System plus bath (S + E)

In order to investigate decoherence in detail, let us consider the system S (a spin-1/2 spin) coupled to a bath E that consists of a chain of L spin-1/2 particles, see Fig. 3.1. We will study how the properties of the bath E and the interaction between S and E affects the evolution of the coherence of S. The Hamiltonian of the complete system (system plus bath) is

H_{S E} =H_S +H_E +H_int, (3.1)

where H_S and H_E are the Hamiltonian of the system S and of the bath E, respectively, and H_int is the coupling term between the system S and the bath E. The complete system com-posed of the system S and the bath E will evolve under this Hamiltonian. Regardless of the actual physical makeup of the qubit and without loss of generality, the single-qubit system is assumed to have a simple self-Hamiltonian,

H_S = −ωS_z= −ω

2(|↑i h↑| − |↓i h↓|), (3.2)

whereωis a constant. States|↑iand|↓i, respectively, are the up eigenstate and the spin-down eigenstate of the qubit. In most discussions of the Loschmidt echo, such assumptions are also used. For some simple cases, the self-Hamiltonian of the system S can be set to zero without changing the evolution of the Loschmidt echo. However, in other cases, the pointer states{|↑i,|↓i}(the states that is not entangled with the bath) [23] can be changed to

Environment E

!

System S

↑"

↓"

Environment E

System S

!

↑"

↓"

Figure 3.1: Schematic diagram of the coupling schemes between system and bath. The spin chain (bath) can have periodic boundary conditions (PBCs) or open boundary conditions (OBCs). One coupling scheme is that the qubit (system) homogeneously couples to each spin in the spin chain (left). If the spin chain has OBCs, the model will always be called the central spin model. Another coupling scheme is that the qubit couples to only one spin or a few arbitrary spins in the chain (right). The qubit may display different decoherence behaviors in the different coupling schemes.

the eigenstates of some special self-Hamiltonians of the system [24, 25, 79]. For example, HS could describe a magnetic field in the x-direction or a time-dependent magnetic field. We will assume the bath Hamiltonian H_E is a one-dimensional spin-1/2 chain with the form

H_E = −2J XL

j=1

[(1+γ)S^x_jS^x_j+1+(1−γ)S^y_jS^y_j+1+ ∆S^z_jS^z_j+1+λS^z_j], (3.3)

where S^x_i, S^y_i, and S^z_i denote the spin operators of the i-th spin site in the chain. For sim-plicity, we will use periodic boundary conditions (PBCs) in analytical calculations and open boundary conditions (OBCs) in the DMRG calculations. The parameters J (J ≥ 0), γ, ∆, andλ, determine the interaction strength in the xy plane between nearest-neighbor spins, the anisotropy in the xy plane and along the z-axis, and a z-direction transverse magnetic field, respectively. We consider the two limiting cases in which γ = λ = 0 (the XXZ chain) and γ = 1,∆ =0 (the transverse-field Ising model). The ground state of the XXZ chain is critical for−1≤ ∆≤1, is in an antiferromagnetic phase for∆< −1, and is in a ferromagnetic phase for∆> 1, see Fig. 3.2 (a). Forλ >0, the ground state of the transverse-field Ising chain has one critical point atλc = 1. For 0 ≤ λ <1, it is in a long-range ordered ferromagnetic phase with a broken Z₂ symmetry, while forλ > 1, it is in a paramagnetic phase, see Fig. 3.2 (b).

XXZ

∆

1

Antiferromagnetic Critical Ferromagnetic

0

−1

(a)

transverse−field Ising

0 1

Magnetic Paramagnetic

λ

(b)

Figure 3.2: Ground-state phase diagram for (a) the XXZ chain and (b) the transverse-field Ising chain.

When the qubit is coupled to a transverse-field Ising chain, the system can be solved exactly for all times so that the long-time behavior can be fully investigated, see Appendix A. For the case of a coupling to an XXZ chain, we compute the time evolution using the adaptive t-DMRG.

We consider two types of coupling Hamiltonians H_int: the Ising coupling and the Heisen-berg coupling. The general anisotropic Ising couping H_int^Ising has the form

H_int^Ising= −ǫ|↑i h↑|S^z_j+ǫ^′|↓i h↓|S^z_j, (3.4) whereǫ and ǫ^′ are the anisotropic Ising coupling constants. The bath site label j indicates that the qubit only couples locally to one bath spin. We will concentrate mainly on the Ising coupling form of H_int^Ising.

The general Heisenberg couping H_int^Heishas the form H_int^Heis = − X

α=x,y,z

ε^αS^α_qubitS^α_j, (3.5)

whereε^α (α = x,y,z) are the coupling constants along three Cartesian axes, and S^α_qubit and S^α_j are the spin operators of the system and the spin operator of the j-th spin in the bath, respectively. Below, we will only consider an isotropic Heisenberg couping in H_int^Heis, i.e., ε^x =ε^y =ε^z.

Notice that, for the Ising coupling form, H_int^Ising, [H_S,H_{S E}] = 0. The interpretation is that the expectation value of the system Hamiltonian HS is constant, and no energy exchange between the system S and the bath E occurs. This gives rise to a purely dephasing form of the time evolution. The system loses its coherence only. However, for a Heisenberg coupling form H_int^Heis, [H_S,H_{S E}], 0, in general, and the system will undergo both dephasing and energy relaxation. However, if one sets H_S = 0, there is also pure dephasing only.

In studying the decoherence of the system S, we assume that the initial state of the

com-plete system (S plus E) is a pure state. That means the wave function of the comcom-plete system is a direct product of the state of the system S and the state of the bath E, i.e., it is given by

|Ψ_{S E}(t =0)i= |ψ_S(0)i|ψ_E(0)i, (3.6) where |ψ_S(0)iand|ψ_E(0)iare the initial states of the system S and the bath E, respectively.

One is relatively free to select an initial state of the system S. Here, in the general case, we choose an arbitrary state |ψ_S(0)i = c_↑|↑i + c_↓|↓i, a superposition of the spin-up state |↑i and the spin-down state |↓i, where|c_↑|² +|c_↓|² = 1. Consider, however, the limiting case of

|ψ_S(0)i = |↑i or |↓i. At first glance, the off-diagonal elements of the density matrix of the system S, which characterize the decoherence (Loschmidt echo), will be zero initially and for all times. It seems that this behavior cannot characterize a general decoherence. However, one can rotate|↑ior|↓ithrough arbitrary angles to obtain a general state|ψ_S(0)iwith nonzero off-diagonal elements of the density matrix. Thus, it is still completely general to choose

|↑i or|↓i as the initial state for calculating the Loschmidt echo. On the other hand, we can choose the specific initial state of the bath |ψ_E(0)i that we are interested in. Building on previous studies [77, 78] that take the ground state of the bath Hamiltonian H_E, Eq. (3.3), as the initial state, we will make more accurate calculations for this case and then will also consider a quenched initial state, which is evolved from the ground state by carrying out a sudden change of the interaction constant, as the initial state. It is useful to mention that when the bath is an XXZ chain, only the case of the anisotropic parameter∆ = 0 in Eq. (3.3) can be exactly solved. These exact results can be used for checking the accuracy and precision of the DMRG results. We apply the static DMRG to compute the ground state of the XXZ chain and then use the adaptive t-DMRG to compute the Loschmidt echo. We choose OBCs for the spin chain in the DMRG calculations because, as is well-known, the classical DMRG method does not converge well under PBCs. Additionally, for small bath size, i.e., L≤ 17, we utilize the exact diagonalization (ED) method.

The complete initial wave function, Eq. (3.6), evolves with the Hamiltonian, Eq. (3.1),

|Ψ_{S E}(t)i=e^{−iHS Et}|Ψ_{S E}(t)i=e^{−iHS Et}|ψ_S(0)i|ψ_E(0)i. (3.7) Since the wave function|ψS(0)iof the initial state of the system S has two branches, the spin up state|↑iand the spin down state|↓i, the complete wave function, Eq. (3.7), also splits into

Ising e^{−iHS Et}|Ψ^{S E}(0)"

|↓"|ψE↓(∆t)"

|↓"|ψE↓(2∆t)"

|↑"|ψE↑(2∆t)"

|↑"|ψE↑(∆t)"

|↓"e^−iH↓t|ψE(0)"

|↑"e^−iH↑t|ψE(0)"

∆t ∆t

∆t

∆t

(a)

Heisenberg e^{−iHS Et}|ΨS E(0)"

e^{−iHS Et}|↑#|ψE(0,Sz)#

|↑"|ψ_E1(2∆t,S_z)"+|↑"|ψ_E2(2∆t,S_z−1)"

|↓"|ψ_E4(∆t,Sz+1)"+|↓"|ψ_E3(∆t,Sz)"

|↑"|ψ_E1(∆t,S_z)"+|↑"|ψ_E2(∆t,S_z−1)"

|↓"|ψ_E4(2∆t,Sz+1)"+|↓"|ψ_E3(2∆t,Sz)"

e^{−iHS Et}|↓#|ψE(0,Sz)#

∆t

∆t

∆t

∆t

(b)

Figure 3.3: Depiction of the manner of the time evolution of the two branch wave functions,

|↑i |ψ_E↑(t)iand|↓i |ψ_E↓(t)iwith two different kinds of interactions between the system and the bath: (a) For the Ising coupling between the system and the bath, two branch wave functions evolve independently. (b) For the Heisenberg coupling between the system and the bath, the total spin of the bath wave function in the z-direction varies over S_z, S_z +1 and S_z− 1 in different time steps. In this case, the two branches of the wave function evolve in a correlated way.

two branches. The result is

|Ψ_{S E}(t)i=e^{−iHS Et}|Ψ_{S E}(0)i (3.8)

=e^{−iHS Et}(c_↑|↑i+c_↓|↓i)|ψ_E(0)i.

For the case of an Ising coupling H_int^Ising, Eq. (3.4), we can simplify the Eq. (3.8) to

|Ψ_{S E}(t)i=c_↑e^{−iHS Et}|↑i |ψ_E(0)i+c_↓e^{−iHS Et}|↓i |ψ_E(0)i (3.9)

=c_↑e^iω2t|↑ie^−iH↑t|ψ_E(0)i+c_↓e^−iω2t|↓ie^−iH↓t|ψ_E(0)i

=c_↑e^iω2t|↑i |ψ_E↑(t)i+c_↓e^−iω2t|↓i |ψ_E↓(t)i,

where the effective Hamitonians H_↑ = −ǫS^z_j+H_Eand H_↓ =ǫ^′S^z_j+H_E. The problem is changed to that of two independent time evolutions of the wave function of the bath E with these two effective Hamiltonians H_↑and H_↓, respectively, see Fig. 3.3 (a). By using the complete wave function |Ψ_{S E}(t)i, Eq. (3.9), and by tracing out the degrees of freedom of the bath E, the

time-dependent reduced density matrix of the system S,

ρ_S(t) =Tr_E|Ψ_{S E}(t)ihΨ_{S E}(t)| (3.10)

=X

n

hn|Ψ_{S E}(t)ihΨ_{S E}(t)|ni

=X

n

hn|(|c_↑|²|↑i |ψE↑(t)ihψE↑(t)| h↑|+|c_↓|²|↓i |ψE↓(t)ihψE↓(t)| h↓|

+c^∗_↓c_↑e^iωt|↑i |ψ_E↑(t)ihψ_E↓(t)| h↓|+c^∗_↑c_↓e^−iωt|↓i |ψ_E↓(t)ihψ_E↑(t)| h↑|)|ni

=X

n

|c_↑|²|↑i h↑| hn|ψ_E↑(t)ihψ_E↑(t)|ni+|c_↓|²|↓i h↓| hn|ψ_E↓(t)ihψ_E↓(t)|ni +c^∗_↓c_↑e^iωt|↑i h↓| hn|ψ_E↑(t)ihψ_E↓(t)|ni+c^∗_↑c_↓e^−iωt|↓i h↑| hn|ψ_E↓(t)ihψ_E↑(t)|ni

=|c_↑|²|↑i h↑|+|c_↓|²|↓i h↓|

+c^∗_↓c_↑e^iωt|↑i h↓| hψ_E↓(t)|ψ_E↑(t)i+c^∗_↑c_↓e^−iωt|↓i h↑| hψ_E↑(t)|ψ_E↓(t)i,

is obtained, where we have taken the states|nito be a complete basis for the bath E. The off-diagonal elements of the reduced density matrix, Eq. (3.10), represented in the basis of the eigenstates|↑i, |↓iof the system S, characterize the decoherence of the system. The matrix form ofρ_S(t) is

ρS(t)=









ρs,↑↑(t) ρs,↑↓(t) ρ_s,↓↑(t) ρ_s,↓↓(t)







=









|c_↑|² c^∗_↓c_↑e^iωthψ_E↓(t)|ψ_E↑(t)i c^∗_↑c_↓e^−iωthψ_E↑(t)|ψ_E↓(t)i |c_↓|²







. (3.11) The off-diagonal elementρs,↑↓(t) decays as^∗

ρ_s,↑↓(t)=c^∗_↓c_↑e^iωthψ_E↓(t)|ψ_E↑(t)i. (3.12) However the diagonal elementsρs,↑↑andρs,↓↓always retain their in the initial values|c_↑|²and

|c_↓|²and will never decay in the time evolution. The overlap between the two branch states of the bath E, hψE↓(t)|ψE↑(t)i, characterizes the Loschmidt echo. We can define the Loschmidt echo as a real number,

LE(t) =|hψ_E↓(t)|ψ_E↑(t)i|² =|hψ_E(0)|e^iH↓te^−iH↑t|ψ_E(0)i|², (3.13) where the wave function of the initial state|ψE(0)iis the ground state or a quenched state of

∗Since,ρ_s,↑↓(t)=ρ^∗_s,

↓↑(t), we will only considerρ_s,↑↓(t).

the bath E. It decays exponentially as a Gaussian at short times [77, 78, 79, 86, 87]:

LE(t) ∼e^−αt2, t∼ 0. (3.14)

We can set one of the coupling constantsǫ,ǫ^′ to zero (actually, in most studies,ǫ^′ = 0). If the initial state|ψ_E(0)iis a ground state of the bath E, Eq. (3.13) is identical to

LE(t)= |hψ_E(0)|e^−iH↑t|ψ_E(0)i|², (3.15) which can also be interpreted as the “survival probability” of the initial state evolved under the effective Hamiltonian H_↑. In the time evolution, the Loschmidt echo quickly deviates from the initial value 1 because of the presence of the interaction, and the system state is converted into a statistical mixture. For some cases, the Loschmidt echo even approaches zero, and the reduced density matrix of the system, Eq. (3.10), becomes diagonal. This means that the coherence of the system is strongly suppressed and gradually leaks to the bath, but the complete system still keeps its coherence. Furthermore, if the Loschmidt echo is close to 1, it indicates that the interaction between the system and the bath is very weak. In particular, the case that the Loschmidt echo always remains 1 simply means the initial system state is a eigenstate of the effective Hamiltonian H_↑, or, in other words, that there is negligible interaction between the system S and the bath E. All previous studies are based on calculations with a finite-sized bath. Since there is, in general, a bath-size-dependent revival time of the Loschmidt echo, this kind of decoherence behavior is called an “echo”. Moreover, the Loschmidt echo can never revive to the full initial value of 1, and the value of the revived Loschmidt echo becomes weaker and weaker with more and more revivals. This is why such decoherence behavior is called a “Loschmidt echo”.

Furthermore, for the case of Heisenberg coupling H_int^Heis, Eq. (3.5), one finds that the two branch wave functions in Eq. (3.8) will evolve correlatively, see Fig. 3.3 (b). Since the Heisen-berg coupling will lead to an exchange of quantum numbers between the system S and the bath E, the quantum number of the bath will be changed. In order to facilitate the discussion, we explicitly write down the total spin to identify different states. We assume that the initial state of the bath E,|Ψ_{S E}(t=0,S_z)i, has a total spin S_z. Thus, Eq. (3.8) can be written as

|Ψ_{S E}(t)i=h

e^{−iHS Et}c_↑|↑i |ψ_E(0,S_z)ii +h

e^{−iHS Et}c_↓|↓i |ψ_E(0,S_z)ii

(3.16)

=

c_↑|↑i |ψ_E1(t,S_z)i+c_↑|↓i |ψ_E4(t,S_z+1)i+

c_↓|↓i |ψ_E3(t,S_z)i+c_↓|↑i |ψ_E2(t,S_z−1)i

=|↑i

c_↑|ψ_E1(t,S_z)i+c_↓|ψ_E2(t,S_z−1)i+|↓i

c_↓|ψ_E3(t,S_z)i+c_↑|ψ_E4(t,S_z+1)i,

where the effective wave function|ψ₁i=|↑i |ψ_E1(t,S_z)i+|↓i |ψ_E4(t,S_z+1)ievolves from the wave function e^{−iHS Et}|↑i |ψE(0,Sz)i, and the effective wave function |ψ2i = |↓i |ψE3(t,Sz)i+

|↑i |ψ_E2(t,S_z−1)ievolves from the wave function e^{−iHS Et}|↓i |ψ_E(0,S_z)i. The wave functions

|ψ_E1(t,S_z)i, |ψ_E2(t,S_z)i, |ψ_E3(t,S_z)iand|ψ_E4(t,S_z)iare not assumed to be normalized. This means that

hψ_E1(t,S_z)|ψ_E1(t,S_z)i+hψ_E4(t,S_z+1)|ψ_E4(t,S_z+1)i=1/|c_↑|², (3.17) hψ_E3(t,S_z)|ψ_E3(t,S_z)i+hψ_E2(t,S_z−1)|ψ_E2(t,S_z−1)i=1/|c_↓|².

The wave function|↑i |ψ_E1(t,Sz)icontributes to the wave function|↓i |ψ_E4(t^′,Sz+1)iat the next time step t^′, and vice versa. The wave functions |↓i |ψ_E3(t,S_z)iand |↑i |ψ_E2(t,S_z− 1)i behave similarly. This behavior is depicted in Fig. 3.3 (b). All four of these four wave functions do not have simple forms; they must be determined by specific calculations. Notice that we can also write Eq. (3.16) as|Ψ_{S E}(t)i=c_↑(t)|↑i |ψ_↑E(t)i+c_↓(t)|↓i |ψ_↓E(t)i[78].

We now examine the Loschmidt echo. It is easy to see that the off-diagonal element of the reduced density matrixρ_s, e.g.,ρ_s,↑↓(t), is given by

ρs,↑↓(t)= c^∗_↓c_↑hψ_E3(t,S_z)|ψ_E1(t,S_z)i. (3.18) Thus, analogously to Eq. (3.13), the Loschmidt echo can be defined as

LE(t)= |hψ_E3(t,S_z)|ψ_E1(t,S_z)i|². (3.19) Here the Loschmidt echo is only related to the overlap between two wave functions that have the same quantum number as the wave function of the initial state.

It is useful to discuss the energy relaxation behavior here. If the Hamiltonian of the system S is still defined by Eq. (3.2), the observablehHSican be written as

hH_Si(t)= Tr(H_Sρ_s(t))= −ω

2ρs,↑↑(t)+ ω

2ρs,↓↓(t)=ω 1

2 −ρs,↑↑(t)

!

= ω 1

2 − |c_↑|²hψ_E1(t,S_z)|ψ_E1(t,S_z)i − |c_↓|²hψ_E2(t,S_z−1)|ψ_E2(t,S_z−1)i

!

(3.20) using Eq. (3.17), whereρs,↑↑(t) andρs,↓↓ are diagonal elements of the reduced density matrix ρs. The energy relaxation behavior is related to the choice of the initial state of the system S.

Different combinations of c_↑and c_↓will lead to different energy relaxation behaviors. To

pre-vent energy relaxation, we can set H_S =0 in our calculations, but it is also easy to generalize to the case of non-zero HS.

The behavior of the Loschmidt echo (i.e., the decoherence) of the system S (the qubit) could be sensitive to the internal dynamics of the bath (i.e., the quantum phases and the quench time). In the following, we will explore how the time evolution of the Loschmidt echo of the system S is affected by the internal dynamics of the bath E.

Im Dokument Explicit Exchange Interaction and Decoherence Dynamics in One-Dimensional Quantum Systems (Seite 71-79)