arXiv subjects

Alexandre Krajenbrink

Publications and source records attributed to Alexandre Krajenbrink.

31 records · Page 2Linked to original sources

arXiv2020

Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line

We consider the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height $h(x,t)$ on the positive half line with boundary condition $\partial_x h(x,t)|_{x=0}=A$. It is equivalent to a continuum directed polymer (DP) in a random potential in half-space with a wall at $x=0$ either repulsive $A>0$, or attractive $A<0$. We provide an exact solution, using replica Bethe ansatz methods, to two problems which were recently proved to be equivalent [Parekh, arXiv:1901.09449]: the droplet initial condition for arbitrary $A \geqslant -1/2$, and the Brownian initial condition with a drift for $A=+\infty$ (infinite hard wall). We study the height at $x=0$ and obtain (i) at all time the Laplace transform of the distribution of its exponential (ii) at infinite time, its exact probability distribution function (PDF). These are expressed in two equivalent forms, either as a Fredholm Pfaffian with a matrix valued kernel, or as a Fredholm determinant with a scalar kernel. For droplet initial conditions and $A> - \frac{1}{2}$ the large time PDF is the GSE Tracy-Widom distribution. For $A= \frac{1}{2}$, the critical point at which the DP binds to the wall, we obtain the GOE Tracy-Widom distribution. In the critical region, $A+\frac{1}{2} = εt^{-1/3} \to 0$ with fixed $ε= \mathcal{O}(1)$, we obtain a transition kernel continuously depending on $ε$. Our work extends the results obtained previously for $A=+\infty$, $A=0$ and $A=- \frac{1}{2}$.

cond-mat.stat-mech↗

arXiv2020

From stochastic spin chains to quantum Kardar-Parisi-Zhang dynamics

We introduce the asymmetric extension of the Quantum Symmetric Simple Exclusion Process which is a stochastic model of fermions on a lattice hopping with random amplitudes. In this setting, we analytically show that the time-integrated current of fermions defines a height field which exhibits a quantum non-linear stochastic Kardar-Parisi-Zhang dynamics. Similarly to classical simple exclusion processes, we further introduce the discrete Cole-Hopf (or Gärtner) transform of the height field which satisfies a quantum version of the Stochastic Heat Equation. Finally, we investigate the limit of the height field theory in the continuum under the celebrated Kardar-Parisi-Zhang scaling and the regime of almost-commuting quantum noise.

quant-ph↗

arXiv2019

Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions

We revisit the Lieb-Liniger model for $n$ bosons in one dimension with attractive delta interaction in a half-space $\mathbb{R}^+$ with diagonal boundary conditions. This model is integrable for arbitrary value of $b \in \mathbb{R}$, the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as $b$ is decreased from the hard-wall case $b=+\infty$, with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar-Parisi-Zhang equation for the growth of a one-dimensional interface of height $h(x,t)$, on the half-space with boundary condition $\partial_x h(x,t)|_{x=0}=b$ and droplet initial condition at the wall. We obtain explicit expressions, valid at all time $t$ and arbitrary $b$, for the integer exponential (one-point) moments of the KPZ height field $\bar{e^{n h(0,t)}}$. From these moments we extract the large time limit of the probability distribution function (PDF) of the scaled KPZ height function. It exhibits a phase transition, related to the unbinding to the wall of the equivalent directed polymer problem, with two phases: (i) unbound for $b>-\frac{1}{2}$ where the PDF is given by the GSE Tracy-Widom distribution (ii) bound for $b<-\frac{1}{2}$, where the PDF is a Gaussian. At the critical point $b=-\frac{1}{2}$, the PDF is given by the GOE Tracy-Widom distribution.

cond-mat.stat-mech↗

arXiv2019

Probing the large deviations of the Kardar-Parisi-Zhang equation at short time with an importance sampling of directed polymers in random media

The one-point distribution of the height for the continuum Kardar-Parisi-Zhang (KPZ) equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the distribution is obtained over a large range of values, down to a probability density as small as $10^{-1000}$ in the tails. The short time behavior is investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations, showing a spectacular agreement with the analytical expressions. The flat and stationary initial conditions are studied in the full space, together with the droplet initial condition in the half-space.

cond-mat.stat-mech↗

arXiv2019

Distribution of Brownian coincidences

We study the probability distribution, $P_N(T)$, of the coincidence time $T$, i.e. the total local time of all pairwise coincidences of $N$ independent Brownian walkers. We consider in details two geometries: Brownian motions all starting from $0$, and Brownian bridges. Using a Feynman-Kac representation for the moment generating function of this coincidence time, we map this problem onto some observables in three related models (i) the propagator of the Lieb Liniger model of quantum particles with pairwise delta function interactions (ii) the moments of the partition function of a directed polymer in a random medium (iii) the exponential moments of the solution of the Kardar-Parisi-Zhang equation. Using these mappings, we obtain closed formulae for the probability distribution of the coincidence time, its tails and some of its moments. Its asymptotics at large and small coincidence time are also obtained for arbitrary fixed endpoints. The universal large $T$ tail, $P_N(T) \sim \exp(- 3 T^2/(N^3-N))$ is obtained, and is independent of the geometry. We investigate the large deviations in the limit of a large number of walkers through a Coulomb gas approach. Some of our analytical results are compared with numerical simulations.

cond-mat.stat-mech↗

arXiv2019

On the qubit routing problem

We introduce a new architecture-agnostic methodology for mapping abstract quantum circuits to realistic quantum computing devices with restricted qubit connectivity, as implemented by Cambridge Quantum Computing's tket compiler. We present empirical results showing the effectiveness of this method in terms of reducing two-qubit gate depth and two-qubit gate count, compared to other implementations.

quant-ph↗

arXiv2018

Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations

The Airy$_β$ point process, $a_i \equiv N^{2/3} (λ_i-2)$, describes the eigenvalues $λ_i$ at the edge of the Gaussian $β$ ensembles of random matrices for large matrix size $N \to \infty$. We study the probability distribution function (PDF) of linear statistics ${\sf L}= \sum_i t φ(t^{-2/3} a_i)$ for large parameter $t$. We show the large deviation forms $\mathbb{E}_{{\rm Airy},β}[\exp(-{\sf L})] \sim \exp(- t^2 Σ[φ])$ and $P({\sf L}) \sim \exp(- t^2 G(L/t^2))$ for the cumulant generating function and the PDF. We obtain the exact rate function $Σ[φ]$ using four apparently different methods (i) the electrostatics of a Coulomb gas (ii) a random Schrödinger problem, i.e. the stochastic Airy operator (iii) a cumulant expansion (iv) a non-local non-linear differential Painlevé type equation. Each method was independently introduced to obtain the lower tail of the KPZ equation. Here we show their equivalence in a more general framework. Our results are obtained for a class of functions $φ$, the monotonous soft walls, containing the monomials $φ(x)=(u+x)_+^γ$ and the exponential $φ(x)=e^{u+x}$ and equivalently describe the response of a Coulomb gas pushed at its edge. The small $u$ behavior of the excess energy $Σ[φ]$ exhibits a change at $γ=3/2$ between a non-perturbative hard wall like regime for $γ<3/2$ (third order free-to-pushed transition) and a perturbative deformation of the edge for $γ>3/2$ (higher order transition). Applications are given, among them: (i) truncated linear statistics such as $\sum_{i=1}^{N_1} a_i$, leading to a formula for the PDF of the ground state energy of $N_1 \gg 1$ noninteracting fermions in a linear plus random potential (ii) $(β-2)/r^2$ interacting spinless fermions in a trap at the edge of a Fermi gas (iii) traces of large powers of random matrices.

cond-mat.stat-mech↗

arXiv2018

Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions

We present a systematic short time expansion for the generating function of the one point height probability distribution for the KPZ equation with droplet initial condition, which goes much beyond previous studies. The expansion is checked against a numerical evaluation of the known exact Fredholm determinant expression. We also obtain the next order term for the Brownian initial condition. Although initially devised for short time, a resummation of the series allows to obtain also the \textit{long time large deviation function}, found to agree with previous works using completely different techniques. Unexpected similarities with stationary large deviations of TASEP with periodic and open boundaries are discussed. Two additional applications are given. (i) Our method is generalized to study the linear statistics of the {Airy point process}, i.e. of the GUE edge eigenvalues. We obtain the generating function of the cumulants of the empirical measure to a high order. The second cumulant is found to match the result in the bulk obtained from the Gaussian free field by Borodin and Ferrari, but we obtain systematic corrections to the Gaussian free field (higher cumulants, expansion towards the edge). This also extends a result of Basor and Widom to a much higher order. We obtain {large deviation functions} for the {Airy point process} for a variety of linear statistics test functions. (ii) We obtain results for the \textit{counting statistics of trapped fermions} at the edge of the Fermi gas in both the high and the low temperature limits.

cond-mat.stat-mech↗

arXiv2018

Large fluctuations of the KPZ equation in a half-space

We investigate the short-time regime of the KPZ equation in $1+1$ dimensions and develop a unifying method to obtain the height distribution in this regime, valid whenever an exact solution exists in the form of a Fredholm Pfaffian or determinant. These include the droplet and stationary initial conditions in full space, previously obtained by a different method. The novel results concern the droplet initial condition in a half space for several Neumann boundary conditions: hard wall, symmetric, and critical. In all cases, the height probability distribution takes the large deviation form $P(H,t) \sim \exp( - Φ(H)/\sqrt{t})$ for small time. We obtain the rate function $Φ(H)$ analytically for the above cases. It has a Gaussian form in the center with asymmetric tails, $|H|^{5/2}$ on the negative side, and $H^{3/2}$ on the positive side. The amplitude of the left tail for the half-space is found to be half the one of the full space. As in the full space case, we find that these left tails remain valid at all times. In addition, we present here (i) a new Fredholm Pfaffian formula for the solution of the hard wall boundary condition and (ii) two Fredholm determinant representations for the solutions of the hard wall and the symmetric boundary respectively.

cond-mat.stat-mech↗

arXiv2018

Coulomb-gas electrostatics controls large fluctuations of the KPZ equation

We establish a large deviation principle for the Kardar-Parisi-Zhang (KPZ) equation, providing precise control over the left tail of the height distribution for narrow wedge initial condition. Our analysis exploits an exact connection between the KPZ one-point distribution and the Airy point process -- an infinite particle Coulomb-gas which arises at the spectral edge in random matrix theory. We develop the large deviation principle for the Airy point process and use it to compute, in a straight-forward and assumption-free manner, the KPZ large deviation rate function in terms of an electrostatic problem (whose solution we evaluate). This method also applies to the half-space KPZ equation, showing that its rate function is half of the full-space rate function. In addition to these long-time estimates, we provide rigorous proof of finite-time tail bounds on the KPZ distribution which demonstrate a crossover between exponential decay with exponent $3$ (in the shallow left tail) to exponent $5/2$ (in the deep left tail). The full-space KPZ rate function agrees with the one computed in Sasorov et al. [J. Stat. Mech, 063203 (2017)] via a WKB approximation analysis of a non-local, non-linear integro-differential equation generalizing Painlevé II which Amir et al. [Comm. Pure Appl. Math. 64, 466 (2011)] related to the KPZ one-point distribution.

cond-mat.stat-mech↗

arXiv2018

Simple derivation of the $(- λH)^{5/2}$ tail for the 1D KPZ equation

We study the long-time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions for the Brownian and droplet initial conditions and present a simple derivation of the tail of the large deviations of the height on the negative side $λH<0$. We show that for both initial conditions, the cumulative distribution functions take a large deviations form, with a tail for $- \tilde s \gg 1$ given by $-\log \mathbb{P}\left(\frac{H}{t}<\tilde{s}\right)=t^2 \frac{4 }{15 π} (-\tilde{s})^{5/2} $. This exact expression was already observed at small time for both initial conditions suggesting that these large deviations remain valid at all times. We present two methods to derive the result (i) long time estimate using a Fredholm determinant formula and (ii) the evaluation of the cumulants of a determinantal point process where the successive cumulants appear to give the successive orders of the large deviation rate function in the large $\tilde s$ expansion. An interpretation in terms of large deviations for trapped fermions at low temperature is also given. In addition, we perform a similar calculation for the KPZ equation in a half-space with a droplet initial condition, and show that the same tail as above arises, with the prefactor $\frac{4}{15π}$ replaced by $\frac{2}{15π}$. Finally, the arguments can be extended to show that this tail holds for all times. This is consistent with the fact that the same tail was obtained previously in the short time limit for the full-space problem.

cond-mat.stat-mech↗

arXiv2017

Exact short-time height distribution in 1D KPZ equation with Brownian initial condition

The early time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension, starting from a Brownian initial condition with a drift $w$, is studied using the exact Fredholm determinant representation. For large drift we recover the exact results for the droplet initial condition, whereas a vanishingly small drift describes the stationary KPZ case, recently studied by weak noise theory (WNT). We show that for short time $t$, the probability distribution $P(H,t)$ of the height $H$ at a given point takes the large deviation form $P(H,t) \sim \exp{\left(-Φ(H)/\sqrt{t} \right)}$. We obtain the exact expressions for the rate function $Φ(H)$ for $H H_{c2}$, which apparently correspond to the symmetric and asymmetric WNT solutions. The rate function $Φ(H)$ is Gaussian in the center, while it has asymmetric tails, $|H|^{5/2}$ on the negative $H$ side and $H^{3/2}$ on the positive $H$ side.

cond-mat.stat-mech↗

arXiv2016

Cold Matter Assembled Atom-by-Atom

The realization of large-scale fully controllable quantum systems is an exciting frontier in modern physical science. We use atom-by-atom assembly to implement a novel platform for the deterministic preparation of regular arrays of individually controlled cold atoms. In our approach, a measurement and feedback procedure eliminates the entropy associated with probabilistic trap occupation and results in defect-free arrays of over 50 atoms in less than 400 ms. The technique is based on fast, real-time control of 100 optical tweezers, which we use to arrange atoms in desired geometric patterns and to maintain these configurations by replacing lost atoms with surplus atoms from a reservoir. This bottom-up approach enables controlled engineering of scalable many-body systems for quantum information processing, quantum simulations, and precision measurements.

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