PHYSICAL REVIEW E 91, 032132 (2015)

Open quantum reaction-diffusion dynamics: Absorbing states and relaxation M erlijn van Horssen* and Juan P. G arrahant School o f Physics and Astronomy, University o f Nottingham, Nottingham NG7 2RD, United Kingdom (Received 7 January 2015; published 20 March 2015) We consider an extension of classical stochastic reaction-diffusion (RD) dynamics to open quantum systems. We study a class of models of hard-core particles on a one-dimensional lattice whose dynamics is generated by a quantum master operator. Particle hopping is coherent while reactions, such as pair annihilation or pair coalescence, are dissipative. These are quantum open generalizations of the A + A 0 and A + A -> A classical RD models. We characterize the relaxation of the state towards the stationary regime via a decomposition of the system Hilbert space into transient and recurrent subspaces. We provide a complete classification of the structure of the recurrent subspace (and the nonequilibrium steady states) in terms of the dark states associated to the quantum master operator and its general spectral properties. We also show that, in one dimension, relaxation towards these absorbing dark states is slower than that predicted by a mean-field analysis due to fluctuation effects, in analogy with what occurs in classical RD systems. Numerical simulations of small systems suggest that the decay of the density in one dimension, in both the open quantum A + A ->■ 0 and A + A -> A systems, behaves asymptotically as t~b with 1/2 < b < 1. DOI: 10.1103/PhysRevE.91.032132

PACS number(s): 05.30.Jp, 03.65.Yz

I. INTRODUCTION N onequilibrium statistical m echanics is a field character­ ized by a rich variety o f dynam ical phenom ena, many o f w hich elude full theoretical understanding. One o f the forem ost the­ oretical challenges is the characterization o f nonequilibrium steady states (N ESS) and the relaxation towards them, w hich rem ains a topic o f current research efforts [1-3]. System s such as particle hopping m odels and directed percolation exhibit phase transitions from fluctuating phases into a particular class o f N ESS, nam ely absorbing states; once reached, such absorbing states cannot be left. This behavior is typical o f reaction-diffusion (RD) m odels [1,4,5], If the diffusive m ixing is not strong enough, asym ptotic decay o f global degrees o f freedom can be slow er than predicted by mean-field approxim ations [1,4,5]. This behavior is explained by fluctua­ tion effects w hich have been confirm ed experim entally [6,7], and this phenom enon continues to be a topic o f current research [8], In this paper we consider dynam ics sim ilar to classical RD m odels, extended to an open quantum spin chain. The theory o f open quantum system s [9,10] is the topic of ongoing theoretical and experim ental research in both quantum optics and cold atom ic system s [11-17]. We consider a class o f one-dim ensional open quantum system s w ith dynam ics analogous to that o f classical RD models: Particle propagation through quantum hopping is coherent, w hile reactions betw een particles are dissipative. We show that, as in the classical RD m odels, these quantum system s exhibit nonexponential decay to absorbing stationary states, w ith m ean-field approxim ations to the dynam ics failing to predict the correct rate o f decay. We consider the algebraic structure o f the system H ilbert space [18] and present a classification o f the dark states w hich generate the absorbing part o f the dynam ics— thus fully describing the stationary states (or N ESS) for the associated

*merlij n.vanhorssen@ nottingham. ac.uk 1j u an.garrahan @nottingham. ac.uk 1539-3755/2015/91(3)/032132(13)

032132-1

quantum m aster operator. We also num erically study the dynam ics o f small system s via quantum ju m p M onte Carlo sim ulations [19] and find evidence for a pow er-law decay of the particle density towards the absorbing state. The power-law exponent appears to be less than that predicted by mean-field analysis and larger than that o f the classical one-dim ensional RD systems. This article is organized as follows. We introduce the quantum reaction-diffusion m odels studied in this paper and discuss their im m ediate properties. We argue that they are sensible open quantum analogs o f classical RD system s by considering conservation o f particle num ber and invariant subspaces and by looking at their quantum ju m p trajectories. This first section concludes with a decom position o f the H ilbert space o f these system s into transient (decaying) and recurrent (absorbing) subspaces. The next section contains our main analytical result as we consider the recurrent subspace in m ore detail. We provide a full classification o f the dark states associated to our quantum RD m odels, and, using spectral properties o f the quantum m aster operator, we argue that these dark states generate the recurrent subspace. We explain the asym ptotic behavior o f the evolution o f the quantum state in term s o f these dark states; we discuss that only a lim ited part o f the recurrent subspace is within reach from an arbitrary initial state. We also provide an argum ent for the equivalence betw een annihilation and coagulation dynam ics in our class o f quantum RD models. T he nontrivial asym ptotic structure o f the state, resulting from a com plex set o f dark states, is a fundam ental feature o f the quantum m odel com pared to classical RD dynam ics. In the final section we characterize the decay o f the density o f particles. We show that a m ean-field approxim ation to the dynam ics, as in the classical RD systems, predicts an asym ptotic t~ ' decay w hich is reaction limited. In contrast, from quantum ju m p M onte Carlo sim ulations on small system s, we find evidence that in dim ension one the dynam ics is instead hopping lim ited (i.e., diffusion lim ited) w ith a pow er-law decay with an exponent sm aller than the m ean-field exponent but larger than that o f the classical RD dynam ics. Published by the American Physical Society

PHYSICAL REVIEW E 91, 032132 (2015)

MERLIJN VAN HORSSEN AND JUAN P. GARRAHAN II. MODELS

In this section we introduce the family of physical models considered in this paper along with a prerequisite mathematical background. We discuss features of the dynamics of the models by considering associated quantum jump trajectories, and we characterize the structure of the system Hilbert space according to conservation of the number of particles. We consider quantum models which are analogous to classical one-dimensional RD models. Each of the models consist of a one-dimensional lattice where a site can be empty, denoted by 0, or occupied by a single particle, denoted by 1. Particles can hop between lattice sites, symbolically (1,0) «-> (0,1). A reaction may only occur when two adjacent sites are occupied. We consider three particular types of reaction: analogous to the classical A + A -*■ 0 reaction, we define pair annihilation, where two neighboring particles annihilate, symbolically denoted by (1,1) —> (0,0). We also define two analogous reactions to the classical A + A —» A reaction: asymmetric coagulation, where two neighboring particles coalesce into one, denoted by (1,1) (1,0), and symmetriccoagulation, where two neighboring particles coalesce in two possible ways, (1,1) — (1,0), (0.1). The quantum nature of the class of analogous models studied in this paper is determined by diffusion due to coherent particle hopping and dissipation due to reactions. With periodic boundary conditions the Hilbert space f t of these models is that of a quantum spin chain of N sites, f t = C2 • • • ® C2. The coherent part of the dynamics is quantum diffusion described by a nearest-neighbour hopping Hamiltonian, H = £2

cr++] +CT,.+a,.+,),

(1)

resolve the QME in a site-basis representation by considering local (two-site) jump operators {LM}. The choice of jump operators {LM} determines the specific RD model. For the annihilation reaction we have one operator for each pair of nearest neighbors, so in one dimension there is one operator per site and the label p coincides with the site label (ann) __

L]

2o

L|ac) = *Jk crjhcr~cr~+].

0

L f c2 = y / ^ j 2 ( J ^ c r i

3,P

=

i [ H, p] +

(LuPL^

(2) We use the notation {% := expfW},^0 for the quantum dynamical semigroup giving the solution to the QME as pit) = T,(po); we denote by X(t) = T*( X q) the corresponding Heisenberg picture time evolution of an observable X. We

±i •

(5)

A. Invariant subspaces

One of the main points of interest of this paper is the time evolution of the density of particles. We define the density operator A to be the global observable A := N ~1 M/> whose expectation value (A(t)) = Tr(Ap(f)) is the density of particles. The operators H, A and the satisfy the algebraic relations [Lf\A] = L * \

(6)

where the superscript (m ) indicates any of the three models (3)—(5) above. The family of jump operators {L^} defines an unraveling of the dynamics associated to the QME (2). The evolution between jumps in the corresponding quantum jump trajectories is governed by the effective Hamiltonian Hcff := H —l £ This operator coincides for the three models and is given by Herr = H

2{Z.|i LM,p}).

7

< ,.

With these definitions these quantum models are intuitively comparable to their classical counterparts. In particular, as we will see below, the QMEs associated to these three models (and therefore the dynamical properties) are equivalent in a well-defined sense.

0

with eigenvectors denoted by cr"|0) = 0, : = £ ( - 1)*2 *7’*_1IV'«+i>-

As we argue in the final section of Appendix B, these are the only dark states in H i. In Tables I and II we summarize our classification of dark states, using the notation for the one-particle eigenvalues and quasimomentum states from Eqs. (11) and (12) and the notation for the two-particle dark states from Eqs. (13) and (14). To argue that there are no dark states to be found in any of the higher sectors, we provide numerical evidence in the remainder of this section. 3. Numerical analysis To further confirm the results summarized in Tables I and II and argue that there are no dark states found in any of the higher subspaces H k, k > 3, we now discuss related numerical results. We approach the problem in two ways: we perform an exhaustive search for dark states within each of the subspaces H k, and we exactly diagonalize the master operator, accounting for all the nondecaying eigenvalues solely using the dark states in Tables I and II. We search for dark states | for N = 4 , . . . ,21. (c) Convergence of the power-law exponent b with increasing system size N [dependence on trajectory length T (arb. units) is also indicated]. With increasing N, oscillations between subsequent values of b become less pronounced, and the value of b in the large system limit appears to stabilize in the range 0.7 < b < 0.9. We conclude that in the large system limit bclass Id < b < bmf, where ^ciass id = 1/2 is the exact classical reaction-diffusion exponent and bmf = 1 is the quantum mean-field prediction for b, respectively (see text).

where a z = 2n — 1. Assuming a homogeneous state, so onesite expectation values are all equal, we arrive at the mean-field equation of motion for the average density «■(«), 3t(n) = —jK (n)2. This equation is readily solved to obtain a power-law decay with exponent —1, . 4=1

respectively. We thus arrive at the expression 1. Odd

N

Let N be odd with N = 2n + 1 and let |^ ) e H 2, then, using the position basis [Tk~l \^i)} introduced above, we may write

//|vp) = E ( 4 - i + 4 3))7,4 - V 2> k= 1

+E E (4+i0+4_,)+4+l)+4 -,'y "1w> /= 3 4=1

1=2 k= 1

where k (^ = (k,k + 1 — l\ty). To determine if I'f') is an eigenvector of H we first consider the action of H on the basis vectors: We find that H\tJ/i) is given by ( T~ l + l ) \ f 3) (7’-1 + l)IVd+i} + (T + l ) \ f i - i ) ( T ~ l + l ) T n+' \ f n+l) + ( T + 1)| f n)

+ E (4+, +4° + 4-+4 +4-t-i)^-1\fn+,>. 4=1

Equating coefficients in the eigenvalue equation | vp) = c | 4Q, we obtain the system of equations for k = 1 , . . . , N ,

i f / = 2, if 3 < / < n, if l = n + 1. (B l)

,(/-D

S t+ i

Using translation invariance of H we obtain the expression

A4+l

H \ V) = J 2 ^ T k~' H \ f 2) + J 2 J 2 X+E(44-, + 4 V _,ito> k=\

1=2 4=1

4=1

+ E

E

(4 + i15 + 4 /_1) + 4 /+ ,)+ 4 - , 1

1=4 4=1

= E E ^ ^ ' i w + E 4 " H)7’t - , ivvn>. 4=1

where we have isolated the / = n + 1 term since the corre­ sponding coefficients are periodic in n rather than N. We first consider the action of H on the generating vectors |ij/i)\ in this case H\i//i) is given by

+ J 2 ( Xt ' ) + Xk ~ ' ) T k- ' \ f n ) k= 1

( T - ' + 1 )|^ 3 >

( T - 1 + l)\irl+l) + (T + l ) \ ^ ) +

E ( 4 + i + 4 " y * _I i ^ + i > 4=1

( Tn + l)(T+l )\ x/ ,n) 032132-11

i f / = 2, if3 < / < n , if/=n+l.

(B2)

PHYSICAL REVIEW E 91, 032132 (2015)

MERLIJN VAN HORSSEN AND JUAN P. GARRAHAN

We solve for the ground states of H first by setting c = 0. Recalling the dark state condition x f ] = 0, the system of ground-state equations simplifies as follows: For A = 1 , . . . , N

We use these expressions to find a suitable expression for / / 1xj/); we find that

Hm

= J 2 ( xk- i + kk )) Tk~l \ M

4 ° + 4 + i = 0,

A=1

/ = 3 , . . . ,n

and for A = l , . . . ,n

+E E /=3 * = ]

+4M)+4,+l)+4&V"11*>

+E 4 k=\

(n) I i («) * + l + AA + 4 + n + 1 + 4 + n ) T k

4 " +1)+ 4++i') = o. From these simple equations we obtain, for / = 3 4 Z) = (-1 f a t for some a/ 6 C while periodicity in n of X^+i)

' I^ « +1) •

means that 4 " +1> = 0 if n is odd but 4 " +l> = (—l)*fln+i if n is even. We conclude that the two-particle ground states of H which are also dark states are linear combinations of the states |«t>3) ____ !„) and, if n is even, |%+i> defined in Eqs. (13) and (14). In the remainder of this appendix we argue why the ground states of H found above are the only dark states in Hi. We consider the eigenstates of H in H 2 corresponding to nonzero eigenvalues: If c ^ 0 the system of eigenvalue equations, written out explicitly, reads

Using this expression and equating coefficients in / / 1xl>) = c|4

4,;,1)+4,_,)+4,+1)+4,-1')=c4).

/ = 3,

and for A- = 1 ,... ,n (n) (n + l) («) i («) A*+i +4°+ ak+n+\ H- Ak-\-n = cXk J

4 4) + 4 - >

(l — 4)

4 5) +

4 - ,

4

4

(1 = 5)

4

+

i +

4

4) +

6) +

- i

=

0, ii Cj >-* *-3

(1 = 3)

4 - i

II

4 3) +

S -*