Introduction to Circuit QED Lecture 2

Transcription

1 Departments of Physics and Applied Physics, Yale University Experiment Michel Devoret Luigi Frunzio Rob Schoelkopf Andrei Petrenko Nissim Ofek Reinier Heeres Philip Reinhold Yehan Liu Zaki Leghtas Brian Vlastakis +.. Introduction to Circuit QED Lecture 2 Theory SMG Liang Jiang Leonid Glazman M. Mirrahimi Marios Michael Victor Albert Richard Brierley Claudia De Grandi Zaki Leghtas Juha Salmilehto Matti Silveri Uri Vool Huaixui Zheng Yaxing Zhang +..

2 Reminder from Lecture 1: Measuring Photon Number Parity - use quantized light shift of qubit frequency q 2aa 2 z e z i2nt ˆ inˆ 2 e 2 z z ˆ 1, 3, 5,... n x ˆ 0, 2, 4,... n 2

3 Lecture 2: Quantum State Manipulation and Measurement in Circuit QED The ability to measure photon number parity without measuring photon number is an incredibly powerful tool. Quantum Optics at the Single Photon Level Measuring Wigner Functions Creating and Verifying Schrödinger Cat States Cat in Two Boxes 3

4 Quantum optics at the single photon level Photon state engineering Goal: arbitrary photon Fock state superpositions a 0 a 1 a 2 a Use the coupling between the cavity (harmonic oscillator) and the two-level qubit (anharmonic oscillator) to achieve this goal. Dispersively coupled cavity-qubit system is fully controllable. 4

5 Previous State of the Art for Complex Oscillator States Expt l. Wigner tomography: Leibfried et al., 1996 ion traps (NIST Wineland group) Rydberg atom cavity QED Phase qubit circuit QED Haroche/Raimond, 2008 Rydberg (ENS) Hofheinz et al., 2009 (UCSB Martinis/Cleland) ~ 10 photons ~ 10 photons Q 5

6 What concepts do we need to know to understand a Schrödinger Cat State? 6

7 Photons in First Quantization 7

8 Coherent state is closest thing to a classical sinusoidal RF signal ( ) ( ) 0 8

9 (normalization is only approximate) 9

10 even odd (normalization approx. only) Novel property: How cats die: a a a a even odd odd even 2n (4 n) 10

11 How do we create a cat? Classical signal generators only displace the vacuum and create coherent states. We need some non-linear coupling to the cavity via a qubit. 11

12 Strong Dispersive Hamiltonian q z z H ra a a ahdamping 2 resonator qubit dispersive coupling, cavity frequency r z g strong-dispersive limit e 2 ~ r r 12

13 Strong Dispersive Limit yields a powerful toolbox g e Cavity frequency depends on qubit state r r Microwave pulse at this frequency excites cavity only if qubit is in ground state Microwave pulse at this frequency excites cavity only if qubit is in excited state g D Engineer s tool #1: Conditional displacement of cavity 13

14 n Engineer s tool #2: Conditional flip of qubit if exactly n photons 2 q z z H ra a a a Hdamping resonator qubit dispersive coupling Reinterpret dispersive term: - quantized light shift of qubit frequency q 2aa 2 z 14

15 - quantized light shift of qubit frequency (coherent microwave state) q 2aa 2 z N.B. power broadened 100X

16 strong dispersive coupling I V DISPERSIVE z a a Qubit Spectroscopy n 2 n 1 n 0 Coherent state in the cavity 2 Conditional bit flip n 16

17 Strong Dispersive Coupling Gives Powerful Tool Set Cavity conditioned bit flip n Qubit-conditioned cavity displacement g D multi-qubit geometric entangling phase gates (Paik et al.) Schrödinger cats are now easy (Kirchmair et al.) Photon Schrödinger cats on demand experiment theory G. Kirchmair M. Mirrahimi B. Vlastakis Z. Leghtas A. Petrenko 17

18 Deterministic Cat State Production Vlastakis et al. Science 342, 607 (2013) 1 g 2 Will skip over details of cat state production; Focus on proving the cat is not an incoherent mixture: - measure photon number parity in the cat - measure the Wigner function (phase space distribution of cat) 18

19 Proving phase coherence via photon number distribution Coherent state: Mean photon number: 4 Even parity cat state: 2 Readout signal 10 Photon number Only photon numbers: 0, 2, 4, ˆP Odd parity cat state: Only photon numbers: 1, 3, 5, ˆP Spectroscopy frequency (GHz) Qubit Spectrum 19

20 Probability (%) Probability (%) ODD CAT Number of parity jumps ODD CAT 12 Probability (%) Probability (%) EVEN CAT Number of parity jumps EVEN CAT Number of parity jumps Number of parity jumps 20

21 We have proven our states have the correct parity and photon number distribution. We have not (strictly) verified all the phases are correct. Need full state tomography via measurement of the Wigner Function. 21

22 Wigner Function Measurement Vlastakis, Kirchmair, et al., Science (2013) Density Matrix: * (, ) ( ) ( ) Define center of mass and relative coordinates:, r 2 2 Wigner Function (definition): iqr r r W ( Q, ) dre (, ) 2 2 Combines position and momentum information by Fourier transforming relative coordinate 22

23 Wigner Function = Displaced Parity Vlastakis, Kirchmair, et al., Science (2013) Simple Recipe: 1. Apply microwave tone to displace oscillator in phase space. 2. Measure mean parity. Handy identity (Luterbach and Davidovitch): W( ) D( ) Pˆ D( ) ˆ ˆ N ( 1) parity P Full state tomography on large dimensional Hilbert space can be done very simply over a single input-output wire. 23

24 Wigner Function of a Coherent State Im Q 4 Re W ( ) D( ) Pˆ D( ) Nˆ P ˆ ( 1) parity 24

25 Wigner Function of a Coherent State Im Q 4 Re W ( ) D( ) Pˆ D( ) Nˆ P ˆ ( 1) parity 25

26 Wigner Function of a Cat State Vlastakis, Kirchmair, et al., Science (2013) Interference fringes prove cat is coherent: Im 0 Q Re Rapid parity oscillations With small displacements 26

27 Deterministic Cat State Production Vlastakis, Kirchmair, et al., Science (2013) Data! 0 Expt l Wigner function 4 27

28 Deterministic Cat State Production Vlastakis, Kirchmair, et al., Science (2013) Data! 0 Expt l Wigner function photons 32.0 photons 38.5 photons 111 photons 111 photons determined by fringe frequency Most macroscopic superposition ever created? 28

29 Deterministic Photon Cat Production Vlastakis, Kirchmair, et al., Science (2013) Three-component cat: Four-component cat: 0 Zurek compass state for sub-heisenberg metrology photons 32.0 photons 38.5 photons 111 photons 111 photons determined by fringe frequency 29

30 Non-Deterministic Cat State Production Using Parity Measurement 30

31 Cat State = Coherent State Projected onto Parity L. Sun et al., Nature (July 2014) even odd x x x time evolve to entangle spin with cat states: e z i2nt ˆ inˆ 2 2 e z x even x odd

32 Wigner Tomography of cats entangled with qubit L. Sun et al., Nature (July 2014) x even x odd 2 2 Wigner function of cavity (tracing out qubit) yields an incoherent MIXTURE of two coherent states and not a cat. (no fringes) Equivalently: mixture of even and odd cats. 32

33 Wigner Tomography Conditioned on Qubit State L. Sun et al., Nature (July 2014) qubit is in +x> Fidelity of produced cats: qubit is in -x> 33

34 Cat In Two Boxes 34

35 Cat in Two Boxes Qubit measures joint parity! P PP e i ( nˆ nˆ ) 1 2 Theoretical proposal by Paris group: Eur. Phys. J. D 32, (2005)

36 Cat in Two Boxes Experiment by Yale group: Science 352, 1087 (2016) Qubit measures joint parity! P PP e i ( nˆ nˆ ) Universal controllability - 3-level qubit can measure P, P, and P

37 Cat in Two Boxes 37

38 Theory 1 2 Experiment Two-cavities: 4-dimensional phase space and Wigner functions. 38

39 Entanglement of Two Logical Cat-Qubits CHSH: (evaluate Wigner at 4 points in 4D phase space) CHSH Bell: 2 B 2 2

40 Summary of Lecture 2: The ability to measure photon number parity without measuring photon number is an incredibly powerful tool. Lecture 2: Using parity measurements for: Wigner Function Measurements Creation and verification of photon cat states Lecture 3: Using parity measurements for: Continuous variable quantum error correction 40

41 For separate discussion offline: Detailed Recipe to Make a 1. Schrödinger Cat 2. Schrödinger Cat State 41

42 Strong Dispersive Coupling Gives Powerful Tool Set Cavity conditioned bit flip n Qubit-conditioned cavity displacement g D multi-qubit geometric entangling phase gates (Paik et al.) Schrödinger cats are now easy (Kirchmair et al.) Photon Schrödinger cats on demand experiment theory G. Kirchmair M. Mirrahimi B. Vlastakis Z. Leghtas A. Petrenko 42

43 Making a cat: the experiment Q cavity M qubit P (*fine print for the experts: this is the Husimi Q function not Wigner) 43

44 Making a cat: the experiment Q cavity M qubit P 44

45 Making a cat: the experiment Q cavity M qubit P 45

46 Making a cat: the experiment Q cavity M qubit P 46

47 Making a cat: Q cavity M qubit after time: qubit acquires phase per photon P t 47

48 Making a cat: Qubit fully entangled with cavity cat is dead; poison bottle open cat is alive; poison bottle closed Q cavity M qubit after time: qubit acquires phase per photon P t 48

49 We have a cat 1 g e 2 We want a cat state 1 2 g Qubit in ground state; cavity in photon cat state How do we disentangle the qubit from the cavity? 49

50 Combining conditional cavity displacements with conditional qubit flips, one can disentangle the qubit from the photons cat 1 g e 2 1 D g 2 e

51 Combining conditional cavity displacements with conditional qubit flips, one can disentangle the qubit from the photons cat 1 g e 2 0 D 1 g 2 g

52 Combining conditional cavity displacements with conditional qubit flips, one can disentangle the qubit from the photons cat 1 g e 2 D 0 D 1 g g 2 52

53 Combining conditional cavity displacements with conditional qubit flips, one can disentangle the qubit from the photons cat 1 g e 2 D g 0 D cat state 1 g 2 53