This page is a blog article in progress, written by Zoltán Zimborás. To see discussions of this article while it was being written, visit the Azimuth Forum.
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If you are a follower of the network theory series on Azimuth, you must feel pretty well-informed in comparing the differences and similarities between stochastic and quantum mechanics. However, we still hope to be able to surprise you by presenting a situation where these two worlds differ in a particularly subtle way: the case of directed (or biased) walks.
One surprising difference is that for quantum walks, biasing a direction $A \to B$ (compared to the reversed $B \to A$ direction) is only possible if the topological structure of the underlying graph is apt for it. This is related to certain obstructions on time-reversal symmetry breaking in quantum evolutions, which we will discuss today.
Sounds like fun math! But before we jump into the details, let’s mention that these considerations are not only mathematical amusements. The discussed topological effects appear in solid state physics and, more intriguingly, they may even be used by plants! Light-harvesting complexes of plants and bacteria, as was mentioned in the previous post on Azimuth, offer one of the main motivations for studying quantum evolutions on graphs with complicated topologies. These complexes are believed to be evolutionarily optimized for the (quantum) transport of energy, see a discussion here. Interestingly, recent theoretical and experimental investigations indicate that quantum directional biasing, the theme of today’s blog post, might also be present in this type of energy transport.
It is an especially good time to consider this topic since we just published a paper in Phys. Rev. A on the subject with experimentalists from the Institute for Quantum Computing
D. Lu et al,Chiral Quantum Walks
This was based on our previous theory paper in Scientific Reports that we wrote together with Zoltán Kádár, James Whitfield and Ben Lanyon:
We discussed the theoretical aspects and also showed by numerics that directional biasing (or time-reversal symmetry breaking) can in principle considerably speed up transport in light-harvesting complexes and in other complex quantum networks.
In this post, we will only concentrate on the basic features. Since some parts of our papers are rather technical, here we will make the exposition more comprehensible with the help of some old friends of quantum mechanics: kets… cats, I mean. Many of you reading this will know that the use of cats in quantum physics is ever so common. The application of cats is credited to Schrödinger; they are nearly always threatened but rarely harmed. In this tradition, today we’ll appeal to these fuzzy quantum felines to conduct our experiments, and illustrate the concepts of the paper.
Imagine our Gedanken-cat sitting on a rung of a horizontal ladder. Being a bit restless, she sometimes jumps to one of the neighboring rungs - with equal probabilities to the left or to the right.
This type of continuous-time random (cat)walk is described by a master equation - as was discussed in parts 16 and 20 of the network theory series, and in the previous post on Azimuth. The main ingredient in this description is the adjacency matrix, which characterizes the topology of the possible elementary jumps. For the depicted six-rung ladder, the actual topology is the following: from the first and the last rung the cat can jump only to one other rung; while from any of the four middle rungs there are two jumping possibilities (to the left and to the right). In a matrix from this neighborhood structure can be encoded as
The adjacency matrix $A$ defines a Laplacian through the formula $L=D-A$, where the entries of the diagonal degree-matrix $D$ are defined by $D_{ii}=\sum_{j=1}^6 A_{ij}$, with the result being:
The probability vector $\psi=(\psi_1, \psi_2, \ldots , \psi_6)$, where the entry $\psi_k$ gives the probability that the cat is on the $k$th rung, evolves according to the master equation generated by $-L$:
It will be
Suppose that while our cat sits on the ladder in the autumn sun, it is approached by the neighbor’s dog from the the left.
As the two species have a different picture of reality, unavoidable conflicts pop up. Hence, as an educated guess, we could assume that the cat’s motion would in this situation be biased towards the left. A biased stochastic motion can be characterized by a weighted and directed adjacency matrix, i.e., with an $A_{d}$ matrix that can have any real nonnegative entries (not only $0$s and $1$s) and that is not symmetric. In the present example, it could be
Here $p$ is between $0$ and $1$. By increasing $p$, the stochastic walk generated by the Laplacian $L_{d}=D_{d} - A_{d}$ would be more and more biased towards the right. The limiting probability distribution, for example, would be more and more skewed towards the right - with $p=1$ being a situation when the cat goes strictly to the right, and the limiting probability distribution would be that the cat is with probability $1$ on the rightmost rung.
How would the previous two scenarios look like if our kitten behaved quantum mechanically? The quantum analogue of the undirected walk, i.e., when the Laplacian is symmetric has been extensively treated in part 16 of our network series. In this case, we could get an analogous quantum walk by the Schrödinger equation:
where the quantum Hamiltonian $H$ is simply the negative Laplacian
We expect that the quantum walk defined this way would not be biased. But what does this mean in exact mathematical terms? Let $P_{A \to B}(T)$ denote the probability that we find our cat on site $B$ at time $t=T$ supposing that she started the walk from site $A$ at time $t=0$. Similarly, let $P_{B \to A}(T)$ be defined by the reverse situation, i.e., the probability of finding the kitten at $A$ when she started from $B$. We call a quantum walk directionally unbiased if
holds for all times $T$ and all pairs of sites $(A,B)$.
In the next section, we will prove that this property holds not only for our particular walk, but for any quantum walk that is generated by a real quantum Hamiltonian. One would expect that such a general result is related to some type of symmetry. This is indeed the case, what we are seeking is time-reversal symmetry.
Our classical intuition tells us already that time reversal-symmetry and directional motion are naturally related:
Jacob’s comment: Example starts by explaining a video played forwards or backwards. I can get the artist to draw the cats motion on what looks like film.
Now let’s take a look at this relation in the quantum world! In this very simple model of one-particle… one-cat quantum mechanics, the operation implementing time-reversal is complex conjugation, which we will denote by $C$. You may recall from our earlier posts, that in the quantum case the vector $\psi=(\psi_1, \psi_2, \ldots)$ has complex entries, and the action of $C$ on such a $\psi$ is simply
where $\overline{\psi} =(\psi_1^{*}, \psi_2^{*}, \ldots)$, i.e., the action is componentwise complex conjugation. $C$ is a so-called anti-linear operator, and is also an involution ($C^{-1}=C$).
But how does $C$ implement time-reversal symmetry?
(To be continued…)
Okay, so $H=-L$, being a symmetric matrix, will not yield directional biasing, and we cannot use $-L_d$ as a quantum Hamiltonian either since it is not self-adjoint. Then purr, purr… purrhaps we should consider using a self-adjoint, but complex Hamiltonian, i.e., setting complex phases into $H$:
(To be continued…)
Blog ends here
Here the draft for the blog post ends. Below you may find the material that we may use for this blog article.
In the standard literature on continuous time quantum walks [FG98,CCDFGS03, MB11,Kempe03, Kendon06], the time-independent walk Hamiltonian is defined by a real weighted adjacency matrix $J$ of an underlying undirected graph,
The condition that the hopping weights $J_{nm}$ are real numbers implies that the induced transitions between two sites are symmetric under time inversion. We can break this symmetry while maintaining the hermitian property of the operator by appending a complex phase to an edge: $J_{nm}\rightarrow J_{nm}e^{i\theta_{nm}}$ resulting in a continuous time chiral quantum walk (CQW) governed by
When acting on the single exciton subspace the Hamiltonian given in Eq. \eqref{eqn:cqw} can be expressed in terms of the spin-half Pauli matrices:
which arises in a variety of physical systems when magnetic fields are considered. We explore a proof-of-concept experimental demonstration of this effect in Supplementary Information, Section S2.
In the CQW framework, we investigate coherent quantum dynamics and incoherent dynamics within the Markov approximation. Both types of evolution are included in the Lindblad equation [Kossakowski72,Lindblad76,Breuer02,Whitfield10]:
where $\rho(t)$ is the density operator describing the state of the system at time $t$ and $L_k$ are Lindblad operators inducing stochastic jumps between quantum states. For example, using the usual terminology of Markovian processes, we call site $t$ a trap if it is coupled to site $s$ by the Lindblad jump operators, $L_k=\ket{t}\bra{s}$. The site-to-site transfer probability, $P_{n\rightarrow m}(t)=\langle m|\rho(t)|m\rangle$, gives the occupancy probability of site $m$ at time $t$ with initial condition $\rho(0)=|n\rangle\langle n|$. Note that the present study, while utilizing open system dynamics, is not related to the enhancement of transport due to quantum noise [SMPE12,MRLA08] which has been well studied in the context of photosynthesis[MRLA08,lloyd2011]. Here the emphasis is instead on the effect the breaking time-reversal symmetry of the Hamiltonian dynamics can have on transport.
To quantify the transport properties of quantum walks, we use the half-arrival time, $\tau_{1/2}$, as the earliest time when the occupancy probability of the target site is one half.
We will also make use of the transport speed, $\nu_{1/2}$, defined as the reciprocal of $\tau_{1/2}$.
The probability for a quantum walker to start from a node $S$ and reach the node $E$ at time $t$ is:
In this settings, the time inversion is given by the complex conjugation operation in the natural vertexes basis:
The time-reversal of a Hamiltonian $H$ is given as $THT^\dag=THT$. The $H\to THT$ action is represented in parameter space by the replacement $J_{nm}\to J_{nm}^*$. Thus exactly the achiral quantum walks (real Hamiltonians) are left invariant by this action.
which can be verified using $T\tho T=\rho$ and the cyclicity of the trace as follows:
A crucial consequence of the above is that in the case of achiral quantum walks, the transition probabilities are the same at time $t$ and $-t$, i.e. $P_{S \to E}(t)=P_{S \to E}(-t)$, and directional biasing is prohibited $P_{S\to E}(t) = P_{E\to S}(t)$. However, $H\neq THT^\dag$ does not necessarily imply that transition rates are asymmetric in time. This is because $THT^\dag$ might be gauge-equivalent to $H$, as will be seen in the next section.
We now introduce a quantum switch which enables directed transport and could, in principle, be used to create a logic gate and offer future implementations of transport devices to store and process energy and information. Fig.~[fig:switch] presents an example of this switch. The value of a phase ($e^{i \theta }$) appended to a single control edge across the junction allows selective biasing of transport through the switch. The maximal biasing occurs at $|\theta|=\pi/2$, and the sign determines the direction. The first maxima of $P_{S\rightarrow E}(t)$ (transfer probability from site S to E)
in the unitary dynamics without traps can be enhanced by 134\% or suppressed to 91\% with respect to the non-chiral case. When considering traps in the Lindbladian evolution, the optimal transport efficiency is 81.4\% in the preferred direction. The switch violates TRS as $P_{S\to E}(-t)\neq P_{S\to E}(t)$. By using $P_{S\to E}(-t)=P_{E\to S}(t)$ and the symmetry of the configuration $P_{E\to S}(t)=P_{S\to F}(t)$, we conclude that transport is biased towards the opposite pole when running backwards in time, see Fig~[fig:switch]. Note that the behaviour of the switch is largely independent of the length of the connecting wires.
We will now utilize the directional biasing of the triangle to give an example of a speed-up of chiral walks. Using the composition of eight triangular switches as depicted in Fig.~[fig:saw+fmo]a, by simultaneously varying all phases along the red control edges to the same value, we examine the effect of time-reversal asymmetry on transport. We find that the occupation probability as a function of $\theta$ is symmetric about $\pm \pi/2$ with the negative value corresponding to maximal enhancement and the positive value to maximal suppression. Unlike the occupation probability maxima in the switch, here the first apexes are separated in time. When we include trapping, the half-arrival time is reduced from the non-chiral value $\tau_{1/2}=38.1$ to $5.2$ which represents a $633$\% enhancement. To conclude this section we focus on suppression of transport by chiral quantum walks. A good example is the polygon with an even number of sites. In this case complete suppression can be achieved by appending a phase of $\pi$ to one of the links in the cycle; thereby rendering it impossible for the quantum walker to move to the diametrically opposite site. This is a discrete space version of a known effect in Aharonov-Bohm loops [Datta]. The proof that the site-to-site transfer probability is zero in this case for all times also in our discrete-space and open-system walks can be found in the Methods Section. However, note that the discrete even-odd effect, which implies that only loops comprised of odd particles can exhibit transport enhancement, and only even loops may exhibit complete suppression, has no known continuous analog.
In natural and synthetic excitonic networks such as photosynthetic complexes and solar cells, we are faced with non-unitary quantum evolution due to dissipative and decoherent interaction with the environment. Studies have shown that dissipative quantum evolution surpasses both classical and purely quantum transport (for interesting recent examples see[Whitfield10,SMPE12]). A widely studied process of such dissipative exciton transport is the one occurring in the Fenna-Matthews-Olsen complex (FMO), which connects the photosynthetic antenna to a reaction centre in green sulphur bacteria[MRLA08,caruso09,fleming10,ringsmuth2012]. Due to the low light exposure of these bacteria, there is evolutionary pressure to optimize exciton transport. Therefore, the site energies and site-to-site couplings in the system are evolutionarily optimized, yielding a highly efficient transport[lloyd2011]. However, it is an open question whether or not there occurs time-reversal asymmetric hoping terms in these systems, and whether these are optimized. Recent 2D Electronic Spectroscopy results lead to the conclusion that , e.g., in the light harvesting complex LH2 hopping terms with complex phases are indeed present [Engel]. Here we ask whether such TRS breaking interactions may further enhance the efficiency of the light harvesting process. We consider the traditional real-hopping Hamiltonian modeling transport on the FMO, and allow for TRS breaking by introducing complex phases and find that the transport speed can be further increased. We study the seven site model of the FMO using an open system description that includes the thermal bath, trapping at the reaction centre, and recombination of the exciton[MRLA08,caruso09, plenio08]. By performing a standard optimization procedure (as outlined in the Supplementary Information, Section S3) that varies the phase on a subset of seven edges, we found a combination of phases where the transport speed, $\nu_{1/2}$, is enhanced by $7.68$\%. In Fig.~[fig:saw+fmo]b, the enhancement of the time dependent occupation probability is shown for the chiral quantum walk. We note that optimization over only three edges already changes the transport speed by 5.92\%, see Supplementary Information, Section S3.
Complex network theory has been used in abstract studies of quantum information science; see for example [Acin, Acin2]. Here we turn to the theory of complex networks to determine if optimization procedures limited to small subsets of edges will generally lead to improved transport in larger and possibly randomly generated networks. We found a positive answer when testing the site-to-site transport between oppositely aligned nodes in the Watts-Strogatz model~[WS98].
This family of small-world networks continuously connects a class of regular cyclic graphs to that of completely random networks (Erd\H{o}s-R'enyi models[ER60]) by changing the value of the rewiring probability.
We numerically investigated graphs with 32 nodes, average degree four and range over rewiring probability $p$ considering 200 different graph realizations for each value of $p$. An example with $p=0.2$ is depicted in Fig.~[fig:WS]a. Here the occupancy of a sink connected to site $E$ is compared between the chiral walk and its achiral counterpart. The particle begins at site $S$ and we perform the optimization of the phases only on edges connected to site $E$. In the case of the chiral quantum walk, the sink reaches half-occupancy in 54.8\% less time on average.
A quantum walk is widely used tool to study or simulate a variety of different quantum systems. A quantum walk, give us an insight on the transport properties of the system and on the dynamics that characterize it’s evolution.
A quantum walk is defined as the one-particle subspace of a quantum system described by some Hamiltonian.
The quantum switch.
(a) Directional biasing: enhanced transport in the preferred direction. (b) The plot shows the occupancy probability $P_{S\to E}$ of site $E$ with the particle initially starting from site $S$ with and without sink (dashed and solid lines, respectively). This evolution is time-reversal asymmetric as replacing $t$ with $-t$ results in the particle moving from site $S$ towards site $F$. When starting at site $E$, the particle evolves towards site $F$.
By replacing $t$ with $-t$, a particle initially at site $E$ evolves towards the initial configuration (b). To recover time-reversal symmetric transition probabilities in the evolution (b), requires that one also performs the antiunitary operation [W31] on the Hamiltonian mapping $\theta$ to $-\theta$. This has the same effect as reflecting the configuration horizontally across the page while leaving the site labels intact.
(a) Triangle chain and (b) the FMO complex. (a) The phase $e^{i \theta}$ is applied to the red edges simultaneously in the triangle chain. The plot illustrates the occupancy probability at the end site $E$ as a function of time for different values of the phase $\theta$ with and without trapping (dashed and solid lines, respectively). (b) shows the occupancy difference with respect to the time reversal symmetric Hamiltonian of the FMO complex. We use an optimization procedure to enhance the transport. While holding the magnitude of the couplings constant, we optimize two sets of phases, $A_1$ and $A_2$, which correspond to seven and three edges with an enhancement at $\tau_{1/2}$ of $3.25$\% and $2.25$\%, respectively.
Transport enhancement of the chiral quantum walk is robust across randomly generated Watts-Strogatz networks. An example of this small-world network, with rewiring probability $p=0.2$, is depicted in (a). The transfer probability $P$ from site $S$ to the sink connected to site $E$ is plotted in a realization of the network. (b) shows the average enhancement of half arrival time ($\Delta\tau_{1/2}$) for different values of $p$.
A list of papers that we might use when discussing the effect of stochastic noise on quantum transport (TODO We have to select from these later, see which fits into our story):
Studying the crossover between stochatic and quantum transport (Verstraete): http://arxiv.org/pdf/0912.0858.pdf
The above work was based on this (Prosen): http://arxiv.org/abs/0801.1257
An important and interesting feature in this respect is “negative differential conductivity” (Prosen) :http://arxiv.org/pdf/0806.2236.pdf
Dephasing enhanced transport (Clark): http://arxiv.org/pdf/1204.1313.pdf
From ballistic to diffusive behavior (in heat transport) (Clark): http://arxiv.org/pdf/1303.6353.pdf
Combined effect of disorder, noise and interaction (Plenio: http://arxiv.org/pdf/1204.1313.pdf
When the the initial state has a momentum (Eisfeld): http://arxiv.org/pdf/1010.4325.pdf
Noise assisted transport - mostly in photosynthetic complexes (Plenio): http://arxiv.org/pdf/1204.1313.pdf http://arxiv.org/pdf/1307.3530.pdf
Here the pictures by federica: