Relaxation times in an open interacting two-qubit system
In a two-qubit system the coupling with an environment affects considerably the entanglement dynamics, and usually leads to the loss of entanglement within a finite time. Since entanglement is a key feature in the application of such systems to quantum information processing, it is highly desirable to find a way to prolonging its lifetime. We present a simple model of an interacting two-qubit system in the presence of a thermal Markovian environment. The qubits are modeled as interacting spin- particles in a magnetic field and the environment is limited to inducing single spin-flip events. A simple scheme allows us to calculate the relaxation rates for all processes. We show that the relaxation dynamics of the most entangled state exhibit critical slowing down as a function of the magnetic field, where the relaxation rate changes from exponentially small values to finite values in the zero-temperature limit. We study the effect of temperature and magnetic field on all the other relaxation rates and find that they exhibit unusual properties, such as non-monotonic dependence on temperature and a discontinuity as a function of magnetic field. In addition, a simple scheme to include non-Markovian effects is presented and applied to the two-qubit model. We find that the relaxation rates exhibit a sharp, cusp-like resonant structure as a function of the environment memory-time, and that for long memory-times all the different relaxation rates merge into a single one.
pacs:03.67.Mn, 03.65.Yz, 03.65.Ud, 42.50.Lc
When considering application of quantum information processing, two main ingredients must be considered - entanglement and decoherence. While entanglement, or non-local coherence, plays a key role in qubit operations review (), decoherence sets the limit to which such operations may be performed Zurek (). Decoherence may result from the interaction of the quantum system with a dissipative environment Mintert1 (), dramatically affecting the dynamics of the quantum system, and its entanglement properties. Specifically, it was recently shown Dodd (); YuEberly () that a (non interacting) two-qubit system can be completely disentangled in a finite time, a phenomenon dubbed ”entanglement sudden death”. This was followed by a plethora of theoretical studies of this phenomenon in various situations, most of them treating non-interacting qubits (i.e. a pair of qubits which interact with each other only in the mediation of the environment), either with a Markovian or non-Markovian environment Zagury (); Ficek (); Lastra (); Fanchini (); Ikram (); Bellomo (); Dajka (); Cao (); Qasimi (). From the experimental side, direct measurements of entanglement have been performed experiment1 (); experiment2 (), and the ”entanglement sudden death” was observed experiment3 ().
The loss of entanglement seems to be a generic feature of two-qubit systems YuEberly (). Our goal is to study a simple system where entanglement sudden death may be avoided. For this aim we study a simple model of an interacting two-qubit system in the presence of a thermal dissipative bath. We model the qubits as interacting spin- particles in a magnetic field Gunlycke () and use the Born-Markov approximation for the system-environment coupling. The environment is assumed to be ohmic and induce thermal transitions, and it only allows for single spin-flip events (in similarity to spin-boson models LeHur ()). We calculate analytically the full dynamics of this system, with emphasis given to the different relaxation rates. We point out here that we use the term ”relaxation rates” loosely, to describe the time-scales of both processes which include energy changes (relaxation) and only coherence loss (decoherence). Both types of processes are inherently present in our calculation scheme. We show that although the coupling with the environment is characterized by a single relaxation rate, different relaxation rates emerge for different coherent states. We study the effect of temperature and magnetic field on the different relaxation rates and find that they may be non-monotonous functions of temperature.
As the main results of this paper, we demonstrate that as a function of the magnetic field the relaxation rate of the highly-entangled states abruptly changes from being finite as the temperature vanishes to being exponentially small. This occurs when one of the states of the entangled pair is in a meta-stable state, and indicates that with a proper tuning of parameters the entanglement may survive very long times even in the presence of a dissipative environment. We demonstrate the long-lifeness of entanglement by calculating the concurrence of a specific entangled state, and show that applying a transverse magnetic field destroys this effect.
Finally, we devise a simple way to account for non-Markovian effects when calculating relaxation rates. Studying the relaxation rates as a function of the environment memory-time, we find that some relaxation rates exhibit a non-monotonic dependence on the memory-time, with a cusp-like resonance. For long memory-times, the different relaxation rates merge into a single rate.
Let us introduce our method for calculating the relaxation times. We consider a quantum system, characterized by a time-independent Hamiltonian with energy levels . The system dynamics are given by the evolution of the density matrix , where are the eigenfunctions of the Hamiltonian. Persh () For the above choice of Hamiltonian, in the Markovian approximation the evolution of the density matrix is given by a quantum master equation VanKampen () (,
where is a superoperator describing the dissipative dynamics. It is commonly takes the Lindblad form Lindblad ()
(where are the anti-commutation relations), which ensures positivity of the density matrix Breuer (). The -operators define the different relaxation processes induced by the environment.
We now follow Ref. Mukamel () and cast the density matrix into a vector form, defining . Here the first elements account for occupation probabilities and the other elements describe coherence between the states in the statistical mixture. It is now a matter of rearranging the master equation into a form , where now is a matrix of dimension which includes both the Hamiltonian and the dissipative part of the evolution.
Due to the semi-group properties the Lindblad equation, (at least) one of the eigenvalues of is exactly zero Mukamel (). The eigenvector corresponding to this eigenvalue is the steady-state of the system, i.e. the limit . The other eigenvalues may have an imaginary part, but all of them have a negative real part, which correspond to the relaxation rate of the corresponding eigenvector. Thus, by calculating the eigenvalues of one obtains the relaxation rates for all possible processes. For the most general initial condition, the smallest non-vanishing eigenvalue of (which we call ) represents the longest relaxation rate.
Iii Application to the two-qubit system
iii.1 Two-qubit system in a perpendicular field
Next we consider the application of the above method to our model two-qubit system. The hamiltonian, with Ising interactions, can be written as
where are the two qubit levels (i.e. spin= particles), describes the interactions and B is the external magnetic field. This choice of Hamiltonian is not only convenient (the Hamiltonian being very simple), but also represents several suggestions for realistic, spin-based quantum computers Kane (); Loss ().
For simplicity we start with magnetic field only in the -direction, i.e. with . Choosing as a basis the four states (which we number from to , respectively), the Hamiltonian can be written (up to a constant energy shift) as
In order to account for the environment, the spins are coupled to one that induces spin-flip processes, represented by the -operators in Eq. 2. Here we make two assumptions, namely (i) the spins are flipped one at a time (i.e. there is no direct relaxation from the state to the state, etc.) and (ii) the relaxation rate between two states is proportional to the Boltzman factor of the corresponding energy difference between the states VanKampen () (these two assumptions on the form of the -operators reflect the properties of the environment as described in the introduction). For example, the relaxation operator from to (taking ) is InfoComp ()
where is some typical relaxation rate which represents the strength of the qubit-environment coupling. For the reverse process the relaxation operator is
so that the condition of detailed balance is maintained, i.e. . Note that one can normalize the transition rate in different ways and still maintain detailed balance. Here we choose such a normalization that keeps all relaxation rates finite (even at and preserves the symmetry (i.e., does not depend on the gauge of the Hamiltonian). However, all the qualitative results presented in this paper equally apply for a different choice of the normalization of the -operators.
Once the form of the -operators is specified, it is now a matter of algebraic manipulation to obtain the -matrix and its eigenvalues and eigenvectors. For the steady state we find
with being the partition function of the system, which results in a pure state (which is the ground state of the Hamiltonian) in the limit of .
For the above example, all the eigenvalues may be calculated analytically. For the lowest rate we find
which is -fold degenerate. Two states contain and (i.e. a coherence between and ), and the third is a mixture of all the diagonal elements and . Inspection of in the limit shows that
This means that for the relaxation time from the coherent state diverges, i.e. the system never reaches its thermal, disentangled, ground state. However, for it relaxes to the ground state in a time scale . This can be easily explained from the following considerations. Note that the state can only relax into one of the degenerate middle states, or . For , its energy is negative (but higher than the ground state energy), and therefore the relaxation rate is exponentially small. Put it differently, the state is a meta-stable state which requires an exponentially rare correlated event to escape from. For , however, the energy of is no longer negative, and hence it can relax into the ground state by a cascade relaxation through the middle states.
While all the other eigenvalues are available analytically, writing them in full form is cumbersome, and thus we present them graphically. In Fig. 1 we plot the eigenvalues (corresponding to the inverse relaxation rates for the different states), as a function of temperature for magnetic field, . In Fig. 1(a) we plot a wide temperature range, and we zoom in on the low-temperature range in Fig. 1(b). For each relaxation rate the corresponding elements of the density matrix are marked. Fig. 1 shows two interesting features: (i) the relaxation rates are nonmonotonic in temperature, and (ii) they break into groups, with only two possible time scales ( and ) at .
For comparison, in Fig. 2 we plot the same for a magnetic field . We now find that the relaxation rates are broken into four groups, i.e. additional time-scales appear. Interestingly, only the -state preserves its limit as the point is crossed, while all other time-scales exhibit a discontinuous change.
In order to demonstrate the above effect on the entanglement of the two-qubit system, we calculate the concurrence, which is a direct measure of the entanglement Hill (). For the two-qubit system (in the basis chosen above) it is defined as , where are the eigenvalues of the matrix , defined by . Clearly, the dynamics of the concurrence depend on the initial condition. For this example we choose the initial density matrix Ikram ()
for which one can easily calculate the concurrence, . The diagonal elements are given by Eq. (7) and . By making this choice we start with a highly entangled state (), but the dynamics are very easy to calculate as the diagonal elements do not change at all, and the off-diagonal ones decay with the rate given by Eq. (9). We point out that one can start with any initial diagonal elements and obtain results similar to those presented below, since the diagonal elements quickly relax to the steady-state (Eq. (7)) and do not contribute to the concurrence time-evolution.
In Fig. 3 we plot the concurrence as a function of temperature and time for magnetic field values (upper panel) and (lower panel). The mesh corresponds to finite , while in the blank regions . For we find that the ”entanglement sudden death” is present at all temperatures. However, for it becomes exponentially suppressed at lower temperatures: the concurrence practically remains finite for all times.
In order to understand this behavior, we note that the formula for the concurrence describes a competition between the off-diagonal elements of the density matrix, which contribute to the entanglement, and the diagonal elements, which ”disentangle” the state. At strictly , the diagonal elements vanish but the off-diagonals survive indefinitely, giving rise to an entangled state. For very low temperature, while in the strict limit the system becomes disentangled, this time is exponentially long.
iii.3 Transverse field
Let us consider the effect of an additional transverse field (the results are identical to an additional field in the -direction). A transverse field can either represent an inherent interaction between the two qubit states, or an actual field (in the qubit relevant Hilbert space), which is used to perform quantum operations. Since both these ingredients appear in any implementation of a physical qubit, it is important to study their effect on the relaxation time-scales.
We thus repeat the above procedure of constructing and diagonalizing the -matrix with an additional term. In Fig. 4 we plot the inverse relaxation rates at as a function of . It is found that the infinite relaxation time becomes finite (i.e. there is no more a meta-stable state) and that the degeneracy is lifted. This means that applying a transverse field might give rise to entanglement ”sudden-death”. Interestingly, one can see that the time-scales are not monotonic functions of .
The conclusion arising from the above calculation is two-fold, (i) in a realistic qubit, coupling between the qubit states should be maximally inhibited, to allow for longer entanglement life-time, and (ii) one has to take into account the fact that performing quantum operations on the qubits will result in faster decay of entanglement.
Iv Non-Markovian effects
In the Markovian approximation, the evolution of the system does not depend on its history. This approximation applies when the correlation times of the thermal bath are much smaller than any timescale associated with the system under consideration. However, there are cases where the bath reacts to the dynamics of the system over a certain memory time, , and the Markov approximation is thus not valid. In such instances, the evolution of the system would depend on its history Breuer (). Non-Markovian effects on the entanglement dynamics have been extensively studied in recent years Dajka (); Bellomo (); Cao ().
In order to include non-Markovian effects, one has to include the history of the system. In the simplest approximation Breuer (), this adds up to a form of the quantum master equation
where the memory kernel defines the response of the bath to the history of the system. Note that not every form of the memory kernel is possible, as positivity of the density matrix may be lost Shabani ().
One can now repeat the procedure described above, and rewrite Eq. (11) in a vector form, which yields the integro-differential vector equation
The time-dependence of the -matrix is such that for elements derived from the Lindblad super-operator one attaches the kernel , and for the elements derived from the Hamiltonian one attaches a -function, . One can now Laplace-transform Eq. (12) and obtain an algebraic vector-equation , where is the (complex) Laplace variable. Note that the Laplace transform has a simple form, as the Hamiltonian elements are multiplied by unity (which is the transform for the -function) and the Lindbladian elements are multiplied by the Laplace transform of . The formal solution of the above equation is thus given by
where is the unit matrix. From Eq. (13) it can be seen that if the secular equation has solutions, the real part of these solutions defines a relaxation time-scale (via a Cauchy-like integration over poles). One can thus obtain the relaxation times from a numerical solution of the secular equation, without the need to solve the full non-Markovian dynamics.
where is the memory-time. The Laplace transform is thus , which yields a polynomial secular equation. Note that in the limit the Markovian limit is exactly obtained. We have solved this equation numerically, and found that it always has solutions with negative real part and a single solution with , corresponding to the steady-state. This means that for an exponential memory-kernel, one can identify different processes which have different relaxation rates. We note that if one takes a more complicated memory kernel (say a power-law), then the secular equation becomes transcendental, with no simple poles. In that kind of environment, one cannot simply attach different time-scales to different processes, and the full dynamics of the system must be calculated Zwolak ().
The above scheme can now be applied to our two-qubit model. In Fig. 5 the inverse time-scales are plotted as a function of the memory-time for . We find that the time-scales have a resonant-structure as a function of , and in fact exhibit a cusp at the resonance. In addition, at relatively large , all the different time-scales merge into a single group, i.e. there is no longer a separation of the time-scale for the relaxation processes of different coherent states.
The long memory-time behavior may be understood by the fact that in this case the bath memory dominates the relaxation processes, giving rise to a single time-scale. The longer the memory-time, the more weight is given in the relaxation process to states far from the steady-state, and hence the relaxation rate diminishes. However, the rise in relaxation rate at small memory-times is a surprising effect, which comes about due to the complex nature of the interaction between the two-qubit system and the environment.
In this paper we have studied the relaxation dynamics of a model interacting two-qubit system in the presence of an ohmic thermal environment. The qubits were modeled as spin- particles with spin-spin coupling and in the presence of a magnetic field. The environment was limited to induce only single spin-flip events. Within this model we analytically calculated the relaxation rates of different processes. Our main result is that disentanglement may be critically slowed down in the limit by varying the magnetic field, and entanglement sudden death may be completely avoided (or at least exponentially suppressed for low temperatures). This was explicitly shown by calculating the concurrence dynamics of a highly entangled state for different magnetic fields. We have also shown that a transverse magnetic field may destroy this effect.
In addition, we have introduced a simple way to include non-Markovian effects in the calculation of the relaxation rates. We have shown that the different relaxation rates exhibit an interesting non-monotonic dependence on the environment memory-time, with a cusp-like resonance. For long memory-times, we have found that the different time-scales merge into a single time-scale.
In order to experimentally verify the results presented here, one needs an experiment with a well-controlled two-qubit system, where the energy-levels can be controlled. A promising candidate is a system composed of two coupled two-quantum-dot qubits DiVincenzo (), where the qubit levels may be controlled by external gates, and a high level of control of the qubit states has been already demonstrated experimentally Petta ().
Acknowledgements.We thank S. Saikin for fruitful discussions and Yu. V. Pershin for valuable comments on the manuscript. This work was funded by the Department of Energy grant DE-FG02-05ER46204.
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