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

Analytical solution of the time evolution of an entangled

electron spin pair in a double quantum dot

nanostructure

M. Blaauboer

Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

共Received 10 May 2005; accepted 27 June 2005; published online 16 August 2005兲 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 elec-tron 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. © 2005 American Institute of Physics. 关DOI: 10.1063/1.2007629兴

I. INTRODUCTION

Master equations are used to describe the quantum evolution of a physical system interacting with some “reservoir,”1and have been applied to a wide variety of physical systems, ranging from two-level atoms in the presence of light fields2to 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 oscillators4or an electron in a single or double quantum dot coupled to external leads,5the 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 particu-lar in fundamental research aimed at investigating the dynamic behavior of qubits, the basic building blocks for quantum computation.7A large theoretical and experimental effort in various fields, e.g., quantum optics, atomic physics, and condensed-matter physics, is presently directed toward 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 formu-las 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 describes the quantum evolution of a particular condensed-matter system, namely the time evolu-tion of an entangled electron spin pair in a double quantum dot nanostructure. Even though our

46, 083518-1

model applies to this specific quantum system, the presented method of solving the master equa-tions 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 共cosine兲 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 Appendixes 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 setup 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 Bzzˆ, which are connected to two quantum point contacts 共QPCs兲 via empty quantum channels. A quantum dot is a small metallic or semiconducting island, confined by gates and connected to electron reservoirs共leads兲 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.10In our system, the gate between the two dots is assumed to be initially open and the dots are occupied by two electrons11关Fig. 1共a兲兴 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 indepen-dently rotated by electron spin resonance共ESR兲 fields 关Fig. 1共b兲兴. The latter are oscillating mag-netic 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 opening gates L and R关Fig. 1共c兲兴 and scattered at quantum point contacts QPC 1 and QPC 2. In a parallel magnetic field and for conductances GQPC1共QPC2兲

艋e2_{/ h these QPCs are spin selective,}13

transmitting electrons with spin-up and reflecting those with spin-down关Fig. 1共d兲兴. The transmitted and reflected electrons are separately detected in the four exits.

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.1The solution presented is exact under three assump-tions:

共i兲 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. 共ii兲 Once the electrons are in a channel they cannot tunnel back into the dots, i.e.,

backreflec-tion of the electrons to the dots during their journey to the detectors is neglected. This corresponds to ballistic transport through the channels.

共iii兲 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 setup 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 ␳共t兲. This set consists of all possible combinations A␴B␴

⬘

, with A苸兵D,C,X其 and ␴ 苸兵↑, ↓其 indicating, respectively, the position 共D=dot, C=channel, and X=exit兲 and the spin direction along zˆ of the electron which started out in the left dot, and B苸兵D,C,X其 and ␴

⬘

苸兵↑, ↓其 representing the position and spin direction of the electron which started out in the right dot. The set is given by

兵D↑D↑,D↑D↓,D↓D↑,D↓D↓,C↑D↑,C↑D↓,C↓D↑,C↓D↓,D↑C↑,D↑C↓,D↓C↑,D↓C↓,C↑C↑, C↑C↓,C↓C↑,C↓C↓,X↑D↑,X↑D↓,X↓D↑,X↓D↓,D↑X↑,D↑X↓,D↓X↑,D↓X↓,X↑C↑,X↑C↓,

X↓C↑,X↓C↓,C↑X↑,C↑X↓,C↓X↑,C↓X↓,X↑X↑,X↑X↓,X↓X↑,X↓X↓其. 共1兲

We number the states in set共1兲 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 together form a macroscopic state. What is meant by the states C and X is the set of all channel modes, respectively, 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. 共9兲. 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␳nm共t兲 is given by the master equations 1 ␳˙n共t兲 = − i ប关H共t兲,␳共t兲兴nn+

兺

m⫽n 关Wnm␳m共t兲 − Wmn␳n共t兲兴, 共2a兲 ␳˙_n,m共t兲 = − i ប关H共t兲,␳共t兲兴nm− Vnm␳n,m共t兲 n ⫽ m, 共2b兲 for n , m苸兵1, ... ,36其. The Hamiltonian H共t兲 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 BxLcos共␻t兲xˆ and BxRcos共␻t兲xˆ applied to the left and right dots, respectively,

H共t兲 = H0− 1 2g*␮Bcos共␻t兲

兺

M,N苸兵L,R其 M⫽N 共BxM+⑀BxN兲␴¯xM. 共3兲

Here, H₀ is a diagonal matrix containing the energies En共n=1, ... ,36兲 of each state, g* the electron g factor, ␮B the Bohr magneton, and␴¯xL共R兲 a 36⫻36 matrix with elements 共␴¯xL共R兲兲ij= 1

for each pair of states共i, j兲 that is coupled by the oscillating field BxL_共R兲and zero otherwise. For

g*_{⬍0 the 4⫻4 upper-left corner H}

dots共t兲 of H共t兲 is then given explicitly as

Hdots共t兲 =

冢

E1 ប⌬RLcos共␻t兲 ប⌬LRcos共␻t兲 0 ប⌬RLcos共␻t兲 E2 0 ប⌬LRcos共␻t兲 ប⌬LRcos共␻t兲 0 E3 ប⌬RLcos共␻t兲 0 ប⌬LRcos共␻t兲 ប⌬RLcos共␻t兲 E4

冣

,

with E₁= 2E_↑+ EC, E2= E3= E↑+ E↓+ EC, and E4= 2E↓+ EC in terms of the single-particle energies

E_↑and E_↓and the charging energy EC= e2/ C, where C is the total capacitance of the quantum dot 共assumed to be equal for both dots兲, ⌬RL⬅⌬R+␦L and ⌬LR⬅⌬L+␦R, with ⌬R共L兲 ⬅关兩g*_兩␮

BBxR共L兲兴/2ប and␦R共L兲⬅关⑀兩g*兩␮BBxR共L兲兴/2ប. The parameter⑀, with 0艋⑀⬍1, represents the relative reduction of the field which is applied to one dot at the position of the spin in the other dot.9The remaining 32⫻32 part of the matrix H共t兲 is diagonal and equal to H0, 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 Wnm共from state m to n兲 in Eq. 共2a兲, we distinguish between two kinds of transitions:共1兲 spin-flip transitions between two quantum states that differ by the direc-tion of one spin only and 共2兲 tunneling 共without spin-flip兲 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,␣⬅W_␣↑↓+ W_␣↓↑, with␣苸兵D,C其 for spin flips in a dot or channel. Here, the W ’ s depend on the Zeeman energy⌬EZ⬅兩g*兩␮BBzand temperature T via detailed balance W_␣↑↓/ W_␣↓↑= e⌬EZ/kBT_{, so that}

W_{␣↑↓共↓↑兲}= 1 T_1,␣

1

1 + e−共+兲⌬EZ/kBT, ␣苸 兵D,C其. 共4兲

The spin decoherence rates Vnmin Eqs.共2b兲 for states n and m with n,m苸兵1, ... ,4其, i.e., the decoherence rate between states in which both electrons are located in a quantum dot, is given by

Vnm= 1 T_2,D+ 1 2j⫽n,m

兺

共Wjn+ Wjm兲 n,m 苸 兵1, ... ,4其, 共5兲

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 T2,Dwhich is caused by, e.g., spin-orbit or

hyper-fine interactions in the dots,14but is also reduced by the共incoherent兲 tunneling processes from dot to channel.15Similarly, Vnmfor all other states n and m is given by

Vnm=

冦

1 T_2,C+

1 2j_⫽n,m

兺

共Wjn+ Wjm兲 n,m 苸 兵1, ... ,16其, but not both n,m 苸 兵1, ... ,4其

⬁ n苸 兵17, ... ,36其 and/or m 苸 兵17, ... ,36其,

冧

共6兲 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 Wnm and decoherence rates Vnm 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, respectively, spin flip and spin decoherence processes, which are not affected by orbital共energy兲 relaxation and decoherence.16Hence, 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共2兲 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.

A. Step 1: ESR applied to the left dot

Initially, at time t = 0, both spins are assumed to be in the singlet state in the quantum dots, so

␳2共0兲 =␳3共0兲 = 1/2; ␳j共0兲 = 0 ∀ j 苸 兵1,4,5, ... ,36其, 共7a兲

␳2,3共0兲 =␳3,2共0兲 = − 1/2; ␳i,j共0兲 = 0 otherwise. 共7b兲 During ESR applied to the left dot, quantum states␳5共t兲−␳36共t兲 remain unchanged, since the

gates between the dots and channels are closed. The coherent evolution of ␳₁共t兲–␳₄共t兲 is then governed by the Hamiltonian

HESR共t兲 =

冢

E1 ប␦Lcos共␻t兲 ប⌬Lcos共␻t兲 0 ប␦Lcos共␻t兲 E2 0 ប⌬Lcos共␻t兲 ប⌬Lcos共␻t兲 0 E2 ប␦Lcos共␻t兲 0 ប⌬Lcos共␻t兲 ប␦Lcos共␻t兲 E4

冣

.

Including spin-flip rates WD_↑↓and WD_↓↑and the decoherence rate⌫⬅1/T2,Dfor both dots,17we

then obtain from Eqs.共2兲 the master equations

␳˙₁= −␦LIm˜␳1,2−⌬LIm˜␳1,3− 2WD_↓↑␳1+ WD_↑↓共␳2+␳3兲, 共8a兲 ␳˙₂=␦LIm˜␳1,2−⌬LIm˜␳2,4+ WD↑↓+共WD↓↑− WD↑↓兲␳1−共2WD↑↓+ WD↓↑兲␳2− WD↑↓␳3, 共8b兲 ␳˙₃= −␦LIm˜␳3,4+⌬LIm˜␳1,3+ WD↑↓+共WD↓↑− WD↑↓兲␳1− WD↑↓␳2−共2WD↑↓+ WD↓↑兲␳3, 共8c兲 Im˜˙␳_1,2= −␦L 2 共␳2−␳1兲 + ⌬L 2 共Re˜␳1,4− Re˜␳2,3兲 − ⌫ Im˜␳1,2, 共8d兲 Im˜˙␳_1,3= −⌬L 2 共␳3−␳1兲 + ␦L 2共Re˜␳1,4− Re˜␳2,3兲 − ⌫ Im˜␳1,3, 共8e兲 Im˜˙␳_2,4= −⌬L 2 共␳4−␳2兲 − ␦L 2共Re˜␳1,4− Re˜␳2,3兲 − ⌫ Im˜␳2,4, 共8f兲 Im˜˙␳_3,4= −␦L 2 共␳4−␳3兲 − ⌬L 2 共Re˜␳1,4− Re˜␳2,3兲 − ⌫ Im˜␳3,4, 共8g兲 Re˜˙␳_1,4= −␦L 2共Im˜␳1,3− Im˜␳2,4兲 − ⌬L 2 共Im˜␳1,2− Im˜␳3,4兲 − ⌫ Re␳˜1,4, 共8h兲 Re˜˙␳_2,3=␦L 2共Im˜␳1,3− Im˜␳2,4兲 + ⌬L 2 共Im˜␳1,2− Im˜␳3,4兲 − ⌫ Re␳˜2,3, 共8i兲

Re˜˙␳_1,2= −⌬L 2 共Im˜␳1,4+ Im˜␳2,3兲 − ⌫ Re˜␳1,2, 共8j兲 Re˜˙␳_1,3= −␦L 2共Im˜␳1,4− Im␳˜2,3兲 − ⌫ Re˜␳1,3 共8k兲 Re˜˙␳_2,4=␦L 2共Im˜␳1,4− Im˜␳2,3兲 − ⌫ Re␳˜2,4, 共8l兲 Re␳˜˙_3,4=⌬L 2 共Im˜␳1,4+ Im˜␳2,3兲 − ⌫ Re˜␳3,4, 共8m兲 Im˜˙␳_1,4=␦L 2共Re˜␳1,3− Re˜␳2,4兲 + ⌬L 2 共Re˜␳1,2− Re˜␳3,4兲 − ⌫ Im␳˜1,4, 共8n兲 Im˜˙␳_2,3= −␦L 2共Re˜␳1,3− Re˜␳2,4兲 + ⌬L 2 共Re˜␳1,2− Re˜˙␳3,4兲 − ⌫ Im␳˜2,3, 共8o兲 with␳˜_i,j共t兲⬅␳_i,j共t兲e−i␻t _{for共ij兲苸 兵共12兲, 共13兲, 共24兲, 共34兲其,␳˜}

1,4共t兲⬅␳1,4共t兲e−2i␻tand␳˜2,3共t兲⬅␳2,3共t兲.

Equations共8兲 are valid on resonance, so ប␻⬅E2− E1= E4− E2=⌬EZand within the rotating wave approximation共RWA兲.18 Here,␳4共t兲 is given by␳4共t兲=1−␳1共t兲−␳2共t兲−␳3共t兲.

Equations 共8兲 can be split into two sets of coupled equations: Eqs. 共8a兲–共8i兲 and Eqs. 共8j兲–共8o兲. The solution of the second set is straightforwardly obtained and given by

Re˜␳_1,2共t兲 =1₂共Re关␳˜_1,2共0兲 −˜␳_3,4共0兲兴cos共⌬Lt兲 − Im关˜␳1,4共0兲 +˜␳2,3共0兲兴sin共⌬Lt兲

+ Re关␳˜_1,2共0兲 +˜␳_3,4共0兲兴兲e−⌫t, 共9a兲

Re˜␳_1,3共t兲 =1₂共Re关˜␳_1,3共0兲 −␳˜_2,4共0兲兴cos共␦Lt兲 − Im关˜␳1,4共0兲 −˜␳2,3共0兲兴sin共␦Lt兲

+ Re关˜␳_1,3共0兲 +␳˜_2,4共0兲兴兲e−⌫t_{, 共9b兲}

Re˜␳_2,4共t兲 =1₂共− Re关˜␳_1,3共0兲 −␳˜_2,4共0兲兴cos共␦Lt兲 + Im关˜␳1,4共0兲 −␳˜2,3共0兲兴sin共␦Lt兲

+ Re关˜␳_1,3共0兲 +˜␳_2,4共0兲兴兲e−⌫t, 共9c兲

Re˜␳_3,4共t兲 =1₂共− Re关˜␳_1,2共0兲 −˜␳_3,4共0兲兴cos共⌬Lt兲 + Im关˜␳1,4共0兲 +␳˜2,3共0兲兴sin共⌬Lt兲

+ Re关˜␳_1,2共0兲 +˜␳_3,4共0兲兴兲e−⌫t, 共9d兲

Im˜␳_1,4共t兲 =1₂共Im关␳˜_1,4共0兲 +˜␳_2,3共0兲兴cos共⌬Lt兲 + Im关˜␳1,4共0兲 −˜␳2,3共0兲兴cos共␦Lt兲

+ Re关␳˜_1,2共0兲 −˜␳_3,4共0兲兴sin共⌬Lt兲 + Re关˜␳1,3共0兲 −˜␳2,4共0兲兴sin共␦Lt兲兲e−⌫t, 共9e兲 Im˜␳_2,3共t兲 =1₂共Im关␳˜_1,4共0兲 +˜␳_2,3共0兲兴cos共⌬Lt兲 − Im关˜␳1,4共0兲 −˜␳2,3共0兲兴cos共␦Lt兲

+ Re关␳˜_1,2共0兲 −˜␳_3,4共0兲兴sin共⌬Lt兲 − Re关˜␳1,3共0兲 −˜␳2,4共0兲兴sin共␦Lt兲兲e−⌫t. 共9f兲 In order to solve the set of equations 共8a兲–共8i兲 we express ␳₁–␳₃, Im˜␳_1,2, Im˜␳_1,3, Im␳˜_2,4, Im˜␳_3,4, Re˜␳_1,4, and Re˜␳_2,3in terms of new variables x1– x8as follows:

␳1共t兲 = 1

␳2共t兲 = 1 2共x1共t兲 − x2共t兲 + x3共t兲兲e−⌫t, ␳3共t兲 = 1 2共− x1共t兲 + x2共t兲 + x3共t兲兲e−⌫t, Im˜␳_1,2共t兲 =1₂共x₄共t兲 + x₆共t兲兲e−⌫t, Im˜␳_1,3共t兲 =1₂共x₅共t兲 + x₇共t兲兲e−⌫t, Im˜␳_2,4共t兲 =1₂共x₅共t兲 − x₇共t兲兲e−⌫t, Im˜␳_3,4共t兲 =1₂共x₄共t兲 − x₆共t兲兲e−⌫t_, Re˜␳_1,4共t兲 = x₈共t兲e−⌫t, Re˜␳_2,3共t兲 = 共− x₈共t兲 + Z兲e−⌫t, 共10兲

with Z⬅Re关˜␳_1,4共0兲+␳˜_2,3共0兲兴. The transformation 共10兲 originates from pairwise adding and sub-tracting those equations among共8d兲–共8i兲 which share a common term on the right-hand side, e.g., the equations for Im˜˙␳_1,2and Im˜˙␳_3,4. The definition of x1– x8then naturally arises. Physically, the

new variables x1, x2, and x3 can be interpreted as x1共2兲= the probability for the spin in the left

共right兲 dot to be up, and x3= the probability for the two spins to be antiparallel, each modulated by

the exponential dependence on the decoherence rate⌫. Using 共10兲, Eqs. 共8a兲–共8i兲 are rewritten in terms of x₁共t兲–x₈共t兲, which leads to three sets of coupled equations. These equations and their solution are given in Appendix A. Equations共10兲 at time t=t₁, where t₁is the time during which the ESR field is switched on, thus represent the density matrix elements for the double-dot states after the ESR rotation applied to the left dot.

B. Step 2: ESR applied to the right dot

Equations共10兲 can also be used directly to obtain the solution after the second ESR rotation applied to the right dot, by substituting ⌬L→␦R and ␦L→⌬R in Eqs. 共9兲 and 共A2兲, and by exchanging x6↔x7in Eqs.共A4兲, using␳1共t1兲 instead of␳1共0兲, etc., as initial conditions. In order

to illustrate this solution, let us consider the initial condition of a singlet in the double dot关Eq. 共7兲兴, and let t2 be the duration of the second ESR rotation. In the case of no dissipation 共all

W ’ s = 0兲 and no influence of ESR applied to one dot on the spin in the other dot, we then obtain from Eqs.共10兲, 共A2兲, and 共A4兲, e.g., the occupation probability␳2共t1+ t2兲 the expression

␳2共t1+ t2兲 = 1 4

冋

1 +

再

冋

cos共⍀ ˜ ⌬t1兲 + ⌫ 2⍀˜ ⌬ sin共⍀˜ ⌬t1兲

册冋

cos共⍀˜⌬t2兲 + ⌫ 2⍀˜ ⌬ sin共⍀˜ ⌬t2兲

册

+ ⌬ ⍀˜ ⌬

sin共⌬ t₁兲sin共⍀˜_⌬t₂兲e−共⌫/2兲t1

冎

_e−共⌫/2兲共t1+t2兲

册

_, 共11兲

with⍀˜_⌬⬅1₂

冑

4⌬2−⌫2 and⌬⬅⌬L=⌬R. In the absence of decoherence共⌫=0兲 the expressions for

␳2共t1+ t2兲 and the other density matrix elements simplify to

␳1共t1+ t2兲 =␳4共t1+ t2兲 = 1

4共1 − cos␪1cos␪2− sin␪1sin␪2兲, 共12a兲

␳2共t1+ t2兲 =␳3共t1+ t2兲 = 1

␳1,2共t1+ t2兲 =␳2,4共t1+ t2兲 = −

i

4共cos␪1sin␪2− sin␪1cos␪2兲, 共12c兲

␳1,3共t1+ t2兲 =␳3,4共t1+ t2兲 =

i

4共cos␪1sin␪2− sin␪1cos␪2兲, 共12d兲

␳1,4共t1+ t2兲 = − 1

4共1 − cos␪1cos␪2− sin␪1sin␪2兲, 共12e兲

␳2,3共t1+ t2兲 = − 1

4共1 + cos␪1cos␪2+ sin␪1sin␪2兲, 共12f兲

with␪1⬅⍀˜⌬t1 and ␪2⬅⍀˜⌬t2. Equation 共11兲 is plotted in Fig. 2 as a function of the amount of

decoherence⌫.

Already for moderate amounts of decoherence⌫共t1+ t2兲=0.001, the occupation probability has

become 0.01% less than its value in the absence of decoherence␳₂⌫=0共t1+ t2兲=0.481, for the set of

parameters chosen in Fig. 2. This increases to 0.1% for⌫共t1+ t2兲=0.01.

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, namely the time evolution of the density matrix elements after the ESR rotations are completed and the gates to the quantum channels are opened; see Fig. 1共c兲. From this moment onward the coherent evolution due to the first term on the right-hand side of Eqs. 共2兲 stops and the time evolution of the matrix elements is solely determined by decay and decoherence rates represented by the second terms on the right-hand side of Eqs.共2兲. The off-diagonal elements␳i,j共t兲 then rotate with 共Ei− Ej兲/ប and decay with rate Vij

␳i,j共t兲 =␳i,j共tESR兲ei共Ei−Ej兲共t−tESR兲/បe−Vij共t−tESR兲 for t艌 tESR, 共13兲

where tESR⬅t1+ t2and Vijis given by Eq.共5兲 for i, j苸兵1, ... ,4其 and Eq. 共6兲 otherwise. The initial values␳i,j共tESR兲 for i, j苸兵1, ... ,16其 are given by␳m,n共tESR兲 for m,n苸兵1, ... ,4其 关Eqs. 共10兲兴 with

the correspondence in indices

i共j兲 苸 兵1,5,9,13其 ↔ m共n兲 = 1, i共j兲 苸 兵2,6,10,14其 ↔ m共n兲 = 2, i共j兲 苸 兵3,7,11,15其 ↔ m共n兲 = 3,

FIG. 2. The occupation probability␳2共t1+ t2兲 关Eq. 共11兲兴 of the quantum state D↑D↓ as a function of the amount of

decoherence关in units of 1/共t1+ t2兲兴. For ⌫=0␳2is given by Eq.共12b兲. Parameters used are ⌬ t1=␲/ 4,⌬ t2=␲/ 8, and all

i共j兲 苸 兵4,8,12,16其 ↔ m共n兲 = 4.

In this way the coherence at time tESRbetween any pair of states i , j苸兵1, ... ,16其 is given by the

coherence at tESR between those dot states m , n苸兵1, ... ,4其 which can 共eventually兲 coherently

evolve into i and j, i.e., the dot states m and n which have the same spin states as i and j, respectively. So, for example, ␳C↑C↑,D↓C↓共tESR兲=␳D↑D↑,D↓D↓共tESR兲⬅␳1,4共tESR兲. Note that ␳i,j共t兲=0 for those states in which at least one electron has reached a detector 共i苸兵17, ... ,36其 and/or j 苸兵17, ... ,36其兲, since for those states Vij=⬁. This corresponds to the assumption of immediate detection.

In the remaining part of this paper we focus on the evolution of the populations ␳₁共t兲–␳₃₆共t兲 for times t艌tESRunder the following conditions:

共i兲 We neglect the possibility of spin flips in the dots, i.e., we set WD↑↓= WD↓↑= 0. This is based on the fact that T1,Dis known to be much longer共0.85 ms at magnetic fields Bz= 8T兲 共Ref. 19兲 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. 共ii兲 We assume that the tunnel rate WTout 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 tdelay in between, and that spin is conserved during this tunneling process.

Typically20 tdelay⬇10−13s, which is much less than the travel time through a channel,

⬃10−10_s.

共iii兲 The tunnel rate WE through the QPCs is taken to be constant and equal for spin-up and spin-down electrons, i.e., the setup 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.

共iv兲 Spin flips in the exits are neglected, i.e., detection is assumed to be faster 共with typical times⬃10−11_{s兲 共Ref. 21兲 than the spin-flip rate 共Ⰷ10}−11_{s兲 共Ref. 9兲 in the detectors.}

The evolution equations for ␳1共t兲–␳36共t兲 for times t艌tESR are then given by the master

equations

␳˙i= − 2WT␳ifor i苸 兵1, ... ,4其. 共14兲

␳˙i= WT␳i−4+ WC↑↓␳i+2−共WE+ WT+ WC↓↑兲␳i, i苸 兵5,6其, 共15a兲

␳˙i= WT␳i−4+ WC_↓↑␳i−2−共WE+ WT+ WC_↑↓兲␳i, i苸 兵7,8其, 共15b兲 ␳˙i= WT␳i−8+ WC↑↓␳i+1−共WE+ WT+ WC↓↑兲␳i, i苸 兵9,11其, 共15c兲 ␳˙i= WT␳i−8+ WC↓↑␳i−1−共WE+ WT+ WC↑↓兲␳i, i苸 兵10,12其. 共15d兲 ␳˙₁₃= WT共␳5+␳9兲 + WC_↑↓共␳14+␳15兲 − 2共WE+ WC_↓↑兲␳13, 共16a兲 ␳˙₁₄= WT共␳6+␳10兲 + WC↓↑␳13+ WC↑↓␳16−共2WE+ WC↑↓+ WC↓↑兲␳14, 共16b兲 ␳˙₁₅= WT共␳7+␳11兲 + WC↓↑␳13+ WC↑↓␳16−共2WE+ WC↑↓+ WC↓↑兲␳15, 共16c兲 ␳˙₁₆= WT共␳8+␳12兲 + WC_↓↑共␳14+␳15兲 − 2共WE+ WC_↑↓兲␳16. 共16d兲

␳˙i共t兲 = WT␳i−8共t兲 + WE␳i−12共t − ttravel+ tESR兲 + WC↑↓␳i+1共t兲 − 共WE+ WC↓↑兲␳i共t兲 for i 苸 兵25,27其, 共18a兲

␳˙i共t兲 = WT␳i−8共t兲 + WE␳i−12共t − ttravel+ tESR兲 + WC↓↑␳i−1共t兲 − 共WE+ WC↑↓兲␳i共t兲 for i 苸 兵26,28其, 共18b兲

␳˙i共t兲 = WT␳i−8共t兲 + WE␳i−16共t − ttravel+ tESR兲 + WC↑↓␳i+2共t兲 − 共WE+ WC↓↑兲␳i共t兲 for i 苸 兵29,30其, 共18c兲

␳˙i共t兲 = WT␳i−8共t兲 + WE␳i−16共t − ttravel+ tESR兲 + WC↓↑␳i−2共t兲 − 共WE+ WC↑↓兲␳i共t兲 for i 苸 兵31,32其. 共18d兲

␳˙i= WE共␳i−8+␳i−4兲 i 苸 兵33, ... ,36其. 共19兲 Here, ttravel⬎tESRdenotes the earliest time at which an electron has traveled through the channels

and reached an exit. For times t艋ttravel,␳1共t兲–␳36共t兲 is thus given by Eqs. 共14兲–共19兲 for WE= 0, since at those times no electron can have arrived at a detector yet. The above sets of coupled equations can be solved one by one: first those for ␳1共t兲–␳4共t兲, then once the latter are known

those for ␳5共t兲–␳12共t兲 关in the pairs 共5,7兲, 共6,8兲, 共9,10兲, and 共11,12兲兴, then ␳13共t兲–␳16共t兲 and

␳17共t兲–␳24共t兲, subsequently ␳25共t兲–␳32共t兲 关in the pairs 共25,26兲, 共27,28兲, 共29,31兲, and 共30,32兲兴 and

finally␳33共t兲–␳36共t兲. Proceeding in this order and using initial conditions

␳i共tESR兲 =

再

Eqs.共10兲 for i = 1, ... ,4,

0 for i = 5, . . . ,36, 兵 共20兲

we obtain for␳1共t兲–␳4共t兲, the states in which both electrons are located in a dot,

␳i共t兲 =␳i共tESR兲e−2WT共t−tESR兲 i苸 兵1, ... ,4其, t 艌 tESR. 共21兲

Next, we find for␳₅共t兲–␳₁₂共t兲, which correspond to the quantum states in which one electron is located in a dot and the other in a channel, from Eqs.共15兲

␳5共t兲 = A5,7,1,3e−WETC共t−tESR兲+ B5,7,1,3e−共WE+WT兲共t−tESR兲+ C1,3e−2WT共t−tESR兲, 共22a兲

␳6共t兲 = A6,8,2,4e−WETC共t−tESR兲+ B6,8,2,4e−共WE+WT兲共t−tESR兲+ C2,4e−2WT共t−tESR兲, 共22b兲

␳7共t兲 = − A5,7,1,3e−WETC共t−tESR兲+ WC↓↑ WC↑↓ B_5,7,1,3e−共WE+WT兲共t−tESR兲_{+ D} 1,3e−2WT共t−tESR兲, 共22c兲 ␳8共t兲 = − A6,8,2,4e−WETC共t−tESR兲+ WC↓↑ WC_↑↓ B_6,8,2,4e−共WE+WT兲共t−tESR兲_{+ D} 2,4e−2WT共t−tESR兲, 共22d兲

␳9共t兲 = A9,10,1,2e−WETC共t−tESR兲+ B9,10,1,2e−共WE+WT兲共t−tESR兲+ C1,2e−2WT共t−tESR兲, 共22e兲

␳10共t兲 = − A9,10,1,2e−WETC共t−tESR兲+

WC↓↑

WC↑↓

B_9,10,1,2e−共WE+WT兲共t−tESR兲_{+ D}

1,2e−2WT共t−tESR兲, 共22f兲

␳12共t兲 = − A11,12,3,4e−WETC共t−tESR兲+ WC_↓↑ WC_↑↓ B11,12,3,4e−共WE+WT兲共t−tESR兲+ D3,4e−2WT共t−tESR兲, 共22h兲 where WETC⬅ WE+ WT+ WC↑↓+ WC↓↑, 共23a兲 Ai,j,k,l⬅ WC↓↑␳i共tESR兲 − WC↑↓␳j共tESR兲 WC↑↓+ WC↓↑ + WT共− WC↓↑␳k共tESR兲 + WC↑↓␳l共tESR兲兲 共WC↑↓+ WC↓↑兲共WE− WT+ WC↑↓+ WC↓↑兲 , 共23b兲 Bi,j,k,l⬅ WC↑↓ WC↑↓+ WC↓↑

冋

␳i共tESR兲 +␳j共tESR兲 − WT WE− WT 共␳k共tESR兲 +␳l共tESR兲兲

册

, 共23c兲 Ci,j= WT WE− WT 共WE− WT+ WC↑↓兲␳i共tESR兲 + WC↑↓␳j共tESR兲 WE− WT+ WC↑↓+ WC↓↑ , 共23d兲 Di,j= WT WE− WT WC↓↑␳i共tESR兲 + 共WE− WT+ WC↓↑兲␳j共tESR兲 WE− WT+ WC↑↓+ WC↓↑ . 共23e兲

For times t艋t_travel, the evolution of ␳₅共t兲–␳₁₂共t兲 is given by Eqs. 共22兲 with WE= 0. For times t 艌ttravel, these populations are given by Eqs.共22兲 with tESR→ttravel.

In order to obtain the solution for␳₁₃共t兲–␳₁₆共t兲, which corrresponds to the situation in which both electrons are located in a channel, we rewrite the equations for␳₁₃–␳₁₆as

␳˙₁₃= WT共␳5+␳9兲 + WC↑↓共␳14+␳15兲 − 2共WE+ WC↓↑兲␳13, 共24a兲

␳˙₁₄+␳˙₁₅= WT共␳6+␳7+␳10+␳11兲 + 2WC↓↑␳13−共2WE+ WC↑↓+ WC↓↑兲共␳14+␳15兲 + 2WC↑↓␳16,

共24b兲

␳˙₁₆= WT共␳8+␳12兲 + WC↓↑共␳14+␳15兲 − 2共WE+ WC↑↓兲␳16, 共24c兲

␳˙₁₄−␳˙₁₅= WT共␳6−␳7+␳10−␳11兲 − 共2WE+ WC↑↓+ WC↓↑兲共␳14−␳15兲. 共24d兲

Equations共24兲 consist of three coupled equations 共24a兲–共24c兲 and a separate one, Eq. 共24d兲. 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 now on WC↑↓= WC↓↑⬅WC,22we obtain

␳13共t兲 = − Ee−2共WE+WC兲共t−tESR兲+ 1 2Fe −2WE共t−tESR兲₋1 2F˜ e −2共WE+2WC兲共t−tESR兲_{+ H} 13e−共WE+WT+2WC兲共t−tESR兲

+ K13e−共WE+WT兲共t−tESR兲+ L13e−2WT共t−tESR兲, 共25a兲

␳14共t兲 = E˜e−2共WE+WC兲共t−tESR兲+ 1 2Fe−2WE共t−tESR兲+ 1 2F˜ e−2共WE+2WC兲共t−tESR兲+ H14e−共WE+WT+2WC兲共t−tESR兲 + K14e−共WE+WT兲共t−tESR兲+ L14e−2WT共t−tESR兲, 共25b兲 ␳15共t兲 = − E˜e−2共WE+WC兲共t−tESR兲+ 1 2Fe −2WE共t−tESR兲₊1 2F˜ e−2共WE +2WC兲共t−tESR兲_{+ H} 15e−共WE+WT+2WC兲共t−tESR兲 + K₁₅e−共WE+WT兲共t−tESR兲_{+ L} 15e−2WT共t−tESR兲, 共25c兲

␳16共t兲 = Ee−2共WE+WC兲共t−tESR兲+ 1 2Fe−2WE共t−tESR兲− 1 2F˜ e−2共WE+2WC兲共t−tESR兲+ H16e−共WE+WT+2WC兲共t−tESR兲 + K16e−共WE+WT兲共t−tESR兲+ L16e−2WT共t−tESR兲. 共25d兲

The coefficients in Eqs. 共25兲 are given in Appendix B. Also here, ␳13共t兲–␳16共t兲 for times t

艋ttravelare given by Eqs.共25兲 with WE= 0, and for times t艌ttravelthese populations are given by

Eqs.共25兲 with tESR→ttravel.

The solution of the next set,␳17共t兲–␳24共t兲, corresponding to the states in which one electron is

located in a dot while the other has reached a detector, is given by

␳i共t兲 = Aie−共WE+WT+2WC兲共t−ttravel兲− Bie−共WE+WT兲共t−ttravel兲− Cie−2WT共t−ttravel兲

+关␳i共ttravel兲 − Ai+ Bi+ Ci兴e−WT共t−ttravel兲 for i苸 兵17, ... ,24其, t 艌 ttravel, 共26兲

and␳i共t兲=0 for t艋ttravel. The coefficients Ai, Bi, and Ciin Eqs.共26兲 are given in Table I. Next, we solve for␳25共t兲–␳32共t兲, the states in which one spin has reached a detector, while the other is still

in a channel, in the pairs ␳i共t兲&␳j共t兲苸兵␳25共t兲&␳26共t兲,␳27共t兲&␳28共t兲,␳29共t兲&␳31共t兲,

and␳30共t兲&␳32共t兲其; see Eqs. 共18兲. For each pair the solution is given by, for times t艌ttravel

␳i共t兲 = Pi,je−WE共t−ttravel兲+ Qi,je−共WE+2WC兲共t−ttravel兲+ Mi,1e−2共WE+WC兲共t−ttravel兲+ Mi,2e−2WE共t−ttravel兲 + Mi,3e−2共WE+2WC兲共t−ttravel兲+ Mi,4e−共WE+WT+2WC兲共t−ttravel兲+ Mi,5e−共WE+WT兲共t−ttravel兲

+ Mi,6e−2WT共t−ttravel兲+ Mi,7e−WT共t−ttravel兲, 共27a兲

␳j共t兲 = Pj,ie−WE共t−ttravel兲+ Qj,ie−共WE+2WC兲共t−ttravel兲+ Mj,1e−2共WE+WC兲共t−ttravel兲+ Mj,2e−2WE共t−ttravel兲 + Mj,3e−2共WE+2WC兲共t−ttravel兲+ Mj,4e−共WE+WT+2WC兲共t−ttravel兲+ Mj,5e−共WE+WT兲共t−ttravel兲

+ Mj,6e−2WT共t−ttravel兲+ Mj,7e−WT共t−ttravel兲, 共27b兲 and␳i共t兲=␳j共t兲=0 for t艋ttravel. The coefficients Pi,j, Qi,j, and Mi,1, . . . , Mi,7 for i , j苸兵25, ... ,32其 are given in Appendix B.

Finally, we obtain the time evolution of the states␳₃₃共t兲–␳₃₆共t兲 in which both electrons have reached an exit. This is given by, for times t艌t_travel

␳j共t兲 = − WE

再

P_m,p+ P_n,q WE e−WE共t−ttravel兲₊Qm,p+ Qn,q WE+ 2WC e−共WE+2WC兲共t−ttravel兲 +Mm,1+ Mn,1 2共WE+ WC兲 e−2共WE+WC兲共t−ttravel兲₊ Mm,2+ Mn,2 2WE e−2WE共t−ttravel兲 + Mm,3+ Mn,3 2共WE+ 2WC兲 e−2共WE+2WC兲共t−ttravel兲₊ Mm,4+ Mn,4 WE+ WT+ 2WC e−共WE+WT+2WC兲共t−ttravel兲

TABLE I. Coefficients Ai, Bi, and Ciin Eqs.共26兲.

i Ai Bi Ci 17 −WEA5,7,1,3

Ⲑ

共WE+ 2WC兲 B5,7,1,3 WE

Ⲑ

WTC1,3 18 −WEA6,8,2,4

Ⲑ

共WE+ 2WC兲 B6,8,2,4 WE

Ⲑ

WTC2,4 19 WEA5,7,1,3

Ⲑ

共WE+ 2WC兲 B5,7,1,3 WE

Ⲑ

WTD1,3 20 WEA6,8,2,4

Ⲑ

共WE+ 2WC兲 B6,8,2,4 WE

Ⲑ

WTD2,4 21 −WEA9,10,1,2

Ⲑ

共WE+ 2WC兲 B9,10,1,2 WE

Ⲑ

WTC1,2 22 WEA9,10,1,2

Ⲑ

共WE+ 2WC兲 B9,10,1,2 WE

Ⲑ

WTD1,2 23 −WEA11,12,3,4

Ⲑ

共WE+ 2WC兲 B11,12,3,4 WE

Ⲑ

WTC3,4 24 WEA11,12,3,4

Ⲑ

共WE+ 2WC兲 B11,12,3,4 WE

Ⲑ

WTD3,4

+Mm,5+ Mn,5 WE+ WT e−共WE+WT兲共t−ttravel兲₊Mm,6+ Mn,6 2WT e−2WT共t−ttravel兲₊Mm,7+ Mn,7 WT e−WT共t−ttravel兲

冎

+ WE

冉

sum of all previous coefficients, so

Pm,p+ Pn,q WE +Qm,p+ Qn,q WE+ 2WC + ¯

冊

, 共28兲 for 共j,m,n,p,q兲 苸 兵共33,25,29,26,31兲,共34,26,30,25,32兲,共35,27,31,28,29兲,共36,28,32,27,30兲其. Special case. In order to illustrate the solution共28兲, we now derive explicit expressions for␳33共t兲

and ␳₃₄共t兲, the probabilities that a spin-up is detected in the left detector and, respectively, a spin-up or a spin-down in the right detector, for the special case of⌫=0 共no decoherence in the dots兲 and WC↑↓= WC↓↑= 0共no relaxation in the channel兲. 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 WT, the tunnel rate from dot to channel, and WE, the tunnel rate from channel to exit.

We are interested in finding ␳33共t兲 and ␳34共t兲 for times t艌ttravel 关since ␳33共t兲=␳34共t兲=0 ∀t

艋ttravel兴. To that end, we first calculate␳j共ttravel兲 for j艋16 from Eqs. 共21兲, 共22兲, and 共25兲 and then

all coefficients entering the expressions for ␳₃₃共t兲 and ␳₃₄共t兲 in Eqs. 共28兲. For ␳₁共t_travel兲 -␳₁₆共t_travel兲 we then obtain

␳1共ttravel兲 =␳4共ttravel兲 =␳1共tESR兲e−2WT共ttravel−tESR兲, 共29a兲

␳2共ttravel兲 =␳3共ttravel兲 =␳2共tESR兲e−2WT共ttravel−tESR兲, 共29b兲

␳i共ttravel兲 =␳1共tESR兲e−WT共ttravel−tESR兲共1 − e−WT共ttravel−tESR兲兲, i 苸 兵5,8,9,12其, 共29c兲

␳i共ttravel兲 =␳2共tESR兲e−WT共ttravel−tESR兲共1 − e−WT共ttravel−tESR兲兲, i 苸 兵6,7,10,11其, 共29d兲

␳13共ttravel兲 =␳16共ttravel兲 =␳1共tESR兲共1 − e−WT共ttravel−tESR兲兲2, 共29e兲

␳14共ttravel兲 =␳15共ttravel兲 =␳2共tESR兲共1 − e−WT共ttravel−tESR兲兲2. 共29f兲

Equation 共29兲 form the initial conditions that appear in the expressions for ␳33共t兲–␳36共t兲 关Eqs.

共28兲兴. We then find ∀t艌ttravel

␳33共t兲 =

冉

␳13共ttravel兲 − 2WT WE− WT ␳5共ttravel兲 + WT 2 共WE− WT兲2

␳1共ttravel兲

冊

e−2WE共t−ttravel兲+

冉

− 2␳13共ttravel兲

−2共WE− 2WT兲 WE− WT ␳5共ttravel兲 + 2WT WE− WT ␳1共ttravel兲

冊

e−WE共t−ttravel兲+ 2WE WE− WT

冉

␳5共ttravel兲 − WT WE− WT ␳1共ttravel兲

冊

e−共WE+WT兲共t−ttravel兲− 2WE WE− WT

共␳5共ttravel兲 +␳1共ttravel兲兲e−WT共t−ttravel兲

+ WE

2

共WE− WT兲2

␳34共t兲 =

冉

␳14共ttravel兲 − 2WT WE− WT ␳6共ttravel兲 + WT 2 共WE− WT兲2

␳2共ttravel兲

冊

e−2WE共t−ttravel兲+

冉

− 2␳14共ttravel兲

−2共WE− 2WT兲 WE− WT ␳6共ttravel兲 + 2WT WE− WT ␳2共ttravel兲

冊

e−WE共t−ttravel兲+ 2WE WE− WT

冉

␳6共ttravel兲 − WT WE− WT ␳2共ttravel兲

冊

e−共WE+WT兲共t−ttravel兲− 2WE WE− WT

共␳6共ttravel兲 +␳2共ttravel兲兲e−WT共t−ttravel兲

+ WE

2

共WE− WT兲2

␳2共ttravel兲e−2WT共t−ttravel兲+␳14共ttravel兲 + 2␳6共ttravel兲 +␳2共ttravel兲. 共30b兲

One can see directly from Eqs.共30兲 that the time dependence of␳33and␳34is determined by five

exponential functions, whose relative magnitude depends on the ratio between WEand WT. This is illustrated in Fig. 3, which shows Eqs.共30兲 as a function of t−ttravelfor various rates WEand WT. For WTⰆWE the time needed to reach the stationary state共the average detection time兲 is domi-nated by the term⬃e−WT共t−ttravel兲_{, whereas for W}

T⬇WE the terms⬃e−2WE共t−ttravel兲, e−共WE+WT兲共t−ttravel兲, and e−2WT共t−ttravel兲_dominate.

IV. CONCLUSION

In summary, we have presented an analytical solution of a set of coupled master equations that describes 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 one to obtain qualitative insight in the competition between time scales in these systems.

ACKNOWLEDGMENTS

Stimulating discussions with D.P. DiVincenzo are gratefully acknowledged. This work has been supported by the Stichting voor Fundamenteel Onderzoek der Materie共FOM兲, by the Neth-erlands Organisation for Scientific Research共NWO兲, and by the EU’s Human Potential Research Network under Contract No. HPRN-CT-2002-00309共“QUACS”兲.

FIG. 3. The probabilities␳₃₃to measure two spin-up electrons and␳₃₄to measure a spin-up and a spin-down electron in the left and right exits, respectively, for times t艌ttravel. Parameters used are␪1=␲/ 2,␪2=␲/ 8关so that␳1共tESR兲=0.154 and

APPENDIX A: SOLUTION OF EQS._{„8a…–„8i…}

Using the substitution Eqs.共10兲, Eqs. 共8a兲–共8i兲 transform into

x˙1= −共WD_↓↑+ WD_↑↓−⌫兲x1−⌬Lx5+ WD_↑↓e⌫t, 共A1a兲 x˙5=⌬Lx1− ⌬L 2 e ⌫t_{, 共A1b兲} x˙2= −共WD↓↑+ WD↑↓−⌫兲x2−␦Lx4+ WD↑↓e⌫t, 共A1c兲 x˙4=␦Lx2− ␦L 2e ⌫t_{, 共A1d兲} x˙3= −共2WD_↓↑+ 2WD_↑↓−⌫兲x3+␦Lx6+⌬Lx7+ 2WD_↑↓e⌫t+共WD_↓↑− WD_↑↓兲共x1+ x2兲, 共A1e兲 x˙6= −␦Lx3+ 2⌬Lx8+ ␦L 2e ⌫t_−⌬ LZ, 共A1f兲 x˙7= −⌬Lx3+ 2␦Lx8+ ⌬L 2 e ⌫t_−␦ LZ, 共A1g兲 x˙₈= −⌬L 2 x6− ␦L 2x7, 共A1h兲

with Z⬅Re关␳˜_1,4共0兲+˜␳_2,3共0兲兴. In deriving Eqs. 共A1兲, we have used that

␳4= 1 −␳1−␳2−␳3,

Re˜␳_2,3= − Re˜␳_1,4+ Ze−⌫t.

Equations 共A1兲 consist of three sets of coupled equations: 共A1a兲–共A1b兲, 共A1c兲–共A1d兲, and 共A1e兲–共A1h兲. The solution of the first two sets is given by

x1共t兲 =

冋

−

冉

共WD_↑↓+ WD_↓↑−⌫兲共x1共0兲 − A1兲 + 2⌬L共x5共0兲 − A2兲 2⍀_⌬

冊

sin⍀⌬t +共x1共0兲 − A1兲cos ⍀⌬t

册

e−共1/2兲共WD↑↓+WD↓↑−⌫兲t+ A1e⌫t, 共A2a兲 x5共t兲 =

冋

冉

2⌬L共x1共0兲 − A1兲 + 共WD_↑↓+ WD_↓↑−⌫兲共x5共0兲 − A2兲 2⍀_⌬

冊

sin⍀⌬t +共x5共0兲 − A2兲cos ⍀⌬t

册

e−共1/2兲共WD↑↓+WD↓↑−⌫兲t+ A2e⌫t, 共A2b兲

x2共t兲 =

冋

−

冉

共WD_↑↓+ WD_↓↑−⌫兲共x2共0兲 − A3兲 + 2␦L共x4共0兲 − A4兲 2⍀_␦

冊

sin⍀␦t +共x2共0兲 − A3兲cos ⍀␦t

册

e−共1/2兲共WD↑↓+WD↓↑−⌫兲t+ A3e⌫t, 共A2c兲 x₄共t兲 =

冋

冉

2␦L共x2共0兲 − A3兲 + 共WD↑↓+ WD↓↑−⌫兲共x4共0兲 − A4兲 2⍀_␦

冊

sin⍀␦t +共x4共0兲 − A4兲cos ⍀␦t

册

e−共1/2兲共WD↑↓+WD↓↑−⌫兲t+ A4e⌫t, 共A2d兲 with ⍀⌬= 1 2

冑

4⌬L 2 −共WD↑↓+ WD↓↑−⌫兲2, 共A3a兲 ⍀_␦=1 2

冑

4␦L 2 −共WD_↑↓+ WD_↓↑−⌫兲2, 共A3b兲 A1= ⌬L2+ 2⌫WD↑↓ 2共⌬L2+⌫共WD_↑↓+ WD_↓↑兲兲 , 共A3c兲 A₂= ⌬L共WD↑↓− WD↓↑兲 2共⌬L 2 +⌫共WD↑↓+ WD↓↑兲兲 , 共A3d兲 A3= ␦L 2_{+ 2⌫W} D↑↓ 2共␦L2+⌫共WD↑↓+ WD↓↑兲兲 , 共A3e兲 A4= ␦L共WD↑↓− WD↓↑兲 2共␦L2+⌫共WD_↑↓+ WD_↓↑兲兲 . 共A3f兲

So far no approximations have been made, apart from assuming the decoherence rate⌫ to be equal for all off-diagonal terms of the density matrix␳关Eqs. 共8兲兴. In order to obtain the solution of the remaining equations共A1e兲–共A1h兲, we assume␦L= 0共no influence of the ESR field on the spin in the right dot兲 and WD↑↓= WD↓↑= 0,23 and find24

x3共t兲 =

冤

⌫

冉

x3共0兲 − 1 2

冊

+ 2⌬Lx7共0兲 2⍀˜_⌬ sin⍀˜_⌬t +

冉

x3共0兲 − 1 2

冊

cos⍀ ˜ ⌬t

冥

e共⌫/2兲t+ 1 2e ⌫t_{, 共A4a兲}

x6共t兲 = x6共0兲cos ⌬Lt +共2x8共0兲 − Z兲sin ⌬Lt, 共A4b兲

x7共t兲 =

冤

− 2⌬L

冉

x3共0兲 −

1

2

冊

−⌫x7共0兲 2⍀˜_⌬

x8共t兲 = 1

2关− x6共0兲sin ⌬Lt +共2x8共0兲 − Z兲cos ⌬Lt + Z兴, 共A4d兲 with⍀˜_⌬=₂1

冑

4⌬L2−⌫2.

APPENDIX B: COEFFICIENTS OF EQS._{„25… and „27…}

The coefficients in Eqs.共25兲 are given by

E =−␳13共tESR兲 +␳16共tESR兲 + H13− H16+ K13− K16+ L13− L16

2 ,

E

˜ =␳14共tESR兲 −␳15共tESR兲 − H14+ H15− K14+ K15− L14+ L15

2 ,

F =␳13共tESR兲 +␳14共tESR兲 +␳15共tESR兲 +␳16共tESR兲 − 2共K14+ K15兲 − L13− L14− L15− L16

2 ,

F

˜ =−␳13共tESR兲 +␳14共tESR兲 +␳15共tESR兲 −␳16共tESR兲 − 2共H14+ H15兲 + L13− L14− L15+ L16

2 , H13= WT关共WE− WT+ WC兲共A5,7,1,3+ A9,10,1,2兲 + WC共A6,8,2,4+ A11,12,3,4兲兴 共WE− WT兲共WE− WT+ 2WC兲 , H14= WT关WC共A5,7,1,3− A11,12,3,4兲 + 共WE− WT+ WC兲共A6,8,2,4− A9,10,1,2兲兴 共WE− WT兲共WE− WT+ 2WC兲 , H15= − H14关共5,7,1,3兲 ↔ 共6,8,2,4兲,共9,10,1,2兲 ↔ 共11,12,3,4兲兴, H16= − H13关共5,7,1,3兲 ↔ 共6,8,2,4兲,共9,10,1,2兲 ↔ 共11,12,3,4兲兴, K13= H13共A → B兲, K14= WT关WC共B5,7,1,3+ B11,12,3,4兲 + 共WE− WT+ WC兲共B6,8,2,4+ B9,10,1,2兲兴 共WE− WT兲共WE− WT+ 2WC兲 , K15= K14关共5,7,1,3兲 ↔ 共6,8,2,4兲,共9,10,1,2兲 ↔ 共11,12,3,4兲兴, K16= H16共A → − B兲, L13= WT关共2共WE− WT+ WC兲2− WC 2_兲共C 1,2+ C1,3兲 + WC 2_共D 2,4+ D3,4兲 + WC共WE− WT+ WC兲 ⫻共C2,4+ C3,4+ D1,2+ D1,3兲兴/关4共WE− WT兲共WE− WT+ 2WC兲共WE− WT+ WC兲兴, L14= WT关WC共WE− WT+ WC兲共C1,2+ C1,3+ D2,4+ D3,4兲 + 2共WE− WT+ WC兲2共C2,4+ D1,2兲 + WC 2_共C 3,4− C2,4− D1,2+ D1,3兲兴/关4共WE− WT兲共WE− WT+ 2WC兲共WE− WT+ WC兲兴, L₁₅= L₁₄共C_2,4↔ C_3,4,D_1,2↔ D_1,3兲,

L16= L13共C1,2↔ D2,4,C1,3↔ D3,4兲. 共B1兲

For共i, j兲=共25,26兲 the coefficients in Eqs. 共27兲 are given by

P25,26= P26,25= − 1 2

兺

_k=1 7 共M25,k+ M26,k兲, Q25,26= − Q26,25= − 1 2

兺

_k=1 7 共M25,k− M26,k兲, M_25,1=共WE+ WC兲E + WCE ˜ WE+ 2WC , M26,1= M25,1共E ↔ − E˜兲, M25,2= M26,2= − 1 2F, M25,3= − M26,3= WE 2共WE+ 2WC兲 F ˜ , M25,4= WE关− 共WT+ WC兲共共WE+ 2WC兲H13− WTA5,7,1,3兲 + WC共共WE+ 2WC兲H14− WTA6,8,2,4兲兴 WT共WE+ 2WC兲共WT+ 2WC兲 , M26,4= M25,4共H13↔ H14,A5,7,1,3↔ A6,8,2,4兲, M25,5= 共WC− WT兲共WEK13− WTB5,7,1,3兲 + WC共WEK14− WTB6,8,2,4兲 WT共WT− 2WC兲 , M26,5= M25,5共K13↔ K14,B5,7,1,3↔ B6,8,2,4兲, M25,6= WE关共WE− 2WT+ WC兲共L13− C1,3兲 + WC共L14− C2,4兲兴 共WE− 2WT+ 2WC兲共WE− 2WT兲 , M26,6= M25,6共L13↔ L14,C1,3↔ C2,4兲, M25,7= WT关共WE− WT+ WC兲关− A17+ B17+ C17兴 + WC关− A18+ B18+ C18兴兴/关共WE− WT+ 2WC兲 ⫻共WE− WT兲兴, M_26,7= M_25,7共17 ↔ 18兲. 共B2兲

The coefficients in Eqs.共27兲 for 共i, j兲=共27,28兲, 共29,31兲, and 共30,32兲 are obtained from Eqs. 共B2兲 by replacing indices as given in Table II.

1_{See, e.g., K. Blum, Density Matrix Theory and Applications共Plenum, New York, 1996兲, and references therein.} 2_{P. Meystre and M. Sargent, Elements of Quantum Optics共Springer, New York, 1999兲.}

3_{Y. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357共2001兲.} 4_{Reference 2, Chap. 14.}

5_{H.-A. Engel and D. Loss, Phys. Rev. Lett. 86, 4648共2001兲; Phys. Rev. B 65, 195321 共2002兲; S. A. Gurvitz, L.}

Fedichkin, D. Mozyrsky, and J. P. Berman, Phys. Rev. Lett. 91, 066801共2003兲, and references therein.

6_{See, e.g., D. Saraga and D. Loss, Phys. Rev. Lett. 90, 166803共2003兲.}

7_{See, e.g., J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73, 565共2001兲.}

8_{Condensed-matter systems such as the one considered here are particularly strongly affected by decoherence due to the}

strong interaction between the system and its environment. But, decoherence and dissipation are also relevant for other quantum systems which may serve as qubits, such as cold atoms关see, e.g., C. F. Roos, G. P. T. Lancaster, M. Riebe, H. Häffner, W. Hänsel, S. Gulde, C. Becher, J. Eschner, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. Lett. 92, 220402 共2004兲兴.

9_{M. Blaauboer and D. P. DiVincenzo, cond-mat/0502060.}

10_{For a review on quantum dots, see L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R. M. Westervelt, and}

N. S. Wingreen, in Proceedings of the Advanced Study Institute on Mesoscopic Electron Transport, edited by L. P. Kouwenhoven, G. Schön, and L. L. Sohn共Kluwer, Dordrecht, 1997兲.

11_{Controlling the number of electrons in a lateral quantum dot down to one or zero has recently been achieved; see M.}

Ciorga, A. S. Sachrajda, P. Hawrylak, C. Gould, P. Zawadzki, S. Jullian, Y. Feng, and Z. Wasilewski, Phys. Rev. B 61, R16315共2000兲; J. M. Elzerman, R. Hanson, J. S. Greidanus, L. H. Willems van Beveren, S. De Franceschi, L. M. K. Vandersypen, S. Tarucha, and L. P. Kouwenhoven, ibid. 67, 161308共2003兲; R. M. Potok, J. A. Folk, C. M. Marcus, V. Umansky, M. Hanson, and A. C. Gossard, Phys. Rev. Lett. 91, 016802共2003兲.

12_{Up to parallel magnetic fields of at least 12 T the singlet state is the ground state of a doubly occupied quantum dot; see}

R. Hanson, L. M. K. Vandersypen, L. H. Willems van Beveren, J. M. Elzerman, I. T. Vink, and L. P. Kouwenhoven, Phys. Rev. B 70, 241304共2004兲.

13_{R. M. Potok, J. A. Folk, C. M. Marcus, and V. Umansky, Phys. Rev. Lett. 89, 266602共2002兲.} 14_{V. N. Golovach, A. E. Khaetskii, and D. Loss, Phys. Rev. B 69, 245327共2004兲, and references therein.} 15_{See Sec. II C in H. A. Engel and D. Loss, Phys. Rev. B 65, 195321共2002兲.}

16_{For the same reason tunneling of an electron from dot to channel—which is an incoherent process and may involve}

inelastic scattering from one orbital to another—does not affect the spin coherence of the entangled pair.

17_{Assuming the same decoherence rate⌫ for both dots is not a crucial assumption and Eqs. 共8兲 can straightforwardly be}

solved for different decoherence rates in the left and right dot. The resulting solution is qualitatively of the same form as Eqs.共9兲 and 共10兲, only with more lengthy expressions.

18_{The RWA approximation共Ref. 2兲 is not essential and one can solve Eqs. 共8兲 without it, which results in a solution that}

is qualitatively similar. Since␻t1Ⰶ1 in our system 共Ref. 9兲, where t1is the time required to rotate one spin, the RWA is

an excellent approximation in practice.

19_{J. M. Elzerman, R. Hanson, L. H. Willems van Beveren, B. Witkamp, L. M. K. Vandersypen, and L. P. Kouwenhoven,}

Nature共London兲 430, 431 共2004兲.

TABLE II. Required substitution of indices and coefficients in Eqs.共27兲 in order to obtain the corresponding coefficients for ␳i共t兲&␳j共t兲 with 共i, j兲

苸兵共27,28兲,共29,31兲,共30,32兲其. 共27,28兲 共29,31兲 共30,32兲 25→28 25→29 25→32 26→27 26→31 26→30 13→16 13→16 14→15 14→15 17→20 17→21 17→24 18→19 18→23 18→22 E↔−E E↔−E E ˜ ↔−E˜ E˜ ↔−E˜ A5,7,1,3↔−A6,8,2,4 共5,7,1,3兲→共9,10,1,2兲 A5,7,1,3→−A11,12,3,4 B_5,7,1,3↔B_6,8,2,4 B_5,7,1,3→B_11,12,3,4 共6,8,2,4兲→共11,12,3,4兲 A6,8,2,4→−A9,10,1,2 B_6,8,2,4→B_9,10,1,2 C1,3→D2,4 C1,3→C1,2 C1,3→D3,4 C2,4→D1,3 C2,4→C3,4 C2,4→D1,2

20_{V. Cerletti, O. Gywat, and D. Loss, cond-mat/0411235.}

21_{R. Deblock, E. Onac, L. Gurevich, and L. P. Kouwenhoven, Science 301, 203共2003兲.} 22_{See the calculation of W}

C↑↓and WC↓↑in Ref. 9.

23_{For the general case the expressions are rather lengthy, but not qualitatively different.} 24_{The solutions for x}