# Quantum State Transmission in a Superconducting Charge Qubit-Atom Hybrid

^{1}

Hybrids consisting of macroscopic superconducting circuits and microscopic components, such as atoms and spins, have the potential of transmitting an arbitrary state between different quantum species, leading to the prospective of high-speed operation and long-time storage of quantum information. Here we propose a novel hybrid structure, where a neutral-atom qubit directly interfaces with a superconducting charge qubit, to implement the qubit-state transmission. The highly-excited Rydberg atom located inside the gate capacitor strongly affects the behavior of Cooper pairs in the box while the atom in the ground state hardly interferes with the superconducting device. In addition, the DC Stark shift of the atomic states significantly depends on the charge-qubit states. By means of the standard spectroscopic techniques and sweeping the gate voltage bias, we show how to transfer an arbitrary quantum state from the superconducting device to the atom and vice versa.

A quantum computer makes direct use of qubits to encode information and perform operations on data according to the laws of quantum mechanics Book:Nielsen2000 (). Due to the properties of superposition and entanglement of quantum states, such a computing device is expected to operate exponentially faster than a classical computer for certain problems. Recently, some basic quantum logic gates have been executed on various quantum systems composed of a small number of qubits, for instances, trapped ions Nature:Kielpinski2002 (), neutral atoms PRL:Brennen1999 (), photons RMP:Kok2007 (), NMR Nature:Vandersypen2011 (), and superconducting (SC) circuits RMP:Makhlin2001 (). However, the development of an actual quantum computer is still in its infancy since no quantum system practically fulfills all DiVincenzo criteria PRA:Loss1998 () for the physical implementation of quantum computation.

Hybridizing different quantum systems could inherit the advantages of each component and compensate the weaknesses with each other PRL:Tian2004 (); PRL:Rabl2006 (); RMP:Xiang2013 (). A promising structure is to combine the SC circuits with neutral atoms. Macroscopic solid-state devices including submicrometer-sized Josephson junctions (JJ) possess the advantages of rapid information processing ( ns), flexibility, and scalability. However, due to the strong coupling to the local electromagnetic environment, the relaxation and dephasing times of the SC circuits, which are of the order of s Science:Mooij2003 (); PRL:Nakamura2002 (); PRB:Duty2004 (); PRA:Koch2007 (); PRB:Rigetti2012 (), are significantly limited by the fluctuations in background charge, flux, and critical current Science:Vion2002 () and even the readout back-action PRB:Schreier2008 (). In contrast, the microscopic atomic systems are characterized by precise quantum-state control and long coherence time (), though they own a relatively long gate operation time because of the weak coupling to external fields PRL:Monroe1995 (); Nat:Monroe2002 (); PRL:Isenhower2010 (); RMP:Saffman2010 () and have limited scalability. Transmitting information between these two distinct quantum realizations could lead to the rapid processing and long-term storage of quantum states, where the SC circuits serve as the fast processor while the atoms play the role of memory AnnuRevCondensMatterPhys:Daniilidis2013 (); PNAS:Kurizkia2015 (); PRA:Petrosyan2009 ().

The SC circuits and atoms can be indirectly coupled by integrating both of them on a microwave SC cavity, such as a LC resonator or a coplanar waveguide (CPW) resonator, which acts as a data bus to transfer the quantum information between the atomic memory and the SC processor PRL:Maitre1997 (); RMP:Raimond2001 (); Nature:Wallraff2004 (); Nature:Majer2007 (). However, the large detunings of the off-resonance SC qubit-resonator and atom-resonator interactions significantly weaken the virtual-photon-mediated SC qubit-atom coupling. Moreover, the fluctuation of intraresonator photon number increases the dephasing rate of qubits PRB:Rigetti2012 ().

The atoms can also directly talk to the SC devices via interacting with the local electromagnetic field. The current relevant research mainly focuses on the information transmission between neutral atoms and flux qubits, where the low-lying atomic states couple to the microwave-frequency alternating magnetic field from SC loops RevMexFis:Hoffman2011 (); PRA:Patton2013 (). Although replacing a single atom by an ensemble of atoms can enhance the magnetic intersubsystem coupling by a factor of , the atomic number fluctuation and the interparticle interaction challenge the experimental implementation. These issues may be solved by employing the electric dipole interface between the highly-excited Rydberg atomic states and local electric field from SC devices PRA:Yu2016 ().

Here, we propose a hybrid scheme, where a charge qubit is electrically coupled to an atomic qubit comprised of the ground and Rydberg states. The neutral atom placed inside the gate capacitor acts as the dielectric medium and affects the gate capacitance, resulting in the modulated charge-qubit energy bands. In addition, the local quasi-static electric field strongly depends on the charge-qubit state, leading to different DC Stark shifts of atomic-qubit states. We show that an arbitrary quantum state can be transmitted between these two distinct qubits. The two-qubit controlled-NOT (CNOT) logic gate and single-qubit Hadamard transform, which are necessary to entangle two qubits with different species and induce a -rotation of the control qubit, respectively, in the state-transmission protocol, can be implemented by means of standard spectroscopic techniques and sweeping the gate charge bias. Our state-transmission protocol also provides a potential for transferring the quantum state between or remotely entangling two distant noninteracting SC qubits via the flying-qubit-linked atoms.

RESULTS

Charge qubit-atom Hybrid. We consider a simple SC charge qubit PRL:Nakamura1997 (); PhysScripta:Bouchiat1998 (); Nature:Nakamura1999 (), where a single Cooper pair box (CPB) is connected to a SC reservoir via a JJ with a low self-capacitance (see Fig. 1a). The Cooper pairs can tunnel into or out of the box at a rate of (the magnetic flux quantum and the critical current of JJ). The CPB is biased by a voltage source via a parallel-plate capacitor with the plate area and the interplate separation . The Hamiltonian describing the dynamics of excess Cooper pairs in the box is written as

(1) |

where gives the Coulomb charging energy, is the offset charge, and is the phase drop across the JJ. The operator counts the number of excess Cooper pairs in the box, . Around the charge-degenerate spot (), two lowest charge states and are well separated from others and implement the charge qubit. We have omitted the work done by the gate voltage, whose effect on the system can be neglected.

A Rb atom placed inside the gate capacitor interacts with the internal electric field and plays a role of dielectric medium. The direction of is chosen as the the quantization axis. Two hyperfine ground states and with an energy spacing of 6.8 GHz are applied to form an atomic qubit (see Fig. 1b), where is the total angular momentum quantum number and gives the corresponding projection along the -axis. The qubit-state flipping of the atom is achieved by the resonant Raman transition via the intermediate state. A highly-excited Rydberg state is employed as an auxiliary state to enhance the charge qubit-atom interaction. In comparison with and , the hyperfine splitting of , which is of the order of several MHz PRA:Tauschinsky2013 (), can be neglected. Here denotes the principle quantum number of Rydberg atom. A resonant -laser pulse at 297 nm transfers the atomic component in or completely to .

The Cooper-pair tunneling through the JJ varies the internal electric field with a frequency typically of the order of (the reduced Planck’s constant ). The energy spacings of any electric-dipole transitions associated with are much larger than . Thus, can be treated as quasi-static. In the weak-field limit, the capacitance with the atom in is expressed as , where gives the empty gate capacitance (without the atom), is the vacuum permittivity, denotes the static polarizability of the atom in , and is the volume of homogeneous atomic distribution over the gate capacitor.

According to , the Coulomb energy , the offset charge , and the internal electric field are rewritten as , , and JLowTempPhys:Pekola2012 (), respectively, where the empty charging energy , the empty gate charge bias , , and the field amplitude . The ratio measures the relative variation of the total box capacitance caused by the single atom. A large reduces but enhances . Combining the energy associated with the atom, the system Hamiltonian is given by

(2) |

where denotes the SC-circuit Hamiltonian with the atom in , is the intrinsic atomic energy of , indicates the DC Stark shift of induced by the electric field , and is the atomic projection operator. We restrict ourselves within the Hilbert space spanned by and obtain a hybrid system consisting of a charge qubit ( and ) and an atomic qubit ( and ). The auxiliary Rydberg state enables the strong interface between SC circuit and atom. Diagonalizing gives us the eigenvalues and eigenstates of the hybrid system,

(3) |

with denoting the different energy bands for a given .

For the zero gate voltage , we have and the hybrid-system eigenenergies are analytically derived as

(4) |

and the corresponding eigenstates are given by

(5) |

In the limit of , we are left with , , , and . For the atom in the hyperfine ground states , whose static polarizabilities Hz/(V/cm) Book:Miller2000 () are extremely small, we obtain , , , and , meaning the atom hardly affects the SC circuit. Thus, the energy difference between and approximates to the intrinsic energy spacing , i.e., , while the energy separation between and is shifted away from , i.e., . Similarly, we have and .

Distinguishing the system energy spectrum with the atom in from that associated with , a large is necessary to induce the apparent variations of and compared with the small Josephson energy and the ratio , respectively, as well as a strong DC Stark shift . Thus, the SC circuit should be carefully designed and the Rydberg state needs to be chosen accordingly. As a specification, we list the structure of CPB in Table 1.

When a Rydberg atom is brought into the vicinity of SC circuit, the inhomogeneous stray electric fields originating from the contaminations on the cryogenic surface are particular detrimental to the quantum hybrid system since they cause the unwanted energy-level shifts and destroy the atomic coherence. However, there might be ways to mitigate or circumvent the effects of stray electric fields. It has been shown that the direction of electric field produced by the adsorbates due to the chemisorption or physisorption depends on the material properties PRL:Chan2014 (). In principle, one can envision to pattern the surfaces with two materials which give rise to opposing dipole moments of adsorbates. Furthermore, as experimentally demonstrated in RPA:HermannAvigliano2014 (), the stray fields can be minimized by saturating the adsorbates film. The remaining uniform electric fields could be canceled by applying offset fields.

The extra measures of reducing the effects of stray fields possibly affect the performance of hybrid system in a different manner. One way to estimate the dependence of the coherence time of a qubit on the surface properties is to investigate the surface-dependent change of the -factor of a cavity. Only a few studies have been done so far investigating the superconducting cavity for various materials absorbed to the surface Knobloch:1998zt (). With the knowledge at hand, it is hard to estimate the effect of a physisorbed layer of rubidium or specific protective coatings on the superconducting system. Here we assume the resulting decoherence time of the atomic qubit close to the surface is much longer than that of the charge qubit. In the following, we discuss the quantum-state transfer between two different qubits.

State transmission from atom to SC circuit. Transferring an arbitrary qubit state from the atom to the charge qubit primarily relies on a two-qubit CNOT logic gate, where the state flipping of the charge qubit is conditioned on the atomic-qubit state, and a one-qubit Hadamard gate acting on the atom Book:Nielsen2000 (). For performing the CNOT operation, the polarizability of Rydberg state should be large enough that the atom is strongly coupled with the SC circuit. In addition, the corresponding internal electric field needs to be smaller than the first avoided crossing field of PRA:Sullivan1985 (). Based on the specification of CPB structure listed in Table 1, we choose , whose relevant physical parameters are derived from Thesis:Pritchard (); JPB:Low2012 (); JPB:Branden2010 () and also shown in Table 1.

We first consider the system energy spectrum. Figure 2a illustrates the shifted eigenenergies with versus the empty charge bias around . For and , are nearly same to that of a common charge qubit due to . An avoided energy-level crossing occurs at , where and approach each other with a minimal energy spacing of . The Cooper-pair tunneling takes effect only around the charge-degenerate point within a narrow region Vion2004 (). In contrast, the energy bands move down relative to with due to the enhanced and large DC Stark shift . The minimal separation between and , however, is still determined by the Josephson energy . The position of the corresponding energy-level anticrossing shifts to the left side of because of the enlarged offset charge . At either avoided crossing, the hybrid system stays in the superposition states and with . We also show the expectation values of excess Cooper-pair numbers, , in Fig. 2b and find that two charge-degenerate spots are separated by larger than the ratio , indicating the shifted energy spectra with the atom in and can be well distinguished.

The dependence of the avoided-level crossing on the atomic state allows us to control the charge-qubit transition via preparing the atom in different states. Setting the empty charge bias at , the hybrid system resonantly oscillates between and with a half period (-pulse duration) of ns while the transitions with are strongly suppressed due to the large detuning as shown in Fig. 2c, where the master equation involving the relaxation and dephasing of charge qubit PRA:Boissonneault2009 () is employed. It is seen that the probability of the system switching between and reaches 0.93 at .

The two-qubit CNOT gate, where the atom acts as the control qubit while the charge qubit plays the target role, can be implemented via three steps: (1) Initially, the gate voltage stays at zero, . Two -light pulses (the time duration ) at 297 nm are applied resonantly on the and transitions to transfer the populations in and completely to and , respectively. Thus, different components and are spectroscopically discriminated (Fig. 2a). (2) The empty charge bias nonadiabatically raises to the charge-degenerate point for two adiabatic energy curves associated with and , i.e., . After staying at this sweet spot for the -pulse duration of , decreases back to zero nonadiabatically. As a result, the populations in and switches with each other while that in and do not change. (3) The -light pulses are used again to bring the populations in and back to and , respectively, without affecting the components of and .

In steps (1) and (3), the intensities of the light pulses need to be strong enough to reduce the pulse duration shorter than the decoherence time of the charge qubit. The experimentally feasible light-pulse length can be as small as 1 ns with a corresponding effective Rabi frequency of the order of 1 GHz PRL:Huber2011 (). However, for such a strong atom-light interaction, the small fine-structure splitting between and , i.e., GHz PRA:Li2003 (), affects the atom transfer between and . To suppress the unwanted population in , the -pulse length should be chosen to fit the experimental conditions. According to Fig. 2d, can be set at 1 ns, much shorter than the decoherence time of the charge qubit, with the corresponding Rabi frequency of GHz. The resulting atom-transfer efficiency is over 0.96. Due to the large ground-state hyperfine splitting of GHz, the light pulses hardly affect the components of and .

We numerically simulate the CNOT operation with different and states via applying the master equation PRA:Boissonneault2009 (). The whole gate operation duration is less that 2.5 ns. The resulting register populations are depicted in Fig. 2e based on the specification. It is seen that the quantum logic gate preserves the charge-qubit states when the atom is prepared in , whereas for the atom in the charge-qubit state switches between and with high probabilities. The standard process fidelity PRA:Gilchrist2005 (); PRA:Bongioanni2010 () reaches .

After performing the CNOT gate, the transmission of quantum state from the atom to SC circuit is straightforward. We assume that the atomic qubit is initially in an arbitrary state while the charge qubit is prepared in , leading to the system state (see Fig. 2e). Passing through the CNOT gate, the system state becomes . Then, a single-qubit Hadamard gate acts on the atom and the hybrid system arrives at . Afterwards, we measure the atomic-qubit state and obtain or depending on the readout which triggers an extra Pauli- (phase-flip) gate JPA:Obada2012 () acting on the SC device. As a result, the final charge qubit is in and the quantum-state transmission is accomplished.

The Hadamard gate for atomic qubit can be performed via the pulsed two-photon Raman process with a Raman detuning and a Raman coupling strength (see Fig. 1b). After adiabatically eliminating the state, one obtains an effective light-driven two-state ( and ) system with the detuning and the Rabi frequency . Choosing and the light-pulse length leads to the time evolution operator of the atom . The operator maps the atomic states and onto and , respectively, achieving the Hadamard transformation.

State transmission from SC circuit to atom. Similarly, the protocol for transferring an arbitrary charge-qubit state to the atom relies on a two-qubit CNOT gate, where the atom flips its state conditioned on the charge-qubit state, and a one-qubit Hadamard gate acting on the SC circuit. We first consider the CNOT operation. The gate voltage is set at zero, resulting in . A large atomic polarizability for , which leads to a strong DC Stark shift , is necessary for spectroscopic distinguishing four , , , and transitions. However, the single-qubit Hadamard operation on the charge qubit, which is performed via adiabatically sweeping from 0 to 0.5, requires that the charge-degenerate spots with the atom in different states approximately locate at , i.e., . Hence, should not be very large. As an example, we employ , whose relevant physical parameters are listed in Table 1. The corresponding approximates zero, indicating the very weak effect of the atom in on the SC circuit, and the avoided crossing between and occurs at . Moreover, at the energy spacings of different transitions are , , , and . For our physical specification, the DC Stark shift of reaches GHz. A -laser pulse resonant to the transition switches the atomic state between and and keeps the charge qubit in , but this pulse weakly interacts with the transition due to the detuning , which is applicable to the CNOT operation.

The two-qubit CNOT gate, where the charge qubit plays the control role while the atom acts as the target qubit, can be simply implemented via three steps (see Fig. 3a): (1) Two -light pulses with the pulse duration are applied resonantly on the and transitions to transfer the atomic population in completely to . (2) A -light pulse with the pulse length is employed to resonantly couple to the transition. The atomic state is flipped between and when the charge qubit is in . By contrast, the populations in and are weakly affected. (3) Two -light pulses with the duration are applied again to map the atomic component in back onto . The extra phase acquired in the gate operation can be canceled by the local operations on the atom PRL:Jaksch2000 ().

The fine-structure splitting between and is GHz PRA:Li2003 (). To suppress the influence of on the atom transfer between and in steps (1) and (3), the -light pulse duration is chosen to be ns, much shorter than the decoherence time of the SC circuit, with the corresponding Rabi frequency of GHz (see Fig. 3b). In step (2), the limited frequency difference between two and transitions extends the -pulse length and, hence, the the relaxation and dephasing of charge qubit reduces the fidelity of two-qubit logic gate as shown in Fig. 3b. We set ns to obtain the optimal CNOT truth table (see Fig. 3c). The total gate duration is ns and the resulting process fidelity is .

After performing the CNOT gate, one can transmit an arbitrary charge-qubit state to the atom via the following three steps (see Fig. 3c): (1) CNOT operation: The hybrid system is initially prepared in at . After the CNOT gate, we arrive at the system state . (2) Hadamard transform: The offset charge is increased to 0.5 adiabatically. Two components and in follow the adiabatic energy bands and , respectively. Actually, it is unnecessary to maintain the sweep rate of constantly from 0 to 0.5. Initially, adiabatically raises from 0 at a large rate. When approaches to 0.5, the sweep rate decreases to a relative low value. After the local operations for canceling the extra accumulated phases JPA:Obada2012 (), the system state becomes . (3) Projective measurement: We measure the excess Cooper pairs in the box. If the readout is 0, we conclude the system state , otherwise, . Then, the offset charge is reduced back to 0 rapidly. After an extra Pauli- operation performed on the atom PRL:Jaksch2000 (), we finally obtain the atomic-qubit state . Thus, the quantum-state transmission is accomplished.

DISCUSSION

The SC circuits operate much faster than the atomic systems. Transmitting the atomic-qubit state to the SC circuit allows the rapid quantum gate operations. Nevertheless, these solid-state devices lose the coherence on a short time scale compared with the atomic systems. Transmitting the quantum state from the SC qubits to the atoms allows a long-time storage. To achieve this reversible state-transmission, we have proposed a hybrid structure composed of a charge qubit and an atomic qubit. Placing the atom inside the gate capacitor results in the atomic-state-dependence energy bands of charge qubit and the charge-state-dependence DC Stark shifts of atomic-qubit states. Applying the standard spectroscopy techniques and sweeping the gate charge bias (gate voltage) enable the quantum-state transmission between two different qubits as well as the universal two-qubit quantum gates.

As is known, the strong coupling to the local electromagnetic environment leads to the short relaxation () and dephasing () times of SC circuits. For the common charge qubit discussed in this paper, the excess Cooper pairs in the box lose their coherence after about 10 ns PRL:Nakamura2002 (); PRB:Duty2004 (); PRA:Koch2007 (). Based on our physical specification, the state transmission can be accomplished within the coherence time of common charge qubit. Nonetheless, the relaxation effects of charge qubit still limit the fidelity of two-qubit gate operations.

In our hybrid system, the atom locates close to the gate-capacitor plates. The inhomogeneous Stark effect from the adsorbate electric fields on the cryogenic surface imposes a severe limitation to the coherence of Rydberg states, reducing the fidelity of state transmission PRL:Chan2014 (). However, the effects of stray fields might be circumvented via coating the surfaces with adsorbates RPA:HermannAvigliano2014 (). Moreover, measuring the distribution of stray fields above the chip surface based on Rydberg-electromagnetically-induced transparency PRA:Thiele2015 () provides a potential of canceling the uniform electric fields by offset fields. We expect the resulting coherence time of Rydberg atom much longer than that of charge qubit.

Our scheme for quantum state transmission in a superconducting charge qubit-atom hybrid opens a new prospect for quantum information processing, where the macroscopic SC devices rapidly process the quantum information which can be saved in the long-term storage composed of a microscopic atomic system. The protocols established in this paper also allow the information transfer between two distant SC qubits. After transmitting the quantum state of a SC qubit to a local atom, the quantum information encoded in this atom can be further transferred to another remote atom via a traveling qubit (photon) Nature:Moehring2007 (); Science:Hofmann2012 (); Nature:Bernien2013 (). Subsequently, the quantum state is transmitted to another distant SC interacting with the remote atom. Moreover, transmitting the quantum states of two entangled atoms to two distant SC qubits, respectively, results in a pair of remotely-entangled SC qubits.

Acknowledgments: This research is supported by the National Research Foundation Singapore under its Competitive Research Programme (CRP Award No. NRF-CRP12-2013-03) and the Centre for Quantum Technologies, Singapore.

Author Contributions: R.D. envisaged the concept of physical model. D.Y. did the calculation and analysis. D.Y. and M.M.V. provided the parameters of suitable Rydberg states. C.H. contributed the experimental realizable parameters. L.C.K., L.A., D.Y., and R.D. contributed the conceptual approach for analyzing the presented system. All authors participated in discussions and writing of the text.

Competing financial interests: The authors declare no competing financial interest.

Charge-Qubit Structure | ||

Physical parameters | Symbol | Value |

Self-capacitance of JJ (aF) | 30 | |

Josephson coupling energy (GHz) | 1.2 | |

Critical current (nA) | 2.4 | |

Empty gate capacitance (aF) | 265.6 | |

Plate area (m) | ||

Interplate distance (m) | 1.2 | |

Empty charging energy (GHz) | 262.1 | |

Electric field amplitude (V/cm) | 9.0 | |

Relaxation time (ns) | 100 | |

Dephasing time (ns) | 10 | |

Atomic Parameters of | ||

Physical parameters | Value | Value |

Principle quantum number | 43 | 28 |

Polarizability [MHz/(V/cm)] | 98.3 | 4.0 |

Ratio | 0.015 | 0 |

Lifetime (s) | 159.3 | 39.5 |

Orbit diameter () | 3255.43 | 1284.7 |

### Footnotes

- thanks: Correspondence and requests for material should be addressed to R.D. (email: rdumke@ntu.edu.sg).

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