Tunable quantum logic gate on photonic qubits with a ladder emitter

Photonic qubits are promising candidates for enabling a universal quantum computer capable of effective integration with long-distance quantum networks because photons offer long coherence times compared to matter based qubits,¹ qubit transmission at light-speed,¹ and trivial realization of single-qubit gates via linear-optical components.² Due to weak photon–photon interactions without high-quality but difficult-to-fabricate cavities,^3–7 the primary challenge in photonic quantum computing is realizing multi-qubit gates.² Linear optical quantum computation (LOQC) has provided a partial workaround by using measurement operations to create effective photon–photon interactions. However, LOQC requires a high resource overhead via pre-prepared ancillary photons and is non-deterministic.^8,9 To create a deterministic multi-photonic qubit gate, multiple schemes using mediating systems to create effective photon–photon interactions, such as atoms in cavities, have been proposed. Many such proposals require active control, such as an external control laser to manipulate the state of a mediating atom, an external electric field to tune via the Stark shift, or a microwave field to control a cloud of atoms; these proposals include gates based on Rydberg atoms and electromagnetically induced transparency as well as the Duan–Kimble proposal and its variants.^10–22 While some photon–photon gates do not require active control, i.e., they are passive, their footprint is large, such as gates that require an array of many interaction sites.²³ For example, Konyk et al. proposed a CPHASE gate by counterpropagating two photons through an array of many two-level-emitters in a waveguide.²³ However, as with similar proposals, a large number of emitters is required for high fidelity.²³ 

In this Letter, we present a scheme for implementing a deterministic, passive, and low-footprint controlled-variable phase gate on photonic qubits. The physical system used is a three-level ladder emitter, historically known for producing quantum correlated photons.^24–29 We couple it to a one-dimensional photon field in the reflection geometry, and we encode the qubits as single-photon pulses in the frequency basis. By adjusting the frequencies of the photons relative to the emitter's transition frequencies, we enable any controlled-phase from 0 to π. This scheme does not utilize any active control and needs only a single ladder emitter, which may be an advantage for more efficient implementations of multi-qubit gates in quantum circuits for photonic quantum computers.

The implementation of the controlled-variable phase gate considered in this Letter is illustrated in Fig. 1. The energy level diagram of the emitter comprises a single ground state $|g⟩$⁠, singly excited state $|e1⟩$⁠, and doubly excited state $|e2⟩$⁠. The control transition $|g⟩↔|e1⟩$ has frequency $ΩC$ and the radiative decay rate $ΓC$⁠, and we assume it couples to horizontally polarized light due to polarization selection rules. The target transition $|e1⟩↔|e2⟩$ has frequency $ΩT$ and the radiative decay rate $ΓT$⁠, and it couples to vertically polarized light. The Hamiltonian, setting $ℏ=c=1$ and assuming the rotating wave approximation, is

where $σij=|ei⟩⟨ej|$ is the atomic transition operator (⁠$|e0⟩≡|g⟩$⁠) and $ĥk$ (⁠$v̂k$⁠) is the annihilation operator for the horizontal (vertical) mode k. We have implicitly assumed that the field-atom coupling is constant since the frequency spread of each single-photon packet is small. The rotation wave approximation is valid because only photon modes near-resonant with the transitions interact strongly with the system. Because the emitter is in its ground state and there is a single horizontal and a single vertical photon at the start of gate operation, the general photon–emitter wavefunction at all time t is

We insert this ansatz into the time-dependent Schrödinger equation to extract the equations of motion for the coefficients in Eq. (2). In addition, at the beginning and end of gate operation, the emitter is in its ground state, and there are two photons. Therefore, we apply the following boundary conditions: only $f(k1,k2,±∞)$ is nonzero, and $g(k,±∞)$ and $s(±∞)$ are zero. Integrating the equations of motion from t = −∞ to t = +∞ (see the supplementary material for details), the output two-photon wave packet $f(k1,k2,+∞)≡fout(k1,k2)$ is related to the input two-photon wave packet $f(k1,k2,−∞)≡fin(k1,k2)$ by

The logical basis state $|1C⟩$ (⁠$|1T⟩$⁠) of the control (target) qubit is assigned to a horizontally polarized (vertically polarized) single-photon pulse that is resonant with the control $|g⟩↔|e1⟩$ (target $|e1⟩↔|e2⟩$⁠) transition, while the $|0i⟩$ logical basis state is strongly detuned and has an identical line shape and polarization as its $|1i⟩$ counterpart. For the target photon, the central frequency of the logical basis state $|1T⟩$ can be detuned slightly by $ΔT$ on the order of $ΓT$ to adjust the controlled-phase shift $ϕ$⁠. After gate operation, the qubits can be trivially separated with a polarizing beam splitter and rotated into the same polarization, should doing so be necessary for the remaining circuit.

Before presenting gate performance results based on the full expression in Eq. (3), we outline the intuition of the gate and make analytical approximations to rationalize the conditions necessary for high fidelity. Generally, if a photon is far off-resonant with a transition, the photon will not interact with the transition and pass unchanged, while if a photon is closely resonant, the photon will interact with the transition and pick up a phase. Because the emitter is initialized in its ground state, the second, vertically polarized target transition is inaccessible to the target photon unless the first and horizontally polarized transition is first excited by the control photon. Therefore, the state of the horizontally polarized control photon controls whether the target photon can interact with the emitter and pick up a phase shift, yielding a controlled-phase gate. Given this framework, we outline the three conditions for high fidelity.

(i) To ensure that an on-resonant control photon will be fully absorbed by the emitter (i.e., the control photon can excite the system to its $|e1⟩$ state fully), it must have a Lorentzian line shape with bandwidth $σC=ΓC$⁠. Intuitively, we can understand this condition as a consequence of time-reversal symmetry: because an emitter initialized in $|e1⟩$ will emit a photon with Lorentzian line shape and bandwidth $σC=ΓC$ as it decays to $|g⟩$⁠, the same line shape and bandwidth are required to fully excite the emitter from $|g⟩$ to $|e1⟩$⁠. While the target photon can have any line shape that can be optimized for maximal gate fidelity, for consistency, we assume it also has Lorentzian line shape. Therefore, we write the two-photon input wave packet as

where $σC=ΓC$⁠. This packet is normalized as $∫dk1∫dk2|fin(k1,k2)2|=1$ and $⟨0|ĥk1v̂k2ĥk1′†v̂k2′†|0⟩=δ(k1′−k1)δ(k2′−k2)$⁠. The bandwidth of the single-photon pulse $σi$ is the same for logical basis states $|0i⟩$ and $|1i⟩$⁠, but ω_i depends on whether the photon is in the $|0i⟩$ or $|1i⟩$ state.

(ii) To ensure that target photon picks up a uniform phase shift across its entire packet, the $|e1⟩↔|e2⟩$ transition must adiabatically follow the target photon pulse. Essentially, the target photon is a weighted superposition of different frequency components, with each frequency component picking up a slightly different phase after interacting with the target transition. For this effect to be negligible, the target photon's pulse length $∼1/σT$ must be much greater than the lifetime of $|e2⟩∼1/ΓT$⁠, or equivalently, the bandwidth of the target photon packet must be much smaller than $ΓT$⁠.

(iii) To ensure that the target photon interacts with the ladder emitter only when control photon has been fully absorbed or, equivalently, when the control transition is fully excited, the pulse length of the target photon $∼1/σT$ must be much smaller than the pulse length of the control photon $∼1/σC$⁠.

Assuming the above three conditions for optimal gate performance, resulting in $σC=ΓC≪ΓT≪σT$⁠, we plug $fin$ from Eq. (4) into Eq. (3) and approximately evaluate it by pulling the slowly varying part of the integrand corresponding to the target packet out of the integral, yielding

With Eq. (5), we confirm our intuition for the gate operation. When the control photon is in the $|0C⟩$ state, we have $|ωC/ΓT−ΩC/ΓT|≫1$ and $|ωC/ΓC−ΩC/ΓC|≫1$⁠, reducing Eq. (5) to $fout(k1,k2)=fin(k1,k2)$⁠, demonstrating that both the control and target photons are left unchanged after interacting with the ladder emitter when the control photon is in its $|0C⟩$ state, regardless of the state of the target qubit. When the control photon is in-state $|1C⟩$⁠, or nearly resonant with the control transition $|g⟩↔|e1⟩$ by $ΔT$⁠, Eq. (5) reduces to

The target photon, therefore, collects a conditional phase shift $ϕ=arg[−(1+i(ωT−ΩT)ΓT/2)/(1−i(ωT−ΩT)ΓT/2)]=π+2arctan(ΔTΓT/2)$ upon re-emission that is dependent on the target photon's detuning $ΔT=ωT−ΩT$⁠. Meanwhile, regardless of the state of the target photon, the control photon is mirrored and rotated in the complex plane from $|1⟩C=∫dk1ΓC/(2π)ΓC/2+i(k1−ΩC)ĥk1†|0⟩$ to $|1′⟩C=−∫dk1ΓC/(2π)ΓC/2−i(k1−ΩC)ĥk1†|0⟩$⁠. Because this transform is deterministic and always occurs for the $|1C⟩$ state, it does not impact the gate's information processing capability; it can be viewed as a re-definition of the logical basis $|1C⟩$ to $|1C′⟩$⁠. Furthermore, it can be trivially rectified via linear optical components or an additional reflection of the control packet;^30–33 these methods for controlling the pulse shape of single photons may also be used to use a photon as the target photon during one operation of the proposed gate mechanism and the control photon during another, although these methods require further development. The overall result of the control and target photons interacting with the ladder system is a passive, low-footprint, and deterministic C-PHASE gate on photonic qubits with tunable phase dependent on detuning $ΔT$⁠.

We plug our Lorentzian two-photon input wave packet described in Eq. (4) into Eq. (3) to calculate the fidelity $f$ and conditional-phase $ϕ$ as a function of the photon–emitter system parameters. The fidelity of the gate is defined as $f=14Tr(Ûideal†Ûgate)$ or equivalently $(⟨00¯|Ûgate|00⟩+⟨01¯|Ûgate|01⟩+⟨10¯|Ûgate|10⟩+⟨11¯|Ûgate|11⟩)/4$⁠, where the state $|ψ¯⟩$ is the ideal output of the gate given the input state $|ψ⟩$⁠. By far, the biggest loss to fidelity is caused by the $|11⟩$ input case where both control and target photon interact with the emitter, so the fidelity is essentially $f=34+14⟨11¯|Ûgate|11⟩$⁠. The $|11⟩$ input basis state is the only case where nontrivial interaction occurs, which may significantly deform the packets of the single-photon pulses. A similar overlap calculation can be done to extract the conditional phase, as detailed in the Supplementary Information, resulting in the following analytic expressions for the fidelity $f$ and conditional phase $ϕ$⁠,

In Fig. 2, we explore assumptions (ii) and (iii) by plotting the fidelity $f$ given by Eq. (7) for varying $σT/ΓC$ and $ΓT/σT$ and constant $ΔT=0$ corresponding to $−ϕ≈π$⁠. Fidelity $f$ increases with both increasing $σT/ΓC$ and increasing $ΓT/σT$⁠, supporting the intuition that the fidelity improves when the target photon pulse length is smaller than the control photon pulse length and when the bandwidth of the target photon is smaller than the radiative decay rate $ΓT$ of the target transition $|e2⟩↔|e1⟩$⁠. The fidelity $f$ is as high as 99.8% for $σT/ΓC=ΓT/σT=103$⁠. Therefore, for a desired gate operation time on the order of 1 μs, control transition timescales on the order of 1 μs and target transition timescales on the order of 1 ps are required.

In Fig. 3, we explore how adjusting the detuning $ΔT$ normalized by $ΓT$ impacts the fidelity and phase of the gate. In (a), we see that increasing $σT/ΓC$ and $ΓT/σT$ increases fidelity, which is consistent with Fig. 2. Additionally, we see that increasing the detuning $ΔT/ΓT$ increases fidelity. In (b), we fix $σT/ΓC=ΓT/σT=103$ and vary $ΔT/ΓT$⁠. Increasing $ΔT/ΓT$ decreases the magnitude of the conditional phase $ϕ$ and increases the fidelity $f$⁠. At $ΔT/ΓT, ϕ$ is nearly $−π$⁠, and $ϕ$ approaches 0 in the $ΔT/ΓT→∞$ limit, indicating a phase lag. The variation of the condition phase $ϕ$ with the detuning is consistent with the previous approximations. Enabling continuous phases with magnitudes up to π enables greater flexibility when designing quantum circuits, especially those for the quantum Fourier transform that would otherwise require dozens to hundreds of gates to approximate each controlled-variable phase gate using only the CNOT, H, S, and T universal set.^34,35

We consider practical implementation based on the constraints of the proposed mechanism for a controlled-variable phase gate on photonic qubits. First, we recall that the difference in central frequencies between $|0i⟩$ and $|1i⟩$ must be $≫Γi$ to ensure that the logical basis states are well defined. Assuming that $σT/ΓC=ΓT/σT=103$ and that $ΓC, σT$⁠, and $ΓT$ are in the MHz, GHz, and THz range, respectively, for $f≥0.998$⁠, then fulfilling this constraint for Ω_i in the optical frequency range is trivial. However, photons must lie in the telecommunications band with a bandwidth on the order of a few THz to be seamlessly transmitted without frequency conversion for quantum networking.^36–39 In this frequency range, while the difference in central frequencies of the logical basis states of the control photon can be trivially $≫ΓC$⁠, the difference in central frequencies of the logical basis states of the target photon is limited by the bandwidth of the telecommunications band to be of a similar order as $ΓT$⁠. As a result, the $|1C⟩|0T⟩$ input basis state is particularly susceptible to acquiring extraneous phase and slight wavepacket deformation but still does not appreciably impact fidelity with a deterministic single-qubit phase gate.

A second critical practical consideration is the feasibility of tuning one emitter to apply the proposed gate with variable phases via detuning $ΔT$⁠. As shown in Fig. 3(c), the phase $ϕ$ changes with $ΔT$ on the order of $ΓT$⁠, therefore requiring the energy level of the ladder emitter to change on the order of THz to be able to apply controlled-variable phase gates on microsecond timescales. In defect systems, often described as “artificial atoms” due to their spatial localization and potentially bright, narrow emission, both strain and electric fields have been shown to shift the emission of defect states.^40,41 While achieving similarly high electric fields to exert similar Stark shifts in conventional atoms or ions may be challenging, it can be more straightforward in Rydberg atoms with larger polarizabilities, resulting in Stark shifts on the order of 0.5 THz for fields of approximately 0.01 MV/m.⁴² 

A third practical consideration is the validity of the bad-cavity approximation,⁴³ where the spontaneous emission rate of the emitter into free space is assumed to be much smaller than the emission into the waveguide. This regime is the bedrock of other leading proposals for photon–photon gates, including a deterministic $SWAP$ gate¹⁶ and the Duan–Kimble proposal¹⁵ that was experimentally realized,¹⁷ where the fidelity was limited by auxiliary optical technologies and not the validity of the bad-cavity approximation. We expect the experimental feasibility of the proposed gate mechanism to be unconstrained by reaching the bad-cavity regime, as it was recently realized in, for instance, several solid-state systems.^44,45

In summary, we propose a deterministic, passive, and low-footprint controlled variable-phase gate on photonic qubits in the frequency basis using a ladder system to mediate effective photon–photon interactions. Specifically, we analytically derive the scattering matrix for two orthogonally polarized photon pulses interacting with the emitter, and we show that this interaction results in high fidelity for the controlled-variable phase gate, given three assumptions of the photon–emitter system. Such a gate enables universal quantum computation when paired with single photon gates as well as efficient decomposition of fundamental quantum circuits. Furthermore, the ability to encode the target qubit in a different frequency range than the control qubit may enable more facile integration with quantum repeaters in quantum networks or coupling quantum systems operating in different energy ranges to each other^46–49 by averting the need for frequency conversion, where optical photons can be stored locally more conveniently, photons with telecommunication wavelength are more easily transmitted over long distances, and microwave photons can interact with both defect-based spin qubits and superconducting qubits.^50,51 The advantages of the present scheme are the complex lack of external control beyond the Stark shift to tune the target transition frequency to change the conditional phase, low-footprint using only a single emitter, and ability to impart any phase from 0 to π, while the potential disadvantages include wavepacket constraints of the control and target photons, the deterministic rotation of the control photon wavepacket, and requirements for the relative transition polarizations and frequencies of the ladder emitter.

The particular level structure and relative transition rates for the ladder emitter, where $σTΓC=ΓTσT∼103$⁠, may be within reach in a variety of physical systems. A particularly interesting potential realization of the ladder emitter is Purcell-enhanced lanthanide ions doped into inert or electro-optical tunable hosts. These lanthanide ions support sub-microsecond long lifetimes, or MHz emission rates, along a ladder of photon-emitting transitions in the optical range that can be Purcell-enhanced,⁵² even dynamically,⁵³ although not yet quite at the six orders of magnitude required for the present proposal. Another potential physical system is defect emitters. In Ref. 54, the authors demonstrate THz emission rates of nitrogen-vacancy centers in diamond in a Purcell-enhanced optical cavity—this technology in conjunction with silicon vacancy defects in diamond with emission rates nearing the MHz range⁵⁵ may be on the cusp of enabling experimental realization of the proposed mechanism, although defect emitters that emit photons along an energy ladder have not yet been discovered. First principles-based computational methods with the potential to incorporate the cavity field^56–58 could also be leveraged to discover new emitters that match the criteria, as has been demonstrated for defects in solid-state materials,⁵⁹ although predicting multiply excited states necessary for the three-level ladder emitter remains challenging.⁶⁰ Another method of producing such a ladder emitter is to dipole-couple two three-level emitters, as described in detail in Ref. 61, with the added stipulation that the transition rate from the ground state to one of the excited states is fast, while the transition rate from the ground to the other excited state is slow; see the supplementary material for further details on this added stipulation. Semiconductor quantum dots may also be engineered to display the correct level structure, as they have already been extensively studied for other two-photon applications.^62,63

We anticipate that our prediction will spur advances in experimental realizations of passive multi-qubit photonic gates and further searches for candidate emitters to realize the required complex light–matter interactions. In addition, one promising direction for further theoretical studies includes coupling the emitters to external fields and sculpted electromagnetic environments to boost the fidelity and improve the practical applicability of the proposed scheme. Another promising direction that warrants further investigation is to generalize the concepts presented in this Letter to multi-qubit gates with more control or target qubits, such as the Toffoli gate, as such gates would enable even less resource-intensive decomposition of quantum circuits. We expect these results to motivate further interest in photonic quantum information processing with designer emitters, such as defect complexes in solid-state materials.

See the supplementary material for the derivation of the general S-matrix, gate performance, and the level structure of a composite emitters with the correct level structure from dipole-coupled single emitters.

We acknowledge Stefan Krastanov, Matthew Trusheim, and Tomáš Neuman for fruitful discussions. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division under No. FWP ERKCK47 “Understanding and Controlling Entangled and Correlated Quantum States in Confined Solid-state Systems Created via Atomic Scale Manipulation.” D.S.W. is supported by a National Science Foundation Graduate Research Fellowship and partially by the Army Research Office MURI (Ab-Initio Solid-State Quantum Materials) under Grant No. W911NF-18-1-0431. P.N. is a Moore Inventor Fellow and gratefully acknowledges support through Grant No. GBMF8048 from the Gordon and Betty Moore Foundation.