Steady state entanglement of two superconducting qubits engineered by dissipation

Steady state entanglement of two superconducting qubits engineered by dissipation

Florentin Reiter1, L. Tornberg, Göran Johansson and Anders S. Sørensen QUANTOP, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
Chalmers University of Technology, SE-41296 Gothenburg, Sweden
11reiter@nbi.dk
August 24, 2019
Abstract

We present a scheme for the dissipative preparation of an entangled steady state of two superconducting qubits in a circuit QED setup. Combining resonator photon loss, a dissipative process already present in the setup, with an effective two-photon microwave drive, we engineer an effective decay mechanism which prepares a maximally entangled state of the two qubits. This state is then maintained as the steady state of the driven, dissipative evolution. The performance of the dissipative state preparation protocol is studied analytically and verified numerically. In view of the experimental implementation of the presented scheme we investigate the effects of potential experimental imperfections and show that our scheme is robust to small deviations in the parameters. We find that high fidelities with the target state can be achieved both with state-of-the-art 3D, as well as with the more commonly used 2D transmons. The promising results of our study thus open a route for the demonstration of a highly entangled steady state in circuit QED.

pacs:
03.67.Bg, 42.50.Dv, 42.50.Lc, 85.25.-j

I Introduction

One of the most peculiar properties a physical system can exhibit is quantum-mechanical entanglement ES . From a fundamental perspective, entanglement is a non-classical effect which is indispensable for the understanding of fundamental quantum physics. From a technological perspective, entanglement is useful for enhanced measurement techniques and is an important element in quantum information processing and quantum communication NC . For the past two decades great effort has been invested into the generation and investigation of entangled states. Inspired by the circuit model of quantum computation, entanglement has predominantly been investigated by means of coherent interactions, i.e. by applying sequences of unitary gates. Today, there is a large number of physical systems where entanglement has been demonstrated and which are considered suitable for the realization of advanced quantum information protocols. Out of these, superconducting systems RevSC have proven to be good candidates for the realization of quantum algorithms involving many gate operations Neeley ; Fedorov ; Reed . Despite impressive reductions of the decoherence in superconducting systems Houck ; Chow ; Rigetti ; Poletto ; Paik ; Sears , any state other than the ground state will deteriorate over time. As a consequence, today’s quantum computation and simulation are still limited to elementary protocols on small scales.

Over the past few years, however, an alternative approach of dissipative state engineering, dissipative quantum computing and dissipative phase transitions VWC ; Kraus ; Diehl has emerged and gained increasing attention. As opposed to unitary quantum computing, where decoherence and dissipation act detrimentally on the state preparation process and on the prepared state, the central idea here is to prepare non-trivial quantum states relevant for quantum information, simulation Diehl ; Muller , memories DissMemory , or communication DissRepeater by means of an engineered interaction of the system with its environment. As opposed to unitary methods, dissipative quantum computation and dissipative state engineering involve steady states. Such states are resilient to the dissipative evolution by which they have been produced. This provides an additional stabilization against other kinds of decoherence. The question of whether this new dissipative paradigm can become an alternative or even superior approach to unitary quantum information processing can, however, not be answered in a single step. Instead, exploration of its capabilities needs to begin at a small scale. Here, an elementary quantum information processing task is found in the preparation of a maximally entangled Bell state as the steady state.

Previous theoretical work on entanglement generation utilizing dissipation has dealt with a number of quantum optical and solid state systems, in particular cavity QED PHBK ; Clark ; VB ; WS ; RKS ; Busch ; KRS , atomic ensembles Parkins ; MPC ; DallaTorre , ion traps PCZ ; CBK ; Muller ; Barreiro , plasmonic systems AGZ ; Gullans ; GP , light fields Kiffner and optical lattices Diehl ; FossFeig . The first experimental demonstrations were achieved in atomic ensembles Krauter and ion traps Barreiro . Several different state preparation tasks involving dissipation have also been considered for superconducting systems Zhang ; Li ; Xia ; Murch . So far, generation of maximally entangled steady states in the widely used setting of two superconducting qubits coupled through a common resonator has not been studied. In this work, we consider the dissipative preparation of a maximally entangled state in this system.

As opposed to previous studies of atomic systems coupled through a common resonator KRS ; RKS the realization of similar effects in superconducting systems raises a number of additional challenges. These are (1) a different energy level diagram, (2) additional, undesired transitions between qubit levels since these are not, as in atomic systems, suppressed by selection rules, and (3) additional decoherence mechanisms acting on the qubit. In addition, the dissipative entangling operation shall be independent of the initial state and reach a highly entangled steady state within reasonable time, also in the presence of imperfections in the setup. We will, in the following, discuss a scheme for superconducting qubits which fulfills these requirements, surmounting the above challenges.

As detailed in Sec. II, our scheme is specifically designed to exploit (1) the level structure of typical transmon qubits Koch , which constitute weakly anharmonic oscillators. The scheme is, however, not particularly restricted to transmons, but can also be applied to phase qubits PhaseQubit coupled to a resonator. Utilizing a coherent two-photon drive of a dipole-forbidden transition with a two-tone microwave field similar to Refs. Kelly ; Poletto , we engineer an effective resonator loss process which deterministically prepares the maximally entangled singlet state , as is described in Sec. III. Here we also show that (2) the coupling of the resonator to several transitions of the transmon is in fact an advantage, as it provides a transfer from the undesired states to the one from which the target state is prepared. Given that is produced by a time-independent loss process and continuous wave fields, it is a steady state of the dissipative evolution.

In Sec. IV, we investigate the performance of our scheme, both analytically, to derive benchmarks for the protocol, and numerically, to verify the mechanisms that underlie the presented dissipative state preparation scheme. Our results show that a maximally entangled state of two superconducting qubits can be prepared rapidly and with a high fidelity, even in the presence of (3) realistic qubit decoherence rates and imperfections. High fidelities are obtained both for state-of-the-art 3D, as well as for the more common 2D transmons. By fulfilling the above requirements our proposal thus opens a route for the dissipative preparation of maximally entangled states of superconducting systems using existing technology.

Figure 1: (Color) Setup. The internal levels of two transmons (a) are coupled by coherent interactions (b) to mimic the system in (c). Two microwave fields provide virtual couplings of the transitions and (b) which combine to an effective two-photon drive of the transition . The transmon-resonator coupling () is resonant with the upper transition and detuned by from the lower transition. Spontaneous emission () and resonator photon loss () are present as decoherence processes. The detunings are defined in the text.
Figure 2: (Color). (a)-(b) Dissipative state preparation mechanisms, (c) loss mechanisms, and (d) effective lower-level decay processes. (a) Effective resonator decay from into involves coherent coupling to . and are strongly coupled () so that these states hybridize and form dressed states (shown here for a choice of , by which the resonator is resonant with the upper transition). By setting the driving from is resonant with the lower dressed state . Population from is thus rapidly excited and decays into via the effective engineered resonator decay . (b) The population of the bright states and is shuffled to by the resonator coupling and successive resonator decay at an effective rate of . (c) The two-photon drive also causes an undesired coupling of the otherwise dark target state to an excited state . , in turn, couples to a number of (resonator-) excited states which form dressed states at different energies (indicated) and eventually decay to other states. These can generally be made off-resonant with the drive from by an appropriate choice of the resonator and microwave detunings so that the effective resonator decay from is suppressed. In addition, since is a dark state of the cavity interaction, the only direct decay mechanism is through the weak qubit decay to . The effective decay processes of the lower levels are summarized in (d).

Ii Setup: coherent and dissipative interactions of two coupled transmons

For our study we consider two superconducting transmons Koch coupled to a common resonator in a circuit QED setup. The coherent dynamics of the system is described by a Hamiltonian . The energy levels are illustrated in Fig. 1 a) and described by the free Hamiltonian

(1)

with levels of transmon and the resonator mode . Here, denotes the level spacing of the two lower levels and the anharmonicity, with . In our analytical discussion we will focus on the first three levels of the transmons, , and . Our numerical assessment will also include the fourth level, .

The transitions of the transmons, and , are coupled by the coherent interactions shown in Fig. 1 b). They are described by a Hamiltonian . Here, represents the coupling of the resonator to the transitions of the transmons,

(2)

with a coupling constant , and a factor of for the matrix element of the upper transition. The coherent drive

(3)

contains several microwave fields which couple the transitions and . We assume that the drive with exhibits an identical phase, whereas the phase of is opposite for the two transmons. This can be achieved by driving the qubits with the field through a common wire and with the field through additional individual wires, similar to Refs. Groot ; Chow11 ; Filipp . As we will see, this choice of phases allows us to break the symmetry of the system and thereby drive certain transitions which play an important role in our proposal.

We choose the frequencies of the two fields in such a way that they combine to an effective two-photon drive of the transition with a coupling constant of that will be derived in Sec. III.1. In doing so, we render the couplings of the system resembling the system shown in Fig. 1 c), with (meta-) stable lower levels and and an “excited” level for each of the transmons. “Excitation” from to is then accomplished by the two-photon drive with . For most of this paper, we will assume that the resonator coupling is resonant with the transition , while being somewhat detuned from the lower transition .

In the following, we will avoid the fast dynamics in the drive by changing into a frame rotating with a Hamiltonian

(4)

where is the mean frequency of the classical driving fields. Applying a unitary we obtain a transformed Hamiltonian in a frame rotating with . The transformed free Hamiltonian can be expressed as

(5)

where , , and denote the energies of the transmons and the resonator in the rotating frame. Furthermore, we obtain the interaction Hamiltonians for the transmon-resonator coupling and

(6)

for the drive. With this choice of the reference frame rotating with the mean frequency, we find the detunings of the microwave fields .

In addition to the coherent dynamics discussed so far, the system also exhibits dissipative couplings, which is essential for the dissipative state preparation mechanisms we would like to engineer. The dissipative dynamics of the open system is determined by its coupling to the bath and the properties of the bath. Assuming the bath to be Markovian, the system dynamics is governed by a master equation of Lindblad form

(7)

with one Lindblad operator for each physical decay process present in the system. As illustrated in Fig. 1 a), we assume that transmon undergoes spontaneous decay which in the transmon regime can be described by

(8)
(9)

For simplicity we restrict ourselves to only considering decay and neglect dephasing in our calculations unless explicitly mentioned. As we will argue and numerically verify below, the exact nature of the decoherence only plays a minor role for our proposal. The photon loss out of the resonator is described by

(10)

where is the photon loss rate.

Due to our choice of the couplings similar to a configuration, most of the dynamics will happen in the two lower levels. To describe them we choose a two-atom basis with triplet states , , , and the singlet state as the desired entangled steady state. For the detailed discussion of the engineered decay processes, we also introduce the excited atomic states , , and . The presence of resonator excitations is indicated by a second ket vector, e.g. . For simplicity we omit this ket vector when the resonator is in the vacuum state. We use this notation to explain the mechanisms of our scheme in Sec. III below.

Iii Mechanisms for dissipative preparation of the maximally entangled singlet state

In this section we will show how to engineer effective decay processes which prepare a steady state close to the maximally entangled singlet state . For now, we will focus our discussion on the physical mechanisms behind the effective decay processes, while Sec. III.1 and III.2 will deal with the derivation of quantitative expressions for the effective operators and rates.

The mechanism of our scheme is illustrated in Fig. 2 a). The working principle is as follows: Since the singlet state is a dark state of the resonator interaction, it can only gain or loose population by effective decay mechanims mediated by the weak coherent drives or through the slow decay by the weak qubit decoherence. A strong asymmetry between the rapid decay into and the slow loss processes out of it results in the dissipative preparation of with high fidelity. In the following we will discuss the physical mechanism for the preparation of .

In the previous section we have introduced a coherent driving . The purpose of it is to drive a two-photon transition . For now, we will assume that we have a coherent drive of with a coupling constant of and defer the derivation to later. Due to the opposite phase of on the two transmons, this drive then couples to an excited state with a detuning of , as can be seen from Fig. 2 a). is in turn coupled to by the resonator coupling . From here, decays into via resonator decay at a rate of . These processes combine to an effective resonator decay process from into with a rate of .

In order to engineer this process to be as strong as possible we have to fulfill two requirements: First, we need to make sure that the coupling of the transmon-excited state to the resonator-excited state is close to resonance, given that only the latter can decay to through resonator photon loss. To this end we set the resonator into or close to resonance with the upper transition of the transmons, . This is reached by choosing (), and results in an equal energy of and , as shown in Fig. 2 a). The two states hybridize and form dressed states

(11)

located at frequencies of (or ).

The second requirement is that the two-photon drive from from is resonant with one of the dressed states in Eq. (11). Choosing a detuning of , we tune the drive into resonance with the transition from to . Population from is then rapidly excited to , which, through its contribution from , decays into . For a strong resonant drive, the resulting effective decay process is only limited by the line width of , the state which mediates it. Thus, the dissipative preparation mechanism of the singlet and its rate can be engineered to be rather large.

Loss from the singlet can occur through the couplings of to any excited state other than by the available microwave fields, e.g. to by . As indicated in Fig. 2 c), these excited states are coupled to a number of other, in particular resonator-excited states. For instance couples to , , , and . Consequently, this establishes a loss channel from through effective resonator decay, e.g. into , which causes losses at a rate from the desired steady state . Fortunately, the photon-number dependent coupling strength between transmons and resonator provides us with a non-equidistant spectrum which consequently makes it possible to have the two-photon drive resonant with the transition from to while keeping it off-resonant with the transitions from to other hybridized excited states. In this way, loss processes from the singlet are suppressed by their detunings.

In order to reach independently from the initial state and to maintain it as the steady state, an additional mechanism is required to transfer population from lower states other than , i.e. from and , to . So far, we have assumed that the resonator is resonant with the upper transition. This means that due to the anharmonicity, the resonator is off-resonant with the lower transition. For reasonable anharmonicities the off-resonant coupling is, however, still sufficient to allow a reshuffling of population from the bright states and to , while as the dark state of the resonator coupling remains unaffected. As is shown in Fig. 2 b), this reshuffling process involves the resonator coupling of the lower transition (), e.g. , and decay of a resonator excitation at a rate of . It can be seen as an effective decay process with a decay rate . This expression contains both limiting cases, where one can either eliminate the resonator-excited states, or where the states can be seen as dressed states with resonator-excited states, for instance the triplet states

(12)

which decay towards at rates . Ideally, the reshuffling mechanism rapidly transfers the population of the triplet states to , from where they decay into by the dissipative preparation mechanism discussed above. The fastest reshuffling is reached by tuning the resonator into resonance with the lower transition, i.e. . This choice is, however, different from the above choice of which optimizes the dissipative state preparation process. With this choice of the resonator frequency we get , from which we see that the reshuffling works best for small anharmonicity . For larger the process becomes less effective. Having both processes, state preparation and reshuffling, simultaneously active might therefore seem problematic for large anharmonicities. However, as we shall see below, the scheme can still be effective for large if we allow for longer time for the reshuffling. Furthermore, as is also addressed below, the two requirements for above are far less critical than the resonant set-up of the two-photon drive. Consequently, both processes, the dissipative state preparation and the reshuffling, can be effective at the same time over a wide parameter range, as we will numerically demonstrate in Sec. IV.

In addition to effective resonator decay, qubit decoherence present in the system can cause loss from the singlet independent of the drives. Most notably, it can cause a loss from into , as shown in Fig. 2 c). The presented mechanisms are summarized in Fig. 2 d): On the left hand side we see the reshuffling mechanisms enabled by the resonator coupling to the lower transition, represented by , and on the right hand side the state preparation () and loss () mechanisms affecting the singlet state, as well as the decay from by qubit decoherence at a rate of .

To sum up this section, we have identified suitable mechanisms for the dissipative preparation of the singlet state and discussed the physical effects behind them. In the following two sections we will analytically derive the couplings and the rates for the effective coherent and dissipative processes in our scheme. Based on these, we derive benchmarks for the performance of the scheme in Sec. IV.

iii.1 Effective coherent driving of the dipole-forbidden transition by a two-photon process

The implementation of the dissipative state preparation scheme discussed above requires a coherent coupling of the transition . Since this transition is dipole-forbidden, such a coupling cannot be accomplished in a single step. One way to overcome this is to use a two-photon process, achieved by the combination of two individual fields. In we have chosen two such fields, and . As we will derive in the following, these provide complementary virtual single-photon excitations which form the desired coupling.

In the following, we will apply the effective operator formalism presented in Ref. [EO, ] to obtain a simple effective Hamiltonian for a single transmon with a two-photon drive. Here, we separate the Hamiltonian into a perturbative part , which contains the fields, and a perturbed part . (Note that the derivation below is for a single transmon only. With this in mind, the reuse of Hamiltonian definitions should not cause any confusion.) While in Ref. [EO, ] only effective processes with an initial excitation are considered, here we also allow for an initial deexcitation. We therefore set up the effective Hamiltonian (cf. Ref. [EO, ]) as with

(13)

Here, we specify the initial state and the field of the perturbation and the unperturbed Hamiltonian . The latter is defined as and contains as the frequency of level and as the detuning of field . We use a projector on the levels to identify coherent drive terms starting from an initial state . The superscript is used to split into for those terms which depend on and for the ones with ; a sign denotes whether the initial process is an excitation , i.e. a term containing a factor , or a de-excitation , with a factor .

Using this formalism we find a considerable number of terms, time-independent and -dependent ones, some closer to resonance and others stronger detuned. Neglecting the time-varying terms rotating at twice a detuning we obtain the effective two-photon Hamiltonian

(14)

Setting the detunings of the fields to we have that and keep a certain virtual character of the single fields by a detuning of , as shown in Fig. 1 b). In this configuration, there exists an effective two-photon drive where the first field (with ) drives the lower transition and the second field (with ) drives the upper transition. Expressing the resulting effective Hamiltonian in terms of the anharmonicity (using ) we obtain

(15)

with an effective two-photon Rabi frequency

(16)

From here we see that for the case of zero anharmonicity , i.e. for harmonic transmons, no effective two photon drive is possible. For , however, there exists a possibility of driving the transition . Note that the remaining diagonal terms in Eq. (15) represent shifts which can be compensated by suitable (minor) detunings of the fields. Their effect on Eq. (15) can be considered very small so that is approximately given by a single coherent coupling of the transition ,

(17)

We have thus obtained the coupling constant of the effective two-photon coupling we introduced in Sec. II. With this result we can turn to the derivation of the effective Lindblad operators for the engineered decay mechanisms used for the preparation of the singlet state.

iii.2 Engineered decay processes and their effective Lindblad operators

To model the effective, dissipative evolution we use the same effective formalism as in the previous section to derive the effective Lindblad operators EO

(18)

with the perturbative coherent excitation from an initial state by a field , and a non-Hermitian Hamiltonian

(19)

with the perturbed Hamiltonian defined previously. We focus on the effective resonator decay process activated by the two-photon drive and followed by decay of a resonator excitation . With , (), and we arrive at an effective Lindblad operator

(20)

with effective decay rates of and . This operator represents the dissipative mechanism we engineer to rapidly prepare the singlet state from . In addition, it includes the loss processes at rates of from into other states . Note that here we have ignored some less important terms as their effect on the population of the singlet is small.

We calculate of Eq. (20), using the driving from to as given by Eq. (17), with a matrix element of . The dynamics of the excited state is described by the non-Hermitian Hamiltonian in Eq. (19) which couples to through the resonator interaction , forming a coupled subspace. For the non-Hermitian Hamiltonian of this subspace which contains and is reached by excitation from with the two-photon drive , we define with

(21)

In order to keep the notation compact, we have written the Hamiltonian in terms of the complex detunings and combining the energy with the imaginary line width of the levels. For the inverted operator we find

(22)

Here, we have introduced effective detunings of and , and an effective coupling constant of . Since the rate for resonator decay from into is given by , we generally find an effective decay of from to , concluding that the effective coupling rate governs the strength of the engineered decay process.

The decay rate is maximized by a parameter choice of and , which corresponds to the two-photon drive from being in resonance with and the resonator being resonant with the upper transition. We then obtain , and thus . In Sec. IV we will make use of this result to derive the error and the speed of the protocol.

We now turn to the effective loss processes as they appear in Eq. (20). Given that is a dark state of the resonator coupling, these rates can be calculated using the same procedure we applied for the derivation of above: As is coupled to by the two-photon drive we need to consider the non-Hermitian Hamiltonian which describes the subspace consisting of and the states coupled to it by . For low anharmonicities , needs to reflect the full complexity of the coupled subspace containing , , , and . For anharmonicities of , however, the subspace of and begins to decouple from the other states so that the dynamics of the excited states can be approximated using only and . The Lindblad operator of Eq. (20) for the effective resonator decay then reduces to

(23)

containing a single loss rate from into .

To derive , we then approximate by the non-Hermitian Hamiltonian of the excited subspace consisting of and ,

(24)

using the complex detunings defined above. The inverted operator is then given by

(25)

Here, we have found effective detunings and , and an effective coupling constant of , which are different from the ones in the previous case of . With a decay rate from into , we obtain an effective decay rate of for the losses from . For the above choice of and , the effective coupling constant becomes which results in . From here we conclude that for the effective loss rate from the singlet is engineered to be much smaller than its preparation rate . These results confirm the explanations in Sec. III.

Note that, on the one hand, the above treatment of the coupled excited subspace where we restrict the excited state subspace to and is quite simplistic, given that it reduces the number of resonances from five to only two. In particular, one needs to ensure that one does not hit an accidental resonance with one of the dressed states of the system. On the other hand, the parameter space consisting of , and is sufficiently big to avoid an excitation of the remaining undesired resonances as there are sufficiently many suitable points in different regions of parameter space for which all of these resonances are off-resonant with the two-photon drive.

Figure 3: (Color). Dressed state energy vs. anharmonicity. An effective two-photon drive from (solid black line) to (green dashed) is implemented as two consecutive single-photon excitations by two microwave fields, and . The individual drives are mediated by (short-dashed red), which is a dressed state of , and made virtual through a detuning of (not shown). The two virtual excitations combine to an effective drive resonant with the transition ; then decays into (indicated). The same field couples to the transition from to the dressed states of (dashed-dotted). By an appropriate choice of the oscillator detuning (here plotted for with ), this coupling to is made off-resonant (left set of arrows). In case that is hit by the drive (right set of arrows), needs to be chosen differently to make the coupling off-resonant.

In Fig. 3, we draw the dressed states of the coupled resonator-transmon system. Here, the single-photon fields are tuned to resonantly excite the transition by a two-photon transition, mediated by the triplet state . The same two-photon drive also couples to a number of dressed states with contributions from . These transitions, however, generally have different frequencies than the desired one from to so that excitation of by the drive is off-resonant and suppressed by its detuning from the dressed states. This can be seen from Fig. 3, where we draw the dressed states together with the two-photon drive for the choice of . In the figure, we show an example near where the driving is off-resonant with the excited states which contain contributions from . We also draw an example at where this is not the case and where a resonance is hit accidentally. Here, it is necessary to choose a different detuning . Below, we will verify by numerical simulation for a broad parameter range that it is always possible to avoid such resonances.

In addition to losses caused by the two-photon drive, also the individual fields and couple to other states. The coupling of the even-phase single-photon drive from to does not cause any significant loss from , since population in is recycled via back into . The odd-phase single-photon drive , on the other hand, couples to and to a superposition state . Both these states are dark states of the resonator coupling. Thus, no exchange excitation to a resonator-excited state can shift them into resonance with the off-resonant drive from and no effective resonator decay process from is established involving them. Accumulation in these states does not occur, either, given that decays through qubit decoherence and decays into as discussed earlier. As a consequence, neither of the two drives causes significant loss from the singlet.

Another source of errors emerges for small anharmonicities from the coherent coupling of to other states like by the single-photon drives and . However, for , these couplings are sufficiently detuned to be ignored. Also, beside effective resonator decay processes, qubit decoherence occurs according to Eqs. (8)-(9). Provided that the decay rate is much weaker than all other physical couplings present in the system, i.e. , effective processes combining qubit decoherence with coherent excitation can be safely neglected.

We conclude that the sources of error originating from effective resonator decay which can cause losses from the singlet state are suppressed for the right parameter choice. These processes are, together with the engineered dissipative state preparation process, contained in the effective resonator decay operator in Eq. (23).

Iv Parameter and performance analysis, imperfections and realization aspects

In the previous section we have identified the effective coherent and dissipative processes which are relevant for our dissipative state preparation scheme and investigated the corresponding Lindblad operators and rates. In this section, we will use these results to derive approximate expressions for the error and speed of the presented protocol as the main benchmarks for our scheme. Later, we will assess the temporal evolution of the system numerically.

iv.1 Error and speed of the protocol

In the previous section we have derived the effective resonator decay operator , given in Eq. (23), which describes both the preparation of the singlet state and the inherent losses of our scheme. The derivation of Eq. (23) was carried out in the limit of weak driving. As we will find numerically below, the dissipative preparation of the singlet at a rate of works well for a driving strength up to , which yields a preparation rate for the singlet state and a loss rate from it. In addition, decays at a rate of , as described by the operators in Eq. (8)-(9).

Figure 4: (Color) Evolution of the system towards an entangled steady state. Initially prepared in an equal mixture of the lower states ( – green, dotted line, – red, dashed-dotted line, – blue, dashed line, – purple, solid line) the system evolves towards its steady state which is close to the maximally entangled singlet state of the two transmons. Part a) and b) show the result for an anharmonicity of and respectively. The remaining parameter values are , and for all plots. The values of , and are obtained through numerical optimization. The inset in a) shows the region in the plane where the singlet fidelity is high, , for . The number on each contour line indicates the preparation time in units of . The inset in b) shows the singlet state fidelity at as a function of anharmonicity.

Based on these rates we can approximate the temporal dynamics for weak driving using rate equations of the populations . We assume that the reshuffling mechanism rapidly transfers all population from the triplet states to the state , which is correct for small anharmonicity , the evolution of the population of the singlet can then be described by a single rate equation for the population of the singlet ,

(26)

formulated in terms of the decay rates specified above. Note that in this limit it is only the total decay rate out of the singlet state which matters, since any population lost from it is rapidly reshuffled to the state regardless of the nature of the loss. Hence additional decoherence mechanisms, e.g. dephasing causing decay from to , can easily be incorporated be replacing by an appropriate total loss rate from the singlet. By simply comparing the gain and loss of the singlet in the steady state, i.e. , we can estimate the steady-state fidelity of the singlet and, consequently, the error of the protocol . Assuming a near unit fidelity we obtain

(27)

From this expression we can readily see that the error of the protocol has a promising scaling with the physical parameters. Specifically, the error depends on the ratios of coupling and noise, and so that it will be small for strong coupling, , and modest qubit decoherence, . Under the assumption that we can vary the resonator decay rate we can minimize the error in Eq. (27) by choosing . Considering , we derive the optimal resonator decay rate . Inserting this yields the optimized error of the protocol,

(28)

From here we conclude that for the inherent error of the protocol can be limited to very small values. We will later confirm this finding numerically.

In addition, the convergence time, i.e. the decay time of the undesired states, can be approximated using Eq. (26), assuming rapid reshuffling of the undesired states to . Given that here the preparation of the singlet at a rate is the dominant process, the convergence time for weak driving is given by

(29)

where we have used and from above.

Note that the above expressions for the error and the convergence time are approximate and are derived using results obtained for the assumption of weak driving in Sec. III.2. In our numerical simulations below we will optimize a number of parameters including the driving strength to achieve highly entangled states within a preparation time as short as possible. In doing so, we arrive at particular choices of the available parameters which allow us to achieve high fidelities in short time. As these optimal parameters are in a regime where the effective Lindblad operators no longer accurately describe the dynamics EO , the findings of Eqs. (27)-(29) deviate from the simulation results below.

iv.2 Numerical results

To verify the findings above as well as to investigate the limitations of the approximation we now depart from the analytical treatment in the previous sections and assess the performance of the scheme numerically QuTIP . To this end we integrate the master equation in Eq. (7) including the three lowest levels of each transmon, , and , considered in the analytics, as well as the fourth level of each transmon, , and up to three photons in the resonator. While level already has a minor effect, the effect of higher excitations is expected to be negligible. Due to the Stark shifts induced by the driving, we have numerically optimized the sum- and difference frequencies and of the drives, as well as the resonator frequency . In Fig. 4 we plot the populations

(30)

between the time evolved density matrix and the four lower states introduced in Sec. II. The results of our simulation are shown in Fig. 4 a)-b), where we plot the populations, starting with an initially equal mixture of all four lower states. In Fig. 4 a), we consider a rather low anharmonicity , which is also what is typically used in experiments Rigetti ; Paik ; Sears . Here, the population of the states and show a fast drop due to the reshuffling into . At the same time, albeit on a slightly longer timescale, the dissipative preparation of the singlet is performed, reaching a fidelity of within a time of about , and a steady state fidelity of . For a transmon experiment with this would allow preparation times of about . For the results in Fig. 4 we have chosen corresponding to a relaxation time of Houck for the above parameter choice. This is much shorter than current state-of-the-art 3D transmon qubits where decoherence times of up to and Chow ; Rigetti have been measured. To accurately simulate this situation we include decay and dephasing rates corresponding to the decoherence times and find that with the numbers for 3D transmons it is possible to reach a steady state fidelity of for . Our analytical results (excluding the negligible effect of pure dephasing) suggest that fidelities of can be achieved for (or, in the presence of dephasing, for a corresponding time). The numbers for the transmon decoherence may, however, be somewhat lower than in the described circuit QED setup, where two qubits need to be tuned into resonance. In the numerical assessment of our scheme we therefore chose to work with a shorter coherence time of for the transmon relaxation time, comparable to the coherence time obtained for 2D transmons. In doing so we show the robustness of our scheme against such imperfections as well as the possibility to demonstrate a maximally entangled steady state not only in state-of-the-art 3D, but also in the more commonly used 2D transmon systems.

iv.3 Anharmonicity of the transmon

As discussed in the previous sections, the coupling of the resonator to the transition for each transmon contributes to the scheme by reshuffling the unwanted populations to . This coupling, however, gets increasingly detuned for higher anharmonicities . In Fig. 4 b) we show the effect of an increasing on the preparation scheme. Here, for a rather high anharmonicity of , the reshuffling of the states and to is slowed down as compared to the result for in Fig. 4 a). This can be seen from the drop in the population of and which is much less pronounced in b) than in a). In addition, we observe an increase in the steady state populations of these states. It is therefore advantageous to work with a rather low anharmonicity, where the coupling to the lower transition is still effective. Such anharmonicities are typical for state-of-the-art experiments Rigetti ; Paik ; Sears .

In the following, we will assess the possibility to operate our scheme for a broader range of anharmonicities, despite the breakdown of the reshuffling. To this end we allow for a rather long preparation time . In the inset in Fig. 4 b) we show results achieved using a numerical optimization routine to optimize the fidelity by fine-tuning the frequencies of the microwave fields and the resonator. These degrees of freedom in the parameter choice are used by the optimization routine to avoid undesired resonances by a slight departure from the resonance conditions of the previous sections. The range of our protocol is then limited by the breakdown of the reshuffling to , as well as to . For lower the effective two-photon drive becomes ineffective and couplings to higher levels of the transmons add shifts to the resonances required for the state preparation mechanism. To reach a high fidelity of the steady state one should therefore work with anharmonicities between and .

Finally, we briefly comment on the possibility for dissipative state preparation with even more anharmonic systems: In this case we choose to have the resonator in (or close to) resonance with the upper transition. Consequently, the lower transition is largely detuned and its effect negligible. We thereby achieve a situation which is very similar to optical cavity QED with atomic schemes – a system where various schemes for dissipative preparation of entanglement are available KRS ; RKS . These schemes can then be mapped to the highly anharmonic circuit QED setup. In those schemes the role of the far-detuned resonator coupling on the lower transition is accomplished by an additional microwave field which takes over the reshuffling of the triplet states. In this way, preparation of a steady state close to the maximally entangled singlet state can be achieved for any anharmonicity. For low anharmonicitiy, however, the coupling of the resonator to the lower transition allows us to avoid this field and thus to simplify the experimental implementation.

iv.4 Experimental imperfections

From the previous discussion it is clear that our scheme relies on the fact that the two transition frequencies of the transmons are identical. Moreover, we have so far only considered the case when the coupling, , is identical for both transmons. In this section, we depart from these assumptions and consider the effect of experimental imperfections. The transmons are characterized by their spectrum which is set by the effective Josephson energy, and the charging energy Koch . Here, we assume that both and the anharmonicity differ between the transmons. We also consider the possibility of having different couplings to the resonator. In Fig. 4a, we focus our analysis on the charging energy (anharmonicity) and the couplings by considering and where the subscript denotes transmon number. In the inset of Fig. 4a, we plot the region in the plane where for . The different contours correspond to the indicated preparation time and we see that there is roughly a error tolerance built into the system with respect to these parameters. The reproducibility of and is set by the precision of the e-beam lithography process and these tolerances are well within the limits of current technology.

In Fig. 5, we consider the effect of different resonance frequencies, , where subscripts denote transmon number. The error tolerance with respect to this parameter is substantially smaller than that for differences in anharmonicity and coupling. We believe that this larger sensitivity is due to the fact that for there is no longer an exact dark state of the transmon-resonator system, and the singlet state begins to suffer from the Purcell enhanced decay, which far exceeds the intrinsic decay rates of the qubits. It is however not necessary to have the same for the two transmons and the tolerance is well within reach of transmon experiments of today.

Figure 5: (Color) The fidelity as a function of the difference in resonance frequency between the two transmons. The parameters are as in Fig. 4 with