The Multimode Character of Quantum States Released from a Superconducting Cavity

  1. Maryam Khanahmadi,
  2. Mads Middelhede Lund,
  3. Klaus Mølmer,
  4. and Göran Johansson
Quantum state transfer by propagating wave packets of electromagnetic radiation requires tunable couplings between the sending and receiving quantum systems and the propagation channel
or waveguide. The highest fidelity of state transfer in experimental demonstrations so far has been in superconducting circuits. Here, the tunability always comes together with nonlinear interactions, arising from the same Josephson junctions that enable the tunability. The resulting non-linear dynamics correlates the photon number and spatio-temporal degrees of freedom and leads to a multi-mode output state, for any multi-photon state. In this work, we study as a generic example the release of complex quantum states from a superconducting resonator, employing a flux tunable coupler to engineer and control the release process. We quantify the multi-mode character of the output state and discuss how to optimize the fidelity of a quantum state transfer process with this in mind.

Random-access quantum memory using chirped pulse phase encoding

  1. James O'Sullivan,
  2. Oscar W. Kennedy,
  3. Kamanasish Debnath,
  4. Joseph Alexander,
  5. Christoph W. Zollitsch,
  6. Mantas Šimėnas,
  7. Akel Hashim,
  8. Christopher N Thomas,
  9. Stafford Withington,
  10. Irfan Siddiqi,
  11. Klaus Mølmer,
  12. and John J.L. Morton
and quantum information"]processors [arXiv:1109.3743]. As in conventional computing, key attributes of such memories are high storage density and, crucially, random access, or the ability to read from or write to an arbitrarily chosen register. However, achieving such random access with quantum memories [arXiv:1904.09643] in a dense, hardware-efficient manner remains a challenge, for example requiring dedicated cavities per qubit [arXiv:1109.3743] or pulsed field gradients [arXiv:0908.0101]. Here we introduce a protocol using chirped pulses to encode qubits within an ensemble of quantum two-level systems, offering both random access and naturally supporting dynamical decoupling to enhance the memory lifetime. We demonstrate the protocol in the microwave regime using donor spins in silicon coupled to a superconducting cavity, storing up to four multi-photon microwave pulses and retrieving them on-demand up to 2~ms later. A further advantage is the natural suppression of superradiant echo emission, which we show is critical when approaching unit cooperativity. This approach offers the potential for microwave random access quantum memories with lifetimes exceeding seconds [arXiv:1301.6567, arXiv:2005.09275], while the chirped pulse phase encoding could also be applied in the optical regime to enhance quantum repeaters and networks.

Measurement of the topological Chern number by continuous probing of a qubit subject to a slowly varying Hamiltonian

  1. Peng Xu,
  2. Alexander Holm Kiilerich,
  3. Ralf Blattmann,
  4. Yang Yu,
  5. Shi-Liang Zhu,
  6. and Klaus Mølmer
We analyze a measurement scheme that allows determination of the Berry curvature and the topological Chern number of a Hamiltonian with parameters exploring a two-dimensional closed
manifold. Our method uses continuous monitoring of the gradient of the Hamiltonian with respect to one parameter during a single quasi-adiabatic quench of the other. Measurement back-action leads to disturbance of the system dynamics, but we show that this can be compensated by a feedback Hamiltonian. As an example, we analyze the implementation with a superconducting qubit subject to time varying, near resonant microwave fields; equivalent to a spin 1/2 particle in a magnetic field.

Normal modes of a superconducting transmission-line resonator with embedded lumped element circuit components

  1. Henrik Lund Mortensen,
  2. Klaus Mølmer,
  3. and Christian Kraglund Andersen
We present a method to identify the coupled, normal modes of a superconducting transmission-line with an embedded lumped element circuit. We evaluate the effective transmission-line
non-linearities in the case of Kerr-like Josephson interactions in the circuit and in the case where the embedded circuit constitutes a qubit degree of freedom, which is Rabi coupled to the field in the transmission-line. Our theory quantitatively accounts for the very high and positive Kerr non-linearities observed in a recent experiment [M. Reh\’ak et.al., Appl. Phys. Lett. 104, 162604], and we can evaluate the accomplishments of modified versions of the experimental circuit.

Closing a quantum feedback loop inside a cryostat: Autonomous state-preparation and long-time memory of a superconducting qubit

  1. Christian Kraglund Andersen,
  2. Joseph Kerckhoff,
  3. Konrad W. Lehnert,
  4. Benjamin J. Chapman,
  5. and Klaus Mølmer
We propose to use a cryogenic nonlinear resonator for the projective readout, classical memory, and feedback for a superconducting qubit. This approach sidesteps many of the inefficiencies
inherent in two-way communication between temperature stages in typical systems with room temperature controllers, and avoids increasing the cryogenic heat load. This controller may find a broad range of uses in multi-qubit systems, but here we analyze two specific demonstrative cases in single qubit-control. In the first case, the nonlinear controller is used to initialize the qubit in a definite eigenstate. And in the second case, the qubit’s state is read into the controller’s classical memory, where it is stored for an indefinite period of time, and then used to reinstate the measured state after the qubit has decayed. We analyze the properties of this system and we show simulations of the time evolution for the full system dynamics.

Multi-frequency modes in superconducting resonators: Bridging frequency gaps in off-resonant couplings

  1. Christian Kraglund Andersen,
  2. and Klaus Mølmer
A Superconducting Quantum Interference Device (SQUID) inserted in a superconducting waveguide resonator imposes current and voltage boundary conditions that makes it suitable as a tuning
element for the resonator modes. If such a SQUID element is subject to a periodically varying magnetic flux, the resonator modes acquire frequency side bands. In this work we calculate the multi-frequency eigenmodes of resonators coupled to periodically driven SQUIDs and we use the Lagrange formalism to propose a theory for their quantization. The elementary excitations of a multi-frequency mode can couple resonantly to physical systems with different transition frequencies and this makes the resonator an efficient quantum bus for state transfer and coherent quantum operations in hybrid quantum systems. As an example of the application of our multi-frequency modes, we determine their coupling to transmon qubits with different frequencies and we present a bi-chromatic scheme for entanglement and gate operations.

Circuit QED flip-flop memory with all-microwave switching

  1. Christian Kraglund Andersen,
  2. and Klaus Mølmer
Microwave electronics constitutes an area of research aimed primarily towards the use of high-speed components and circuits for communication and sensing, while digital logic is difficult
to implement with all-microwave technologies. We introduce a microwave driven circuit composed of superconducting resonators and qubits which shows a bistable behaviour, and we present a simple mechanism that allows single- or few-photon microwave pulses to work as Set- and Reset-signals that switch the circuit between its stable modes. The resulting system constitutes an ultra-low-energy Set-Reset flip-flop, and we show that its memory lifetime far exceeds the lifetime of states stored in any of its separate components.

Study of a quantized resonator field coupled to a current-biased Josephson junction in circuit QED

  1. Christian Kraglund Andersen,
  2. Gregor Oelsner,
  3. Evgeni Il'ichev,
  4. and Klaus Mølmer
A Lagrangian formalism is used to derive the Hamiltonian for a λ/4-resonator shunted by a current-biased Josephson junction. The eigenstates and the quantum dynamics of the system
are analyzed numerically, and we show that the system can function as an efficient detector of weak incident microwave fields.

Quantum memory for microwave photons in an inhomogeneously broadened spin ensemble

  1. Brian Julsgaard,
  2. Cécile Grezes,
  3. Patrice Bertet,
  4. and Klaus Mølmer
We propose a multi-mode quantum memory protocol able to store the quantum state of the field in a microwave resonator into an ensemble of electronic spins. The stored information is
protected against inhomogeneous broadening of the spin ensemble by spin-echo techniques resulting in memory times orders of magnitude longer than previously achieved. By calculating the evolution of the first and second moments of the spin-cavity system variables for realistic experimental parameters, we show that a memory based on NV center spins in diamond can store a qubit encoded on the |0> and |1> Fock states of the field with 80% fidelity.