Dynamics and thermodynamics of a central spin immmersed in a spin bath
An exact reduced dynamical map along with its operator sum representation is derived for a central spin interacting with a thermal spin environment. The dynamics of the central spin shows high sustainability of quantum traits like coherence and entanglement in the low temperature regime. However, for sufficiently high temperature and when the number of bath particles approaches the thermodynamic limit, this feature vanishes and the dynamics closely mimics Markovian evolution. The properties of the long time averaged state and the trapped information of the initial state for the central qubit are also investigated in detail, confirming that the non-ergodicity of the dynamics can be attributed to the finite temperature and finite size of the bath. It is shown that if a certain stringent resonance condition is satisfied, the long time averaged state retains quantum coherence, which can have far reaching technological implications in engineering quantum devices. An exact time local master equation of the canonical form is derived . With the help of this master equation, the non-equilibrium properties of the central spin system are studied by investigating the detailed balance condition and irreversible entropy production rate. The result reveals that the central qubit thermalizes only in the limit of very high temperature and large number of bath spins.
pacs:03.65.Yz, 42.50.Lc, 03.65.Ud, 05.30.Rt
In the microscopic world, physical systems are rarely isolated from environmental influence. Systems relevant for implementation of quantum information theoretic and computational tasks like ion traps Cirac and Zoller (1995),quantum dots Loss and DiVincenzo (1998), NMR qubits Cory et al. (1997), polarized photons Knill et al. (2001), Josephson junction qubits Bouchiat et al. (1998) or NV centres Gordon et al. (2013); Doherty et al. (2013) all interact with their respective environments to some extent. Therefore it is necessary to study the properties of open system dynamics for such quantum systems immersed in baths. For quantum systems exposed to usual Markovian baths, their quantumness gradually fades over time, thus negating any advantage gained through the use of quantum protocols over classical ones. Even in thermodynamics, the presence of quantum coherence (Brask and Brunner, 2015; Mitchison et al., 2015) or entanglement (Brunner et al., 2014) enhances the performance of quantum heat machines. Thus, it is imperative to engineer baths in such a way so as to retain nonclassical features of the system for large durations.
Baths can be broadly classified into two different classes, namely Bosonic and Fermionic. Paradigmatic examples for Bosonic baths include the Caldeira-Leggett model Caldeira and Leggett (1981) or the spin Boson model Leggett et al. (1987a).
Lindblad type master equations for these models can be found in the literature Breuer and Petruccione (2002). However, in the Fermionic case, where one models the bath as a collection of a large number of spin- particles,
the situation is generally trickier and one often has to rely on perturbative techniques or time nonlocal master equations Breuer et al. (2004); Fischer and Breuer (2007). Far from from being a theoretical curiosity, the solution of such systems
is of paramount importance in physical situations such as magnetic systems (Parkinson and Farnell, 2010), quantum spin glasses (Rosenbaum, 1996) or superconducting systems (Leggett et al., 1987b).
One specific example of a qubit immersed in a Fermionic bath is the Non-Markovian spin star model (schematic diagram in Fig. 1) Breuer et al. (2004); Fischer and Breuer (2007); Yu et al. (2015); Mandal et al. (2017), which is relevant for quantum computing with NV centre Dutt et al. (2007) defects within a diamond lattice. We show that it is possible to preserve coherence and entanglement in this system for quite a long time by choosing bath parameter values appropriately. Even more interestingly we confirm the presence of quantum coherence in the system for the long time averaged state for certain resonance conditions, which is an utter impossibility for the usual Markovian thermal baths. Such strict and fragile resonance conditions underlie our emphasis on the need for ultra-precise engineering of the bath. We also investigate the amount of information trapped (Smirne et al., 2012) in the central spin system and draw a connection of the same with the process of equilibration.
A time nonlocal integrodifferential master equation was set up for the central spin model using the correlated projection operator technique in Ref. (Fischer and Breuer, 2007). An exact time local master equation for this system was derived in the limit of infinite bath temperature in Ref. Bhattacharya et al. (2017) from the corresponding reduced dynamical map. In this paper, we considerably extend the scope of previous results by deriving the exact reduced dynamics and the exact Lindblad type master equation for arbitrary bath temperature and system bath coupling strength. Our formalism allows us to study the approach towards equilibration in sufficient detail.
The paper is organised as follows. In Section II we introduce the central spin model and find the exact reduced dynamics for the system and the corresponding Kraus operator representation. We use the solution for the exact reduced dynamics to study the evolution of quantum coherence and entanglement. In Section III we study the long time averaged state and its properties. We analyse the resonance condition for the existence of quantum coherence even in the long time averaged state and the phenomenon of information trapping in the central qubit. In Section IV, we begin with the derivation of the exact time-local master equation for this system and use this master equation to investigate the non-equilibrium nature of the dynamics through a thorough study of the deviation from the detailed balance condition as well as the temporal dependence of irreversible entropy production rate. We finally conclude in Section V.
Ii Central spin model and its reduced dynamics
In this section we present the model for the qubit coupled centrally to a thermal spin bath. Then we derive the exact dynamical map for the qubit. We also derive the Kraus operators for the reduced dynamics.
ii.1 The model
We consider a spin- particle interacting uniformly with other mutually non-interacting spin- particles constituting the bath.
The total Hamiltonian for this spin bath model is given by
with () as the Pauli matrices of the i-th spin of the bath and () as the same for the central spin and is the system-bath interaction parameter. Here , and are the system, bath and interaction Hamiltonian respectively. is the number of bath atoms directly interacting with the central spin. The bath frequency and the system-bath interaction strength are both rescaled as and respectively. By the use of collective angular momentum operators for the bath spins (where ), we rewrite the bath and interaction Hamiltonians as
where and are the bosonic annihilation and creation operators with the property . Then the Hamiltonians of Eq. (2) can be rewritten as
ii.2 Dynamical map of the central spin
In the following, we derive the exact reduced dynamical map of the central spin after performing the Schrödinger evolution for the total system and bath and then tracing over the bath degrees of freedom. It is assumed that the initial system bath joint state is a product state , which ensures the complete positivity of the reduced dynamics (Gorini et al., 1976; Lindblad, 1976). The initial bath state is considered as a thermal state , where , and are the Boltzman constant, temperature of the bath and the partition function respectively. Consider the evolution of the state , where is the system excited state and is an arbitrary bath state. After the unitary evolution , let the state is . let us now define two operators and corresponding to the bath Hilbert space such that and . Then we have . Now from the Schrödinger equation , we have
By substituting and , we have
where is the number operator. The operator equations (6) can be straight forwardly solved and the solutions will be functions of and . Then , where . Therefore the evolution of the reduced state of the qubit () can now be found by tracing over the bath modes as
where from the solution of (6), we have and .
Similarly we define and . Following the similar procedure and with the substitution , we find
From the solution of (8), we find
with and . For the off-diagonal component of the reduced density matrix, we have
Therefore the reduced state of the system after the unitary evolution of the joint system-bath state, can be expressed as
where the components of the density matrix are given by
where the partition function is .
ii.3 Operator sum representation
A very important aspect of general quantum evolution, represented by completely positive trace preserving operation is the Kraus operator sum representation, given as . The Kraus operators can be constructed Leung (2003) from the eigenvalues and eigenvectors of the corresponding Choi-Jamiolkowski (CJ) state Choi (1975); Leung (2003). The CJ state for a dynamical map acting on a dimensional system is given by , with being the maximally entangled state in dimension. For the particular evolution considered here, we find the CJ state to be
From the eigensystem of the CJ state given in (15), we derive the Kraus operators as
One can check that the Kraus operators satisfy the condition .
ii.4 Coherence and Entanglement dynamics of the central spin
Having obtained the exact reduced dynamics of the central spin, in the following we study the temporal variation of non-classical properties, viz. quantum coherence and entanglement of the system. It is well known that for usual Markovian systems, such non-classical quantities decay monotonically over time and eventually disappear (Baumgratz et al., 2014; Chanda and Bhattacharya, 2016; Yu and Eberly, 2004). However, the central spin system is strongly non-Markovian in nature and therefore, a natural and pertinent question is to ask whether it is possible to preserve quantum features for long periods of time for this system. The following subsections are devoted to answering that question for various parameter regimes of the spin bath model.
Quantum Coherence: In this article we consider -norm of coherence as a quantifier of quantum coherence. For a qubit system, the -norm of coherence Baumgratz et al. (2014) is simply given by twice the absolute value of any off-diagonal element, i.e., . The evolution of coherence is then given by
This is a straightforward scaling of the initial quantum coherence. One immediate consequence is that we cannot create coherence over and above the coherence present in the system initially, even though this is a strongly non-Markovian system. In subsequent analysis, we can thus take the initial coherence to be unity, i.e. the maximally coherent state without loss of generality.
Quantum Entanglement: Operationally, quantum entanglement is the most useful resource in quantum information theory (Ekert, 1991; Shor, 1997; Bennett and Wiesner, 1992; Bennett et al., 1993; Horodecki et al., 2009). However, it is also a fragile one (Yu and Eberly, 2009) and decays quite quickly for Markovian evolution (Yu and Eberly, 2004). We suppose a scenario in which the central spin qubit is initially entangled to an ancilla qubit in addition to the spin bath. There is no subsequent interaction between the ancilla qubit and the central spin. Our goal is to investigate the entanglement dynamics of the joint two-qubit state . From the factorization theorem for quantum entanglement Konrad et al. (2008), we have
where is the CJ State in (15) and the entanglement measure is concurrence Hill and Wootters (1997). Concurrence of a two qubit system is given as where are the square roots of the eigenvalues of in decreasing order, . Here the complex conjugation is taken in the computational basis, and is the Pauli spin matrix. From now on, we mean concurrence by entanglement throughout the paper. Then the entanglement of the CJ state can be written as . Since the initial entanglement is simply a constant scaling term, we take this to be unity, i.e. consider a maximally entangled initial state without loss of generality and study the subsequent dynamics.
We now present the results for time evolution of quantum coherence and entanglement with the bath temperature , the strength of system-bath interaction and number of spins () in the spin bath attached to the central spin. If the spin bath is in a very high temperature, we expect the thermal noise to swamp signatures of quantumness, which is broadly confirmed in Fig. 2(a) and 2(d). However, small fluctuations in quantum coherence continue to occur testifying to the non-Markovianity of the dynamics. On the contrary, for low bath temperature, as demonstrated in Fig. 2(a), quantum coherence does not decay noticeably and for the timespan we considered, it does not dip below a certain value that is in itself quite high. For intermediate temperatures, coherence broadly decays with increasing decay rate as we increase the bath temperature, but along with small fluctuations due to non-Markovianity. The dynamics of entanglement as shown in Fig. 2(d), is quite similar to that of coherence. At the high temperature limit, the difference with dynamics for quantum coherence lies in the fact that entanglement encounters a sudden death and never revives. This is entirely consistent with the usual observation for many physical systems where quantum coherence turns out to be more robust against noise than entanglement Pei et al. (2011); Cianciaruso et al. (2015); Bromley et al. (2015). In the opposite regime, for low enough temperatures, entanglement dynamics is very much similar to that of coherence. Another parameter we can tune is the system-bath interaction strength , which depending upon the species of the central spin as well as the bath spins, may differ. In case the interaction parameter is too small, the system evolves almost independently from the bath and therefore the coherence and entanglement of the system decay quite slowly as shown in Fig. 2(b) and 2(e). In the opposite limit, if the system-bath interaction is comparable to the energy difference of the spin levels of the central spin, we observe a rapid decay in quantum coherence with the presence of usual non-Markovian fluctuations. Whereas, entanglement decays to zero almost immediately with no revival detected in the time span considered in Fig. 2(e). Eq. (13) also allows us to study the dynamics of coherence for varying number of bath spins. If the number of spins in the bath is large, we observe from Fig. 2(c), that the coherence rapidly decays and only small fluctuations are subsequently detected. In case the number of spins in the bath is not very large, the evolution of coherence undergoes periodic revivals. The magnitude of such revivals decreases with increasing bath size, eventually reducing to being indistinguishable with smaller fluctuations for large enough number of spins in the bath. As seen in Fig. 2(c), revivals themselves occur in periodic packets, magnitudes of which decrease steadily with time. On the other hand, if the number of bath particles is quite large, entanglement decays very quickly to zero. However for smaller number of spins in the bath, the entanglement dynamics depicted in Fig. 2(f) is quite similar to the corresponding dynamics of coherence captured earlier in Fig. 2(c).
Iii Analysis of time averaged dynamical map
In this section we probe the behaviour of long time averaged state of the central spin qubit. We study under what condition the long time averaged state is coherent. We further investigate whether or under what conditions the long time averaged state is a true fixed point of the dynamical map, i.e. independent of initial condition. In connection to that we further study what role the finite size of the environment plays in this context. The long time averaged state of the central spin qubit is given by
Following this definition, we find
where , and are long time averages of , and respectively. When we integrate a bounded periodic function over a long time and divide by the total time elapsed, we can consider the integral being over a large integer number of periods without loss of generality. Now,
where the result follows from the fact that average of over any integer number of time periods = . Similarly we get
The equation for population dynamics shows Eq (III) that even the very long time averaged state retains the memory of the initial state, which is a signature of the system being strongly non-Markovian. This initial state dependence is captured in Fig. 3(a). It is observed that the parameter which captures the population distribution for long time averaged state is heavily dependent on the initial ground state population. If the initial population of the ground state increases, so does the population of the ground state for long time averaged state. However, in case the bath is very large, the population statistics for the long time averaged state is markedly less sensitive to the initial population. This leads us to posit that the only true fixed point independent of the initial conditions for this system exists only in the limit . We also observe that in the limit , tends towards 1 regardless of bath size indicating the dynamics is almost unital. Also we should mention that in the thermodynamic limit (), when the temperature of the bath is infinite, the state is not only the fixed point of the dynamics but the canonical equilibrium state also. Thus we can conclude that in the limit and , the present open system dynamics is ergodic. Moreover, we see that the system-bath coupling strength not only affects the timescale of evolution but also plays significant role in the population statistics of the time averaged state. This we can see from Eq.s (20) and (21), which is also depicted in Fig. 3(b). Also for most of the cases, we have . It is interesting to note that the long-time averaged state is incoherent in general. This implies, even though quantum coherence or entanglement persists for quite a long time if the bath temperature is very low, as depicted in Fig. 2(a) or Fig. 2(d) respectively, they must eventually decay. It is important to mention that there are specific resonance conditions under which can have finite value, which will be analysed in the following section.
iii.1 Resonance Condition for long lived quantum coherence
We have mentioned previously that the long time averaged state is in general diagonal, but for very specific choices of parameter values, this is not true and there indeed is long lived quantum coherence even in the long time averaged state. This can be of significant interest for theoretical and experimental purposes. For the off-diagonal component, the real and imaginary parts of , defined as and respectively equals to
We always have
For each of the rest of the terms, it can be shown that the criteria for non-zero time averaged coherence reads
For the condition to hold, it is easily shown that
This, given that and are usually of the same order of magntitude, we feel is a rather unrealistic demand on N, since we are concerned with a heat bath, albeit finite sized. We thus concentrate on the other condition . The equation can be explicitly expanded out and the following quadratic equation in is obtained
By solving this quadratic equation and noting that the value of must be an integer, we reach the following equation, which is the resonance condition.
where is the set of positive integers . Taking and in the limit , we have the resonance condition as
Thus, if we are interested in obtaining non zero amount of quantum coherence in the long time averaged state, we have to tune the interaction parameter exactly in such a way that is a positive integer. This is a nice example where precise bath engineering can help us achieve long sustained coherence.
iii.2 Information trapping in the Central Spin System
Let us now investigate whether or under what condition the dynamical map considered here does have a true fixed point; i.e. the existence of a state which is invariant under the particular dynamics. In order to do that, define the time-averaging map as the map which takes any initial state to the corresponding time averaged state as given by Eq. (III). Now suppose the system is initially in a state . Then a natural question to ask is the following - “Is the corresponding time averaged state invariant under the map ?” This can only happen when the map is an idempotent one, i.e. .
Clearly, if the time averaged state did not retain the memory of the initial state, this would be the case. Therefore the deviation from idempotence of the map can serve as a useful measure of the initial state dependence of the system in the long run, which is termed as Information Trapping Smirne et al. (2012) and defined by
where D[.,.] is a suitable distance measure on the Hilbert space of the system. Choosing the trace norm as our distance measure, the expression for in the central spin model is computed as
We immediately note that this quantity vanishes iff , which is the case only in the limit , i.e. the thermodynamic and high temperature limit. The above statement is confirmed in Fig. 4. As we increase the temperature of the bath, the trapped information asymptotically vanishes. It is also observed that at any given temperature, the amount of information trapped is greater for a smaller sized bath. This is consistent with the observation that a very large bath is required for to vanish.
Iv Canonical master equation and the process of equilibration
Finding the generator of a general dynamical evolution of a quantum system is one of the fundamental problems in the theory of open quantum systems, which leads to a better understanding of the actual nature of decoherence. It is our aim here to derive a canonical master equation without resorting to weak coupling and Born-Markov approximation for the reduced dynamics presented in Eq. (12), by virtue of which we will later analyse various thermodynamic aspects of the qubit system. Using the formalism of (Andersson et al., 2007), we obtain the following exact time local master equation for the central spin in the Lindblad form.
where , and are the rates of dissipation, absorption and dephasing processes respectively, and corresponds to the unitary evolution, respectively, given as
For the detailed derivation of the master equation, one can look into the Refs. (Bhattacharya et al., 2017; Andersson et al., 2007). Note that the system environment interaction generates a time dependent Hamiltonian evolution in the form of . This is analogous to the Lamb-shift correction in the unitary part of the evolution. Complete positivity Breuer et al. (2016); Rivas et al. (2014); Usha Devi et al. (2011, 2012); Laine et al. (2010); Rivas et al. (2010) is one of the important properties of a general quantum evolution, following the argument that for any valid quantum dynamical map, the positivity must be preserved if the map is acting on a system which is correlated to an ancilla of any possible dimension. For a Lindblad type evolution, this is guaranteed by the condition Chruściński and Kossakowski (2010), which can be easily verified for the specific decay rates given in (30). However since the dynamical map here is derived starting from an initial product state, complete positivity is always guaranteed Alicki (1979); Pechukas (1994).
iv.1 The principle of detailed balance
Here we investigate the process of approach towards steady state for the open system dynamics considered in this paper. There are various different approaches to explore the process of equilibration in an open system dynamics, each of which has their own merit (Gogolin and Eisert, 2016). In this work we carry out this investigation for the specific system considered here from a few different aspects, one of which is the quantum detailed balance first introduced by Boltzmann, who used it to prove the famous H-theorem (LIFSHITZ and PITAEVSKI, 1981). When two or more irreversible processes occur simultaneously, they naturally interfere with each other. If due to the interplay between those different processes, over a sufficient period of evolution time, a certain balance condition between them is reached, then the system reaches a steady state. Consider the Pauli master equation for the atom undergoing such processes (Breuer and Petruccione, 2002) given by
where is the diagonal matrix element of the density operator and is the transition probability for the process . The well known detailed balance condition Kubo (1957); Martin and Schwinger (1959) for Pauli master equation is given as , where is diagonal density matrix element at the steady state. We first derive a rate equation of the form of Eq.(31) from the master equation (29) in order to study the detailed balance for our particular system (Kawamoto and Hatano, 2011; Esposito and Mukamel, 2006). Let us consider the unitary matrix , which diagonalizes the system density matrix () as . Then we can straightforwardly derive the equation of motion for the diagonalized density matrix as
where . Considering , we get the rate equation similar to the Pauli equation as