Generation and detection of a spin entanglement in nonequilibrium quantum dots

Stefan Legel¹, Jürgen König² and Gerd Schön^1,3

¹ Institut für Theoretische Festkörperphysik and DFG-Center for Functional Nanostructures (CFN), Universität Karlsruhe, 76128 Karlsruhe, Germany
² Institut für Theoretische Physik III, Ruhr-Universität Bochum, 44780 Bochum, Germany

³ Author to whom any correspondence should be addressed.

E-mail: gerd.schoen@phys.uni-karlsruhe.de

Received 19 December 2007
Published 30 April 2008

Contents

• 1. Introduction
• 2. Model
• 3. Preparation of stationary entangled states
• 4. Probing the singlet-triplet imbalance
• 5. Conclusions
• Acknowledgments
• References

1. Introduction

The controlled generation and detection of entanglement of quantum states remains one of the fundamental challenges of quantum physics. Experiments with entangled photons have already entered the realm of advanced quantum communication and cryptography. In solid-state systems electron spins are considered as prime candidates for the demonstration of such effects. Spin degrees of freedom are only weakly coupled to the environment, which leads to long decoherence times [1, 2], and coherence lengths well exceeding the micrometer range. Several set-ups involving quantum dots have been proposed to create and detect pairs of spatially separated, spin-entangled electrons. The schemes rely, e.g. on the extraction of Cooper pairs from a superconductor [3], or the separation of a pair of entangled electrons from a singlet ground state of single [4, 5] or double quantum dots [6, 7]. Evidence of entanglement in these systems can be obtained from noise measurements [8] or coincidence detection [7]. In [9], a detection scheme was proposed where pure singlet and triplet states can be distinguished by Aharonov–Bohm (AB) interferometry.

In this paper, we analyze a set-up with two spatially separated quantum dots, illustrated in figure 1, which allows the creation and detection of entanglement of spatially separated electron spins. The two dots are coupled coherently to a joint source electrode on the left and to two independent drain reservoirs at the top and bottom. When a bias voltage is applied, electrons are driven from the source via the dots to the drain electrodes. A nonequilibrium, mixed electronic state is created. For appropriate values of the voltage, due to the strong onsite Coulomb repulsion, a state with two electrons (one on each dot) has a high probability. It turns out to be of the form of a Werner state, with a strong enhancement of the singlet component, although singlet and triplet states are degenerate [10].

In order to detect the entanglement, an additional joint reservoir, closing the AB geometry, is coupled to the system. The current to this reservoir is studied in the cotunneling regime. Under certain conditions the entanglement leads to a suppression of the AB oscillations. Hence this part of the set-up serves as a probe of the state of the system and as a detector for entanglement. Exploring the results in a wide parameter range we identify regimes where the double dot has a large probability to be in a singlet state. The predicted behavior provides a proof of concept for the entanglement generation in coherently coupled, nonequilibrium quantum dots.

2. Model

We consider the system shown in figure 1. It consists of two single-level quantum dots, i = u and d, with strong on-site Coulomb repulsion U described by the Hamiltonian

The dots are coupled to joint reservoirs on the left and right forming an AB loop, and to two separate leads `up' and `down'. The total system is then modeled by H = H_dots + H_res + H_t with the four leads, r = L, R, U and D, with spin degenerate energies ε_rk,

and the four tunnel Hamiltonians H_t = H_tL + H_tR + H_tU + H_tD. Tunneling between the upper dot and reservoir is simply described by

and similar for the lower pair. Tunneling between the joint leads on the left and the right, on the other hand, is sensitive to the phases of the matrix elements. Without loss of generality we can distribute them in a symmetric gauge,

All tunneling matrix elements are taken to be independent of spin and energy, and they can be chosen real and positive. The leads are assumed to be in local equilibrium, described by electrochemical potentials μ_r.

To keep the discussion transparent, we assume some symmetries. We set the upper and the lower reservoirs to the same potential, μ_U/D = μ_U = μ_D, and restrict ourselves to degenerate dot levels, ε = ε_u = ε_d. We, furthermore, assume symmetric coupling strengths of the upper and lower dots to the left reservoir, t_Lu = t_Ld = t_L, and similarly for the right reservoir, t_Ru = t_Rd = t_R, and we assume t_U = t_D = t_U/D. Defining the tunneling strength by Γ_r = 2π t_r² N_r, where N_r is the density of states of the relevant reservoir, we are left with three parameters Γ_L, Γ_R and Γ_U/D, which gives us enough freedom to tune the system into the interesting regimes. Double occupancy of each dot is suppressed by a strong Coulomb repulsion U, for simplicity assumed here to be larger than all other energy scales. However, the generalization to finite U is straightforward. We apply two bias voltages measured relative to μ_L = 0: a strong bias voltage between the left and the upper and lower reservoirs, eV_U/D = μ_U/D, and a small transport voltage between the left and right reservoir, eV_R = μ_R, used to probe the state of the system.

In essence, the preparation and measurement scheme analyzed in the following is based on two observations. First, the components of the density matrix which describe the doubly occupied system with one electron on each dot is a so-called Werner state, a mixture of a singlet and three equivalent triplet states. The distribution between singlet and triplet probabilities can be tuned in a certain range by the coupling strengths and the potentials applied to the system. For illustration, we consider the state obtained immediately after charging the double dot with two electrons with opposite spin from the left reservoir

The minus sign in the last term is a direct consequence of the Fermi statistics. In view of this sign and the Pauli principle it is obvious that the equivalent expression for a pair of electrons with equal spin vanishes. A strong onsite Coulomb interaction prohibits double occupancy of each dot. Hence the state reduces to a perfectly entangled, pure singlet state

In equilibrium, however, the state has relaxed to a completely unentangled mixed state with uniformly distributed probabilities for singlet and triplets.

The second observation is that for the cotunneling rates the phases of the transport contributions for singlet and triplet states are shifted relative to each other by π. Hence, they yield different interference patterns in the AB interferometer.

The strategy, therefore, is to use the subsystem composed of the left, upper and lower reservoirs, and the dots to prepare the state of the system. In particular, we tune the ratio of singlet and triplet probabilities via the bias voltage V_U/D. The signatures of this state in the transport are monitored by the AB interferometer, which is operated in the cotunneling regime for the tunneling to the right reservoir. The main message of this paper is that for a set-up with cascading coupling strengths, Γ_L  Γ_U/D  Γ_R, an enhanced probability for singlet over triplet formation is associated with a suppression of the oscillations in the conductance of the AB probe.

3. Preparation of stationary entangled states

We investigate the system in the frame of the real-time diagrammatic technique  [11]–[13] which allows us to go beyond an orthodox master equation approach. In particular, we account for the quantum coherent time evolution of the double dot and focus on correlations between the electrons. The elements of the reduced density matrix are given by p_χ^χ ' = χ '|tr_res ρ|χ, where the reservoir degrees of freedom of the full density matrix are traced out, and χ and χ ' label general many-body states of the double dot. For strong Coulomb repulsion, the reduced Hilbert space is spanned by nine basis states |χ_u, χ_d with χ_i = 0, ↑ and ↓.

To describe the symmetries implied by tunneling we introduce a tailored basis. The state of the empty system and its probability are denoted by |0, 0 and p₀, respectively. A single electron of spin σ can occupy, with overall probability for single occupancy , either the upper or the lower dot, or, in general, any coherent superposition of these two. Thus, the corresponding block of the density matrix describes a quantum two-state system, which we express as an isospin for each spin component, I_σ. A natural basis are the two states with one electron in the upper or in the lower dot. However, in this basis the joint left and right reservoirs couple via the Hamiltonian (4) to the non-diagonal components I_σ·m_L/R of the isospin, with

which introduces a non-diagonal time evolution of the reduced density matrix. Electrons entering the double dot from the left reservoir generate an isospin polarization in the direction of m_L, whereas electrons entering from the right are polarized in the direction of m_R. We further note that due to the spin symmetry of the Hamiltonian the two spin components are completely equivalent. We will formulate equations for the expectation values of the isospin for either spin component, I_↑ = I_↓ = I_↑/↓. To simplify expresssions we dropped the angular brackets in the last form.

The doubly occupied subspace with one electron on each dot is naturally spanned by the spin singlet |S, which we find with probability p_S, and the three triplet states |T ₊ , |T₀ and |T_–. Due to the spin symmetry of the Hamiltonian the three triplet states are equivalent and are occupied with equal probability p_T0 = p_T± = p_T/3. The corresponding block of the density matrix can be represented in the Werner form [14]

where the coefficient F = p_S/(p_S + p_T) denotes the Werner fidelity. Values in the range of 1/2 < F≤1 indicate a quantum correlated mixed state from which arbitrary entanglement can be distilled by suitable purification protocols [15, 16]. As we will show below, the Werner fidelity of the stationary state can be tuned in a certain range by the tunnel couplings and the applied bias voltage. Regimes with F > 1/2 are feasible if the double dot is predominantly charged from a common reservoir.

The kinetic equations for the reduced density matrix can be arranged in vector form with the diagonal probabilities collected in p = (p₀, p₁, p_S, p_T) and the state of the singly occupied subspace specified by the isospin I_↑/↓ = (I_x, I_y, I_z). In lowest order in the coupling strengths Γ_L, Γ_R and Γ_U/D we obtain

where the Fermi distributions f_r = 1/(e^{β(ε–μ_r)} + 1) in the respective reservoirs are evaluated at the dots' energy ε. The equations (7) extend our earlier work [10], where we considered a double-dot system coupled to three reservoirs, but without the AB probe and right reservoir.

To generate and detect a singlet–triplet imbalance, the system parameters have to be chosen specifically. Firstly, the tunnel coupling to the right lead should be much weaker than the other ones, Γ_R  Γ_L, Γ_U/D, in order to ensure that the state of the double dot is not affected by the measurement via the interferometer. Furthermore, only a small bias voltage is applied across the AB ring, and the dot levels are kept off resonance such that the flux-sensitive linear conductance is dominated by cotunneling processes. Finally, in order to find with large probability two electrons in the system, the rate for charging should be much larger than for discharging, Γ_L  Γ_U/D.

In the following, we compare two situations:

4. Probing the singlet–triplet imbalance

Cases (i) and (ii) lead to significantly different interference signatures in the transport through the AB ring. While the first one shows a strong flux dependence and large amplitude of the AB oscillations, they are suppressed in the second case, where the Werner fidelity approaches 1/2 if the double dot is predominantly occupied with two electrons. To demonstrate this difference we calculate the linear conductance through the AB interferometer. The contribution to first-order in the coupling strength is evaluated within the diagrammatic technique. It turns out to be small due to our choice of parameters (compare also figure 3). For the dominant second-order contribution, we calculate the stationary density matrix according to (7) and obtain the cotunneling rates from Fermi's golden rule [17]–[19]

Here the initial double-dot state is denoted by χ_i, and all possible final states are summed over. The energy E_n of the intermediate virtual state |n is either 0, ε or 2ε and ω is the energy of the incoming lead electron.

Gathering all cotunneling terms corresponding to the four scenarios where the dot system is empty (χ = 0), occupied with a single electron with isospin I_↑/↓ (χ = 1), or occupied with two electrons either in a singlet (χ = S) or a triplet (χ = T) state, we find the second-order contribution to the linear conductance G_R = ∂I_LR/∂V_R|_VR = 0,

The AB flux dependence is included in the dimensionless conductance , with

The flux sensitivity of the conductance is twofold. First, the phase factors from the tunneling Hamiltonian are taken into account in the coherent summation of processes in the cotunneling rates (8). In particular, as discussed in [9], the phase dependences of the contributions from an initial singlet and a triplet state (equations (10c) and (10d)) are shifted relative to each other by π. We emphasize that these results rely on the degeneracy of singlet and triplets. All four constitute equivalently possible final states of cotunneling processes with an initially doubly occupied system. If there was a large singlet–triplet energy splitting with transitions between singlet and triplet states being suppressed, then g_{S, T}() = (1 + cos ) p_{S, T}/4 without any distinction between the contributions.

The second kind of flux dependence results from the stationary state distribution of the system. In general nonequilibrium situations, the reduced density matrix, represented by p₀, p₁, I_↑/↓, p_S and p_T, is influenced by the AB-probe and becomes phase sensitive. This effect is weak for weak tunnel coupling Γ_R  Γ_L and Γ_U/D where the state of the system is only weakly influenced by the AB-probe. In the limit Γ_L  Γ_U/D  Γ_R and the double dot charged from the left reservoir the isospin points approximately in the direction of m_L such that g₁()≈2 (3–cos  ) p₁–16 (1–cos ) |I_↑/↓|.

We compare now the two scenarios (i) and (ii). For low dot energies in an equilibrium situation, ε  μ_r, the double dot is mainly occupied with two electrons with p_T≈3/4 and p_S≈1/4. As a consequence, in this equilibrium reference situation there is good visibility of the AB oscillations,

In contrast, for strong bias voltage, the phase dependence is suppressed. To see this, we expand the density matrix for a strong asymmetry in the coupling to the left, and the upper and lower reservoirs, x = Γ_U/D/Γ_L  1, and obtain , , , and . Thus, the dimensionless conductance approaches

For strong asymmetry, Γ_U/D  Γ_L, the oscillations of the singlet and triplet terms cancel each other and the phase dependence of the conductance vanishes. This suppression of the AB amplitude allows us to detect the singlet–triplet asymmetry generated in the system.

We remark that if the system were prepared in a state with maximal entanglement, p_S = 1 or p_T = 1, as considered in [9], then the visibility of the AB oscillations would be even twice as large as in equilibrium, . For a steady-state scenario considered in this paper, however, it is not possible to have both a large overall probability of double occupancy of the double dot and a maximal singlet–triplet imbalance at the same time. To optimize the probability for double occupancy we choose Γ_L  Γ_U/D, for which p_S = p_T = 1/2, which indicates finite but not maximal entanglement.

The plots in figure 2 display the linear conductance up to second order in the coupling strength based on a full solution of the stationary equations (7). If the system is close to equilibrium, |eV_U/D|  k_BT, the triplet probability is large (F≈1/4), and the conductance is dominated by a positive cos  kind of oscillation (compare equation (11)). In contrast, the interference is suppressed if the system is driven into a singlet–triplet imbalance by charging the double dot from the joint left lead, F≈1/2.

Figure 2. The linear conductance of the AB subsystem up to second order in the coupling strength is plotted versus the AB phase. The parameters are ΓL = 1 kBT, ΓU/D = 0.1 kBT, ΓR = 0.01 kBT, ε = –10 kBT and |eVR|  kBT. For strong bias voltage between the left and the upper/lower reservoirs, eVU/D = –20 kBT (left plot), and in equilibrium (right plot), the ideal system (i) is compared to results for singlet–triplet decay rates of (ii) ΓST = 0.5 ΓU/D and (iii) ΓST = 10 ΓU/D.

A finite singlet–triplet relaxation reduces the imbalance between singlet and triplet in the stationary state, and the AB oscillations are restored. To estimate the influence of spin flip and dephasing we introduce phenomenological transition rates between singlets and triplets. For simplicity, we choose all of them equal to Γ_ST. To observe a significant suppression of the AB oscillations a relaxation rate smaller than the current rate between source and drain is required, i.e. Γ_ST < Γ_U/D in our case.

In figure 3, the first- and second-order conductance contributions are plotted individually. The transport supported by the equilibrium state is dominated by the second-order triplet contribution and the AB oscillations are well visible. The singlet–triplet relaxation does not affect the behavior. If the electrons are strongly driven into the double dot from the joint left reservoir, a singlet–triplet imbalance forms and the oscillations of the singlet and triplet contributions cancel each other. A finite relaxation rate partially destroys the imbalance and the cancellation. The conductance supported by the singly occupied states becomes quite strong for a large bias voltage, however, its phase dependence is weak since for strongly asymmetric coupling the isospin is parallel to m_L with |I_↑/↓|≈1/8 p₁ such that g₁≈4 p₁.

Figure 3. The individual contributions to the linear conductance are plotted with and without singlet–triplet relaxation. The first order, GR(1), and second order terms, GR, χ(2), are specified for equilibrium (right plots) and for strong bias voltage, eVU/D = –20 kBT (left plots). The parameters are ΓL = 1 kBT, ΓU/D = 0.1 kBT, ΓR = 0.01 kBT, ε = –10 kBT and |eVR|  kBT.

5. Conclusions

Here, we proposed a set-up for the generation and detection of spin entanglement between two spatially separated electrons. A bias voltage applied across a double-dot system with strong onsite Coulomb repulsion, coherently coupled to a joint source electrode and individually coupled to separate drains, creates an imbalance between singlet and triplet probabilities. An asymmetry in the coupling of source and drain increases the overall probability to find two excess electrons in the double dot system and for them to form a Werner state. The underlying mechanism of entanglement generation relies on the specific nonequilibrium situation and is fundamentally different from those schemes in which two electrons are extracted from a singlet ground state.

To detect the singlet–triplet imbalance we added an AB probe, obtained by coupling a second joint reservoir coherently to the double dot. The enclosed flux controls the interference pattern which depends on the state of the double dot. In particular, the cotunneling transport through the AB ring is sensitive to an imbalance in the singlet–triplet distribution.

We compared two situations: for vanishing bias voltage between the joint source and the drain reservoirs the double dot is in equilibrium, and singlets and triplets are uniformly distributed, and the AB conductance is strongly flux dependent. On the other hand, by applying a strong bias voltage a singlet–triplet imbalance is generated, and the AB oscillations are suppressed. By choosing asymmetric couplings, we can ensure that the double dot was predominantly occupied by two electrons.

For an experimental realization one needs a geometry with spatial separation of the dots shorter than the phase coherence length. Finite decoherence due to spin–orbit effects, hyperfine coupling or coupling to some other bath will reduce the overall visibility of the AB oscillations. Nevertheless, the extra suppression of the AB oscillations for strong bias voltage as compared to equilibrium will indicate a singlet–triplet asymmetry. The probability of finding the system occupied with two electrons can be enhanced by tuning the dot levels well below the Fermi energy of the probe and choosing a strong asymmetry in the coupling of source and drain. For the detection of a suppressed phase dependence caused by a singlet–triplet imbalance a tunneling rate between source and drain much larger than the spin decoherence time is required.

Acknowledgments

We acknowledge stimulating discussions with G Burkard, P W Brouwer, J Weis, J Martinek, Y Gefen, T Löfwander, and E Prada. This work was supported by the Landesstiftung Baden-Württemberg via the Kompetenznetz Funktionelle Nanostrukturen and DFG via SFB 491 and SPP 1285.

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