A cavity-mediated quantum CPHASE gate between NV spin qubits in diamond
While long spin coherence times and efficient single-qubit quantum control have been implemented successfully in nitrogen-vacancy (NV) centers in diamond, the controlled coupling of remote NV spin qubits remains challenging. Here, we propose and analyze a controlled-phase (CPHASE) gate for the spins of two NV centers embedded in a common optical cavity and driven by two off-resonant lasers. In combination with previously demonstrated single-qubit gates, CPHASE allows for arbitrary quantum computations. The coupling of the NV spin to the cavity mode is based upon Raman transitions via the NV excited states and can be controlled with the laser intensities and relative phase. We find characteristic laser frequencies at which the scattering amplitude of a laser photon into the cavity mode is strongly dependent on the NV center spin. A scattered photon can be reabsorbed by another selectively driven NV center and generate a conditional phase (CPHASE) gate. Gate times around 200 ns are within reach, nearly three orders of magnitude shorter than typical NV spin coherence times of around 10 s. The separation between the two interacting NV centers is only limited by the extension of the cavity.
Nitrogen-vacancy (NV) centers in diamond have emerged as powerful and versatile quantum systems with applications as sources of non-classical light, as high-precision sensors, and as qubits for quantum information technology Dobrovitski2013 (). The electron spin of the NV center unites several essential properties required for quantum information processing (QIP). Its quantum coherence is preserved over long times, even at elevated temperatures, and it allows for optical preparation and read-out, as well as quantum gate operations via radio-frequency (rf) excitation, at the level of a single-NV center. One of the remaining challenges on the way towards diamond-based QIP is the establishment of a scalable architecture allowing for the coherent coupling between NV spins. A controlled coupling is required to realize a two-qubit gate such as controlled-phase (CPHASE) or controlled-not (CNOT) which forms a universal set of quantum gates in combination with single-qubit gates. Controlled operations between the NV electron spin and a nearby nuclear spin have been performed using a combination of rf and microwave pulses Gaebel2006 (), whereas entanglement generation can be achieved between the electron spins of two nearby NV centers on the basis of static dipolar interactions Dolde2013 (), and between NV center spins separated by several meters Pfaff2012 (), and subsequently over more than one kilometer Hensen2015 () via a non-deterministic coincidence measurement protocol. Here, we propose and theoretically analyze a fully controllable and switchable coupling between the spins of distant NV centers coupled to the same mode of a surrounding optical cavity (Fig. 1).
A variety of optical cavity systems for cavity quantum electrodynamics (QED) coupled to defect centers in diamond exist. The advantage of whispering gallery modes of silica microsphere is their ultrahigh quality factors Park2006 () , whereas photonic crystals fabricated within the diamond crystal Wang2007 (); Faraon2012 (); Riedrich2012 () or on top Englund2010 () allow for the embedding of the NV centers directly into the optical cavity structure, but comprise (so far) somewhat lower factors. However, photonic crystal cavities in diamond with have recently been fabricated Burek2014 (). The architecture to be proposed here can in principle be used with any realization of NV-cavity coupling, provided sufficiently high and dipole matrix element of the ground state (GS)-excited state (ES) transition in the cavity field.
The basic working principle of the quantum gate operation proposed here is as follows. We restrict ourselves to two of the three GS spin triplet states, and , which will serve as the qubit basis in our scheme (Fig. 2a). Near the GS level crossing around a magnetic field of about , these two states are nearly degenerate, and separated by several GHz from the third () state. Off-resonant coupling of the GS-ES transition to the cavity mode combined with off-resonant laser excitation can be used to generate Raman-type two-photon transitions starting and ending in the GS, accompanied by the scattering of a laser photon into the cavity mode, or vice versa (Fig. 2b). The off-resonant coupling is the main distinguishing feature from resonant schemes which are limited by spontaneous emission Zagoskin2007 (). The proposed two-qubit coupling mechanism relies on a spin-dependent scattering of laser photons into the cavity and back which is possible because of the difference in zero-field splittings in the GS and ES. More specifically, the and states in the ES are not degenerate at , which leads to unequal scattering matrix elements for the and states. To produce an entangling quantum gate between two NV spin qubits, we find it to be sufficient if the laser-cavity photon scattering rate is different for the two spin states. If two NV centers are simultanously coupling in this way to the same cavity mode, they will exchange a virtual cavity photon, thus generating a conditional phase shift; once the accumulated relative phase amounts to , a CPHASE gate on the two NV spin qubits has been achieved.
In contrast to cavity-mediated spin interactions proposed for semiconductor quantum dots Imamoglu1999 () where the spin-orbit splitting in the valence band can be used for spin-selective excitation with polarized radiation and Raman-type spin flip transitions, we propose here to use another mechanism based on the different zero-field splittings of the NV ground and excited states to perform phase and controlled-phase operations. Earlier work on cavity-mediated quantum gates for defect qubits in diamond makes use of spectral hole burning Shahriar2002 () or a series of systemsSolenov2013 (). The latter requires a sequence of at least three two-color pulses, while our scheme manages on just one single-color laser pulse for a CPHASE gate. A model for three NV centers coupled to a whispering-gallery mode in a silica microsphere cavity using polarized excitation has been studied with the goal of achieving a three-qubit CPHASE gate Yang2010 (). Our scheme relies on spectral selectivity and thus does not require polarized excitation. The effect studied here produces an elementary, universal two-qubit CPHASE gate.
Ii Single NV center in a cavity
The NV center in its ground state (GS) and excited state (ES) spin triplet will be described by the Hamiltonian
where the first term describes the Zeeman splitting of the spin with eigenvalues in a magnetic field applied along the NV () axis with identical electronic Landé g-factor for the GS and ES ( denotes the Bohr magneton). See Appendix A for a discussion of a possible magnetic field misalignment. The second term in Eq. (1) includes the GS-ES energy gap and the distinct GS and ES zero-field spin splittings and . The off-diagonal terms describe laser excitation at a frequency , with the spin-independent dipole matrix element . We assume that the ES orbital state energies are strongly split by the strain in the diamond crystal, and we can concentrate on one of the two orbital ES triplets. The prerequisite for this to be a reasonable approximation is that the strain splitting exceeds the ES spin-orbit coupling . Strain splittings in excess of this value and up to 20 GHz have been observed Batalov2009 (); Bassett2014 (). Taking only one orbital ES into account, we can view the Hamiltonian in Eq. (1) as a 6x6 matrix consisting of four 3x3 blocks. The Zeeman splitting described by the first term in Eq. (1) is independent of the orbital state. Using Pauli matrices to describe the GS-ES orbital state, i.e., for the GS and for the ES, and working in a rotating frame with the frequency , we can write
where and denote the mean and difference between the GS and ES zero-field splittings, describe transitions between the GS and ES, and is the laser detuning. We have so far neglected the spin-spin couplings in the ES, but will discuss their effect further below.
We now consider a single NV center coupled to a near-resonant mode of a surrounding optical cavity which we describe, using the rotating-wave approximation, with the following Hamiltonian,
where denotes the detuning of the cavity mode from the laser excitation frequency and () creates (annihilates) a cavity photon. The dipole matrix element of the cavity field can be made real-valued by an appropriate phase convention in the excited state. However, can in general not be made real-valued at the same time; its phase depends on the phase of the laser field.
The magnetic field is chosen at a working point around the GS level crossing where we focus our description on the nearly degenerate and levels (the level will be included further below). This approximation is justified because the level is split off by the zero-field splitting which is much larger than the spin-spin splittings coupling it to the other two spin levels. We describe here the situation of an initially empty cavity, which subsequently holds at most one virtual photon. Starting from an empty cavity, and assuming sufficiently large detunings and of the cavity and laser frequencies, we can further reduce the relevant states to , , and , where and denote the GS and ES, respectively, and denotes the cavity photon number. Including the two remaining spin projections, this leaves us with six states for a single NV and the cavity.
The combined action of the coupling to the laser and cavity fields can scatter a photon from the laser into the cavity or vice versa, via an intermediate virtual ES. Starting from the Hamiltonian Eq. (3), and assuming that the electric dipole couplings are much smaller than the detuning from the one-photon resonances, we can derive an effective GS Hamiltonian for such second-order processes (see below),
where denotes the projection operator on the spin state with projection ,
the effective coupling strength, and the magnetic field detuning from the GS level crossing. The last term in Eq. (4) describes spin dependent scattering processes at the NV center of a cavity photon into a laser photon or vice versa. Generally, we find that in order to construct a CPHASE gate, it is sufficient if (see also below). A possible extreme case where is described in the Appendix B. In Eq. (4), we have suppressed optical Stark and Lamb shifts of order and , which will not play an essential role in what follows.
We now give a more detailed derivation of Eqs. (4) and (5), starting from Eq. (3). To describe the combined action of the coupling between the NV center to the laser and cavity fields we write Eq. (3) as with the perturbation Hamiltonian
generated by the antihermitian operator
such that , and obtain the effective GS interaction Hamiltonian
Iii Two NV centers coupled to a common cavity mode
The scattering of a photon from the laser to the cavity field and vice versa, conditional on the spin (qubit) state of an NV center can be used to construct a cavity-photon mediated quantum gate between two NV spin qubits coupled to a common cavity mode. Starting from two NV centers (), each coupled to the same cavity mode as described above (Fig. 1), we derive the effective coupling Hamiltonian for two NV spins by eliminating the virtual cavity photon.
It is important to recognize that the cavity mediated interaction between the NV centers is a fourth-order process in the coupling strengths which prevents us from using the second-order Hamiltonian Eq. (4) directly to calculate the coupling between the NV center spins. In order to systematically account for all contributions up to the fourth order, we perform a fourth-order Schrieffer Wolff transformation of the Hamiltonian describing two NV centers coupled to a common cavity mode,
where we have restricted ourselves to the and states near the GS level crossing where . As this Hamiltonian commutes with the operators and of the NV centers, we can treat it separately for each of the four ground-state spin configurations, which represent the logical basis for our two-qubit system. For each spin configuration, we consider the five states , , , , and , where denotes the state with NV () in the ground () or excited () state, while the cavity mode is occupied with photons. In analogy with the previous section we are only interested in the effective interaction between the NV centers and the cavity in the NV ground state. To derive an effective spin Hamiltonian for the NV ground states, we decouple the two states and from the remaining three states by performing a Schrieffer-Wolff transformationDVincenzo (); Winkler (). In analogy with Eq. (7), and expanding to fourth order, we have
We then expand the matrix S as a series , where each term is derived using Eq. (11) under the requirement that there is no coupling between the () subspace and the excited states of the NV centers up to -th order in the coupling constants and . In the sum Eq. (11), we then calculate all the residual terms and obtain the effective Hamiltonian in the basis , ,
Introducing the phases of the lasers as , we find for the eigenenergies of this effective Hamiltonian
and , whereas for the off-diagonal matrix element we obtain
We present only up to the fourth order corrections, as only these terms will be important for the following discussion. We have also calculated using conventional perturbation theory, rather than a Schrieffer-Wolff transformation, with identical results (see Appendix C). The expression for in Eq. (13) consists of two parts, where each term of the first part depends on the spin state of only one NV center and thus only leads to single qubit dynamics. Entanglement can be generated by the second part (last term) of Eq. (13),
as it depends on the spin state of both NV centers. Calculating this term for each spin configuration leaves us with the diagonal spin Hamiltonian
This Hamiltonian generates a quantum gate which up to single qubit operations, is the CPHASE gate with
Equation (17) proves that the interaction of two NVs through the cavity can give rise to an entangling gate. This gate can be controlled both by the amplitude and phase of the lasers and by the detuning of the laser frequency from the cavity mode .
The results of this section can only be considered a qualitative proof of the entangling gate. They are valid as long as the perturbation analysis works, which implies that the couplings are much smaller than the detunings . Moreover, to make predictions one should take into account the spin-spin interaction in the excited state of the NVs, which will be done in the next section in the description of our numerical results.
Iv Spin-spin interaction
To make quantitative predictions, we need to include the spin-spin interactions in the ES which have been studied both experimentally Fuchs2008 (); Batalov2009 (); Bassett2014 () and theoretically Doherty2011 (); Maze2011 (),
where and .
The Hamiltonian of the system will then take the form
where and have been introduced in the previous section. In the spin Hamiltonian the term mixes the spin states and , while the term mixes and , as well as and . Therefore, we can no longer treat each of the four logical states separately.
It is important to note that both cavity photon creation and spin-spin interaction are only possible when one of the NVs is in the excited state. To achieve this and thus create a quantum gate, laser excitation can be used to transform the initial ground state of the NVs. But it is also important that after the excitation is switched off, the system should remain in a final state that is the coherent superposition of the logical basis states. Thus the probability to have an excited NV after the laser pulse is turned off should be very low. This will be the case if the intensity of the lasers changes slowly, such that the adiabatic theorem provides that the system remains in the same eigenstate of the time-dependent Hamiltonian. The final state of the system after the pulse is turned off will correspond to the ground state of the NVs and zero cavity photons the logical basis of the two qubit system.
We now introduce our numerical results obtained for this system, including spin-spin interactions. The laser detuning and the cavity detuning are asumed to be and respectively. The distance between the ground state and the lower excited state of the NV would then be for spin state and for spin state. The energy of the cavity excitation would be . The inverses of these values ( respectively) define the internal dynamical rate of the system, with respect to which one has to choose the ramp time of the pulse. To stay within the adiabatic regime we took the pulse to be a convolution of a Gaussian and a rectangle with the widths (FWHM) and respectively. The coupling at the maximum of the pulse is assumed to be . The coupling between the NV and the cavity is assumed to be . We consider both NVs to be identical and driven by two identical and synchronized lasers with the same amplitude, phase and the pulse form described above. Note that the two-qubit gate operation requires neither the NV centers nor the driving fields to be identical; this choise is made here only to simplify the analysis. Under these assumptions we numerically propagate each of the four logical states of the system. This results in a unitary in the logical space of the two-qubit system, corresponding to a CNOT gate, as shown by the Makhlin invariants and (Fig. 3), for which the values 1 and 0 respectively were obtained, which is a characteristic of a CNOT gateMakhlin2002 ().
We have shown that virtual exchange of photons in an optical cavity can mediate the two-qubit CPHASE gate between two NV spin qubits in diamond. Combined with single-qubit operations, produced by rf excitation or by laser fields Yale2013 (), the CPHASE gate allows for arbitrary (universal) quantum computations. Therefore, optical cavity QED with NV centers in diamond represents a realistic path towards spin-based quantum information processing. The cavity-mediated quantum gate proposed here could be applied to other defects with a similar level structure, i.e., comprising spin triplet ground and excited states with deviating zero-field splittings. For example, we expect that the gate protocol would also work for certain divacancy centers in silicon carbide.
As a further prerequisite for the scheme to work, the NV spin coherence time and average time between cavity photon loss must be longer than the gate operation time . The NV spin coherence time can reach , even at elevated temperatures. The photon loss rate can be estimated as where is the probability for the cavity mode to be occupied by a virtual photon during the gate operation, and is the photon loss rate in the cavity with quality factor . For the parameters used above, a Q factor of is needed to achieve . Because while , increasing the detuning allows the use of cavities with lower at the expense of slower gates, which in turn are admissible for sufficiently long . The limit of this scaling can be described in terms of a (coherent) cooperativity factor Kolkowitz2012 () .
Finally, we expect this scheme to work below a temperature of about where the excited state levels are stable. It is an open question whether a variation of this scheme will also work at higher temperatures.
In a scalable qubit architechture, pairs of qubits need to be selectively coupled within a large array. A possible architecture comprises single NV centers in optical cavities linked via optical fibers Nemoto2014 (). The coupling mechanism described here lends itself to another architechture where many NV centers are embedded in a single cavity. In an array with separations between NV centers on the order of 10 to 100 nm, selective pairwise coupling can be accomplished with a combination of spatial and spectral selectivity of the laser excitation.
We thank Adrian Auer, Christopher Chamberland, Mikhail Lukin, and Chris Yale for helpful discussions. We acknowledge funding from AFOSR and NSF (DDA), and from CAP, DFG SFB767, and BMBF Q.com-HL (GB).
Appendix A Magnetic field alignment
In our model, we have so far assumed that the magnetic field is perfectly aligned with the NV axis of both defects involved in the CPHASE gate. This raises two important issues: (1) how to treat NV centers with different oriantations with respect to the diamond crystal, and (2) to what extent will the CPHASE operation be disturbed by any small misalignment of the magnetic field? As for (1), we note that there are four distinct NV orientations (up to small misalignments which we discuss below). Only the NV centers with their orientation along the external B field will be near resonance and will participate in the CPHASE gate operation while the NV centers oriented along the three other axes can be safely ignored. Regarding (2), the field misalignment will add a term to the Hamiltonian Eq. (1) where is the transverse (misalignment) field (chosen to point in direction) and denotes the misalignment angle. The effect of the misalignment field is small if . For a misalignment of one degree, the NV center should be operated at least away from the level anticrossing.
Appendix B Minimal model for spin-dependent cancellation of laser-cavity photon scattering
Here, we provide a minimal model to explain the spin-dependent cancellation of laser-cavity photon scattering. Neglecting spin-spin coupling and assuming and to be real, we can treat the two spin states and separately, with the Hamiltonian
in the basis , , . For , we find for the state,
Note that in the rotating frame, the excited state now lies exactly in between the states with zero and one cavity photon. We introduce the dressed states ,
and note that up to corrections cubic in they form an orthonormal basis of the space spanned by , , and . In this new basis, the Hamiltonian Eq. (21) takes the diagonal form
The absence of any effective coupling between and in the state for can be traced back to the equal and opposite contributions from coupling the excited state to the two states and which are symmetrically arranged in energy around in the rotating frame. In contrast to this result, we find for the state that
with a non-zero amplitude for emitting or absorbing a cavity photon,
Note that for , the destructive interference of the two terms in Eq. (24) leads to a decoupling, .
Using our minimal model, we can also discuss the validity of the effective Hamiltonian derived using the Schrieffer-Wolff transformation. The realization of a quantum gate (CPHASE) operation leads to a time-dependent problem, because the control lasers need to be switched on and off to perform the quantum gate. The Schrieffer-Wolff transformation and use of the obtained effective Hamiltonian for this time-dependent problem are appropriate if the following two conditions are satisfied: (i) Weak coupling (also mentioned above in the text), more specifically, , (ii) adiabatic switching on and off of the laser fields (sufficiently long ramp time ) compared to the separation of ground and excited states (for in the rotating frame), .
Appendix C Perturbation analysis
In this section we will give an alternative derivation of equation (13), using conventional time independent perturbation theory. We are interested in the shift of the ground state of , induced by the perturbation . The matrix element of , that causes the transition from the initial state to the final state , is
Thus, the matrix elements of the perturbation are:
that account for the interaction between the NV centers and the laser, and
that account for the interaction between the NV centers and the cavity. There are also six inverse transitions with the conjugate matrix elements. We consider only the first five energy levels of , as we use fourth-order perturbation theory and higher energy levels are not excited under this approximation.
One can think of the perturbation to the particular eigenenergy level of as arising from transitions that start and end at this level. First order processes are thus absent as we have no diagonal terms in the perturbation. The second order processes are
and the second order energy correction will be
There are no third order processes that would start and end in the ground state, and therefore, the third order correction to the energy is zero. Now we include all of the fourth order processes, described by the formula
where we have used that the orbital energy of the state can be set to zero. Also, we have omitted all the terms that contain the diagonal perturbation elements, as those are zero for our system. The first term in this equation contains all eight fourth-order processes that exist for this system. The second term is responsible for renormalization of the perturbed wavefunction. After the calculation, we find
Appendix D Makhlin invariants
We are interested in producing a two-qubit gate (e.g. ) only up to single-qubit operations, i.e.,
with and arbitrary single-qubit unitaries. To test whether and are equivalent in this sense, one can use two invariants Makhlin2002 () of a two-qubit unitary , defined as
where and , with the transformation into the Bell basis,
For the identity operation , we find , , whereas the CPHASE gate lies in the same class as the CNOT gate, with , . Finding the latter values for and with for some time therefore proves that we have generated the CPHASE gate (and with this also CNOT gate) up to single-qubit operations.
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