Effect of randomness on quantum data buses of Heisenberg spin chains

A strongly coupled spin chain can mediate long-distance effective couplings or entanglement between remote qubits and can be used as a quantum data bus. We study how the fidelity of a spin-1/2 Heisenberg chain as a spin bus is affected by static random exchange couplings and magnetic fields. We find that, while nonuniform exchange couplings preserve the isotropy of the qubit effective couplings, they cause the energy levels, the eigenstates, and the magnitude of the couplings to vary locally. On the other hand, random local magnetic fields lead to an avoided level crossing for the bus ground-state manifold and cause the effective qubit couplings to be anisotropic. Interestingly, the total magnetic moment of the ground state of an odd-size bus may not be parallel to the average magnetic field. Its alignment depends on both the direction of the average field and the field distribution, in contrast with the ground state of a single spin which always aligns with the applied magnetic field to minimize the Zeeman energy. Lastly, we calculate sensitivities of the spin bus to such local variations, which are potentially useful for evaluating decoherence when dynamical fluctuations in the exchange coupling or magnetic field are considered.

Figure 1

Left panels: Two external qubits $A$ and $B$ (red circles) are weakly coupled to an (a) odd-size or (b) even-size spin bus (blue circles). The right-hand panels show the corresponding effective Hamiltonians in the low-energy limit. An odd-size spin bus in its ground state acts as an effective spin-$1/2$ particle denoted by $C$. It is coupled to the external qubits with strength $J_{\alpha i}^{\left(1\right)}$ at first order in the perturbation theory, and induces an RKKY-like coupling $J_{i,j}^{\left(2\right)}$ between the external qubits at second order. The even-size spin bus only mediates an RKKY-like coupling between the external qubits at second order.

Figure 2

(a) Four lowest energy levels $E_0,E_1,E_2$ and $E_3$ and (b) two energy gaps ${\Delta }_{01}=E_1-E_0$ and ${\Delta }_{12}=E_2-E_1$ are plotted as a function of sample number $N_S$ of odd-size chains with size $N=7$ with random exchange couplings $J_i$ with the standard deviation ${\sigma }_J/J_0=0.1$ at $g{\mu }_B{B}_0/J_0=0.1$. In (c) and (d), two energy gaps ${\Delta }_{01}$ and ${\Delta }_{12}$ are plotted as a function of $B_0$ for the samples $N_s=100$.

Figure 3

Dimensionless local magnetic moments $m_i=\langle 0;\frac{1}{2}|{\sigma }_{iz}|0;\frac{1}{2}\rangle$ are plotted (red dots) for all lattice sites on an odd-size spin bus with $N=9$ for an ensemble of $N_S=100$ random samples, with ${\sigma }_J/J_0=0.01$ and $B_0=0$. The height of the blue bars represents the local magnetic moment $m_i^{\left(0\right)}$ for uniform exchange couplings ($\delta J=0$). The spread of the red dots at each bus node illustrates the variations in local magnetic moments due to the random exchange couplings, with larger variations apparent at the end sites.

Figure 4

For odd-size chain with $N=5$, the ensemble averages of (a) the gap ${\langle {\Delta }_{01}\rangle }_{\mathrm{en}}$ in units of $J_0$, (b) the dimensionless local magnetic moment ${\langle m_5\rangle }_{\mathrm{en}}$ at the end of the chain, and (c) the normalized second-order effective coupling ${\langle K_{i,j}\rangle }_{\mathrm{en}}={\langle J_{1,5}^{\left(2\right)}\rangle }_{\mathrm{en}}{J}_0/\left(J_{A,1}{J}_{B,5}\right)$ are plotted (open red circles) as a function of the standard deviation ${\sigma }_J$ of $\delta J_i$. The blue filled circles in each panel indicate the standard deviations of each data point, which is obtained by averaging over 2000 random configurations. Here $B_0=0$.

Figure 5

Ensemble average of the gap ${\langle {\Delta }_{01}\rangle }_{\mathrm{en}}$ in units of $J_0$ between the two lowest states of a single spin as a function of the external magnetic field $B_0$. The height and width of the rectangle, touching the bottom of the curve, indicates ${\langle {\Delta }_{01}\rangle }_{\mathrm{en}}/J_0$ and $g{\mu }_B{\sigma }_B/J_0$ at $B_0=0$, respectively.

Figure 6

(a) Total magnetic moment $m$ of ground state $|0\rangle$, (b) local magnetic moment $m_9$ for ground state $|0\rangle$ (orange solid circles) and for first excited state $|1\rangle$ (green open circles), and (c) the fidelity $F$ of the ground state in the presence of random magnetic fields with respect to that in the uniform field as a function of the sample number. Here the size of spin buses is $N=9$, $g{\mu }_B{B}_0/J_0=0.1$, and $g{\mu }_B{\sigma }_B/J_0=0.065$.

Figure 7

Fraction of flipped ground states for $N_s=10000$ random realizations of odd-size chains with $N=5$, as a function of $B_0$ and ${\sigma }_B$.

Figure 8

Local magnetic moments of the ground and first excited states for several random field configurations in a 9-node spin bus. Here, sample #0 corresponds to the case of a uniform applied field. The others correspond to the samples in Fig. 6. The total magnetic moment of the ground states of samples #0 and #6 is $m^0=1$ (i.e., $S_z=1/2$). On the other hand, samples #10, #19, and #36 have the spin-flipped ground states, $m^0=-1$ (i.e., $S_z=-1/2$).

Figure 9

For an odd-size spin bus with $N=5$, the ensemble averages of (a) the gap ${\langle {\Delta }_{12}\rangle }_{\mathrm{en}}$ in units of $J_0$, (b) the local magnetic moment ${\langle m_5\rangle }_{\mathrm{en}}$ on one end of the chain, and (c) the normalized second-order effective coupling ${\langle K_{1,5}\rangle }_{\mathrm{en}}$ are plotted as a function of the standard deviation ${\sigma }_B$ of $\delta B_i$. In (b), the labels “$X$” (symbol: red open circles) and “$Z$” (symbol: orange open diamonds) stand for the $x$- and $z$ components of the first-order effective coupling, respectively. In (c) the labels “$XX$” (symbol: red open circles) and “$ZZ$” (symbol: orange open diamonds) refer to the $xx$ and $zz$ component of the second-order effective coupling, respectively. Here ${\mu }_B{B}_0/J_0=0.05$. In panels (b) and (c), the solid diamonds and solid circles represent standard deviations for the data labeled by the open diamonds and open circles, respectively.

Figure 10

For $N_s=100$ even-size chains with random exchange couplings, the two lowest energy gaps (a) ${\Delta }_{12}=E_2-E_1$ and (b) ${\Delta }_{01}=E_1-E_0$ are plotted as functions of $B_0$. The size of each chain is $N=8$ and the standard deviation ${\sigma }_J/J_0$ of $\delta J_i$ is 0.025.

Figure 11

For an even-size bus with $N=6$, (a) the ensemble averages of the ground energy gap ${\Delta }_{01}$, (b) the fidelity $F$ with respect to the uniform ground state, and (c) the normalized effective qubit-qubit coupling $K_{1,6}=J_{1,6}^{\left(2\right)}{J}_0/J_{1,A}{J}_{6,B}$ are plotted as functions of the fluctuation ${\sigma }_J$ of the random exchange coupling (all data are represented by red open circles). Here $B_0=0$ and the sample size is $N_s=100$. The blue solid circles represent the standard deviation of ${\Delta }_{01}$, $F$, and $K_{1,6}$, respectively.

Figure 12

Effect of random magnetic field for an even-size bus with $N=6$. (a) The average ground energy gap ${\Delta }_{01}$ in units of $J_0$, (b) the average fidelity $F$ of the ground states with respect to that without random $B_j$, and (c) the average of the normalized second-order effective coupling $K_{1,6}$ are plotted (in open red circles or open orange diamonds) as a function of the standard deviation of the random magnetic field ${\sigma }_B$. The standard deviations of the averages are given by the blue solid circles or blue solid diamonds, respectively. Here we take $B_0=0$.

Figure 13

(a) A few of the lowest energy levels of an odd-size chain with $N=7$ as a function of $B_0$. The two lowest levels around $B_0$ form up and down states of an effective single spin. Two gaps, ${\Delta }_{01}$ and ${\Delta }_{12}$ are indicated. (b) A magnified plot of the two lowest levels, offset against the ground-state energy $E_0/J_0=-2.83624$ at $B_0=0$, shows a Zeeman splitting of a single spin $1/2$.

Figure 14

Two energy gaps, ${\Delta }_{01}$ and ${\Delta }_{12}$ measured in units of $J_0$ are plotted as a function of $B_0$ for $N=3,5,7,9,11$.

Figure 15

Four lowest energy levels of an even-size chain with $N=8$ as a function of $B_0$. The two lowest energy gaps ${\Delta }_{01}$ and ${\Delta }_{12}$ are indicated in units of $J_0$.

Figure 16

Two lowest energy gaps ${\Delta }_{01}$ and ${\Delta }_{12}$ of even-size chains with $N=4,6,8,10,12$, as functions of $B_0$.

Figure 17

(a) Ground-state energy $E_0/J_0$ per bond of open chains (open red circles) and rings (solid blue circles), (b) gap $\Delta$ between the ground state and the excited state, (c) local magnetic moment $m_N$ at the end of the open-chain as a function of size $N$, and (d) log-log plot of $m_N$ for odd-size chains. The dashed line in panel (a) indicates $E_0/\left(J_{0}N\right)=\frac{1}{4}-\ln 2\approx -0.4431$ obtained from the Bethe ansatz.[33] The dashed line in panel (b) represents $\Delta /J_0={\pi }^2/N$ from Ref. 22. The gray lines in panels (c) and (d) represent an $m_N\propto 1/\sqrt{N}$ fit.