λ) . (52)
and rearranging equation 51 leaves us with δG L = Z ∞ 0 dλ[δρ(λ)log(λ 2 c2( 1 4+ λ2 c2))−δρ(λ)log(1+ ρh(λ) ρ(λ) )−δρ h(λ)log(1+ ρ(λ) ρh(λ))]−h Z ∞ 0 dλδρ(λ) (53)
Bethe equation says (12): ρ(λ) + ρh(λ) = 1 2π + Z ∞ −∞ dµ 2πK(λ − µ)ρ(µ) (54)
This gives us, after taking the partial derivative form, the following equation: δρ(λ) − Z ∞ −∞ dµ 2πK(λ − µ)δρ(µ) = −δρ h(λ) (55) which we will use to substitute δρh(λ) in equation 53:
δG L = Z ∞ 0 dλ δρ(λ)log λ 2 c2 1 4 + λ2 c2 − δρ(λ)log1 + ρ h(λ) ρ(λ) + δρ(λ) − Z ∞ −∞ dµ 2πK(λ − µ)δρ(µ) ! log(1 + ρ(λ) ρh(λ)) − h Z ∞ 0 dλδρ(λ) (56) We use the fact that:
log(1 + ρρ(λ)h(λ)) log(1 + ρρ(λ)h(λ)) = log ρ h(λ) ρ(λ) (57) δG L = Z ∞ 0 dλδρ(λ) log λ 2 c2 1 4 + λ2 c2 − log(ρ h(λ) ρ(λ) ) − Z ∞ −∞ dµ 2πK(λ − µ)log 1 + ρ(λ) ρh(λ) − h (58) The following variables are introduced:
x = λ/c (59)
α(x) = ρ(λ)
ρh(λ) (60)
K(x) = 2
x2+ 1 (61)
where K was defined in (10).
This allows us to rewrite equation 58 into: δG L = Z ∞ 0 dλδρ(λ) log x21 4 + x 2 + log(α(x)) − Z ∞ −∞ dµ 2π 2 c(x2+ 1)log(1 + α(x)) − h (62)
At this point we can finally use the saddle point method. Hence by equating the integrand to zero, we obtain the following saddle point condition:
log x2 1 4 + x 2 + log(α(x)) − Z ∞ −∞ dµ 2π 2 c(x2+ 1)log(1 + α(x)) − h = 0 (63)
Introducing y = µc allows us to rewrite it into: log(α(x)) = −log x21 4+ x 2 + Z ∞ −∞ dy 2π 2 (x2+ 1)log(1 + α(x)) + h (64)
Further we introduce the variable T = eh2 such that h = log(T2). At the end of the day, we
are left with the equation which was calculated in (12): log(α(x)) = log(T2) − log
x21 4 + x 2 + Z ∞ −∞ dy 2π 2 (x2+ 1)log(1 + α(x)) (65)
The expression for log(α(x)) we found in equation 65 contains all the useful physical informa-tion about our steady state, since it is the ratio of the density ρ(λ) and the hole density ρh(λ). Evaluating this expression gives us the following equation for the density ρ(x) of the steady state.
2πρ(x) = α(x) 1 + α(x)T
∂T
2 log(α(x)) (66)
B
Popular scientific summary (Dutch)
De afgelopen 10 jaar hebben wetenschappers belangrijke ontdekkingen gedaan op het gebied van gesloten quantum systemen die uit evenwicht gebracht zijn. Er is gekeken naar de on-twikkeling van een gas, nadat het een zogenaamde ’quantum quench’ doorgemaakt heeft. Een quantum quench drukt het systeem samen, waardoor het energieniveau veranderd. Na de quench is het systeem niet meer in een grondtoestand, maar in een aangeslagen toestand. Omdat het systeem zijn energie niet kwijt kan aan de omgeving (het is immers een gesloten systeem), vroegen wetenschappers zich af wat er ging gebeuren. Zou het systeem, na de quench, weer terugvallen naar een grondtoestand? En wat voor toestand zou dit zijn?
Er werden experimenten uitgevoerd met extreem koude gassen en vele theoretische berekenin-gen later ontdekte men iets interessants: er is een groot verschil tussen integreerbare en niet-integreerbare systemen. Een niet-integreerbaar quantum gas vond na een quench zijn weg terug naar een nieuw thermisch evenwicht. Een integreerbaar systeem vertoonde dit gedrag echter niet. Er zijn veel verschillende definities voor quantum integreerbaarheid. Degene die voor deze situatie relevant is, is het feit dat een integreerbaar systeem veel behouden grootheden bevat. Natuurlijk moeten in een gesloten systeem de energie en het aantal deeltjes behouden zijn, ze kunnen immers niet ineens verdwijnen. Maar bij een integreerbaar systeem zijn er nog andere behouden grootheden van hogere orden. Deze grootheden zorgen ervoor dat het systeem niet zomaar een nieuw evenwicht kan vinden na de quench. Om het zo te zeggen, het systeem ’onthoudt’ de waarden van deze grootheden en kan dus niet zomaar een nieuw evenwicht aan-nemen.
In deze scriptie beschouwen we een specifiek geval van een quantum quench: de quench van een Bose-Einstein Condensaat naar een Lieb-Liniger gas. Een Bose-Einstein Condensaat (BEC) is een extreem koud, niet-interacterend gas. Tijdens de quench worden er interacties toegevoegd, wat het systeem verandert naar een Lieb-Liniger gas, wat wel interacteert en inte-greerbaar is. We kijken naar de ontwikkeling van dit gas na de quench. Deze quench was een uitdaging voor wetenschappers omdat de ’gebruikelijke’ methode niet gebruikt kon worden. Een nieuwe techniek (de Quench Action) maakte het alsnog mogelijk de tijdsevolutie van het systeem te beschrijven. Deze techniek maakt gebruik van de overlap tussen de energie van voor en na de quench. We zien dat er inderdaad een niet-thermisch evenwicht gevonden wordt, zoals te verwachten was van een integreerbaar quantum systeem. Dit betekent dat de energieverdeling van de deeltjes in het systeem niet voldoen aan een normale thermische verdeling. Er zijn iets te veel deeltjes met hoge energie dan bij een thermisch evenwicht.
Nu we weten dat de Quench Action een goede methode is om de tijdsevolutie van een systeem te beschrijven, kan deze methode worden toegepast om bepaalde experimentele phe-nomenen te beschrijven, bijvoorbeeld in extreem koude bosonische gassen.
