Fermion parity measurement and control in Majorana circuit quantum electrodynamics
We investigate the quantum electrodynamics of a device based on a topological superconducting circuit embedded in a microwave resonator. The device stores its quantum information in coherent superpositions of fermion parity states originating from Majorana fermion hybridization. This generates a highly isolated qubit whose coherence time could be greatly enhanced. We extend the conventional semiclassical method and obtain analytical derivations for strong transmon-photon coupling. Using this formalism, we develop protocols to initialize, control, and measure the parity states. We show that, remarkably, the parity eigenvalue can be detected via dispersive shifts of the optical cavity in the strong-coupling regime and its state can be coherently manipulated via a second-order sideband transition.
Advances occurring over the past decade have given rise to a new generation of single-qubit solid-state architectures which hold the promise of compatibility with existing electronics and fabrication techniques. Among these devices, superconducting circuit processors based on the transmon qubit Koch et al. (2007); Schreier et al. (2008); Houck et al. (2008) have shown great potential in terms of coherent control, measurement, and scalability. These provide a unique opportunity to study fundamental quantum phenomena in engineered macroscopic two-level systems which are controlled by coherent microwave photons. Of particular interest is the study of hybrid devices where a microscopic or a mesoscopic system is embedded within the superconducting circuit. The properties of the constituent devices can contribute to the optimization of the qubit, including the processes related to its preparation, manipulation, and readout, to its coherence properties, and to its prospects for scaling up.
The integration of Majorana fermions Read and Green (2000); Kitaev (2001) into the superconducting circuit architecture Hassler et al. (2011); Bonderson and Lutchyn (2011); van Heck et al. (2011a); Schmidt et al. (2013); Pekker et al. (2013); Müller et al. (2013); Cottet et al. (2013); Ginossar and Grosfeld (2014); Ohm and Hassler (2015) can potentially lead to improved qubits, with the ultimate goal being the realization of a topologically protected information storage and high-coherence processing device Hassler et al. (2011); Bonderson and Lutchyn (2011). The recently introduced Majorana transmon (MT) (Ginossar and Grosfeld, 2014) sacrifices full topological protection by directly exploiting a weak interaction between two neighboring Majorana fermions, but gains a highly anharmonic spectrum, composed of well-separated nearly degenerate doublets, originating from parity states hybridization. In order to facilitate both detection and control, the circuit should be embedded within a microwave resonator Blais et al. (2004); Gambetta et al. (2006); Rigetti et al. (2012); Majer et al. (2007); Pellizzari et al. (1995); Goppl et al. (2008) where the strong interaction with the cavity field will provide the means for coherent qubit control and readout. In the proposed device, the lowest doublet of states is analogous to the familiar ion hyperfine qubit Benhelm et al. (2008) and does not couple directly to the cavity and the radiative environment owing to both a small matrix element and its small frequency. The consequence is an increased qubit lifetime and therefore potentially very high readout and control fidelities.
In this paper we develop the hybrid circuit quantum electrodynamics of the MT system strongly coupled to a single-mode electromagnetic field (see Fig. 1). First, we use a semiclassical approximation to obtain the eigenstates and spectrum of the device. We then use these to find expressions for the dipole matrix elements, demonstrating that the doublet forming the logical qubit is coupled to a higher doublet which can serve as a control. When the microwave transitions of the MT device are detuned from the cavity resonance, we show that a dispersive interaction arises between the fermionic parity, the transmon oscillator, and the cavity photon degrees of freedom. We use this regime to propose a scheme for measuring the state of the qubit, which is revealed via a fine structure in the cavity dispersive frequency shifts that are, remarkably, sensitive to the different fermionic parities associated with the two levels. In addition, we discuss protocols for (i) qubit cooling and (ii) implementing a single qubit rotation, demonstrating qubit control. Together, these provide the minimal ingredients required for establishing the relevance of the device for quantum information processing Barenco et al. (1995).
The measurement and control scheme that we develop here is based on an indirect coupling of photons to the parity doublet mediated through transitions to higher transmon levels. Another approach relies on the presence of a dipole coupling between the doublet states Ohm and Hassler (2015), which is, however, extremely small in our model when taken to the charge-noise resilient transmon regime. Other proposed qubit measurement and control schemes require electronic shuttling of Majorana fermions between superconducting islands using a series of depletion gates, relying on the adiabaticity of this process Hassler et al. (2011); Müller et al. (2013).
ii.1 Eigenstates of the Majorana-transmon device
The system we study consists of a Josephson junction capacitively coupled to a gate creating an offset charge between the superconducting islands. A nano wire which can support Majorana fermions Lutchyn et al. (2010); Oreg et al. (2010) is placed along the junction. Another realization could be based on the recent discovery of Majorana fermions in a chain of magnetic impurities Nadj-Perge et al. (2014), where here the chain should cross the Josephson junction. The zero-energy Dirac fermions composed of these Majorana fermions allow the relative number of Cooper pairs, , to admit both integer and half-integer values Fu (2010). Such a set-up can be described by the model Hamiltonian , where () are Pauli matrices operating in parity space. The first term in , containing the transmon Hamiltonian , has the eigenstates , , which correspond to an even and odd fermionic parity. These wave functions obey the boundary conditions , , which were chosen to ensure the correct quantization of the charge. Since the transmon operates in the regime , there is a close resemblance to an anharmonic oscillator which suggests using a semiclassical derivation of its eigenvalues and eigenstates if the parity-related boundary conditions could be accounted for (see the Appendix). The second term describes the interaction between the adjacent Majorana fermions van Heck et al. (2011a); Ginossar and Grosfeld (2014); Kwon et al. (2004) and creates a condensate “parity flip” accompanying a single electron tunneling process across the junction. In the setup, we consider , and are all independent energy scales. In the following, we assume .
We proceed with the diagonalization of the Hamiltonian which consists of approximately independent transmon bands, with two interlaced parity subbands split due to the Majorana interaction. We define the overlap between the odd and even states due to , , where we used the harmonic-oscillator states as an approximation for and . The intraband coupling is independent of the specific band and it is dominated by the interaction energy, to a leading order. The coupling between the bands decreases as , for even , and vanishes for odd . By neglecting terms of order and higher, the Hamiltonian matrix takes a block diagonal form . In the , basis, these blocks are given by
where are the energies of the harmonic oscillator with a first-order anharmonic correction and is the transmon dispersion (see the Appendix). The matrix can be diagonalized by a rotation around the axis, , , where and is the quadrant-dependent arctangent. The eigenvectors of Eq. (1) are
and have corresponding eigenvalues
where the quantum number corresponds to the rotated parity according to Eq. (2). It can be seen that the MT eigenvalues are further flattened by the presence of , improving on the charge-noise resilience of the transmon Koch et al. (2007). The eigenstates are superpositions of transmon wavefunctions with only a parametric dependence on . As we now discuss, this dependence leads to interference effects in the dipole transitions, and thus can be exploited to control the qubit using a cavity.
ii.2 Generalized Jaynes-Cummings model
We now couple the MT to a single mode of a quantized electromagnetic field in the microwave range, confined within a cavity consisting of a transmission-line resonator Majer et al. (2007); Wallraff et al. (2004). The full quantum description of this system is given by the Rabi Hamiltonian Koch et al. (2007); Blais et al. (2004) () where is the frequency of the photons created (annihilated) by the operator (). The interaction between the MT and the cavity is achieved via the dipole coupling , where is the dipole coupling strength, is the distance between the superconducting islands, is the rms field at the ground state of the resonator, and is the ratio between the gate capacitance and the capacitance of the junction. In the harmonic-oscillator approximation, the parity of the wave functions ensures that the dipole transitions are non zero only between neighboring bands. Projecting on the eigenstates given in Eq. (2) and using the rotating-wave approximation (RWA), the Hamiltonian takes the generalized Jaynes-Cummings (JC) form
where . We proceed to analyze the Hamiltonian, given by Eq. (4), in the combined qubit-cavity basis , with associated energies , where is the photon number in the cavity. The doublet states , which will serve as the qubit, decouple from the cavity interaction and, within the semiclassical approximation, intradoublet transitions vanish. For a nonzero photon number in the cavity, the qubit states are dressed to first order in the dipole coupling, only with the first excited transmon level , according to the th excitation sector of the Hamiltonian (here, ):
The relevant dipole matrix elements are
ii.3 Spectroscopy and parity state detection
By coupling the MT to a high-Q superconducting resonator and tuning it to the strong dispersive regime Frunzio et al. (2005); Geerlings et al. (2012); Wallraff et al. (2004); Niemczyk et al. (2010); Zueco et al. (2009); Ashhab and Nori (2010) , it would be possible to probe the telling features of the spectrum Ginossar and Grosfeld (2014) and to verify its dependence on the parameters and that were discussed in the previous sections. Experimentally these would involve a homodyne measurement setup for detecting changes in the complex amplitude of a coherent microwave tone which is transmitted at the resonance of the cavity, while spectroscopic pulses are driving the four MT microwave resonances. When a spectroscopic pulse excites the qubit, the transmission through the cavity resonance is diminished, indicating the transition frequency. The pulse can be emitted from a dedicated transmission line terminating in the vicinity of the qubit Leek et al. (2010). This measurement only requires the usual dispersive interaction between the cavity occupation and the transmon state Schuster et al. (2005). Extending the applicability of the dispersive measurement to the parity state requires an additional dependence of the resonator frequency on the fermionic parity.
To see how these interactions arise here we return to the system Hamiltonian, given by Eq. (5), and assume that dominates over the transmon dispersion. We diagonalize to get with , where and operate in the rotated parity () and the dressed transmon () degrees of freedom, respectively; and is the cavity-transmon detuning. We next express with the quantum numbers of the eigenstates () and in the dispersive regime, and for small we expand it in the small parameters , . The resulting diagonal matrix in the basis represents the effective Hamiltonian
The dispersive shift , characteristic of the transmon, is accompanied by an additional higher-order shift , which dispersively couples the photon, transmon, and parity degrees of freedom. Focusing on the sector, we obtain the Hamiltonian for the qubit with the parity-dependent dispersive interaction We realize that the cavity resonance frequency of photon transition is dependent on the parity of the ground state. This interaction, which scales as , can be used in a homodyne measurement setup to determine which of the two ground states the MT occupies Gambetta et al. (2007); Schuster et al. (2005); Johnson et al. (2010); Fragner et al. (2008); Wallraff et al. (2005), opening the way to use this pair of states as a qubit. The signal-to-noise ratio of such a readout scheme depends also on the strength of the decoherence processes, the scale of , and the quality factor of the cavity.
ii.4 Qubit initialization and control
In the presence of a cavity-MT interaction, we denote by and the dressed states of the excitation sector, which are approximate eigenstates of Eq. (5) in the dispersive regime. We drive the qubit via the cavity with , written in terms of the itinerant electric field at the port of the resonator with the drive amplitude . The drive mixes the dressed states with the bare states according to
To first order in , this scaling does not depend on and is similar to the transmon-cavity case. The general dependence on the small parameters and is different, with higher-order terms depending on . Therefore, in the first order, this form of the drive would evidently lead to a qualitatively similar spectroscopic pattern of the transition strengths on .
In addition to the logical qubit , (which lacks direct dipole couplings between its two states), we denote the lowest dressed doublet of the sector as the control doublet [see Fig. 3(c)], forming together a double system (with shared ground states). Due to the small energy splitting between the logical qubit’s levels, the equilibrium thermal state of the system generically mixes the two levels. To prepare the qubit in a pure state, a cooling procedure should initially be performed. This involves an external drive operating at the frequency , where corresponds to transitions. Photonic decay channels exist between the control states and the qubit states, with the decay rates coinciding with the dipole transitions. For transitions that preserve , the decay rates are (neglecting the small energy difference between the two transitions), and for transitions that flip , we take , with as the decay rate associated with the transmon. The process results in the relaxation of the system to the ground state, as depicted in Fig. 3(a).
Following the initialization to the pure state , single-qubit quantum gates can be performed on the qubit. To demonstrate a simple gate operation, we focus here on a population flip, taking . The trick is to use a two-tone microwave photon drive operating in the frequencies and sharing the same detuning , fixed between the control doublet levels. This results in a coherent population transfer between the states of the qubit; see Fig. 3(b). A complete transition is achieved by taking , in which case the probability for the occupation of takes the form cohen1998atom (). The transition is a coherent evolution where a dynamical phase is accumulated, which can be described as a combination of and rotations.
The MT may show a remarkable resilience to major forms of decoherence which affect the transmon. It can be protected from photon-induced dephasing by tuning to the point following each qubit operation. At this point, the effective coupling to the cavity is turned off as , leading to the vanishing of the qubit-photon interaction term in the dispersive Hamiltonian. A similar effect is achieved by increasing the ratio , as seen in Fig. 2. In addition, by operating in the ground-state sector of the transmon, with no discernible direct dipole coupling, the MT is not affected by spontaneous emission or transmon relaxation processes. Future research is needed into the influence of nonequilibrium quasi-particles and topological protection. Finally, recent experimental progress indicates that the global parity has a very long lifetime, exceeding ms Higginbotham et al. (2015). This process will be setting the upper bound for the MT coherence time.
Acknowledgements.K.Y. and E.Gr. acknowledge the support from the Israel Science Foundation (Grant No. 401/12) and the European Unionâs Seventh Framework Programme (FP7/2007-2013) under Grant No. 303742. E.Gi. acknowledges support from EPSRC (Grant No. EP/I026231/1). E.Gr. and E.Gi. acknowledge support from the Royal Society International Exchanges programme, Grant No. IE121282. Details of the data and how to request access are available from the University of Surrey publications repository doi: 10.15126/surreydata.00808295.
We are interested in the transmon regime where the fluctuations of are mostly localized around and the energy has a small deviation from the harmonic-oscillator values: , where and . By inserting this expression into Eq. (1) and rearranging, we get
where . As increases, approaches an integer value.
In deriving the solution, we focus for convenience on the domain , which contains two barriers centered at . To describe tunneling through the barriers, the wave function should be a linear combination of two independent functions: one exponentially increasing and one exponentially decaying. For these, we assume the form , where is the action through the barrier, for which we take the ansatz . By inserting into Eq. (2) and neglecting terms of order , we obtain a first-order equation
which is readily solved to find the two independent solutions
These are valid mainly close to . Close to , we can rewrite Eq. (2) approximately as
where . Equation (5) is the Weber equation, which has two independent solutions and , i.e., the parabolic cylinder functions (following the notation of Abramowitz and Stegun Abramowitz and Stegun (1970)). For positive integer , this equation simply describes the harmonic oscillator.
We can completely satisfy the solution in the domain using the three functions
where and (for simplicity we omit the band index from the basis functions). Here, () is the solution to the left (right) of and is valid in the region close to . In order to impose the boundary condition on Eq. (6), we need to represent the coefficients of as a linear combination of the coefficients of . This is achieved by comparing to and in their common region of validity.
Using an asymptotic approximation for the parabolic cylinder functions Abramowitz and Stegun (1970), for any and we obtain the form
In addition, Eq. (4) takes the approximate form near ,
Using the above, can be expressed approximately using , for . A similar method can be applied in the region , but the approximations in Eq. (7) cannot be used directly since they are valid only for . Instead we use the identities (Abramowitz and Stegun, 1970)
Writing in the region as , we get
Using Eq. (10), we can now also represent using , . By comparing the coefficients of this basis in to the coefficients in , we acquire the connection matrix
where we neglected the wave-function terms containing . The factor is
Next we construct a matrix which represents the boundary condition via the coefficients by using the property of Eq. (4): . Together with the required boundary condition, we obtain
Combining Eq. (12) with Eq. (14) gives us a system of equations for the coefficients, which has a nontrivial solution only when det. Using the approximation [everywhere except in , which we deal with separately below], the condition can be written as . In order to retrieve the value of , we next use the identity Abramowitz and Stegun (1970) to get, expanding to first order in in the denominator,
By plugging this expression into Eq. (13) we obtain the tight-binding-like spectrum with the “tunneling amplitude” defined as
The functions described in Eq. (6) outline the entire solution in the region . Although we only found the coefficients of and , a similar process can be applied to find the connections to . In keeping with the spirit of our approximation, we argue that in cases where the tunneling process between the barriers is negligible, , which need not satisfy the relevant boundary condition, can be used as the wave function of the transmon to a good approximation. Finally, since is a bounded function in the region and the ratio is exponentially small, we can approximate , where is the th harmonic-oscillator wave function. Since the dependence of on appears only as a relative phase between its two coefficients, in this approximation the offset charge is absent from the wave function and the domain of integration can be taken as . Using this wave function, we form the basis for the rest of our analysis.
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