maps are invariant under such operations. If the dynamical maps do not change, it means that all the dynamical variables of the plant will follow the same equations of motion, before and after applying the unitary operations. Solely from the dynamical point of view, the dynamics cannot be treated as non-Markovian, as it is difficult to reveal the memory effect by studying dynamics of the plant variables alone. In the following, we will use two examples to illustrate this new criterion, and show that it can reveal non-Markovianity in systems inwhich the unperturbed dynamics (before applying the unitary operation) is indistinguishable from the Markovian one.
The first example—the first such example is the interesting spin-cavity system as shown in Fig. 5.2—a two-level atom coupled to a cavity mode which in turn couples to external continuum—a quantum Wiener process that is equivalent to a zero-temperature Markovian bath [13]. If we view the cavity mode and the external continuum together as the bath, the two-level atom—the plant—
is effectively coupled to a damped cavity mode which is a non-Markovian dissipative bath. The corresponding Hamiltonian for this system is given by [14]:
Hˆ =~(ωq/2)ˆσz+~∆ˆa†ˆa+~g(ˆσ−ˆa†+ ˆσ+a)ˆ +~√
γ[ˆaˆb†in(t) + ˆa†ˆbin(t)]. (5.5) Here ˆσzis the Pauli matrix and ˆσ±= ˆσx±iˆσy; ˆaand ˆbinare the annihilation operators of the cavity mode and the in-going field of the external continuum with [ˆa,ˆa†] = 1 and [ˆbin(t),ˆb†in(t0)] =δ(t−t0);
ωq is the atom transition frequency and ∆ is detune frequency of the cavity mode; g and γ are the corresponding coupling constants. After tracing over the external continuum, the joint density matrix of the atom and cavity satisfies the following Markovian master equation:
˙ˆ
ρ(t) =−i
(ωq/2)ˆσz+ ∆ˆa†ˆa+g(ˆσ−aˆ†+ ˆσ+a),ˆ ρ(t)ˆ
+γ[2ˆaˆρ(t)ˆa†− {ˆa†a,ˆ ρ(t)}].ˆ (5.6)
To further obtain the master equation for the atom by eliminating the cavity mode, we need to know the initial state of the atom and the cavity mode. In the simplest case when they initially are separable and the cavity mode is in the vacuum state, namely|ψi=|ψai ⊗ |0i, as shown in Ref. [14],
Figure 5.2: (Color online.) A schematic showing the atom-cavity system. The cavity mode is coupled to the external continuum field (a Markovian bath), and they together form an effective non-Markovian bath for the atom. (Adapted from Ref. [14]).
the reduced density matrix of the atom ˆρa(t) satisfies a time-local master equation:
˙ˆ
ρa(t) =−ihωq
2 σˆz+g={f(t)}σˆ+σˆ−, ρˆa(t)i
+g<{f(t)}[2 ˆσ−ρˆa(t)ˆσ+− {ˆσ+σˆ−, ρˆa(t)}], (5.7) where the time-dependent function f(t) satisfies a Riccati equation: f˙(t)−i(ωq −∆ +iγ)f(t)− g f2(t) =gwith an initial condition f(0) = 0, of which the solution is well-known. In the case with ωq = ∆ and strong dissipationγ≥2g,f(t) is real and positive, and we simply have
˙ˆ
ρa(t) =−ihωq
2 σˆz,ρˆa(t)i
+g f(t) ˆLˆρa(t) (5.8) with ˆLρˆa(t) = 2 ˆσ−ρˆa(t)ˆσ+− {ˆσ+σˆ−,ρˆa(t)}. Such a master equation can also describe the case when the atom is directly coupled to the Markovian bath but with a time-dependent coupling rate, of which the Hamiltonian is
Hˆ =~(ωq/2)ˆσz+~p
gf(t)[ˆσ−ˆb†in(t) + ˆσ+ˆbin(t)]. (5.9)
Basically, from this reduced dynamics alone, we cannot tell whether the underlying dynamics is Markovian or not, even though the atom-cavity interaction is highly non-Markovian when the cavity decay rate γ becomes comparable to the atom transition frequency ωq. This master equation is simply an artifact of a specially-chosen initial state for the atom and the cavity. If we perturb the atom, e.g., by applying aπ-pulse, the dynamics of the atom will deviate from the one described by Eq. (5.7) due to the memory of the cavity mode.
To show the change in the dynamical map for the plant after applying local operations, we
Figure 5.3: (Color online.) The left panel shows the change in the trace distance after applying a local unitary operation; the right panel shows the concurrence for the entanglement between the atom and a two-level ancilla. The black line shows the Markovian dynamics described by Eq. (5.8). We have chosen ωq = ∆ = g = 1, and γ = 2g. For evaluating the trace distance Tr|ρˆ1(t)−ρˆ2(t)|/2, the two initial quantum states of the atom are ˆρ1(0) =|1ih1|and ˆρ2(0) =|0ih0|.
For evaluating the concurrence, the atom and ancilla are initially in the maximally entangled state:
√1
2[|0i| ⊗ |0i+|1i| ⊗ |1i].
Figure 5.4: (Color online.) Plot showing the time evolution ofhˆσziaccording to Markovian dynamics given in Eq. (5.10) (black) and to non-Markovian dynamics (red) with ˆσz⊗Iˆc applied att= 1. The initial atom-cavity state is (|0i+|1i)/√
2⊗ |0i.
numerically solve the master equation in Eq. (9.54a) for the joint atom-cavity density matrix. We use the same initially-separable quantum state for deriving Eq. (5.8) and in addition imposeγ= 2g, so that when there is no unitary operation on the atom, the reduced density matrix of the atom simply follows Eq. (5.8) withf(t)>0, which is a time-local Markovian master equation. In Fig. 5.3, we show the change of the trace distance and also the concurrence with and without introducing a local unitary operation on the atom ˆU = ˆσz⊗Iˆc at t= 1 ( ˆIc is the identity operator of the cavity mode). As we can see, after introducing a local operation on the atom, there is an increase in both the trace distance and the concurrence, which manifests the memory effect of the cavity mode. In contrast, if the atom were coupled to a Markovian bath with Hamiltonian given by Eq. (5.9), a local perturbation as ˆσz⊗Iˆc will change neither the trace distance nor the concurrence.
Apart from the change of the trace distance or the concurrence, the change in the dynamical map can also show up in the expectation values of plant dynamical variables. Here, we takehˆσzi ≡Tr[ ˆρσˆz] for illustration. In Fig. 5.4, we show the non-Markovian evolution Tr[ ˆρˆσz] due to the full dynamics described in Eq. (9.54a) and the Markovian evolution Trp[ ˆρpσˆz] from Eq.(5.8) which gives:
hσ˙ˆzi=−2gf(t)[1 +hˆσzi]. (5.10)
Before applying the local operation there is no difference between them, as the reduced dynamics is indistinguishable from the Markovian case. They start to deviate from each other after the local unitary operation.
The second example—this new criterion can also be applied to study the recent experiment by Liu et al.[5]. In their setup, the polarization degree of freedom of photons acts as the plant, and it couples to the frequency degree of freedom which acts as the bath. They jointly undergo the following unitary evolution:
Uˆ(t)|λi ⊗ |ωi=einλωt|λi ⊗ |ωi. (5.11) Hereλ=H, V representing the horizontal and vertical polarization states and|ωiis the frequency
Figure 5.5: (Color online.) The left panel shows the time evolution of the trace distance of directly decoupling at t = 1 by applying a sequence of unitary operations on the plant in a short time interval, or a delay decoupling after applying ˆσz⊗Iˆc in the atom-cavity example; the right panel shows that the maximal recovering is achieved when the atom and cavity mode are disentangled [solid curve is the trace distance (same as left panel) and the dashed curve is the concurrence for the atom-cavity entanglement].
eigenstate. In the experiment, the intial polarization state|φi= (|Hi ± |Vi)/√
2 and the bath state
|χi=R
dω g(ω)|ωi. After evolving for duration t, the joint state is ˆU(t)|φi ⊗ |χiwhich forms an entangled state:
√1 2
Z
dω g(ω)(einHωt|Hi ±einVωt|Vi)⊗ |ωi. (5.12) If we trace out the bath state|ωi, the polarization state starts decohering and its dynamics depends on the actual form ofg(ω). By controllingg(ω), the time evolution of the trace distance can either monotonically decrease or oscillate. This was claimed to be the signature of switching between the Markovian and the non-Markovian regime in Ref. [5]. Interestingly, if we apply the unitary operation ˆσx⊗Iˆωon the polarization attand let it evolve for another timet, the final state is given by ˆU(t)(ˆσx⊗Iˆω) ˆU(t)|φi ⊗ |χiwhich is equal to
|φi ⊗ Z
dω g(ω)ei(nH+nV)ωt|ωi. (5.13)
It is a separable state and the polarization state returns to its initial value |φi, independent of the actual form of g(ω). This indicates that the dynamics is indeed non-Markovian regardless of the initial state of the bath, otherwise we would not recover the plant initial state at later times after applying the local unitary operation.
Dynamical recoverin—the increase of quantum coherence, when the plant is perturbed with local unitary operations, can be important for quantum-information processing. This allows us to recover information concerning the plant that is stored in the bath, which we can call dynamical recovering.
More importantly, by combining local operations with the dynamical decoupling protocols [6–8], we can characterize the bath dynamics by studying the maximal amount of information that we can recover at any given moment, which can help us find the optimal strategies for maintaining quantum
coherence. Take the atom-cavity system for example, in the left panel of Fig. 5.5, we show the time evolution of the trace distance for two different dynamical decoupling procedures—the first one is a direct decoupling att= 1 while the second one is a delayed decoupling after a local operation at t= 1. The maximal difference in the trace distance not only can tell us the memory time scale of the bath but also information about the plant-bath entanglement dynamics. Indeed, we find that in this atom-cavity case, the maximal difference is achieved when the atom becomes disentangled with the cavity mode as shown in right panel of Fig. 5.5. In general, starting from any time t in the evolution, there should exist an optimal sequence of local operations on the plant for maximally recovering its information, which critically depends on the details of the plant-bath interaction.
Operationally, one can also define a measure for non-Markovianity based on such dynamical recovering. We introduce the set of plant state pairs with the initial trace distanceD that is equal to 1: M={{ρˆ1(0),ρˆ2(0)},∀D{ρˆ1(0),ρˆ2(0)}= 1}. Suppose at momentt, D{ˆρ1(t),ρˆ2(t)}=α <1, the measure is:
Nα= max
∀τ,Uτ
D{ΦˆUτρˆ1(t),ΦˆUτρˆ2(t)} −α
1−α , (5.14)
where ˆΦUτρ(t)ˆ ≡Trb[ ˆUp(τ)⊗Iˆbρˆpb(t) ˆUp†(τ)⊗Iˆb] is a sequence of unitary maps for the plant from t to t+τ. Obviously for Markovian systemsN is equal to 0, and for the non-Markovian systems in which the plant can recover its initial states via local operations, the measure is equal to 1. In general, Nα ranges between 0 and 1 depending on how strong the bath memory is, and also the momenttat which we start to apply unitary operations.
Experimental test—this new criterion can be tested experimentally, e.g. using the existing setup of Liuet al.[5]. The first step is the same as the one outlined by Breueret al.[3]—one needs to carry out state tomography of the plant to measure the trace distance as a function of time. If there is an increase of trace distance, one can assert that the dynamics is non-Markovian by using the criterions of Breueret al. However, if the trace distance decreases monotonically, one will need to take the next step by applying different unitary operations onto the plant at different times, and then repeat the tomography procedure to see whether the trace distance increases.