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Analytical solution of the time evolution of an entangled electron spin pair in a double quantum dot nanostructure

Using master equations we present an analytical solution of the time evolution of an entangled electron spin pair which can occupy 36 different quantum states in a double quantum dot nanostructure. This solution is exact given a few realistic assumptions and takes into account relaxation and decoherence rates of the electron spins as phenomenological parameters. Our systematic method of solving a large set of coupled differential equations is straightforward and can be used to obtain analytical predictions of the quantum evolution of a large class of complex quantum systems, for which until now commonly numerical solutions have been sought.


I. INTRODUCTION
Master equations are used to describe the quantum evolution of a physical system interacting with some "reservoir" 1 , and have been applied to a wide variety of physical systems, ranging from two-level atoms in the presence of light fields 2 to solid-state nanostructures such as quantum dots and Josephson junction devices 3 . For simple systems, such as a two-level atom damped by a reservoir consisting of simple harmonic oscillators 4 or an electron in a single or double quantum dot coupled to external leads 5 , the set of master equations that describes the quantum dynamics of the system is small and its solution can be obtained analytically in a straightforward way. If the system is more involved, however, due to the presence of quite a few atomic levels or because the nanostructure is composed of various coherent parts, its quantum state space consists of a large number of quantum states with various coherent and incoherent couplings between them and the analytical solution of the corresponding large set of coupled master equations does not spring to the eye. Hence often a numerical solution is sought 6 . Understanding the quantum evolution of such "complex" quantum systems -where "complex" refers to a system which is described by a large number of coupled quantum states -has recently become increasingly important, in particular in fundamental research aimed at investigating the dynamic behavior of qubits, the basic building blocks for quantum computation 7 . A large theoretical and experimental effort in various fields, e.g. quantum optics, atomic physics and condensed matter physics, is presently directed towards investigating possibilities to use two-level systems such as polarized photons, cold atoms, electron spins and superconducting circuits as qubits and finding ways to couple these qubits together. In the latter three systems, one of the major questions involved is how the desired coherent evolution of the system will be affected by coupling to the environment, which is necessary to manipulate and measure the states of the qubits but invariably introduces undesired decoherence of their quantum states. A master equation model of the quantum evolution of one or more qubits interacting with their environment allows one to construct transparent general formulas and is therefore very suitable to give both qualitative and quantitative insight into the dynamics of these complex quantum systems.
In this paper we present an analytical solution of a large set of coupled master equations that describe the quantum evolution of a particular condensed-matter system, namely the time evolution of an entangled electron spin pair in a double quantum dot nanostructure. Even though our model applies to this specific quantum system, the presented method of solving the master equations is general and can be applied to study the dynamics of many other complex quantum systems. The time evolution of the electron spins is governed by several coherent and incoherent processes, each of which depends on time in a simple way as either oscillatory (cosine) or exponential functions. The solution we obtain shows how these simple ingredients combine to describe the evolution of the entangled spins in a complex nanostructure which consists of several coherent parts. It can be used to predict the occupation probability of all quantum states at any given time and to provide analytical estimates of the important time scales in the problem, such as the time at which decoherence of the entangled pair becomes substantial 8 .
The paper is organized as follows. In Sec. II the quantum nanostructure and the assumptions made are described.
Section III contains the master equations and their solution, with technical details given in Appendices A and B. A summary of the results and their range of applicability is presented in Sec. IV.

II. THE DOUBLE QUANTUM DOT NANOSTRUCTURE
The system we consider consists of a double quantum dot nanostructure, which is occupied by two entangled electron spins and operated as a turnstile. We studied this system in an earlier paper as a suitable set-up for the detection of entanglement between electron spins 9 . Here we focus on the dynamic evolution of the electron pair in the system, which is depicted in Fig. 1. In detail, the structure consists of two adjacent quantum dots in a parallel magnetic field B zẑ which are connected to two quantum point contacts (QPCs) via empty quantum channels. A quantum dot is a small metallic or semiconducting island, confined by gates and connected to electron reservoirs (leads) through quantum point contacts.
If the gates are nearly-closed and form tunnel barriers, the dot is occupied by a finite and controllable number of electrons which occupy discrete quantum levels, similar to atomic orbitals in atoms 10 . In our system, the gate between the two dots is assumed to be initially open and the dots are occupied by two electrons 11 [ Fig. 1(a)] in their lowest energy state, the singlet state 12 . The gate between the two dots is then adiabatically closed, so that the electrons become separated and one dot is occupied by an electron with spin-up and the other by one with spin-down. The two spins do not interact anymore and are independently rotated by electron spin resonance (ESR) fields [ Fig. 1(b)].
The latter are oscillating magnetic fields which, if the frequency of oscillation matches the energy difference between the two spin-split single-electron energy levels, cause coherent rotations of a spin between these levels, analogous to Rabi oscillations in a two-level atom. After spin rotation, the electrons are emitted into empty quantum channels by In the next section we analyze the dynamics of the two spins from the moment they are separated and each occupies one of the two dots, until both have been detected in one of the four exits. We use a master equation approach in which the effects of relaxation and decoherence are included as phenomenological decay rates 1 . The solution presented is exact under three assumptions: • The time evolution during ESR in the dots is decoupled from the time evolution in the channels and exits.
Physically, this means that the gates between the dots and channels are closed during the ESR rotations, so no tunneling occurs out of the dots during that time.
• Once the electrons are in a channel they cannot tunnel back into the dots, i.e. backreflection of the electrons to the dots during their journey to the detectors is neglected. This corresponds to ballistic transport through the channels.
• Once the electrons are in one of the exits they cannot return to the channels, i.e. the electrons are immediately detected and absorbed into the detectors.

III. THE MASTER EQUATIONS AND THEIR SOLUTION
In the set-up as depicted in Fig. 1 each electron is assumed to be either in a dot, in a channel or detected. This leads to a set of 36 possible quantum states represented by a 36×36 density matrix ρ(t). This set consists of all possible combinations AσBσ ′ , with A∈{D, C, X} and σ∈{↑, ↓} indicating resp. the position (D=dot, C=channel and X=exit) and the spin direction alongẑ of the electron which started out in the left dot, and B∈{D, C, X} and σ ′ ∈{↑, ↓} representing the position and spin direction of the electron which started out in the right dot. The set is given by: We number the states in set (1) by the numbers 1 to 36, so 1=D↑D↑, 2=D↑D↓ etc. The states labeled by C and X do not refer to individual quantum states in the channels and detectors, since in a channel many longitudinal modes exist and the detectors consist of many quantum states which form together a macroscopic state. What is meant by the states C and X is the set of all channel modes resp. of all quantum states in the detectors. These states thus describe the probability of an electron to occupy any one of these channel modes or detector states. We come back to why this definition is useful and appropriate in the paragraph below Eq. (6). For long times, the only states that are occupied are 33-36, in which both electrons have entered into an exit and the channels and dots are empty.
The time evolution of the density matrix elements ρ nm (t) is given by the master equations 1 : for n, m ∈ {1, . . . , 36}. The Hamiltonian H(t) describes the coherent evolution of the spins in the quantum dots due to the ESR fields and is given by, for two oscillating magnetic fields B xL cos(ωt)x and B xR cos(ωt)x applied to the left and right dots respectively, Here H 0 is a diagonal matrix containing the energies E n (n = 1, . . . , 36) of each state, g * the electron g-factor, µ B the Bohr magneton andσ xL(R) a 36×36 matrix with elements (σ xL(R) ) ij = 1 for each pair of states (i, j) that are coupled by the oscillating field B xL(R) and zero otherwise. For g * < 0 the 4×4 upper left corner H dots (t) of H(t) is then given explicitly as terms of the single-particle energies E ↑ and E ↓ and the charging energy E C = e 2 /C, where C is the total capacitance of the quantum dot (assumed to be equal . The parameter ǫ, with 0 ≤ ǫ < 1, represents the relative reduction of the field which is applied to one dot at the position of the spin in the other dot 9 . The remaining 32×32 part of the matrix H(t) is diagonal and equal to H 0 , since the ESR fields are applied when both electrons are located in a dot and the quantum channels do not contain any electrons whose spin might otherwise also be rotated by these fields.
Turning to the transition rates W nm (from state m to n) in Eqs. (2a), we distinguish between two kinds of transitions: 1) spin-flip transitions between two quantum states that differ by the direction of one spin only and 2) tunneling (without spin-flip) between quantum states that involve adjacent parts of the system, i.e. from dot to channel and from channel to exit. The latter are externally controlled by opening and closing the gates between the dots and channels. The former are modeled by the phenomenological rate 1/T 1,α ≡ W α↑↓ + W α↓↑ with α ∈ {D, C} for spin flips in a dot or channel. Here the W ′ s depend on the Zeeman energy ∆E Z ≡ |g * |µ B B z and temperature T via detailed The spin decoherence rates V nm in Eqs. (2b) for states n and m with n, m ∈ {1 . . . 4}, i.e. the decoherence rate between states in which both electrons are located in a quantum dot, is given by: where the W's refer to tunnel rates out of a dot. The coherence between state n and m thus not only depends on the intrinsic spin decoherence time T 2,D which is caused by e.g. spin-orbit or hyperfine interactions in the dots 14 , but is also reduced by the (incoherent) tunneling processes from dot to channel 15 . Similarly, V nm for all other states n and m is given by with the W's tunnel rates from a channel to an exit. Note that energy relaxation processes between different modes in the channels, i.e. between modes that contribute to the same set of channel states C, do not affect the transition rates W nm and decoherence rates V nm for the states where either n or m or both refer to a channel state. The reason for this is that these rates refer to resp. spin flip and spin decoherence processes, which are not affected by orbital (energy) relaxation and decoherence 16 . Hence our definition of the channel states as sets of all modes with the same spin does not interfere with the definition of spin relaxation and decoherence of the quantum states.
With the above ingredients, the coupled equations (2) can be solved analytically. We proceed in three steps: ESR applied to the left dot, ESR applied to the right dot and the time evolution after the gates to the quantum channels have been opened. During each step only part of the quantum states are evolving in time, while the others remain unchanged. This simplifies the procedure to obtain an analytical solution.
Initially, at time t = 0, both spins are assumed to be in the singlet state in the quantum dots, so During ESR applied to the left dot quantum states ρ 5 (t) − ρ 36 (t) remain unchanged, since the gates between the dots and channels are closed. The coherent evolution of ρ 1 (t)-ρ 4 (t) is then governed by the Hamiltonian Including spin-flip rates W D↑↓ and W D↓↑ and the decoherence rate Γ ≡ 1/T 2,D for both dots 17 we then obtain from Eqs.

C. Step 3: Time evolution after the gates to the channels have been opened
We now turn to the next step in the evolution of the entangled pair in Fig. 1 where t ESR ≡ t 1 + t 2 and V ij is given by Eq. and/or j ∈ {17, . . . , 36}), since for those states V ij = ∞. This corresponds to the assumption of immediate detection.
In the remaining part of this paper we focus on the evolution of the populations ρ 1 (t)-ρ 36 (t) for times t ≥ t ESR under the following conditions: • We neglect the possibility of spin flips in the dots, i.e. we set W D↑↓ = W D↓↑ = 0. This based on the fact that T 1,D is known to be much longer (0.85 ms at magnetic fields B z = 8T ) 19 than the time required to travel through the channels to the exits. This assumption is not essential to obtain an analytical solution; it only simplifies the resulting equations.
• We assume that the tunnel rate W T out of the dots into the channels is equal for spin-up and spin-down electrons, i.e. the two electrons tunnel out of the singlet state with a negligible time delay t delay in between, and that spin is conserved during this tunneling process. Typically 20 t delay ≈ 10 −13 s, which is much less than the travel time through a channel ∼ 10 −10 s.
• The tunnel rate W E through the QPCs is taken to be constant and equal for spin-up and spin-down electrons, i.e. the set-up is assumed to be constructed in such a way that the detection time for spin-up and spin-down electrons once they have reached the QPCs is the same.
• Spin flips in the exits are neglected, i.e. detection is assumed to be faster (with typical times ∼ 10 −11 s) 21 than the spin-flip rate (≫ 10 −11 s) 9 in the detectors.
In order to obtain the solution for ρ 13 (t)-ρ 16 (t), which corrresponds to the situation in which both electrons are located in a channel, we rewrite the equations forρ 13 -ρ 16 aṡ Eqs. (24) consist of 3 coupled equations (24a)-(24c) and a separate one, Eq. (24d). We first solve the latter and then the first three. In each case the solution is a combination of a homogeneous and a particular solution. Taking from B11,12,3,4 W E W T D3,4 The coefficients in Eqs. (25) are given in Appendix B. Also here, ρ 13 (t)-ρ 16 (t) for times t ≤ t travel are given by Eqs. (25) with W E = 0, and for times t ≥ t travel these populations are given by Eqs. (25) with t ESR → t travel .
The solution of the next set, ρ 17 (t) − ρ 24 (t), corresponding to the states in which one electron is located in a dot while the other has reached a detector, is given by and ρ i (t) = 0 for t ≤ t travel . The coefficients A i , B i and C i in Eqs. (26) are given in Table I. Next, we solve for ρ 25 (t)-ρ 32 (t), the states in which one spin has reached a detector, while the other is still in a channel, in the pairs ρ i (t)&ρ j (t) ∈ {ρ 25 (t)&ρ 26 (t), ρ 27 (t)&ρ 28 (t), ρ 29 (t)&ρ 31 (t), and ρ 30 (t)&ρ 32 (t)}, see Eqs. (18). For each pair the solution is given by, for times t ≥ t travel : and ρ i (t) = ρ j (t) = 0 for t ≤ t travel . The coefficients P i,j , Q i,j and M i,1 . . . M i,7 for i, j ∈ {25, . . . , 32} are given in Appendix B.
Finally, we obtain the time evolution of the states ρ 33 (t) − ρ 36 (t) in which both electrons have reached an exit. This is given by, for times t ≥ t travel , Special case. In order to illustrate the solution (28), we now derive explicit expressions for ρ 33 (t) and ρ 34 (t), the probabilities that a spin-up is detected in the left detector and resp. a spin-up or a spin-down in the right detector, for the special case of Γ = 0 (no decoherence in the dots) and W C↑↓ = W C↓↑ = 0 (no relaxation in the channel). This corresponds to the situation in which the time evolution occurs in the absence of any decoherence and dissipation mechanisms in the dots and channels and only depends on W T , the tunnel rate from dot to channel, and W E , the tunnel rate from channel to exit.

IV. CONCLUSION
In summary, we have presented an analytical solution of a set of coupled master equations that describe the time evolution of an entangled electron spin pair which can occupy 36 different quantum states in a double quantum dot nanostructure. Our method of solving these equations is based on separating the time evolution in three parts, namely two coherent rotations of the electron spins in the isolated quantum dots and the subsequent travel of the electrons through two quantum channels. As a result of this separation, the total number of master equations is split into various closed subsets of coupled equations. Our analytical solution is the first of its kind for a large set of coupled master equations and the same method can be used to study and predict the quantum evolution of other quantum systems which are described by a large set of quantum states. This type of analysis complements numerical approaches to study the dynamic evolution of complex quantum systems and allows to obtain qualitative insight in the competition between time scales in these systems.
So far no approximations have been made, apart from assuming the decoherence rate Γ to be equal for all off-diagonal terms of the density matrix ρ [Eqs. (8)]. In order to obtain the solution of the remaining equations (A1e)-(A1h) we assume δ L = 0 (no influence of the ESR field on the spin in the right dot) and W D↑↓ = W D↓↑ = 0 23 and find 24 : x 6 (t) = x 6 (0) cos ∆ L t + (2x 8 (0) − Z) sin ∆ L t (A4b)