Dynamics of operation and a problem with the transmon

In the previous sections I have introduced a model for a quantum switch using su- perconducting qubits. The proposed device (FIG.2.6) consists of a semi-infinite incoming transmission line coupled to a system composed of four SQUIDs and a transmon. Two outgoing transmission lines couple the SQUIDs to the next quan- tum switch as incoming transmission lines in a tree-like network. The model I propose is described by a Hamiltonian that can be divided in three parts. The first part of that Hamiltonian [H1, see Eq. (3.13)], describe the energy level of each of

the artificial atoms. All the levels are supposed to have a different energy. The second part is more interesting [H2, see Eq. (3.19)]. It describes a density-density

interaction between the energy levels of different atoms. That is, when a trans- mon level is excited, this part of the Hamiltonian contributes to modify the energy levels of all the other atoms. Ideally, an incoming photon will have the energy to excite only the first or third level of the transmon (the second level is not coupled to the transmission lines, as shown by the third part of the Hamiltonian,H₃). If a second photon comes in, it will not be possible to excite further levels of the transmon –the photon will not have enough energy to do so. Instead, the photon will excite one of the SQUIDs. Since the excitation of the transmon modifies the energy spectrum of the SQUIDs, now this will be possible. We have to make sure that, if the first photon excites the first transmon level (or the third) the second only excites the SQUIDs leading to the second transmission lines (or the third). This way, the first photon decides which paths the other photons will follow.

Up to this point I have assumed that the lifetimes of the transmon excitations are large compared to those of the SQUIDs. Otherwise the transmon will decay to its ground state and the other photons will not be routed forward, but this is some- thing that will be determined byH₃, the third and last part of the Hamiltonian [see Eq. (3.23)]. This last expression tells us that the transmon is only coupled to the incoming transmission lines (both levels) thus, a photon absorbed by the trans-

mon can only be reflected back into the incoming transmission line. It cannot be transmitted. The SQUIDs, on the other side, are coupled to both the incoming transmission lines and to the closest outgoing transmission line, but not the other. Therefore, the second photon –which cannot be absorbed by the transmon– will be absorbed by the SQUID. After some time this photon will be emitted into a trans- mission line, either back or forward to the next node in the network of switches. The constraints I have imposed before force the excited SQUID to emit a photon into the outgoing transmission lines.

3.2.1

Operation

The operation of the quantum switch is represented in FIG. 3.1. This scheme shows how a register of photons is routed by the device. In the ideal case, a register of photons sent through the incoming transmission line will arrive at the first node of the network, where the first photon of the register is absorbed by the transmon. The photons in the register can be in two different states (or a superpo- sition), each with energiesωaandωb. When the first photon arrives at the switch

it encounters a system with the energy spectrum shown in the top left box (A). The switch is in the ground state, with the paths closed (i) and the first incoming photon can be absorbed by the device by exciting the first level of the transmon (red) if it has energyωa=ωT1or the third level (green) if it has energyωb=ωT3.

The other levels depicted in this box are not accessible for a photon that can only be in the statesωaorωb. In the box (A), the left upright arrow contains the energy

of the incoming photon. The levels next to it describe the spectrum of the trans- mon in the range of possible energies the incoming photons can have. The lowest level, labeled with|GSi, is the ground state.

Once the transmon is excited, a new set of energies is accessible for the next incoming photons (B). Although ω2a6=ωT1, the Hamiltonian H2 gives an extra

contribution to the energy levels (Ji,k) such that if the first level of the transmon

is excited, an incoming photon with energyωacan be absorbed by the SQUID 2a

(or the SQUID 3a, if the third level of the transmon is excited) or by the SQUID 2bif the energy of the photon isωb. Thus, the transmon opens one or the other

path (ii) according to the state of the first photon absorbed.

After being absorbed by the corresponding SQUID, the second photon is emit- ted into the open outgoing transmission line and the system goes back to the state with just one excited state (either|ωT1ior|ωT3i), waiting for the next photon of

the register. If the first incoming photon has energyωa, then all the successive

ω3b ω2b ωT3 ω3a ω2a ωT1 ωa ωb E |GSi ω3b ω2b ωT3 ω3a ω2a ωT1 ωa ωb E |GSi ωT1+ω2a−J12a ωa ωb E ωT1 ωT2 ωT4 ωT1+ω3a−J12a ωT1+ω3b−J12b ωT1+ω2a−J12b

A

B

C

Figure 3.1: In this figure I schematically show the expected behavior of the quantum switch based on the interactions contained in the Hamiltonians. The upper part of the figure contains a set of diagrams with the energy levels of the device at each step of the process of routing a register of photons. The lower part contains a graphic representation of the behavior of the device at each step of the process. In (A) the system is in its ground state and an incoming photon (i) comes in. This can excite any of the energy levels and “open” the path (ii) to any (or both) of the outgoing transmission lines. Once the transmon is excited, a second incoming photon can only excite a selection of states (B) and decay into the path opened by the transmon (iii). A third, fourth, etc. photon experiences the same process while the transmon is excited. At the end of the process (iv), when all the photons have gone through, the transmon emits a photon into the first transmission line (where the photons came in) and the system goes back to the ground state (C), waiting for the process to start over again. For a more detailed explanation see the text.

same transmission line, independently of their state (iii).

In the inset (B), the third level of the transmon does not appear. This is be- cause in order to excite this level –assuming that the first level of the transmon is already excited–, the photon would have to create two elemental excitations, but according toH₃, it can only create either one or three. That is why the second and a hypothetical fourth level is displayed. Nevertheless, these levels do not have enough energy to become excited.

Finally, when the whole register has gone through, the first photon is emitted into the ingoing transmission line and the rest of the register continues its way to the next quantum switch (iv). The device, after being relaxed, comes back to the ground state, with all the paths closed (iv), waiting for another register to come in (C).

During all of this process, if the first photon is in a superposition of states, say

ψ= (|ωai+|ωbi)/ √

2, then the photons that follow will all be forwarded through both the second and third outgoing transmission lines.

3.2.2

Scalability

In order to make a system composed of a tree-like network with one of these de- vices in each node, the transmon energy levels must coincide with those of the “open” SQUIDs (ωT1=ω2a−J12a). Unfortunately, this is not the case with the

present system. The energies of the transmon excitations are some orders of mag- nitude larger than the SQUID excitations and the energy separation is also larger. To solve this problem an external device can be added before each switch that in- creases the energy of only the first photon without measuring it. Another solution would be to add another element in the SQUIDs that increases the energy of its excitations. This could be a simple inductor connected to the point labeled asϕ2a

in FIG.2.6on one end, and to the ground at the other end. Since it is an inductive element, it can be straightforwardly added to the Hamiltonian. This element will generate a term raising the energy of the SQUID, but also another term contain- ing a†_2aa†_2a and a2aa2a. These two last elements cannot be suppressed using the

rotating wave approximation because they are large compared to the energy of the excited levels of the SQUID –the main contribution to its energy would come from this expression. Moreover, the density-density interaction strength (in the expressionH₂) will be too small compared to the energy of the excitations.

continue with the analysis (I will come back to this in Section5.1). In the worst case, it will not be possible to construct a QRAM using a network of quantum switches, but a fully operational quantum switch that routes many-photons regis- ters can still be found.