Stabilizing Open Quantum Systems by Markovian Reservoir Engineering

Stabilizing Open Quantum Systems by Markovian Reservoir Engineering

S. G. Schirmer sgs29@cam.ac.uk Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom    Xiaoting Wang xw233@cam.ac.uk Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom
July 14, 2019
Abstract

We study open quantum systems whose evolution is governed by a master equation of Kossakowski-Gorini-Sudarshan-Lindblad type and give a characterization of the convex set of steady states of such systems based on the generalized Bloch representation. It is shown that an isolated steady state of the Bloch equation cannot be a center, i.e., that the existence of a unique steady state implies attractivity and global asymptotic stability. Necessary and sufficient conditions for the existence of a unique steady state are derived and applied to different physical models including two- and four-level atoms, (truncated) harmonic oscillators, and composite and decomposable systems. It is shown how these criteria could be exploited in principle for quantum reservoir engineeing via coherent control and direct feedback to stabilize the system to a desired steady state. We also discuss the question of limit points of the dynamics. Despite the non-existence of isolated centers, open quantum systems can have nontrivial invariant sets. These invariant sets are center manifolds that arise when the Bloch superoperator has purely imaginary eigenvalues and are closely related to decoherence-free subspaces.

pacs:
03.65.Yz,42.50.-p,42.50.Dv

I Introduction

The dynamics of open quantum systems and especially the possibility of controlling it have attracted significant interest recently. One of the fundamental tasks of interest is the stabilization of quantum states in the presence of dissipation. In recent years a large number of articles have been published on control of closed quantum systems or, more precisely, on systems that only interact coherently with a controller, with applications from quantum chemistry to quantum computing Schirmer (2006). The essential idea in most of these articles is open-loop Hamiltonian engineering by applying control theory and optimization techniques. Although open-loop control design is a very important tool for controlling quantum dynamics, it has limitations. For instance, while open-loop Hamiltonian engineering can be used to mitigate the effects of decoherence, e.g., using dynamic decoupling schemes Viola et al. (1999), or to implement quantum operations on logical qubits, protected against errors due to environmental interactions by a redundant encoding Nigmatullin and Schirmer (2009), Hamiltonian engineering has intrinsic limitations. One task that is difficult to achieve using Hamiltonian engineering alone is stabilization of quantum states.

Alternatively, we can try to engineer open quantum dynamics described by a Lindblad master equation Gorini et al. (1976); Lindblad (1976) by changing not only the Hamiltonian terms but also the dissipative terms. Various ideas along these lines have been proposed in several articles Wiseman (1994); Wang and Wiseman (2001); Wang et al. (2005a); Wang and Wiseman (2005); Carvalho and Hope (2007); Carvalho et al. (2008). There are two major sources of dissipative terms in the Lindblad equation: the interaction of the system with its environment, and measurements we choose to perform on the system. Accordingly, we can engineer the open dynamics by either modifying the system’s reservoir or by applying a carefully-designed quantum measurement. In this sense, the quantum Zeno effect is a simple model for reservoir engineering Misra and Sudarshan (1977). In addition, the open dynamics can be modified by feeding the measurement outcome (e.g. the photocurrent from homodyne detection) back to the controller. This idea was first proposed in Wiseman (1994), where a feedback-modified master equation was derived and it was shown in Wang and Wiseman (2001) that such direct feedback could be used to stabilize arbitrary single qubit states with respect to a rotating frame. More recently, there have been several attempts to extend this work to stabilize maximally entangled states using direct feedback Wang and Wiseman (2001); Wang et al. (2005a); Wang and Wiseman (2005); Carvalho and Hope (2007); Carvalho et al. (2008). The idea of reservoir engineering can also be used to stabilize the system in the decoherence-free subspace (DFS) Lidar and Whaley (2003). In Beige et al. (2000), it is illustrated that atoms in a cavity can be entangled and driven into a DFS. In Kraus et al. (2008), several interesting physical examples are presented showing how to design the open dynamics such that the system can be stabilized in the desired dark state.

Such stabilization problems are a motivation for thorough investigation of the properties of a Lindblad master equation. Important questions include, for instance, which states can be stabilized given a certain general evolution of the system and certain resources. There are a number of classical articles discussing the stationary states and their (asymptotic) stability, as well as sufficient conditions for the existence of a unique stationary state Spohn (1976, 1977); Frigerio (1977, 1978); Davies (1970); Evans (1977). More recently, a detailed analysis of the structure of the Hilbert space with respect to the Lindblad dynamics was carried out in Baumgartner et al. (2008a); Baumgartner and Narnhofer (2008), implying that all stationary states are contained in a subspace of the Hilbert space that is attractive. Necessary and sufficient conditions for the attractivity of a subspace or a subsystem have been further considered in Ticozzi and Viola (2008a); arxiv.0809.0613. Nonetheless there are still important issues that deserve further study. One is the issue of asymptotic stability of stationary states. It is often assumed that uniqueness implies attractivity of a steady state. Although this turns out to be true for the Lindblad equation, it does not follow trivially from the linearity of the master equation, and a rigorous derivation of this result is therefore desirable, as is a summary of various sufficient conditions for ensuring uniqueness of a stationary state. Similarly, linear dynamical systems can have invariant sets or center manifolds surrounding the set of steady states. The existence of such invariant sets usually precludes converges of the system to a steady state, but criteria for the existence of non-trivial invariant sets are also of interest as they are natural decoherence-free subspaces. Finally, many investigations of the steady states have been based on considering the dynamics on the Hilbert space of the system, e.g., giving criteria for the attractivity of a subspace of the Hilbert space. However, since the steady states are points in the convex set of positive operators on this Hilbert space, such criteria are not always useful. For instance, only systems with steady states at the boundary of the state space (e.g., pure states) have (non-trivial) attractive subspaces of the Hilbert space. While these states may be of special interest, since the states at the boundary form a set of measure zero, most systems will have steady states in the interior. We may not be able to engineer a steady state at the boundary, but perhaps we could stabilize a state arbitrarily close to it, which may be entirely sufficient for practical purposes. Thus, complete characterization of the steady states requires considering the set of positive operators on the Hilbert space rather than the Hilbert space itself.

The purpose of this article is twofold: (i) to further investigate the properties of the stationary states of the Lindblad dynamics and the invariant set of the dynamics generated by imaginary eigenvalues, including the relationship between uniqueness and asymptotic stability and (ii) to present several sufficient conditions for the existence of a unique steady state, apply them to different physical models, and show how these criteria could in principle be used to stabilize an arbitrary quantum state using Hamiltonian and reservoir engineering. In Sec. II, we introduce the Bloch representation of Lindblad dynamics, which will be used throughout the article. In this representation, the spectrum of the dynamics can be easily derived and stability analysis can be conveniently presented. In Sec. III, we characterize the set of all stationary states as a convex set generated by a finite number of extremal points, analyze the properties of the extremal points and give several sufficient conditions for the uniqueness of the stationary state. We also state a theorem that uniqueness implies attractivity, which is proved in the appendix. In Sec. IV these conditions are applied to different systems including two and four-level atoms, the quantum harmonic oscillator, and composite and decomposable systems, and several useful results are derived, including: (i) if the Lindblad terms include the annihilation operator, then the system has a unique stationary state regardless of the other Lindblad terms or the Hamiltonian; (ii) for a composite system, if the Lindblad equation contains dissipation terms corresponding to annihilation operators for each subsystem, then the stationary state is also unique; (iii) how any pure or mixed state can be stabilized in principle via Hamiltonian and reservoir engineering. Finally, in Sec. V, we discuss the invariant set generated by the eigenstates of the dynamics with purely imaginary eigenvalues, and its relation to decoherence-free subspaces (DFS), including examples how to find or design a DFS.

Ii Bloch Representation of Open Quantum System Dynamics

Under certain conditions the evolution of a quantum system interacting with its environment can be described by a quantum dynamical semigroup and shown to satisfy a Lindblad master equation

(1)

where is positive unit-trace operator on the system’s Hilbert space representing the state of the system, is a Hermitian operator on representing the Hamiltonian, is the commutator, and , where are operators on and

(2)

In this work we will consider only open quantum systems governed by a Lindblad master equation, evolving on a finite-dimensional Hilbert space .

From a mathematical point of view Eq. (1) is a complex matrix differential equation (DE). To use dynamical systems tools to study its stationary solutions and the stability, it is desirable to find a real representation for (1) by choosing an orthonormal basis for all Hermitian matrices on . Although any orthonormal basis will do, we shall use the generalized Pauli matrices, suitably normalized, setting , and , where

(3a)
(3b)
(3c)

The state of the system can then be represented as a real vector of coordinates with respect to this basis ,

and the Lindblad dynamics (1) rewritten as a real DE:

(4)

where , are real matrices with entries

(5a)
(5b)

being the usual anticommutator. As , we have , and (5) can be reduced to the dynamics on an -dimensional subspace,

(6)

This is an affine-linear matrix DE in the state vector . is an real matrix with and a real column vector with . Notice that this essentially is the -dimensional generalization of the standard Bloch equation for a two-level system, and we will henceforth refer to as the Bloch operator. The advantage of this representation is that all information of and is contained in and and it is easy to perform a stability analysis of the Lindblad dynamics in matrix-vector form Wang and Schirmer (2009a). Defining , we have the following relation:

Since for any physical state , the Bloch vector must satisfy , i.e. all physical states lie in a ball of radius . Note that for the embedding into of the physical states into this ball is surjective, i.e., the set of physical states is the entire Bloch ball, but this is no longer true for .

Iii Characterization of the Stationary States

A state is a steady or stationary state of a dynamical system if . Steady states are interesting both from a dynamical systems point of view, as well as for applications such as stabilizing the system in a desired state. Let be the set of steady states for the dynamics given by (1). As (1) is linear in , inherits the property of convexity from the set of all quantum states. includes special cases such as the so-called dark states, which are pure states satisfying . For some systems it is easy to see that there are steady states, and what these are. For a Hamiltonian system () it is obvious from Eq. (1), for instance, that the steady states are those that commute with the Hamiltonian, i.e., . Similarly, for a system with subject to measurement of the Hermitian observable , the master equation (1) can be rewritten as , and we can show that . In general, assuming is the Bloch vector associated with a particular steady state, the set of steady states for a system governed by a LME (1) can be written as in the Bloch representation. This is a convex subset of the affine hyperplane in , where satisfies . Moreover, using Brouwer’s Fixed Point Theorem, we can show that the set of steady states is always non-empty (see Appendix A) and we have:

Proposition 1.

The Lindblad master equation (1) always has a steady state, i.e., the Bloch equation always has a solution and , where is the matrix horizontally concatenated by the column vector .

As any convex set is the convex hull of its extremal points, we would like to characterize the extremal points of . A point in a convex set is called extremal if it cannot be written as a convex combination of any other points. See Fig. 1 for illustration of convex sets and extremal points. To this end, let be the smallest subspace of such that , where is the projector onto the subspace and is the projector onto the orthogonal complement of in .

Figure 1: (Color online) (a) Non-convex set as a line segment connecting two points in the set is not contained in the set. (b) Convex set spanned by five unique extremal points given by the vertices of the polygon. (c) Convex set with infinitely many extremal points comprising the entire boundary.
Proposition 2.

The steady state of is extremal if and only if it is the unique steady state in its support.

Proof.

Since any convex set is the convex hull of its extremal points, the rank of the extremal point is the smallest among its neighboring points, and the rank of boundary points is smaller than that of points in the interior. Suppose that besides the extremal steady state , there is another steady state in the subspace . Then any state which is a convex combination of and must also be in . However, since is an extremal point, the rank of , which is equal to the dimension of , must be lower than the rank of , which is impossible. Conversely, let be the unique steady state in its support. Suppose it is not an extremal point, which means that there exist and with , . From Lemma 1 in Appendix B, and also lie in , a contradiction to uniqueness of steady states in . ∎

We call a subspace invariant if any dynamical flow with initial state in remains in . It has been shown that if is a steady state then is invariant Baumgartner and Narnhofer (2008); arxiv.0809.0613. Furthermore, Proposition 1 shows that any invariant subspace contains at least one steady state. Thus, if is an extremal point of then is a minimal invariant subspace of the Hilbert space , i.e., there does not exist a proper subspace of that is invariant under the dynamics. It can also be shown that is attractive as a subspace of , and has been called a minimal collecting subspace in Baumgartner and Narnhofer (2008).

Different extremal steady states generally do not have orthogonal supports. For example, for a two level-system governed by the trivial Hamiltonian dynamics , is equal to the convex set of all states on , all pure states are extremal points, and it is easy to see that two arbitrary pure states generally do not have orthogonal supports. Just consider the pure states and , which are extremal states but . However, in this case there is another extremal steady state with and . In general, given two extremal steady states and , we have either , or there exists another extremal steady state with such that . That is to say, given an extremal steady state , if there exist other steady states, then we can always find another extremal steady state whose support is orthogonal to that of , . Finally, let be the union of the supports of all steady states . It can be shown (see, e.g., Baumgartner and Narnhofer (2008)) that we can choose a finite number of extremal steady states with orthogonal supports, such that . This decomposition is generally not unique, however. In the above example, any two orthonormal vectors of provide a valid decomposition of , and no basis is preferable. Therefore, such a decomposition of is not necessarily physically meaningful, but it does give the following useful result:

Proposition 3.

If a system governed by a LME (1) has two steady states, then there exist two proper orthogonal subspaces of that are both invariant.

In addition to the characterization of from the supports of its extremal points, it is also useful to characterize the steady states from the structure of the dynamical operators and in the LME (1).

Proposition 4.

If is a steady state at the boundary then its support is an invariant subspace for each of the Lindblad operators .

Proof.

A density operator belongs to the boundary of if it has zero eigenvalues, i.e., if . In this case, there exists a unitary operator such that

(7)

where is an matrix with full rank, and and are and matrices with zero entries and , and

(8)

with and . Partitioning

(9)

accordingly, it can be verified that a necessary and sufficient condition for to be a steady state of the system is that , where

(10a)
(10b)
(10c)

Since is a positive operator with full rank and hence strictly positive, the third equation requires for all . The second equation is for , which shows that it will be satisfied if and only if the matrix vanishes identically, which gives the equivalent conditions

(11a)
(11b)
(11c)

The last equation implies that if is a steady state at the boundary then all have a block tridiagonal structure and map operators defined on to operators on , i.e., is an invariant subspace for all . ∎

The following theorem (proved in Appendix C) shows furthermore that uniqueness implies asymptotic stability:

Theorem 1.

A steady state of the LME (1) is attractive, i.e., all other solutions converge to it, if and only if it is unique.

The fact that only isolated steady states can be attractive restricts the systems that admit attractive steady states. In particular, if there are two (or more) orthogonal subspaces of the Hilbert space , which are invariant under the dynamics, i.e., for , then the dynamics restricted to either invariant subspace must have at least one steady state on the subspace, and the set of steady states must contain the convex hull of the steady states on the subspaces. Thus we have:

Corollary 1.

A system governed by LME (1) does not have a globally asymptotically stable equilibrium if there are two (or more) orthogonal subspaces of the Hilbert space that are invariant under the dynamics.

The previous results give several equivalent useful sufficient conditions to ensure uniqueness of a steady state.

Condition 1.

Given a system governed by a LME (1) with an extremal steady state , if there is no subspace orthogonal to that is invariant under all then is the unique steady state.

We compare Condition 1 with Theorem 2 in Kraus et al. (2008), which asserts that if there exists no other subspace that is invariant under all orthogonal to the set of dark states, then the only steady states are the dark states. To prove that a given dark state is the unique stationay state, Theorem 2 in Kraus et al. (2008) requires that we show (i) uniqueness of the dark state, and (ii) that there exists no other orthogonal invariant subspace. Since the dark states defined in Kraus et al. (2008) are extremal steady states, Condition 1 shows that (ii) is actually sufficient in that it implies uniqueness and hence attractivity of the steady state.

Condition 2.

If there is no proper subspace of that is invariant under all Lindblad generators then the system has a unique steady state in the interior 111Proper subspace means we are excluding the trivial cases and ..

Equation (11) also shows that if there are two orthogonal proper subspaces of the Hilbert space that are invariant under the dynamics, then and there exists a basis such that

for all , and , i.e., in particular both subspaces are invariant for all . Hence, if there are no two orthogonal proper subspaces of that are simultaneously invariant for all , then the system does not admit orthogonal proper subspaces that are invariant under the dynamics. Thus we have:

Condition 3.

If there do not exist two orthogonal proper subspaces of that are simultaneously invariant for all then the system has a unique fixed point, either at the boundary or in the interior.

The following applications show that these conditions are very useful to show attractivity of a steady state.

Iv Applications

iv.1 Two and Four-level Atoms

Let us start with the simplest example, a two-level atom governed by the Lindblad master equation

with . This model describes a two-level atom subject to spontaneous emission, or a two-level atom interacting with a heavily damped cavity field after adiabatically eliminating the cavity mode. Noting that the Lindblad operator corresponds to a Jordan matrix , the previous results guarantee that this system has a unique (attractive) steady state. More interestingly, the previous results still guarantee the existence of a unique steady state if the atom is damped by a bath of harmonic oscillators

where is the average photon number. It suffices that one of the Lindblad term corresponds to an indecomposable Jordan matrix. In this simple case we can also infer the uniqueness of the steady state directly from the Bloch representation. We can decompose the Bloch matrix into an antisymmetric matrix corresponding to the Hamiltonian part of the evolution and a diagonal and negative-definite matrix . Since for any , it follows that is invertible and the Bloch equation has a unique attractive stationary state.

On the other hand, if the atom is subjected to a continuous weak measurement such as then we can easily verify that the pure states and are steady states. Hence, there are infinitely many steady states given by the convex hull of these extremal points, with . Of course, this is the well-known case of a depolarizing channel, which contracts the entire Bloch ball to the axis, which is the measurement axis.

In the previous examples uniqueness of the steady state followed from similarity of at least one Lindblad operator to an (indecomposable) Jordan matrix. When is decomposable then the last example shows that the system can have infinitely many steady states, but similarity of a Lindblad operator to an indecomposable Jordan matrix is only a sufficient condition, i.e., it is not necessary for the existence of a unique steady state. If has two or more Jordan blocks, for example, then each Jordan block defines an invariant subspace, but provided these subspaces are not orthogonal to each other, Condition 3 still applies, ensuring the uniqueness of the steady state.

For instance, a system governed by a LME with , and

has a unique steady state because, although has two eigenvalues and and two proper eigenvectors, the respective eigenspaces are not orthogonal and there are no two orthogonal subspaces that are invariant under . Perhaps more interestingly, for a system with a nontrivial Hamiltonian, e.g., , uniqueness of the steady state can often be guaranteed even if has two (or more) orthogonal invariant subspaces, if suitably mixes the invariant subspaces.

Consider a four-level system with energy levels as illustrated in Fig. 2 and spontaneous emission rates , and satisfying . This is a simple model for a laser. To derive stimulated emission we require population inversion, a cavity and a gain medium composed of many atoms. For simplicity, we only consider one atom and try to describe the dynamics in the time scale such that the spontaneous decay can be neglected. On this scale the Hamiltonian optical-pumping term and the spontaneous decay term are:

There are two invariant subspaces under and : and . Hence, when , we have two metastable states and in addition to the ground state , which is a steady state. However, for the pumping Hamiltonian mixes up those two invariant subspaces, and through calculation we can easily find the unique steady state: . Thus, on the time scales considered, population inversion between states and can be realized, but eventually spontaneous emission from to will kick in, resulting in the stimulated emission characteristic of a laser. (Of course, this is only the first stage of the whole process and it is far from the threshold of the laser.)

Figure 2: Schematic plot of the four energy levels of one atom in a prototype system for a laser. The atom is pumped by an external field. The spontaneous decay rates satisfy . On the time scale when decay from level to can be ignored is the unique steady state of the system, realizing the population inversion.

This is just one example of optical pumping, a technique widely used for state preparation in quantum optics. Although the principle of optical pumping is easy to understand intuitively for simple systems in that population cannot accumulate in energy levels being pumped, forcing the population to accummulate in states state not being pumped and not decaying to other states, it can be difficult to intuitively understand the dynamics in less straightforward cases. For example, what would happen if we applied an additional laser field coupling and . Would the system still have a unique steady state? If so, what is the steady state? These questions are not easy to answer based on intuition, but we can very easily answer them using the mathematical formalism developed, especially the Bloch equation. In fact, we easily verify that the system

with has a unique steady state

independent of and , provided . For this state becomes , as intuition suggests.

iv.2 Quantum Harmonic Oscillator

The harmonic oscillator plays an important role as a model for a wide range of physical systems from photon fields in cavities, to nano-mechanical oscillators, to bosons in the Bose-Hubbard model for cold atoms in optical lattices. Although strictly speaking the harmonic oscillator is defined on an infinite-dimensional Hilbert space, the dynamics can often be restricted to a finite-dimensional subspace. For many interesting quantum processes the average energy of the system is finite and we can truncate the number of Fock states from to a large but finite number. In many quantum optics experiments, for example, the intracavity field contains only a few photons, or has a number of photons in some finite range if it is driven by a field with limited intensity. In such cases the truncated harmonic oscillator is a good model for the underlying physical system provided is large enough, and we can apply the previous results about stationary solutions and asymptotic stability.

Consider a harmonic oscillator with where is the annihilation operator of the system, which on the truncated Hilbert space with , takes the form

(12)

If there is a Lindblad term of the form then we can infer from the previous analysis that the system has a unique and hence asymptotically stable steady state, regardless of whatever Hamiltonian control or interaction terms or other Lindblad terms are present. To see this note that the matrix representation of is mathematically similar to the Jordan matrix

(13)

It is easy to verify that has a sole proper eigenvector whose generalized eigenspace is all of and thus does not admit two orthogonal proper invariant subspaces. Hence we can conclude from Condition 3 that for any dynamics governed by a LME (1) with a dissipation term , there is always a unique stationary solution to which any initial state will converge. In general, if (1) contains a Lindblad term with similar to a Jordan matrix , then (1) always has a unique stationary state, no matter what the other terms are. For example, the Lindblad equation for a damped cavity driven by a classical coherent field is

showing that the system has a unique steady state. For the steady state is

with real, , and . When , i.e. there is no driving field, we get is the ground (vacuum) state, as one would expect for a damped cavity, while for a nonzero driving field we stabilize a mixed state in the interior.

iv.3 Composite Systems

Many physical systems are composed of subsystems, each interacting with its environment, inducing dissipation. For example, consider two-level atoms in a damped cavity driven by a coherent external field. Assuming the atom-atom and atom-cavity interactions are not too strong, and the main sources of dissipation are independent decay of atoms and the cavity mode, respectively, we obtain the Lindblad terms , , and in the LME (1), where is the decay operator for the th atom and is the annihilation operator of the cavity. Simulations suggest systems of this type always have a unique steady state, and this can be rigorously shown using the sufficient conditions derived.

A composite quantum system whose evolution is governed by a LME containing terms involving annihilation operators for each subsystem has a unique steady state, regardless of the Hamiltonian and any other Lindblad terms that may be present. This property can be inferred from Condition 3. Assume the full system is composed of subsystems with Lindblad terms , and let be an invariant subspace for all . Then must contain the ground state of the composite system as for all . Hence, any simultaneously -invariant subspace must contain the state and there cannot exist two orthogonal proper subspaces of that are invariant under all . By Condition 3, the system has a unique steady state.

Thus, a system of atoms in a damped cavity subject to a Lindblad master equation

has a unique steady state, regardless of the Hamiltonian . The steady state need not be , however. In general, this will only be the case if is an eigenstate of . Similarly, the presence of the two dissipation terms in the LME for the two-atom model in Wang et al. (2005a)

with ensures that there is a unique steady state provided . This is no longer the case for . In particular, in the regime where and the last two terms can be neglected as in Wang et al. (2005a), the reduced dynamics no longer has a unique steady state.

iv.4 Decomposable Systems

Figure 3: Schematic diagram of an atomic system under direct quantum feedback control. A laser beam is split to generate the local oscillator for homodyne detection as well as the driving field which is then modulated by the feedback photocurrent to derive the final field .

A system is decomposable if there exists a decomposition of the Hilbert space such that for any where is an (unnormalized) density operator on . Decomposable systems cannot have asymptotically stable (attractive) steady states by Corollary 1.

One class of systems that are always decomposable and hence never admit attractive steady states, are systems governed by a LME (1) with a single Lindblad operator that is normal, i.e., , and commutes with the Hamiltonian. This is easy to see. Normal operators are diagonalizable, i.e., there exists a unitary operator such that with diagonal, and since , we can choose such that it also diagonalizes . Thus the system is fully decomposable, and it is easy to see in this case that every joint eigenstate of and is a steady state, and therefore there exists a steady-state manifold spanned by the convex hull of the projectors onto the joint eigenstates of and . In the absence of degenerate eigenvalues this manifold is exactly the dimensional subspace of consisting of operators diagonal in the joint eigenbasis of and .

A more interesting example of a physical system that is decomposable, and thus does not admit an attractive steady state, is a system of indistinguishable two-level atoms in a cavity subject to collective decay, and possibly collective control of the atoms as well as collective homodyne detection of photons emitted from the cavity, as illustrated in Fig. 3. Let be the single-qubit annihilation operator and define the single-qubit Pauli operators , and . Choosing the collective measurement operator

being the -fold tensor product whose th factor is , all others being the identity , and the collective local control and feedback Hamiltonians

where for , the evolution of the system is governed by the feedback-modified Lindblad master equation Wiseman (1994)

assuming local decay of the atoms is negligible. It is easy to see from the master equation above that the system decomposes into eigenspaces of the (angular momentum) operator

i.e., both the measurement operator and the control and feedback Hamiltonians and (and hence ) can be written in block-diagonal form with blocks determined by the eigenspaces of . Therefore, the system is decomposable and we cannot stabilize any state, no matter how we choose . For this system was studied in Wang et al. (2005b) in the context of maximizing entanglement of a steady state on the subspace using feedback, although the question of stability of the steady states was not considered. Although the system does not admit an attractive steady state in the whole space, we can verify that contains a line segment of steady states that intersects both the and subspaces in a unique state. Thus subspace has a unique steady state determined by , to which all solutions with initial states in this subspace converge.

iv.5 Feedback Stabilization

An interesting possible application of the criteria for the existence of unique, attractive steady states is the possibility of engineering the dynamics such that the system has a desired attractive steady state by means of coherent control, measurements and feedback. An special case of interest here is direct feedback. Systems subject to direct feedback as in the previous example, can be described by a simple feedback-modified master equation Wiseman (1994):

(14)

where is composed of a fixed internal Hamiltonian , a control Hamiltonian and a feedback correction term . This master equation is of Lindblad form, and hence all of the previous results are directly applicable. Setting

(15a)
(15b)
(15c)
(15d)

we see immediately that if the control and feedback Hamiltonian, and , and the measurement operator are allowed to be arbitrary Hermitian operators, then we can generate any Lindblad dynamics. This is also true for a non-Hermitian measurement operator as arises, e.g., for homodyne detection, since the anti-Hermitian part of can always be canceled by the effect of the feedback Hamiltonian in . Given this level of control, it is not difficult to show that we can in principle render any given target state , pure or mixed, globally asymptotically stable by choosing appropriate and or, equivalently, by choosing appropriate , and .

To see how to do accomplish this in principle, let us first consider the generic case of a target state is in the interior of the convex set of the states with . A necessary and sufficient condition for to be an attractive steady state is

  • and

  • no (proper) subspace of is invariant under (1).

The first condition ensures that is a steady state, and the latter ensures that it is the only steady state in the interior by Corollary 2. It is easy to see that choosing and such that

(16)

where is unitary, ensures that (i) is satisfied as

To satisfy (ii) we must choose such that has no orthogonal invariant subspaces, or equivalently mixes up any two orthogonal invariant subspaces may have. If , where is the projector onto the th eigenspace then the invariance condition implies that must not commute with any of the projection operators , or any partial sum of such as . To see this, suppose commutes with , a projector onto an eigenspace of . Then is a simultaneous eigenstate of and with and , where must be real and positive as is a positive operator, and we have , , is an eigenstate of

and thus and , i.e., is a steady state of the system at the boundary. Hence, the steady state is not unique, and cannot be attractive. In practice almost any randomly chosen unitary matrix such as , where is a random matrix, will satisfy the above condition, and given a candidate it is easy to check if it is suitable by calculating the eigenvalues of the superoperator in (6). Of course, choosing and of the form (16) is just one of many possible choices for condition (i) to hold. It is possible to find other suitable sets of operators in terms of when the class of practically realizable control and feedback operators or measurements is restricted. For example, we can easily verify that is the unique attractive steady state of a two-level system governed by the LME (1) with

even though and do not satisfy (16). Thus, there are generally many possible choices for the control, measurement, and feedback operators that render a particular state in the interior asymptotically stable.

If the target state is in the boundary of the convex set of the states, i.e., , then the proof of Proposition 4 shows that we must have

(17a)
(17b)
(17c)

with and defined as in Eq. (9), to ensure that is a steady state. To ensure uniqueness we must further ensure that there are no other steady states. This means, by Corollary 3, that (a) we must choose and such that is the unique solution of (17a), and thus no subspace of is invariant, and (b) we must choose the remaining operators and and such that (17b) is satisfied and no subspace of is invariant, because if such a subspace exists, then and will be two proper orthogonal invariant subspaces and will not be attractive.

One way to construct such a solution is by choosing such that and setting , where is a suitable unitary operator defined on as discussed in the previous section. Then we choose such that no proper subspace of is invariant. Finally, we must choose and such that (17b) is satisfied and is itself not invariant. Although these constraints appear quite strict, in practice there are usually many solutions.

For example, suppose we want to stabilize the rank-3 mixed state at the boundary. Then we partition , and as above, setting with and a suitable unitary matrix such as

Then we choose such that , e.g., we could set , a choice, which ensures that is the unique steady state on the subspace . Next we choose such that is not an invariant subspace. Any choice other than will do in this case, e.g., set . Finally, we set , and to ensure that is the unique globally asymptotically stable state.

Note that the Hamiltonian, which was not crucial for stabilizing a state in the interior and could have been set to , does affect our ability to stabilize states in the boundary. We can stabilize a mixed state in the boundary only if . If then Eq. (17b) implies , and there are two possbilities. If but has a zero eigenvalue, then the system restricted to the subspace has a pure state at the boundary and thus cannot be the unique attractive steady state on . Alternatively, if then is decomposable with two orthogonal invariant subspaces and , and cannot be attractive either, consistent with what was observed in Baumgartner et al. (2008b).

Target states at the boundary include pure states. in this case is a one-dimensional subspace of , and Eq. (17a) is trivially satisfied as and have rank 1, and the crucial task is to find a solution to Eq. (17b) such that no subspace of is invariant. If then this is possible only if and thus if has a zero eigenvalue, as was observed in Baumgartner et al. (2008b), but again, if then there are many choices for and , that stabilize a desired pure state. For example, we can easily check that the pure state is a steady state of the system if is the irreducible Jordan matrix with eigenvalue and .

V Invariant Set of Dynamics, Decoherence-free subspaces

Having characterized the set of steady states, the question is whether the system always converges to one of these equilibria. The previous sections show that this is the case if the system has a unique steady state, as uniqueness implies asymptotic stability. In general, however, this is clearly not the case for a linear dynamical system. Rather, all solutions converge to a center manifold , which is an invariant set of the dynamics, consisting of both steady states and limit cycles Glendinning (1994). Although we have seen that the Lindblad master equation (1) does not admit isolated centers, limit cycles often do exist for systems governed by a LME. This is easily seen when we consider the special case of Hamiltonian systems. In this case any eigenstate of the Hamiltonian is a steady state but no other dynamical flows converge to these steady states. For the Bloch equation (6) can be characterized explicitly. Consider the Jordan decomposition of the Bloch superoperator, , where is the canonical Jordan form. Let be the eigenvalues of and be the projector onto the (generalized) eigenspace of the eigenvalue , and let be the set of indices of the eigenvalues of with .

Definition 1.

Let be the affine subspace of consisting of vectors of the form , where is a solution of , and , where is the direct sum of the eigenspaces of corresponding to eigenvalues with zero real part. Then the invariant set .

It is important to distinguish the invariant set , which is a set of Bloch vectors (or density operators), from the notion of an invariant subspace of the Hilbert space . In particular, as contains the set of steady states , it is always nonempty. Although