Sequential generation of matrix-product states in cavity QED

We study the sequential generation of entangled photonic and atomic multiqubit states in the realm of cavity QED. We extend the work of C. Schön et al. [Phys. Rev. Lett. 95, 110503 (2005)], where it was shown that all states generated in a sequential manner can be classified efficiently in terms of matrix-product states. In particular, we consider two scenarios: photonic multiqubit states sequentially generated at the cavity output of a single-photon source and atomic multiqubit states generated by their sequential interaction with the same cavity mode.

Figure 1

A trapped $D$-level atom is coupled to a cavity qubit, determined by the energy eigenstates ∣0⟩ and ∣1⟩. After bipartite source-qubit operations, photonic time bins are sequentially and coherently emitted at the cavity output, creating a desired entangled multiqubit stream.

Figure 2

Sequential generation of a multiqubit state $\mid \psi ⟩$. In each step the $D$-dimensional ancilla produces one qubit. This process is described by a $(2\mathrm{D}\times \mathrm{D})$-dimensional isometry $V_{\left[i\right]}$. The achievable multiqubit states are instances of matrix-product states with $D$-dimensional bonds and open boundary conditions specified by $\mid {\phi }_I⟩$ and $\mid {\phi }_F⟩$.

Figure 3

Restricted interaction between ancilla and qubit. Each atomic transition from $\mid a_i⟩$ to its respective $\mid b_i⟩$ is accompanied by the generation of a photon in a certain time bin.

Figure 4

The box represents a single step in the sequential generation scheme. Since tag- and time-bin qubit are always in state ∣0⟩ after the preceding step, they can be ignored at the input. After an arbitrary unitary operation $U_{\mathcal{A}}$ on the atom, tag- and time-bin qubit are interchanged by the standard map. Ignoring the tag qubit also at the output (since it is always in ∣0⟩), the process can be described by an arbitrary isometry $V$ applied on the effective ancillary system ${\mathcal{A}}^{\prime }$.

Figure 5

(a) Two-qubit gates $V_{\left[i\right]}$ are sequentially applied between nearest neighbor qubits. This situation can be simulated by a two-dimensional ancilla as demonstrated in (b). Instead of applying the gate between neighboring qubits $k$ and $k+1$, we use qubit $k$ and the ancilla and swap the ancilla state afterwards with the qubit $k+1$. Since we can merge the swap operations and the arbitrary unitary operations $V_{\left[i\right]}$ to arbitrary unitary operations, we know that the class of achievable states $\mid \psi ⟩$ is equivalent to the class of MPS with open boundary conditions and two-dimensional bonds.

Figure 6

Repeating the procedure $m$ (here $m=3$) times defines a class of states described by MPS with $\left(2^{2m-1}\right)$-dimensional bonds. Note that in between the three sequences we also allow an arbitrary interaction between the first and the last qubit.

Figure 7

The $D$-level atom can be viewed as a set of $M$ qubits with $D⩽2^M$. For an arbitrary operation we need local unitaries for the atomic qubits and a universal two-qubit gate between the cavity qubit and one specific atomic qubit.

Figure 8

(a) Atomic level structure: levels $\mid a⟩$ $\left(\mid b⟩\right)$ and $\mid e⟩$ are coupled by a laser (cavity mode) off resonance. (b) After adiabatic elimination of the upper state $\mid e⟩$, we are left with a Jaynes-Cummings type of Hamiltonian, where states $\mid a,n⟩$ and $\mid b,n+1⟩$ are coupled. Both, the energy difference of those levels and the corresponding Rabi frequency depends on $n$. The reason for the first is the ac-Stark shift, whereas the second is due to the Jaynes-Cummings coupling.

Figure 9

Levels $\mid a^{\prime }⟩$ and $\mid b^{\prime }⟩$ ($\mid a⟩$ and $\mid b⟩$) are coupled off-resonance to $\mid e_a⟩$ and $\mid e_b⟩$ by two lasers (cavity modes).

Figure 10

Atomic level structure: level $\mid a_i⟩$ $\left(\mid b_i⟩\right)$ are coupled to $\mid f_i⟩$ via an adiabatic passage driven by a laser with Rabi frequency ${\Omega }_a$ $\left({\Omega }_b\right)$ and the cavity mode $a$ $\left(b\right)$ with coupling strength $g_a$ $\left(g_b\right)$.

Figure 11

A stream of uncorrelated atoms crosses a cavity. The atomic qubits couple sequentially to the cavity mode, which acts as a two-dimensional ancillary system. If an arbitrary unitary operation can be realized between the cavity qubit and the atomic qubits, the class of entangled multiqubit atomic states at the output are equivalent to the class of ${\mathrm{MPS}}_2$.

Figure 12

Cluster state generation. (a) An iSWAP gate is performed sequentially between the atomic qubits and the cavity mode. (b) One can as well assume that the local unitaries are performed on the atomic qubit $k$ after the $k\text{th}$ SWAP gate. Therefore they can be compensated in a final step.

Figure 13

A stream of uncorrelated atoms passes a cavity. Two atomic qubits couple to each other via the common cavity mode. If an arbitrary unitary operation can be realized between them, the class of entangled multiqubit atomic states at the output are quivalent to the class of ${\mathrm{MPS}}_2$.

Figure 14

The iSWAP gate between neighboring qubits is equivalent to a SWAP gate followed by a CNOT gate up to local unitaries. The latter can be compensated since local operations can be applied to the atomic qubits with Ramsey zones.