Quantum gates in rare-earth-ion doped crystals

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1 Quantum gates in rare-earth-ion doped crystals Atia Amari, Brian Julsgaard Stefan Kröll, Lars Rippe Andreas Walther, Yan Ying Knut och Alice Wallenbergs Stiftelse

2 Outline Rare-earth-ion doped crystals as quantum computer hardware Experimental results Current status and outlook

3 Requirements for quantum computing Coherent two-level systems acting as qubits Possibility to manipulate the qubits individually (single qubit operations) Coupling between any two qubits (two-bit gates) Possibility for reliable read-out of the individual qubits Scalability

4 The rare-earth-ions hyperfine states are used as qubit states Long coherence times of the optical transitions (100 μs to 6 ms) At 4 Kelvin the ground state hyperfine levels have ms-s coherence times and very long lifetimes (~ hours) The duration of a laser π-pulse, ~ 1 μs Ground state with hyperfine splitting exc> Laser driven 4f-4f transitions 0> 1> aux>

5 Requirements for quantum computing Coherent two-level systems acting as qubits Possibility to manipulate the qubits individually (single qubit operations) Coupling between any two qubits (two-bit gates) Possibility for reliable read-out of the individual qubits Scalability

6 4f-4f absorption line from dopant ions in a rare earth doped inorganic crystal ν 1 ν 2 Conceptual picture of crystal with dopant ions exc intensity ν 1 ν 2 Γ inhom Γ hom frequency Narrow homogeneous line-widths (1-10 khz) Large inhomogeneous line-widths (1-200 GHz)

7 Addressing two different qubits in a rare-earth quantum computer exc> exc> ν 1 ν 2 0> 1> aux> 0> 1> aux>

8 Requirements for quantum computing Coherent two-level systems acting as qubits Possibility to manipulate the qubits individually (single qubit operations) Coupling between any two qubits (two-bit gates) Possibility for reliable read-out of the individual qubits Scalability

9 Dipole-dipole interaction 1. Two ions absorbing at different frequencies are located close to each other in the crystal lattice. In a non-centrosymmetric site the ions will have a permanent electric dipole moment and ground and excited state dipole moments can be different ν 1 ν 1 ν 2 +δν 2. One of the ions is excited on its optical transition 3. This change in dipole moment is sensed by the other ion causing its absorption frequency to change.

10 Dipole-dipole interaction strength in rare-earth crystals Approximate numbers Ion distance frequency shift 100 nm 1 line width 10 nm 1000 line widths 1 nm line widths

11 Controlled-NOT quantum gate Control ν 1 e Target ν 2 e ν 1 ν

12 Controlled-NOT quantum gate Control ν 1 e ν 1 δν Target ν 2 e ν 1 ν 2 +δν

13 Requirements for quantum computing Coherent two-level systems acting as qubits Possibility to manipulate the qubits individually (single qubit operations) Coupling between any two qubits (two-bit gates) Possibility for reliable read-out of the individual qubits Scalability

14 All ions in a randomly selected small volume will interact strongly Small volume ions All ions interact strongly, but detecting single ions is difficult

15 Absorption line from dopant ions in a rare earth doped inorganic crystal ν 1 ν 2 Conceptual picture of crystal with dopant ions exc intensity ν 1 ν 2 Γ inhom Γ hom frequency Narrow homogeneous line-widths (1-10 khz) Large inhomogeneous line-widths (1-200 GHz)

16 Challenges with multiple instance approaches All ions in a qubit (i.e. all QC instances) must have identical wave functions Both optical and hyperfine transitions are inhomogeneously broadened, ions in different instances may experience different laser field strengths and couple differently to the field Ion-ion interaction may differ between instances It may be possible to construct pulses which compensate for this but this adds to the operation time and increase decoherence

17 Requirements for quantum computing Coherent two-level systems acting as qubits Possibility to manipulate the qubits individually (single qubit operations) Coupling between any two qubits (two-bit gates) Possibility for reliable read-out of the individual qubits Scalability

18 Qubit distillation Arbitrary volume in the crystal ion absorbing at frequency ν 1 ion absorbing at frequency ν 2

19 Inhomogeneous absorption profile is tailored to create qubit structures Absorption Absorption Frequency Frequency

20 Qubit distillation Arbitrary volume in the crystal Ions interacting strongly enough for mutual control. Potential QC ion absorbing at frequency ν 1 ion absorbing at frequency ν 2

21 Selecting strongly interacting ions 1. Control Target 2. Control Target 0> 1> 0> 1> Frequency 3. Control Target Absorption Absorption 0> 1> 0> 1> Frequency 4. Control Target Absorption Absorption 0> 1> 0> 1> Frequency 0> 1> 0> 1> Frequency

22 Requirements on excitation pulses

23 How to interact with the qubit ions without interacting with ions at nearby absorption frequencies Excitation 1 ν Frequency

24 Excitation with Gaussian π-pulses Transferred population Transferred population Detuning / MHz Detuning / MHz 1 μs pulse 0.25 μs pulse

25 Pulse shapes for coherent transfer of population This work was carried out by Ingela Roos together with Klaus Mølmer Robust quantum computing with composite pulse and coherent population trapping, Phys Rev A69, (2004) Requirements Complete transfer of the peak of ions No excitation of surrounding ions

26 Complex hyperbolic secant pulse real sechyp amplitude envelope tanhyp frequency chirp Rabi frequency / MHz frequency / MHz time / μs time / μs

27 Excitation with complex hyperbolic secant pulse Evolution on the Bloch sphere Transferred population Detuning / MHz Further more, above a certain threshold intensity the operation is insensitive to different ions having different Rabi frequencies

28 We must have same wave function for all QC instances Hyperbolic secant pulses can compensate for Finite channel widths Variation in detunings Field inhomogeneities & ion orientation variations Variations in laser coupling

29 Excitation with complex hyperbolic secant pulse Evolution on the Bloch sphere Transferred population Detuning / MHz The right hand figure illustrates that the ions are driven coherently on the Bloch sphere

30 High fidelity qubit operations require coherent laser systems Rare earth ion coherence times Pr:YSO, optical ~100 μs, qubit ~100 ms Eu:YSO, optical ~1 ms, qubit? Er:YSO, optical (1.5 μm) several ms Dye lasers Commercial, coherence time <1 μs Dortmund system coherence time ~10 μs Lund system coherence time ~100 μs Drift <1 khz/s

31 Photo: Tomas Svensson

32 Photo: Tomas Svensson

33 Phase stabilization against a cavity E Pound - Drever - Hall E LASER ω c Freq ω c -ω m ω c E ω c +ω m λ/4 REFERENCE CAVITY ω c -ω m ω c ω c +ω m ~ ω m = 50MHz PS LP ERROR Frequency f

34 Locking to spectral hole Cavity with fixed transmission linewidth is replaced against a spectral hole with a linewidth that dynamically adapts to the laser linewidth Different optimum modulation index Optimum absorption New locking regimes system locks leading to constant drift

35 Free induction decay Pr:YSO Beat signal between laser and Pr ions Am plitude (m V ) Time (μs)

36 Free induction decay Amplitude (mv) Time (μs) Amplitude (mv) Time (μs)

37 Laser phase drift <5 over 10 μs Coherence time > 100 μs Phase (D eg) Time (μs)

38 Qubit creation

39 Inhomogeneous absorption profile is tailored to create qubit structures Absorption Absorption Frequency Frequency

40 1. Creating a wide pit Pr:Y 2 SiO MHz 4.6 MHz e 17 MHz 10.2 MHz 17.3 MHz 0 1 aux

41 2. Isolating one type of ions Pr:Y 2 SiO 5 e 0 10 MHz 1 17 MHz aux 10 MHz 17 MHz 0 1 aux

42 Experimental data

43 Pit & peak creation Absorption (αl) MHz 4.6 MHz ±5/2 ±3/2 ±1/2 Absorption (αl) Thursday March 1 Absorption (αl) Friday March MHz 17.3 MHz ±1/2 ±3/2 ±5/2 Frequency (MHz)

44 Readout procedure Absorption (αl) Frequency Wednesday March 7 Absorption (αl) Frequency chirp rate 1MHz/μs Time (μs) Frequency (MHz)

45 Experimental set-up λ meter CW dye laser rf Arbitrary waveform generator single mode fibre acousto-optic modulator beam pick-off λ/2- plate sample in cryostat signal gate AOMs reference

46 Absorption (αl) Absorption (αl) State to state transfer Single state-to state transfer efficiency 97.5% 1/2 g 1/2 e 1/2 g 3/2 e Preparation of 1/2 g state Single transfer to 3/2 g state Thursday March 8 3/2 g 1/2 e 3/2 g 3/2 e Absorption (αl) Absorption (αl) /2 g 1/2 e 1/2 g 3/2 e 3/2 g 3/2 e Transfer to 3/2 e state 14 consecutive transfers 3/2 g 1/2 e 3/2 g 3/2 e Frequency (MHz)

47 Selecting strongly interacting ions 1. Control Target 2. Control Target 0> 1> 0> 1> Frequency 3. Control Target Absorption Absorption 0> 1> 0> 1> Frequency 4. Control Target Absorption Absorption 0> 1> 0> 1> Frequency 0> 1> 0> 1> Frequency

48 With Dieter Suter and Robert Klieber in Dortmund Qubit distillation experiment 1. TARGET TARGET CONTROL 2. TARGET TARGET CONTROL Absorption, αl ν-ν 0 / MHz e aux 0 Absorption, αl ν-ν 0 / MHz e aux 0 3. TARGET TARGET CONTROL 4. TARGET TARGET CONTROL Absorption, αl ν-ν 0 / MHz e aux 0 Absorption, αl ν-ν 0 / MHz e aux 0

49 Selecting strongly interacting ions 1. Control Target 2. Control Target 0> 1> 0> 1> Frequency 3. Control Target Absorption Absorption 0> 1> 0> 1> Frequency 4. Control Target Absorption Absorption 0> 1> 0> 1> Frequency 0> 1> 0> 1> Frequency

50 Arbitrary single qubit operations Secant hyperbolic pulses compensate for the inhomogeneous broadening in the qubit for transitions going from one pole to the other pole on the Bloch sphere Thus they are not readily applicable to arbitrary single qubit operations

51 Excitation with complex hyperbolic secant pulse Evolution on the Bloch sphere Transferred population Detuning / MHz Further more, above a certain threshold intensity the operation is insensitive to different ions having different Rabi frequencies

Pulse sequences for arbitrary single qubit operations This work was carried out by Ingela Roos together with Klaus Mølmer Robust quantum computing with composite pulse and coherent

52 Pulse sequences for arbitrary single qubit operations This work was carried out by Ingela Roos together with Klaus Mølmer Robust quantum computing with composite pulse and coherent population trapping, Phys Rev A69, (2004) Requirements Find pulses compensating for instances having different transition frequencies and different ion-field coupling strength

53 Three-level system e> 0> 1> Δ ν 0 ν ) ( 2 ) ( c e t i c e t i c i c i R i R e e ϕ ϕ Ω + Ω + Δ =

54 Three-level system ΩR 0 =ΩR 1 Ω R c e = iδc e + i ΩR ( t) 2 ( iϕ ) 0 iϕ1 e c + e c 0 1 = 0 Coherent trapping ϕ ϕ = φ 1 0 d> = b> = ( 0> -exp(-iφ) 1>) ( 0>+exp(-iφ) 1>) d> is a dark state b> is a bright state

55 Single qubit operations A two-colour sechyp pulse with phases ϕ 0 and ϕ 1 will drive the system into the excited state along route A. A new two-colour sechyp pulse with phases ϕ 0 +π+θ and ϕ 1 +π+θ will drive the system to the bright state along route B. This corresponds to the operation iθ b > e b > Which in the original 0>, 1> basis corresponds to the unitary operation unitary rotation about any axis on the equator of the Bloch sphere representing the qubit x state zbasis e> B θ A y U e iθ / 2 = cos( θ / 2) iφ ie sin( θ / 2) iφ ie sin( θ / 2) cos( θ / 2) b>

56 Rabi frequency selection Ions in different parts of the beam experience different Rabi frequencies 1 Nπ excitation pulse Upper state Lower state Rabi frequency Lower state Rabi frequency

57 Rabi frequency distillation Friday March 9 3π 5π 7π 9π Time (μs) 3π+ 7π + 9π

58 Requirements for quantum computing Coherent two-level systems acting as qubits Possibility to manipulate the qubits individually (single qubit operations) Coupling between any two qubits (two-bit gates) Possibility for reliable read-out of the individual qubits Scalability

59 Qubit distillation Arbitrary volume in the crystal Ions interacting strongly enough for mutual control. Potential QC ion absorbing at frequency ν 1 ion absorbing at frequency ν 2

60 Dipole-dipole interaction δν frequency shift due to the dipole-dipole interaction δν ( Δμ) 3 r Δμ - difference in dipole moment between ground and excited state r distance between interacting ions Fraction of controllable ions in a qubit (η) scale as η Ν(Δμ) 2 N = dopant concentration Higher dopant concentration, or larger difference in dipole moments between ground and excited state would lead to increased probability to find ions that interact 2

61 Scaling Arbitrary volume in the crystal Ions interacting strongly enough for mutual control. Potential QC ion absorbing at frequency ν 1 ion absorbing at frequency ν 2

62 Readout: Single instance with search approach Minimal laser focus Read out ion of different species only read out one with specialised readout ion

63 Single instance with search approach q4 q3 q5 q1 q2 Readout ion q6 q7 q1-qn: Qubit ions

64 Summary Selected results Qubits are initiated in well defined states Qubit state-to-state transfer efficiency is 97.5%, simulations predict 98.5% Qubit distillation >90% Scalable schemes have been developed

65 Outlook Our dye laser system has been frequency stabilised to a few khz linewidth to carry out high fidelity: Single qubit operations Two-qubit gate operations Potential read out ion candidates are investigated

66 Atia Amari Lars Rippe Andreas Walther Yan Ying Former members Brian Julsgaard Mattias Nilsson Nicklas Ohlsson Ingela Roos

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