Dynamics of non-equilibrium steady state quantum phase transitions

Dynamics of non-equilibrium steady state quantum phase transitions

Patrik Hedvall    Jonas Larson Department of Physics, Stockholm University, AlbaNova University Center, Se-106 91 Stockholm, Sweden
July 9, 2019
Abstract

In this paper we address the question how the Kibble-Zurek mechanism, which describes the formation of topological defects in quantum systems subjected to a quench across a critical point, is generalized to the same scenario but for driven-dissipative quantum critical systems. In these out of equilibrium systems, the critical behavior is manifested in the steady states rather than in the ground states as in quantum critical models of closed systems. To give the generalization we need to establish what is meant by adiabaticity in open quantum systems. Another most crucial concept that we clarify is the characterization of non-adiabatic excitations. It is clear what these are for a Hamiltonian systems, but question is more subtle for driven-dissipative systems. In particular, the important observation is that the instantaneous steady states serve as reference states. From these the excitations can then be extracted by comparing the distance from the evolved state to the instantaneous steady state. With these issues resolved we demonstrate the applicability of the generalized Kibble-Zurek mechanism to an open Landau-Zener problem and show universal Kibble-Zurek scaling for an open transverse Ising model. Thus, our results support any assumption that non-equilibrium quantum critical behavior can be understood from universal features.

pacs:
05.30.Rt, 03.75.Gg, 03.75.Kk

I Introduction

The theory for describing systems at equilibrium, and especially what drives transitions between different phases, is well developed and the picture rather complete, whether it is for finite temperature (thermal phase transition (PT)) Goldenfeld (1992), or at zero temperature (quantum phase transition (QPT)) Sachdev (2007). A great leap in our understanding was taken with Landau’s mean-field theory of phase transition, which in particular tells that a continuous (i.e. second order) PT can be ascribed a spontaneous breaking of some symmetry Landau et al. (1937). The type of symmetry being broken also predicts what kind of excitations one can predict in the symmetry broken phase, so called Higgs or Goldstone modes representing respectively massive or massless excitations Altland and Simons (2010). The non-analytic behavior at a quantum critical point implies that the spectrum is necessarily gapless at this point, and more importantly the physics in the vicinity of the critical point, for example how the gap closes, is universal. This means that microscopic details become irrelevant and the physics can be described by a set of critical exponents that only depend on global symmetries and dimensions Goldenfeld (1992). The fact that the gap closes means that the typical time-scale (inverse of the gap) diverges, what is called critical slowing down. An interesting and important result of critical slowing down is that if we drive the system through a critical point, by externally varying some system parameter, no matter how slow this quench is the system will always get excited due to non-adiabatic contributions. Kibble and independently Zurek suggested that the density of these generated defects also behaves universal – it can be estimated from some critical exponents Kibble (1976); Zurek (1985). While originally for thermal PT’s, the idea of Kibble and Zurek can be applied to QPT’s as well Zurek et al. (2005). This idea is simple, the evolution of the quench can be divided into being either adiabatic (typically when the gap is large) or diabatic (in the vicinity of the critical point where the gap closes). This has been termed the Kibble-Zurek mechanism (KZM), and its applicability has been demonstrated in numerous experiments by now Bowick et al. (1994); Ruutu et al. (1996); Ducci et al. (1999); Monaco et al. (2002); Maniv et al. (2003); Weiler et al. (2008); Bakr et al. (2010); Chen et al. (2011); Braun et al. (2015); Ulm et al. (2013); Pyka et al. (2013); Mielenz et al. (2013).

Lately a new type of critical behavior in non-equilibrium systems has gained much attention thanks to great experimental advances especially in the AMO (atomic, molecular and optical) community. These are driven-dissipative systems, in which, in principle, both the drive and the sort of dissipation can be tailored Diehl et al. (2008); Verstraete et al. (2009a); Eisert and Prosen (2010); Hoening et al. (2012). The system relaxes to some steady state that will not be an equilibrium one, i.e. a non-equilibrium steady state or short NESS. Criticality in these models are correspondingly defined via non-analyticities in the system’s steady state upon varying some model parameter, similar to how the ground state shows non-analytic behavior at the critical point for a QPT. For AMO experiments, the evolution of the system in contact with an environment can often be accurately described by a Markovian Lindblad master equation Gardiner and Zoller (2015); Breuer and Petruccione (2002)

(1)

The influence of the environment is included in the last term containing a sum over possible Lindblad jump operators , and where the ’s are the corresponding decay rates. When tailoring the system-environment couplings these are constructed with the purpose of reaching a desired steady state of Eq. (1). Typically the jump operators are local but in principle need not be so Schneider and Milburn (2002); Hannukainen and Larson (2017). For to be non-analytic in the thermodynamic limit, the gap in the spectrum, now of the Liouvillian , must close at the critical point Kessler et al. (2012). Since the time-evolution governed by is in general not unitary, the spectrum is normally complex with the real parts representing relaxation to the NESS’s Albert and Jiang (2014); Albert et al. (2016).

It is clearly so that our understanding of NESS critical behavior is far less developed than what it is for thermal or quantum PT’s. It has been shown, for example, that new universality classes are possible in these systems Sieberer et al. (2013); Marino and Diehl (2016); Zamora et al. (2017), as is the presence of continuous PT’s lacking any spontaneous symmetry breaking Hannukainen and Larson (2017). Hence, qualitative difference between equilibrium and non-equilibrium QPT’s may indeed exist. A very famous example from classical physics is the ‘flocking transition’ which describes build-up of long-range order in for example flocks of birds Vicsek et al. (1995). At equilibrium, long-range order is prohibited by the Mermin-Wagner theorem Auerbach (2012), but the theorem does not apply for non-equilibrium situations and thereby the possibility for the birds to order.

In the present paper we analyze how NESS critical systems, described by some Lindblad master equations, respond to slow quenches across a critical point. As pointed out above, for equilibrium systems we know that under rather general circumstances the KZM accurately explains such scenarios. Thus, the corresponding question is whether some KZM can be applied also for these models? We will see that there is indeed a generalization of the traditional KZM to open systems, but much care needs to be taken into account to settle this correspondence. For example, how the concept of adiabaticity is translated to open quantum systems, and more importantly how to quantify the amount of non-adiabatic excitations generated during the quench. It is clear that since the criticality manifests in the system’s steady state, this state should be the reference when determining the amount of excitations. We will argue that the natural measure for the amount of excitations is the trace distance for density operators.

It is in general harder to solve dynamical than statical or equilibrium problems, and in particular it is more difficult to simulate the evolution governed by a Lindblad master equation than that deriving from a Hamiltonian. This is easily understood due to the simple observation that one needs many more values to describe a general state than a pure state . Indeed, for a Hilbert space dimension , the number of eigenstates of is to be compared to eigenstates of . A zero eigenvalue eigenstate is obviously a steady state of the Liouvillian. Adiabatic evolution should imply that a system initialized in some steady state at time remains in the instantaneous steady state throughout the duration of the quench. In the past, several works have studied how an environment affects the KZ-type quench through a quantum critical point Fubini et al. (2007); Cincio et al. (2009); Dutta et al. (2016); Dziarmaga (2006). This is a conceptually different question than the one we address: In our setting we cannot in general characterize the system from some underlying Hamiltonian – it is truly the system plus environment that determines the properties of the state . In the Refs. Fubini et al. (2007); Cincio et al. (2009); Dutta et al. (2016); Dziarmaga (2006) the environment serves mainly as an additional source of fluctuations, and the general finding is that the amount of excitations created when quenched through the (Hamiltonian) critical point is increased. It is, in particular, assumed that the presence of an environment will not qualitatively alter the critical behavior of the Hamiltonian. Moreover, excitations are measured with respect to the Hamiltonian ground state, and not from the steady state as in the present work. Another crucial observation is that when the ground state is the reference state, as in the mentioned references, the amount of excitations will not, in a strict sense, only depend on the quench rate and critical exponents but also on the initial and final times, and . This is irrespective of whether the quench end close or far from a critical point, since even far from the critical point, where the evolution is presumably adiabatic, there is an environment induced relaxation of the system towards some steady state. In our analysis of quenches through NESS critical points, the results do not depend on and , and thereby only rely on system parameters and properties. In the respect, the approach we take is more appropriate when the goal is to explore universal properties of the critical behavior.

The paper is organized as follows. In the following section we summarize some earlier results such that they can be put in context with ours, and we also classify different scenarios that can emerge in NESS critical systems. More precisely; in Sec. II.1 we reproduce the arguments (so called ‘adiabatic-impulse approximation’) for the KZM when applied to quantum critical points of closed systems, then in Sec. II.2 we continue with defining NESS criticality and introduce different classes arising from the competing terms in the Lindblad master equation, and in Sec. II.3 we review earlier works related to our results. Section III presents the general results. We start in Sec. III.1 by introducing the Bloch equations of Lindblad master equations and discuss their structure in rather general terms. This allows us to explain what is meant with adiabaticity for open quantum systems in Sec. III.2. To use the trace distance as a measure of excitations is argued for in Sec. III.2 by demonstrating that it correctly predicts the amount of excitations in the limiting situation of a closed quantum system. In Sec. III.4 we generalize the adiabatic-impulse approximation to our Bloch equations and thereby derive the KZM for open quantum systems. The proceeding Sec. IV is devoted to two examples, the open Landau-Zener model in Sec. IV.1 and the dephasing transverse Ising model in Sec. IV.2. The two examples verify the applicability of our general results from Sec. III. We conclude in Sec. V with a summary and possible future continuations.

Ii Preliminaries and earlier results

ii.1 Dynamics of quantum phase transitions

In this Subsection we reproduce the idea of the adiabatic-impulse (AI) approximation, and how the KZM is applied to QPT’s Zurek et al. (2005). A QPT occurs at zero temperature and result from quantum fluctuations Sachdev (2007), contrary to classical PT’s which are driven by thermal fluctuations. Typically for a system showing critical behavior, its Hamiltonian can be decomposed as

(2)

with and is a coupling parameter determining the relative strength between the two terms. For , the ground state is governed by , while in the opposite limit, , it is dictated by . The characteristics of the ground state may be very distinct in the two limits, and in particular in the thermodynamic limit it may become non-analytic for some finite critical coupling . When the transition is continuous this marks the critical point. The non-analytic behavior necessarily implies that the spectrum of is gapless at , i.e. the energy gap from the ground state . In particular, for the ground state is non-degenerate and the spectrum gapped (normal phase), while the ground state is degenerate for (symmetry broken phase) and the spectrum may either be gapped or not. At the critical point the ground state cannot be said to be ruled by either or , and furthermore we cannot assign any finite characteristic length scale to the state. In the vicinity of the critical point, the length scale diverges algebraically Goldenfeld (1992)

(3)

for some universal critical exponent that does not depend on microscopic details but only on global symmetries and dimensions. The inverse of the gap, , sets an intrinsic time-scale for how fast the system responds to small parameter changes. Since the behavior of the gap closing is universal we must have that this time-scale also diverges with some dynamical exponent defined as

(4)
Figure 1: The idea behind the AI-approximation which forms the basis for the KZM applied to QPT’s. Far from the critical point, the energy gap is large such that the reaction time (marked by solid black lines) is small, i.e. the system responds fast to any external changes. In this adiabatic regime the inverse transition rate (dashed red lines), giving the time-scale for the external driving, is much larger than the reaction time. As the critical point is approached, the reaction time becomes longer and at the freeze-out time the system cannot stay adiabatic and it enters the impulse regime. Here the evolution is assumed frozen, i.e. diabatic. After traversing the critical point, the system can reenter into an adiabatic regime at the second freeze-out time .

We assume now that the system is quenched through the critical point such that Zurek et al. (2005)

(5)

where is the ‘quench rate’, i.e. it sets the rate of change of the Hamiltonian. At the time we are exactly at the critical point, and we assume that for negative times the instantaneous ground state is the gapped normal phase. Even though the energy gap closes at the critical point, we may assume that sufficiently far from the critical point the evolution is adiabatic. As we move closer to the critical point, non-adiabatic excitations cannot be overlooked. While the Hamiltonian changes smoothly, in the AI-approximation one assumes that there is a certain time for which the system swaps from following the quench adiabatically to diabatically. Thus, after this freeze-out time the population transfer is hindered – the evolution is frozen. The design is thereby such that the quench can be split into an adiabatic regime followed by an impulse regime – the AI-approximation. Within the AI-approximation, the problem becomes essentially one of determining . The breakdown of adiabaticity should occur in the vicinity of the point when the rate of change equals the response time . Using (4) and the explicit form of the quench (5), one finds

(6)

The validity of the AI-approximation hinges on that the window where the evolution is neither adiabatic nor diabatic is narrow. Typically, the quench rate should not be too small for good predictions from the KZM, since then this window is not narrow any more.

Up to the instant we have assumed adiabatic following, and thereafter frozen evolution. The quench thereby imprints a characteristic length scale

(7)

to the system state. At the system is still approximately in its instantaneous ground state, but this is not the ground state at later times meaning that the system gets excited as time progresses. These are the non-adiabatic excitations. Depending on the dimensionality and the symmetries, the excitations can have different character like domain walls/kinks, or vortices. Hence, local topological defects in the otherwise uniform ground state. The length scale (7) determines the density of such defects, i.e.

(8)

where is the dimension.

The KZM predicts universal behavior away from equilibrium, the scaling of the defects is determined by the critical exponents and  Del Campo and Zurek (2014). For a continuous PT that breaks a discrete symmetry, in the symmetry broken phase the ground state is degenerate but there is a gap to higher excited states (Higgs mode) Altland and Simons (2010). The adiabatic-impulse scenario is then that you cross from adiabatic to diabatic at , and then back to adiabatic at , see Fig. 1. The imprinted excitations are still the same in this scheme. We may note that nothing restricts us to linear quenches (5), and neither to the case where the two freeze-out times are symmetric, i.e. and , which could for example occur when the critical exponents are different above and below the critical point Liu et al. (2009); Mumford et al. (2015). Furthermore, the transition does not have to be a proper continuous PT, but can be a crossover that appears in finite systems Zurek et al. (2005); Dziarmaga (2005); Cincio et al. (2007); De Grandi et al. (2010). Indeed, the KZM can successfully be applied also to avoided crossing models Damski (2005); Damski and Zurek (2006), which ca be seen as an approximation of a QPT in a finite system Zurek et al. (2005).

Figure 2: Adiabatic (solid black lines) and diabatic (dashed red lines) energies of the LZ model (9) with . The slope of the curves away from the crossing is determined by the velocity , while the gap at is . A large coupling thus implies a large gap which favors adiabaticity, and likewise a small means a gradual change and more adiabatic evolution.

In Refs. Damski (2005); Damski and Zurek (2006) Damski implemented the KZM on the solvable LZ problem Zener (1932); Landau (1932) that describes an avoided level crossing. The LZ Hamiltonian is given by ()

(9)

such that the quench rate , and is the coupling between the two diabatic states and . If we assume that the system starts in its ground state for , then the probability that it ends up in the excited state at is given by the LZ formula Zener (1932); Landau (1932)

(10)

The adiabaticity parameter determines how adiabatic the process is; the larger it is the more adiabatic quench. The instantaneous eigenvalues of (adiabatic energies), as well as the ‘bare energies’ (diabatic energies) are displayed in Fig. 2. Damski found that the KZM works very well to reproduce the correct analytical result (10) for the excitations. The agreement was particularly good for fast quenches, i.e. in the non strictly adiabatic regime Hwang et al. (2015). Naturally, there is no length scale in the LZ problem, and instead the amount of excitations replaces the density of defects (8).

ii.2 NESS criticality and the Lindblad master equation

The conventional wisdom is that dissipation and/or decoherence decimate quantum properties Schlosshauer (2007). This is, for example, the main hindrance when it comes to quantum information processing. One possibility to circumvent this problem for quantum computing is to employ decoherence-free subspaces Lidar et al. (1998), which uses states that are insensible for the decoherence. Relatedly, one could imagine to monitor the dissipation/decoherence channels such that the system approaches a steady state manifold where the computations take place Kraus et al. (2008); Diehl et al. (2008); Verstraete et al. (2009b). Typically, these are controlled driven-dissipative systems with desirable non-equilibrium steady states .

Naturally, the NESS may also show non-analytic behavior in some proper thermodynamic limit, just like equilibrium states can become critical Diehl et al. (2010); Eisert and Prosen (2010); Tomadin et al. (2010); Hoening et al. (2012); Sieberer et al. (2013). As manifestly non-equilibrium, these NESS phase transitions need not necessarily obey laws applicable to equilibrium PT’s like the Mermin-Wagner theorem Auerbach (2012). There are three possible scenarios we can imagine:

  1. The model is critical at the closed level, i.e. the Hamiltonian supports different phases. We exclude here the possibility of new phases appearing due to the openness, see below. The questions are, how does the inclusion of dissipation/decoherence (noise) affect the types of transitions and the properties of the phases? It has been demonstrated that the presence of environmental noise can indeed change the critical exponents of the PT Nagy et al. (2011); Baumann et al. (2011), or even lead to new universality classes Sieberer et al. (2013); Marino and Diehl (2016); Kardar et al. (1986); Zamora et al. (2017), as well as altering the properties of the phases Joshi et al. (2013); Ludwig and Marquardt (2013); Altman et al. (2015). In certain cases, the transition becomes classical and the fluctuations induced by the environment can be described as an effective temperature Mitra et al. (2006); Diehl et al. (2008); Dalla Torre et al. (2010); Sieberer et al. (2013); Kessler et al. (2012), which may prohibit build-up of long-range order in lower dimensions according to the Mermin-Wagner theorem. In realistic experimental settings considering finite systems, the length scale may, however, be larger than the system size and the characteristics of the PT could in these cases still be accessible Altman et al. (2015); Dagvadorj et al. (2015).

  2. Much less studied is the situation where the criticality emerges as an interplay between different dissipative channels, and not due to the Hamiltonian. Then, depending on which channel that dominates the state approaches different steady states. Verstrate et al. showed that it is possible to perform quantum computing tasks using solely dissipative evolution Verstraete et al. (2009b). The target state is then a steady state and the dissipation is monitored in such a way that the desired target state is reached.

  3. The last scenario is when the criticality is only possible in the presence of both a Hamiltonian and dissipative channels Diehl et al. (2008, 2010); Eisert and Prosen (2010); Hoening et al. (2012); Hannukainen and Larson (2017); Carmichael (2015). Here, the emerging phases are typically resulting from either dissipation or unitary Hamiltonian evolution. The separate parts alone, Hamiltonian or dissipator, may well be trivial. It was also recently shown that the criticality in these systems can be conceptually different from equilibrium situations where a continuous PT is generally accompanied by a spontaneous symmetry breaking Hannukainen and Larson (2017).

AMO systems have the advantage that they can be fairly versatile and offer high controllability Bloch et al. (2008); Cirac and Zoller (2012) in comparison to other material systems. The atoms or ions are manipulated with optical light, and controlled dissipation is accomplished by combination of excitations and spontaneous emission. In these optical settings, a Born-Markovian master equation is typically applicable Gardiner and Zoller (2015). Under such circumstances and invoking the secular (also called rotating wave) approximation, the system is described by the Lindblad master equation of Eq. (1Breuer and Petruccione (2002). The first commutator term on the right hand side of Eq. (1) comprises the Hamiltonian evolution, with the notification that in principle also includes Lamb shifts steaming from coupling with the environment. The second term accounts for the coupling to the environment, where are decay rates or coupling strengths, and are the Lindblad jump operators. We have also defined the Liouvillian which is the total generator of the time-evolution. In what follows we will always assume that the system evolution is generated by a master equation on the form (1).

For our purpose when studying NESS criticality, the steady state

(11)

is the main objective, i.e. the kernel of the Liouvillian or the eigenstate with zero eigenvalue. It serves the same role as what the ground state does in equilibrium QPT’s. The manifold of steady states forms a connected convex set. If the Liouvillian is time-independent, any initial state approaches a steady state in the infinite time limit Alicki and Lendi (); Rivas and Huelga (2012). Thus, if the steady state is unique it is attractive in the meaning that any initial state will eventually path its way to it. Attractiveness implies robustness – if for some reason the steady state is perturbed, the dynamics will bring it back. For systems with more than one steady state, attractiveness is lost as you can be brought back to another steady state. As will become clear, this observation is very important for our work. The uniqueness of steady states has been explored in the past Spohn (1977); Schirmer and Wang (2010); Frigerio (1978); Fagnola and Rebolledo (2001, 2002); Baumgartner and Narnhofer (2008). To say something general about whether the steady state will be unique or not is not trivial apart from special cases. For example, if the only operator commuting with all jump operators is the identity then the steady state is unique Spohn (1977); Schirmer and Wang (2010). It can be proven that there must exist at least one steady state Rivas and Huelga (2012). Reversely, given a pure state it is always possible to construct a Liouvillian with at least one jump operator that has this pure state as unique steady state. If the steady state is unique the evolution is ofter called relaxing.

A general Liouvillian eigenstate has a complex eigenvalue , i.e. , such that time-evolution results in , or that we can expand a general state as

(12)

The eigenstates may not be physical states, i.e. positive semi-definite, and the set is over-complete. Of course, Liouvillian evolution preserves positivity and norm of the state, but nevertheless the sum (12) may well contain unphysical states even though is physical. In fact, the eigenvalues/eigenstates typically come in complex/hermitian conjugated pairs which assures that the expansion (12) always exists for any physical state.

It follows that the eigenvalues must obey  Kessler et al. (2012). The quantity

(13)

with the minimum taken over all eigenvalues with non-zero real parts, defines the Liouvillian gap that sets the inverse time-scale for reaching the steady state Albert and Jiang (2014); Albert et al. (2016). The subscript is introduced to distinguish this gap from the Hamiltonian energy gap . In order for to become non-analytic in the thermodynamic limit, and thereby allow for critical behavior, one must have that  Kessler et al. (2012), analogously to the gap closening for equilibrium QPT’s. Scaling (i.e. universality) of is not known, and, in general, it seems that the mechanisms behind NESS QPT’s can be qualitatively different from those of equilibrium QPT’s Hannukainen and Larson (2017).

In order to discuss various scenarios for different Liouvillians, and thereby try to classify them, let us assume that the sum in (1) is restricted to a single term, i.e. we have just a single jump operator . This is, of course, a special case of the general situation but, as we will see, it can be a relevant case for engineered driven-dissipative systems. Even when confine the discussion to this special case, it will provide insight also for the general cases. We note also that for several jump operators , the decomposition of the Liouvillian is known to be not unique Baumgartner et al. (2008), which makes a general classification much more complicated. For single jump operators we can construct the following four classes:

Class I

Energy dephasing. The jump operator is hermitian, , and commutes with the Hamiltonian, . Any energy eigenstate will also be a steady state, and so will any mixed state , i.e. . Thus, the steady states are diagonal in the energy eigenbasis . For any initial state, evolution will cause to become diagonal in the energy eigenbasis without dissipation of energy, but an increase of entropy Nielsen and Chuang (2010). The probability distribution determines the steady state, and, in particular, this exemplify how the manifold of steady states is convex and simply connected.

Class II

General dephasing. The jump operator is hermitian, , and does not commute with the Hamiltonian, . The maximally mixed state , with the hilbert space dimension, is a trivial steady state in this class. In many physically relevant situations, the maxiamally mixed state is also the unique steady state. With the eigenstates of , any diagonal state is transparent to the environment, but the Hamiltonian will drive you out of this manifold which implies relaxation to some true steady state.

Class III

Dissipation I. The jump operator is non-hermitian, , but commutes with the Hamiltonian, . This class is probably the least physically relevant; we know of no non-trivial physical scenarios were it occurs. A trivial situation is that the jump operator acts in a space with only degenerate energy eigenstates, e.g. .

Class IV

Dissipation II. The jump operator is non-hermitian, , and does not commute with the Hamiltonian, . Non-hermitian operators typically appear in cases of spontaneous decay (e.g. for decay of photons/phonons and for spontaneous decay of the upper level of a two-level system) or for incoherent pumping (e.g. for the situation of a single boson mode). A dark state is a pure state that is transparent to the dissipation/decoherence, e.g.  Arimondo (1996). It is clear that if further is an eigenstate of the Hamiltonian, then is also a steady state. The most common example being a system coupled to a zero temperature bath which cools the system down to its ground state. More generally, if the thermal bath is at some non-zero temperature, detail balance between incoherent loss and gain of particles leads to a thermal steady state provided the system is not driven.

As a final remark on properties of steady states, it is rather straight forward to show Dietz (2003) that one steady state solution falling in the classes I and III (and possibly also in the other classes in certain cases) is

(14)

where .

ii.3 Earlier studies

ii.3.1 Dynamics across equilibrium quantum critical points

For thermal PT’s, the KZM have been well explored both numerically Laguna and Zurek (1997); Yates and Zurek (1998); Hindmarsh and Rajantie (2000) and experimentally Bowick et al. (1994); Ruutu et al. (1996); Ducci et al. (1999); Monaco et al. (2002); Maniv et al. (2003); Weiler et al. (2008). It is only recently, however, that the dynamics emerging from quenches through quantum critical points has been investigated Zurek et al. (2005); Dziarmaga (2005); Damski (2005); Polkovnikov (2005); Schützhold et al. (2006); Cherng and Levitov (2006); Cincio et al. (2007); Cucchietti et al. (2007). Laser cooling and manipulation of atoms or ions have opened up a new avenue for studying quantum many-body systems out of equilibrium Bloch et al. (2008). Especially the high control of system parameters make these systems good candidates for exploring critical dynamics like the KZM and its predicted scaling Dziarmaga and Rams (2010); Del Campo and Zurek (2014).

The KZM was theoretically studied for the superfluid-to-Mott insulator transition of an ultracold atomic gas in an optical lattice Schützhold et al. (2006); Cucchietti et al. (2007), and a KZ scaling was suggested. Quenching across this transition has also been studied in several experiments  Bakr et al. (2010); Chen et al. (2011); Braun et al. (2015). In the first experiment the KZ scaling was however not analyzed. In Ref. Chen et al. (2011) the scaling after quenching from the Mott to the superfluid was mapped out, but deviations from simple theory was found which was explained as a result of the inhomogeneity of the atomic cloud Dziarmaga and Rams (2010); Bernier et al. (2011); Dziarmaga and Zurek (2014). It is also known that the KZM can fail as the quench becomes too slow, where instead different scaling exponents can be predicted from adiabatic perturbation theory Hwang et al. (2015). The behavior after quenching across a PT in a spinor BEC has also been experimentally explored Sadler et al. (2006); Nicklas et al. (2015); Anquez et al. (2016), where indeed scaling in agreement with theory was found Anquez et al. (2016).

For trapped ions, it was suggested that the KZM could be ideally studied in the so called ‘zig-zag’ transition in which a linear chain of trapped ions reorganize to form a zig-zag structure as the confining trapping is weakend Del Campo et al. (2010); De Chiara et al. (2010). Experiments following the theory proposals confirmed the KZM scaling with very good accuracy Ulm et al. (2013); Pyka et al. (2013); Mielenz et al. (2013).

ii.3.2 Influence of dissipation/decoherence

For open quantum systems the focus has been on how decoherence/dissipation affects the creation of defects during the quench. The rule of thumb is that fluctuations stemming from any environment will increase the amount of defects Fubini et al. (2007); Cincio et al. (2009); Dutta et al. (2016). For the random Ising model, the characteristics may also be qualitatively different from those predicted by the KZM for closed systems Dziarmaga (2006). In particular, even at infinitely slow quenches the density of defects is non-vanishing also for finite systems where the gap is always non-zero. In Refs. Fubini et al. (2007); Dutta et al. (2016), studying a system belonging to Class II in the classification above, it was especially found that as the quench became slower the number of excitations increased, something that was termed ‘anti-Kibble-Zurek mechanism’. Such a behavior is explained from the fact that a slower process implies a longer duration for which the system interacts with its environment, such that the environment induced excitations become essential. The same phenomenon is known both in ‘quantum coherent control’ Ivanov et al. (2004); Mathisen and Larson (2016) and adiabatic quantum computing Sarandy and Lidar (2005a), i.e. there is an optimal process time for minimizing the amount of excitations.

ii.3.3 Preparation of NESS’s

Using environments and dissipation as resources have been especially discussed in the realm of quantum information processing. The Liouvillian gap (13) plays the role for open systems as what the energy gap (above the ground state) does for Hamiltonians. For a large gap, the steady state or ground state is robust to external fluctuations. However, steady states are not only protected against external imperfections due to the large size of the gap, the gap also implies that if you are taken out of the steady state manifold you will relax back to it. If your desired steady state is unique you are guaranteed to be projected back onto it if the disturbances are temporary.

For quantum information processing the natural candidate for steady states are non-classical entangled states. Several proposals have been put forward how dissipation can be harnessed for preparation of entangled states, e.g. Bell states Clark et al. (2003); Kraus and Cirac (2004); Paternostro et al. (2004); Cho et al. (2011); Kastoryano et al. (2011). It has also been experimentally demonstrated when it comes to the creation of Bell states of trapped ions, either via discrete gates Barreiro et al. (2011) or continuous relaxation Lin et al. (2013), or to entanglement between macroscopic atomic clouds Krauter et al. (2011). Extending the schemes for the generation of multi-qubit entangled states have also been suggested Kraus et al. (2008) and experimentally verified Barreiro et al. (2010); Schindler et al. (2013), or other exotic many-body quantum states Diehl et al. (2008); Witthaut et al. (2008); Diehl et al. (2010, 2011); Müller et al. (2012); Bardyn et al. (2013).

As discussed already in the previous Subsection, in driven systems dissipation may cause critical behavior of the steady states Diehl et al. (2008, 2010); Eisert and Prosen (2010); Hoening et al. (2012); Sieberer et al. (2013); Eisert et al. (2014); Hannukainen and Larson (2017). Criticality in driven-dissipative systems has recently been experimentally explored in quantum optical systems like the Dicke ‘normal-superradiance’ PT Baumann et al. (20010, 2011); Klinder et al. (2015). Other examples of steady state criticality are ‘optical bistability’ Bonifacio and Lugiato (1978); Drummond and Walls (1980); Fink et al. (2017) and the onset of lasing in the laser DeGiorgio and Scully (1970); Rice and Carmichael (1994); Haken (2012).

Iii General description of the Kibble-Zurek mechanism for NESS phase transitions

iii.1 The Bloch representation for Lindblad master equations

The KZM relies on the AI-approximation, and while it is clear what adiabatic vs. diabatic (impulse) evolutions mean for closed quantum systems it is not completely evident what it implies for open quantum systems. We therefor must define what we mean by adiabaticity for open quantum systems. As we will see there are various approaches one may take, but before that we need to say something about the Liouvillian of Eq. (1).

The Lindblad master equation (1) is linear, i.e. . If we parametrize the density operator and represent it as some vector, the linearity implies that the Lindblad equation is cast in a simple matrix form. There are different choices for how to parametrize , but here we employ one related to the Bloch vector representation. Given that the Hilbert space dimension is , any density operator obeying positivity and normalization can be expressed as Kimura (2003); Bertlmann and Krammer (2008)

(15)

Here, is the generalized (real) Bloch vector and the vector is composed of the generalized Gell-Mann matrices Hioe and Eberly (1981). The matrices are generators of the Lie algebra corresponding to the group , and in particular they are traceless, hermitian, and mutually orthogonal such that given any density operator its Bloch vector is obtained from . For (qubit) and (qutritt) the matrices are the standard Pauli and Gell-Mann matrices respectively. The Bloch vector length such that the state space can be represented by a ‘Bloch hyper-sphere’. However, whenever not all points in the Bloch sphere represent a physical state Kimura (2003); Goyal et al. (2016). The larger dimension, the sparser the Bloch sphere is in terms of physical states. The non-physical states are not proper density matrices as they are not positive semi-definite.

In the Bloch representation we have parametrized the density operator in terms of , and the Lindblad master equation is given by Schirmer and Wang (2010)

(16)

The Liouvillian matrix is of dimension and in general not hermitian. In fact, for a closed system is skew-symmetric. For now, let us assume that both and are time-independent. The term is a column vector of elements and represents some sort of pumping that prevents the state to be a trivial steady state. The general steady state is given by , or if is invertible . If is not invertible the system of equations is under-determined and the steady state need not be unique which will give more interesting situations when discussing the KZM for NESS critical models. When and is not invertible we note that the steady state defines a connected manifold of steady states due to the ambiguity of the norm of . The continuity and linearity of (16) warrant that there must exist at least one steady state (or fixed point) Schirmer and Wang (2010). In the general case of a non-vanishing pump term , by solving the homogeneous equation the solution of the inhomogeneous problem is . Or if we introduce the matrices and that diagonalizes the Liouvillian matrix, i.e. with diagonal, then the right eigenvectors of evolve as where is the corresponding eigenvalue of . Thus, the eigenvalues of the Liouvillian matrix determines the characteristic time-scales, and note that we must have in order to preserve normalization. The Liouvillian gap is defined as before in Eq. (13). Provided that the Liouvillian matrix is diagonalizable (for example if it is normal, , as will be relevant for us in the following Section), its eigenvectors (or if ) of form an over-complete basis, i.e. they are not linearly independent. Furthermore, given one eigenvector, its corresponding density operator need not be physical. Nevertheless, any initial physical will evolve into a new physical Bloch vector .

The fact that is not hermition implies that it might not be diagonalizable. There exists, however, a similarity matrix that puts on a Jordan block form, i.e.

(17)

where the Jordan blocks of have identical values on the diagonal and ones on the superdiagonal, and the remaining values are all zero. Thus, if is diagonalizable all Jordan blocks has dimension one. If is not diagonalizable, the spectrum shows an exceptional point Heiss (2004, 2012), where the real parts of at least two eigenvalues of coalesce. Furthermore, at the exceptional point the corresponding eigenvectors are identical. The presence of exceptional points will however not be relevant for us in the examples discussed in the following section.

iii.2 Adiabaticity for open quantum systems

Having introduced the Bloch representation, it is rather straightforward to generalize the ideas of adiabaticity from quantum mechanics Ballentine (2014) to open quantum systems Sarandy and Lidar (2005b, a). In order to make the analysis transparent we here consider the cases with vanishing pump terms (the generalization to is direct), and we call the instantaneous right eigenvectors of for . Adiabaticity then implies that the dynamics does not generate any transfer of population between the instantaneous eigenvectors. Thus, if we initialize the system in one eigenvector , and the system evolves adiabatically the state at a later time is . The condition warranting adiabatic evolution for an open quantum system described by a Liouvillian matrix takes a very similar form to that for Hamiltonian systems Sarandy and Lidar (2005b). That is, if represents the “gap” and a left eigenvector of , then the ‘rate of change’ , with dot representing time derivative, should be small in comparison to the “gap” for all times and all .

The similarities between adiabatic evolution generated by a hermitian matrix and a non-hermitian matrix are many, but there are still important differences that should be appreciated; () we already mentioned that the set of eigenvectors forms an over-complete basis, () most of the eigenvectors do not represent physical states , () contrary to state vectors, the norms of the Bloch vectors are arbitrary between 0 and 1, and () since even under adiabatic evolution the norm of the adiabatic Bloch vector typically decreases. If the instantaneous eigenstates define the adiabatic states , the ambiguity of the Bloch vector norm means that for every there is a set comprised of infinitely many adiabatic states. The last point in the list above then implies that even under adiabatic evolution an adiabatic state evolve within the ’th set of adiabatic states.

The generalization of the adiabatic theorem, going from a Hamiltonian to a non-hermitian matrix as sketched above, is natural from a mathematical perspective. It provides, however, a less clear physical picture, especially since most eigenvectors are non-physical. Naturally, we are more interested in how physical states evolve, and what is meant by adiabaticity for those. The physical state of special interest is the instantaneous steady state

(18)

where the subscript marks that the Liouvillian is explicitly time-dependent and the superscript (ad) denotes an adiabatic steady state. Thus, is an exceptional example of the adiabatic states introduced in the previous paragraph. Note that if we imagine a fixed time , is a corresponding steady state. If the time-dependence of the Liouvillian is weak we may expect that the relaxation time of the system is short on the overall time-scale such that throughout the driving the system stays close to a steady state. This then resemblance adiabatic evolution.

Analyzing adiabaticity in terms of deviations from the instantaneous steady state has been the subject of several papers Davies and Spohn (1978); Joye (2007); Avron et al. (2012); Venuti et al. (2016). In particular, if the quench can be assigned a time the deviations from deep in the adiabatic regime scales as , with for a gapped system and for a system in which the gap closes , where the exponent determines how fast the Liouvillian gap (13) closes Venuti et al. (2016). Since , any gap closening worsen the adiabaticity. should be compared to , and to for the closed system discussed in Sec. II.1.

Figure 3: Schematic picture of the time-evolution for open quantum systems. The yellow area represents the space of all physical states , and the light blue the manifold of instantaneous steady states . Since the model is explicitly time-dependent, the steady state manifold could change in time. In (a) the evolution is adiabatic and an initial steady state is adiabatically propagated by the Liouvillian to a final steady state . The steady state manifold is continuously deformed as time progresses and the propagated state remains in the manifold throughout. There is a one-to-one mapping between every steady states in the different manifolds at different times. For a non-adiabatic evolution, the initial steady state is taken out from its instantaneous steady state into some state , typically lying outside the steady state manifold, as depicted in (b). When the driving stops at time , the state relaxes down to a steady state in the steady state manifold (c). The distance is then a measure of how non-adiabatic the whole process was.

iii.3 Measure of non-adiabatic excitations

The KZM for quantum systems estimates the density of defects in terms of the quench rate and the critical exponents according to Eq. (8). These defects are topological by nature and describe local excitations. Thus, the sum of them gives the amount of excitations above the ground state energy. The more excited the system gets, while quenched through the critical point, the higher number of defects.

When we deal with an open system there is nothing like the ground state energy. We can of course consider the instantaneous energy where is the state at time . This energy could be compared to the ‘adiabatic’ energy to give some sort of ‘excitation measure’. However, such a comparison only makes sense if the influence of the environment is modest and the full system is mainly prescribed by the Hamiltonian. In fact, previous works have had this idea in mind; how is the quench affected by a weak coupling to an environment Dziarmaga (2006); Fubini et al. (2007); Cincio et al. (2009); Nalbach et al. (2015); Dutta et al. (2016)? The general answer to this question is that the environment causes additional fluctuations that increase the amount of excitations. When the system plus environment cannot be thought of as separate subsystems, for example when the criticality crucially depends on both parts, has little to do with any excitations. Learning from the discussion above about adiabaticity in open quantum systems, the proper measure for adiabaticity should be the distance from the instantaneous steady state . Indeed, is the state of relevance and any non-adiabatic excitations should be measured relative to this state. There are many possible choices for such a distance, but as we will see the natural one here is the trace distance Nielsen and Chuang (2010), which for two density matrices is defined as

(19)

where is the ’th eigenvalue of . Thus, the quantity of interest for us is . The idea behind the trace distance as the appropriate measure is pictured in Fig. 3, and we will visualize it further in the following section when we discuss particular examples.

Since any steady state has eigenvalue zero of the Liouvillian , under adiabatic evolution the Liouvillian does not generate any evolution of . That is, given that we started in the instantaneous steady state , then for all times. In particular, the length of the corresponding Bloch vector is constant. Any non-zero trace distance can only be the result of non-adiabatic transitions ocurring during the evolution. Typically, transitions out from will put the system in a non-steady state. The real parts of the Liouvillian eigenvalues will push the state back towards the steady state manifold, simultaneously as the driving may cause further non-adiabatic excitations. As the driving stops, according to the discussion above regarding fixed points of (16), the relaxation maintains and eventually the system is to be found in a steady state. The distance of this steady state to the adiabatic steady state,

(20)

is a direct result from non-adiabatic transitions developed during the quench and hence serves as our measure of excitations. This is one of the key results of the present work.

We typically consider a situation where the final times are far from the impulse regime. For the quench in the closed system this implies that the evolution is adiabatic and no further excitations develop. For the open case, during the later stages far from the critical point when the system relaxes to its instantaneous steady state – the relaxation time-scale is typically the short one. Thus, we normally have that is to a good approximation an instantaneous steady state as long as is large in comparison to any freeze-out time. The relaxation depicted in Fig. 3 (c) therefor occurs already before we reached the final time . With this in mind it should be clear that the trace distance is not only the suitable measure in terms of quantifying the amount of non-adiabatic excitations, but it also has the advantage that it does not depend on as long as it is much larger than the freeze-out time . This observation is important since our aim is to describe universal features of the quench for open quantum critical systems. As a universal quantity we do not want it to depend on nor . This is different from earlier works exploring quenches through critical points in open quantum systems Dziarmaga (2006); Fubini et al. (2007); Cincio et al. (2009); Nalbach et al. (2015); Dutta et al. (2016).

In the absence of an environment we wish that the trace distance (20) should in some sense be directly related to the amount of non-adiabatic excitations out from the ground state. As we already pointed out, in certain situations one cannot even talk about energy excitations since there is no energy spectrum. However, in the cases when one can it is of course desirable that the trace distance measures such energy excitations. Class I of the above classification is such a scenario. Here criticality cannot arise from coupling to the environment since the jump operator commutes with the Hamiltonian – the environment induces a dephasing in the energy basis. In particular, in this Class all steady states are diagonal in the energy eigenbasis. The instantaneous ground state of is thereby also an instantaneous steady state. With the instantaneous (adiabatic) eigenstates we envision the situation where the system is initialized in the Hamiltonian ground state, and hence for adiabatic evolution . The actual time-evolved state, after relaxation to its final steady state, is on the form . The trace distance at the final time becomes

(21)

Thus, the distance gives the probability to be excited. Furthermore, if we shift the instantaneous ground state energy , the amount of excitations in terms of energy is simply .

iii.4 The KZM for open quantum systems

As we discussed in Sec. III.2, the adiabatic concepts for time-dependent hermitian matrices (i.e. Hamiltonians) can be generalized to in principle any square matrix like for example the Liouvillian matrix . Regardles of the properties of the matrix the idea is that the variations of the change should be small in comparison to the squared gap in order to warrant adiabatic evolution. Since the eigenvalues of are in general complex, the absolute value of the gap should be considered, i.e. . Remember that even if the pump term is non-zero, a transformation can cast the Bloch equation into a homogenous one such that we can limit the discussion to homogenous equations.

We already pointed out that there is, however, one crucial aspects of adiabaticity in open quantum systems differing from the standard setting of closed quantum systems. As long as the adiabatic state is not a steady state, even under adiabatic evolution the norm of will decrease (keep in mind that we assume , and if this is not the case it is the norm of that is shrinking). Thus, on the Bloch sphere any symmetry axis defines a class of connected adiabatic states.

As for QPT’s, criticality of open quantum systems is also accompanied by a gap closening in the spectrum of  Kessler et al. (2012). Thus, at the critical point the Liouvillian gap of Eq. (13) . This is the equivalence of the critical slowing down for open quantum PT’s. Far from the critical point we expect instead that , for any and , is large compared to the inverse rate-of-change, such that the dynamics is adiabatic. In this respect we are lead to introduce the AI-approximation. The picture is then very similar to that of Fig. 1, where the evolution is divided into an adiabatic, an impulse, and an adiabatic regime. Recall that the freeze-out time when the system goes from adiabatic to diabatic is determined from equalizing the relaxation time with the inverse transition rate. For a linear quench we thereby have , where is some parameter that could depend on the system parameters Damski and Zurek (2006). When considering a quench through a critical point of an open quantum system we should replace the Hamiltonian reaction time with the Liouvillian reaction time (note that the Liouvillian gap sets the relaxation rate to the steady state, while determines total response), and furthermore the parameter may well be altered by the environment and especially is expected to depend on the loss rates . In fact, it is not a priori clear that the characteristic time for the rate-of-change will be linear even though the quench is linear. One could, for example, imagine Lindblad jump operators that are explicitly time-dependent. Another possibility is that the impulse regime need not be symmetric around the critical point such that there is a left and right freeze-out time with . Nevertheless, with the above argument we hope that it is cleatr that the general idea of the AI-approximation can be applied to critical open quantum systems. In the following section we will verify this by analyzing two examples.

Iv Examples

For a non-trivial situation we need a model that supports a manifold of many steady states. For a single unique steady state it is clear that by taking large enough the system always ends up in this state and we cannot conclude how adiabatic the quench was. We know that Class I of our classification supports a connected manifold of steady states. This Class is also physically relevant as it describes energy dephasing. In fact, small fluctuations in experimental parameters should at first cause a dephasing in the energy basis. We thereby look for Lindblad jump operators that commute with the Hamiltonian for all times. One choice is to take the instantaneous projectors onto the adiabatic states; . Alternatively one can take that . In both examples we have that the jump operators are explicitly time-dependent and in a strict sense the resulting master equation is not on a Lindblad form. Nevertheless, this is not so important for us since it is still a ‘completely positive trace preserving map’ that guarantees that the density matrix stays physical under time-evolution. In both examples discussed in this section we consider the case when the jump operators equals the system Hamiltonian. Using the adiabatic state projectors instead does not change the results qualitatively in any way.

iv.1 Dephasing LZ model

As Damski pointed out, the simplest model supporting the KZM is the LZ problem Damski (2005). It is not a model describing true criticality, but rather a smooth transition between two orthogonal states, see Fig. 2. For the dephasing LZ problem the Lindblad equation takes the form

(22)

with the LZ Hamiltonian (9). The manifold of steady states comprises those along a symmetry axis in the Bloch sphere between the adiabatic states, and . For large negative or positive times these states approximately coincide with the diabatic states and , i.e. the north and the south pole on the sphere. Starting in say the south pole, regardless of whether vanishes or not, adiabatic evolution implies that the state stays pure and traverses the Bloch sphere and ends up on the north pole. In the adiabatic basis, the state is frozen under adiabatic evolution. Non-adiabatic excitations takes the state away from the symmetry axis. Simultaneously, the dephasing shrinks the length of the Bloch vector and pushes the state back towards the symmetry axis (steady state manifold). This relaxation is clearly absent when . The result of a numerical simulation demonstrating this behavior is presented in Fig. 4.

Figure 4: (Color online) Comparison between the Bloch vector evolution for the closed, , and open, , LZ model. The thick blue arrow represents the manifold of steady states (symmetry axis); the arrow head is the adiabatic state and the other end of the blue arrow is the orthogonal adiabatic state . The thick black arrow shows the final state at (big enough such that the relaxation is complete). Adiabatic evolution would mean that the black and blue arrows completely overlap. The turquoise thin lines give snapshots of the Bloch vector at different times . The decreased Bloch vector for is evident, as is the relaxation down to the steady state manifold. The amount of excitations is determined from . For this numerical example, , , and the initial and final times and are taken to warrant convergence of the population transfer.

In Appendix A we give the general expression for the Bloch equation for two-level systems, from which we can extract the corresponding equations for (22). The Liouvillian matrix takes the form

(23)

and the pump term , as follows from that the jump operator is hermitian. The Liouvillian matrix is normal, , which implies that it is unitarilly diagonalizable Arfken and Weber (2005). The instantaneous eigenvalues are

(24)

where, as before, . The corresponding, orthonormal, instantaneous eigenvectors are

(25)

The first Bloch vector is the steady state with the accompanying zero eigenvalue. The remaining two Bloch vectors are both complex, and hence cannot represent physical states. The spectral gap . For we regain the LZ gap . The absolute values of the eigenvalues are shown in Fig. 5, from where it is also evident that the gap grows with .

Figure 5: (Color online) Absolute values of the instantaneous eigenvalues (24) of the Liouvillian matrix (23). There is a time-independent zero eigenvalue corresponding to the steady states. The difference between the zero eigenvalue and the remaining two reflects the spectral gap function which sets the inverse response time. In the figure and we show two examples; for the solid black line and for the dashed red line. Note, in particular, that the gap increases with implying that the response time gets shorter the larger is.

Turning to the KZM for the open LZ problem, we note that the response time is a decreasing function of . This suggests that the non-vanishing loss rate makes the quench more adiabatic. In return, this should result in a shorter impulse region, i.e. the value of the freeze-out time decreases. But this is in contrast to the general knowledge that the coupling to an environment causes further excitations stemming from the additional fluctuations induced by the environment Fubini et al. (2007); Cincio et al. (2009); Dutta et al. (2016). Indeed, one finds that the amount of excitations do increase for a non-zero (see below). Does this mean that the idea of the KZM, as a result of an AI approximation, breaks down for open systems? As pointed at in Sec. III.4, the answer to this question lies in the fact that the rate-of-change also depends on , i.e. the slope of the red dashed line in Fig. 1 is -dependent. Thus, even if the solid black lines of Fig. 1 move downward (suggesting a shorter impulse region), the change in the slope of the red dashed line can compensate this to keep the impulse region large. If the amount of excitations increases we even expect the impulse region to grow with .

For the closed LZ problem it was motivated that the slope of the red dashed line in Fig. 1 is given by  Damski and Zurek (2006). In particular, in the regime of relatively fast quenches where we expect the AI-approximation to be applicable, an expansion argument, and comparing to the known analytical result (10), give this slope value. To analyze the open LZ problem we rewrite the Liouvillian matrix in powers of ;

(26)

We note: () the last term, quadratic in , generates relaxation of and but does not directly affect , () for any state on the symmetry axis between the north/south poles is a steady state, () for the time-dependence is linear and the second term contains the elements which result in identifying the slope  Damski and Zurek (2006), and () the -dependent elements of the second term clearly describe non-unitary evolution as they appear with the same sign. The fact that contains both a linear and quadratic time-dependence could hint that the rate-of-change should not be taken linear as for the closed case, or in the visual example of Fig. 1. However, the quadratic time-dependence only enters on the diagonals and does not generate direct couplings between the Bloch vector components. For the linear term we expect that , and furthermore that . We also know that we must have for non-zero .

Since the open LZ problem studied here is not analytically solvable (to the best of our knowledge), we cannot use the same arguing as in Ref. Damski and Zurek (2006) to determine . Assuming a linear dependence, a least square fit gives

(27)

Here it is understood that this formula is valid for . The freeze-out time is obtained from

(28)

The analytic expression for is long and not very informative. We note, however, that is an increasing function of , which explains the larger amount of excitations generated through the quench in the open compared to the closed LZ problem. The applicability of the AI-approximation, using the parameter (27), is demonstrated in Fig. 6. The regime of study is for relatively fast quenches where the KZM is believed to reproduce quantitatively correct predictions. For these parameters, the agreement is very convincing, even for as large loss rates as . Indeed, for the parameters of the figure the agreement is improved with larger . For even larger , beyond , the behavior is qualitatively different as the evolution is then described by a Zeno effect, see discussion in the concluding remarks in Sec. V.

Figure 6: (Color online) The amount of excitations, measured by the trace distance(21), as a function of for three different loss rates . Solid lines give the results according to the AI-approximation with the rate-of-change(27), while the dashed ones are the corresponding numerical results obtained from direct numerical integration of (22). Note that the agreement gets better the larger is. The closed case represents the situation studied in Ref. Damski (2005); Damski and Zurek (2006). The interval of integration is the same as in Fig. 4.

iv.2 Dephasing transverse Ising model

The open LZ model of the previous subsection is a most simple example demonstrating the ideas of the KZM for open quantum systems, but it is not, in a true sense, describing a proper quantum phase transition. In this subsection we turn to a model, the transverse Ising model Sachdev (2007); Suzuki et al. (2012), that indeed hosts a quantum phase transition, both in the open and closed version of the model. Thanks to a set of clever transformations proposed by Dziarmaga in Ref. Dziarmaga (2005), the transverse Ising model can be mapped to a set of decoupled LZ problems. Each LZ system describes the dynamics of a single momentum mode, and for the of limit long wavelengths () the effective LZ velocity diverges marking the presence of a critical point and the unavoidable breakdown of adiabaticity.

Figure 7: Schematic picture showing the quench order of the transverse Ising model. At we have and the system is deep in one of the paramagnetic phases. The ground state is unique and the spectrum gapped. For the system passes the first critical point and enters the symmetry broken ferromagnetic phase. In this phase the parity symmetry is broken and the ground state is doubly degenerate. Local excitations consists in domain walls between the two ferromagnetic ground states. At the system goes through the second critical point and ends up in the other paramagnetic phase.

The transverse Ising model in one dimension for sites is

(29)

where is the typical energy scale, the transverse field strength, and () the Pauli matrices at site . Throughout we use periodic boundary conditions, . For the Hamiltonian is diagonal in the computational basis with the doubly degenerate ferromagnetic ground states and . In the opposite limit , the ground state is paramagnetic (or polarized) or . Here and are the eigenstates of with eigenvalues , while and are the eigenstates of . At there is a continuous QPT between the para- and ferromagnetic phases. Thus, a quench dictated by the time-dependent field strength

(30)

drives the system from a paramagnetic phase through a critical point at (i.e. ) into the symmetry broken ferromagnetic phase and through a second critical point at (i.e. ) back into a symmetric paramagnetic phase. The quench scheme is depicted in Fig. 7.

iv.2.1 Mapping to a LZ problem

Before entering the problem of how dephasing affects the evolution we first discuss the closed system. In Appendix B we present the details of the mapping of the quenched Ising model to a LZ problem. Here we only record the main results of the mapping. The simplest approach in solving the transverse Ising model is by employing the Jordan-Wigner transformation that casts the Hamiltonian into a problem of free fermions Sachdev (2007). In the momentum representation the Hamiltonian reads

(31)

where , , and are the creation and annihilation operators of a fermion with momentum , and the sum runs over only positive momentum modes. As a quadratic Hamiltonian it can be diagonalized by a Bogoliubov transformation which introduces the new fermion operators according to

(32)

and equivalently for , and to warrant the correct fermionic commutation relations . The Heisenberg equations give two coupled equations for the Bogoliubov amplitudes as

(33)

which defines a - and time-dependent Hamiltonian . With the substitution Dziarmaga (2005)

(34)

the equations of motions for the amplitudes become

(35)

i.e. they attain the familiar form of a LZ problem with a corresponding Hamiltonian . Thus, we have reduced the analysis of the full time-dependent transverse Ising model to a problem of solving a set of LZ models, one for each momentum mode .

iv.2.2 Universal scaling of excitations

For the regular LZ sweep from to , as already mentioned, the system crosses two critical points. We thereby consider the time interval to start with, and briefly discuss the second scenario later. The final time is assumed to be within the adiabatic regime such that the LZ formula (10) should be applicable. One point to notice is that the LZ crossing instant does not coincide with the actual crossing of the critical point at ; it is shifted by . Furthermore, note that for the low energy modes, corresponding to small , the rate diverges, marking the breakdown of adiabaticity in the vicinity of the critical points. It is mainly these long wave-length modes being excited during the quench, and we may expand . By identifying we estimate the excitation from the LZ formula (10),

(36)

The total amount of excitations is then evaluated as

(37)

Going back to the KZ prediction (8), and using the transverse Ising exponents , we see that it exactly reproduce the above result (37Dziarmaga (2005).

With the energy dephasing jump operator , the transformation to fermions in the momentum representation is still straightforward. The total state , where, for each ,

(38)

Thus, for each momentum mode the amount of excitations is derived as in the previous subsection, and their sum gives the total excitations . The results of a numerical calculation are presented in Fig. 8 showing for different ’s and as a function of . The amount of excitations increases with as anticipated. Universality implies a power-law scaling, which is also found as is evident from the log-log plot, Fig. 8 (b). The interesting result is that the exponent is altered by the dephasing. For mean-field models Nagy et al. (2011); Baumann et al. (2011) and quantum critical models Patane et al. (2008, 2009) it has been demonstrated that critical exponents may change due to coupling to an environment. The numerically extracted exponents, , for the examples of Fig. 8 are listed in Tab. 1.

Figure 8: (Color online) The density of defects (a) accumulated through the quench from to , and for different decay rates (see inset). Clearly, the larger the more excitations. The right panel (b) displays the same but as a log-log plot. The defect density scales with an exponent in all cases (see Tab. 1), which interestingly is modified by the dephasing. For the numerics, the initial time which is deep in the adiabatic regime, and the number of sites .

 

                   
0 -0.5
0.05 -0.504
0.1 -0.523
0.4 -0.541
0.8 -0.544

 

Table 1: Numerically estimated KZ exponents , for different . The ’s have been calculated from a least square fit from the data of Fig. 8 (b), and the numerically obtained error is around 1 for all examples. Convergence of the results have been checked, both in respect to the integration interval and the system size.

iv.2.3 Correlation function

For a quench ending at in the ferromagnetic phase we expect excitations in terms of domain walls (also called kinks in one dimension). According to the result above, the KZM predicts a correlation length for the closed Ising model. For non-zero we still expect domain wall excitations as the Lindblad jump operator commutes with the Hamiltonian for every time instant. Nevertheless, we saw that the exponents are modified by the dephasing and should accordingly affect the correlation length.

The characteristic length between the domain wall excitations is extracted from the correlator