QUANTUM PARAMETRIC PHENOMENA IN CIRCUIT- QED

Transcription

1 QUANTUM PARAMETRIC PHENOMENA IN CIRCUIT- QED Vitaly Shumeiko Dept of Microtechnology and Nanoscience Chalmers University of Technology, Göteborg Sweden Capri School on Transport in Nanostrucures, 24 April 2017

2 Superconduc:ng electrical circuits Architecture of SC quantum processors = network of Josephson elements (qubits) + resonators (coupling) + resonators (readout) + parametric amplifiers 5- qbit processors (DelJ) qubit Ultra- low T setup coupling 9- qbit processors (UCSB) readout to paramp UCSB Typical qubit characteris6cs Qubit frequencies 2-10 GHz Currents 10 na 1 mka Temperature < 50mK Decoherence 6me > 50 mksec 2

3 Content v Parametric resonance in electrical circuits: linear theory parametric amplification quadrature squeezing v Quantum parametric resonance amplification of quantum noise quantum squeezing BCS analogy Dynamical Casimir Effect v Nonlinear parametric resonance in tunable cavity parametric oscillator photon condensation v Non-degenerate parametric resonance frequency conversion optomechanics 3

4 PARAMETRIC RESONANCE Michael Faraday (1831) was the first to no6ce oscilla6ons of one frequency being excited by forces of double the frequency, in the crispa6ons (ruffled surface waves) observed in a wine glass excited to "sing". Phil. Trans. Royal Soc. (London), vol. 121, pages Melde (1859) generated parametric oscilla6ons in a string by employing a tuning fork to periodically vary the tension at twice the resonance frequency of the string. Ann. Phys. Chem. vol. 109, pages Parametric oscilla6on was first treated as a general phenomenon by Rayleigh ( ). Phil. Mag. vol.24, pages Floquet theory One of the first to apply the concept to electric circuits was George Francis FitzGerald, who in 1892 tried to excite oscilla6ons in an LC circuit by pumping it with a varying inductance provided by a dynamo. Parametric amplifiers (paramps) were first used in for radio telephony from Berlin to Vienna and Moscow, and were predicted to have a useful future. The early paramps: varied inductances, varactor diodes, klystron tubes, Josephson junc:ons.

5 LINEAR PARAMETRIC RESONANCE 5

6 LC oscillator a(t) F(t) L C superconducting phase slow varying complex amplitudes a 2 pic oscillator response 0 Δ 6

7 Parametric LC oscillator F(t) JJ Φ(t) JJ ε << ω 0 δ << ω 0 ω SQUID >> Ω : SQUID is variable inductance Resonance approxima:on 7

8 Parametric instability JJ Φ(t) JJ Instability region ε Instability threshold a = 0 0 δ 8

9 Driven parametric oscillator JJ Φ(t) JJ F(t) idler a ε < γ reduced damping divergence at threshold 0 δ a ε > γ -δ th δ th δ 9

10 Oscillator in environment a Φ(t) b (input = signal + noise) c (output) Interes:ng new aspect: external response of the oscillator c Langevin equa:on total losses external losses Input- output rela:on Walls and Milburn, Quantum Optics Clerk, RMP 82, 1155 (2010)

11 Bogoliubov transforma3on Parametric amplifica:on Signal gain Idler gain Iden33es (no internal losses) 11

12 Parametric amplifica:on loss resonance Signal gain resonance off resonance G S =1 Idler off resonance G I =0 12

13 Quadrature squeezing (Δ = 0) Parametriza:on r = squeezing parameter rotated amplitudes quadratures Y amplified de-amplified b c X 13

14 Two- mode squeezing Δ 0 Amplifica:on of detuned signals 14

15 Quan:za:on v Quantum Noise in electrical circuits, Callen, Welton 1951 v Quantum description of linear electrical circuits, Weber 1953,Widom 1979 Yrke & Denker 1984 Devoret 1995 Quantum electrical circuits? Kirchhoff s equations = lumped element version of Maxwell equations EM field is quantum Quantization rule: [Φ, Q] = iħ Low noise resonators are feasible at microwaves: ω/γ = Q ~ 1000 Low temperature is available: T << ω ( ω ~ 10 GHz T < 100 mk ) Nonlinear circuit is required JJ (Leggett 1981) 15

16 Quan:za:on Quan:za:on is straighsorward in linear theory bosonic input Bogoliubov transformation properties of coefficients guarantee bosonic output 16

17 Crea:on of photons vacuum input contains no real photons spectral density of created photons Parametric amplifica:on in quantum regime appears as crea:on of real photons from vacuum divergence at threshold N(0;δ,ε) Wustmann, PRB 87, (2013) 17

18 Amplifica:on of noise Compare amplifica:on of signal and noise signal noise (vacuum) signal to noise ratio Minimum added noise: quantum limited amplifica:on Clerk, RMP 82, 1155 (2010) 18

19 Squeezing operator BT can be wriuen through squeezing operator Squeezing operator explicitly indicates correla:on of photon pairs 19

20 Squeezed vacuum Output of parametric amplifier = squeezed vacuum is pure state set of photon pairs (photon pairing) Weak squeezing limit: vacuum is a superposition of no photon pairs and one photon pair per mode 20

21 BCS analogy Hamiltonian Bogoliubov transforma:on Bosonic / fermionic commuta:on rela:ons Vacuum state 21

22 Single mode detec:on Two-mode density matrix of squeezed vacuum Detection of one of the modes produces reduced density matrix Reduced density matrix describes the Gibbs state In large squeezing limit (at threshold) T = 22

23 Two- photon entanglement Photon entanglement in squeezed vacuum is quantified with entropy At threshold entanglement increases without bound in linear theory In fact it is bounded due to nonlinear effect Entanglement entropy for different pump detuning δ = 0 23

24 Dynamical Casimir Effect (MW) Φ(t) φ(0,t) b (vacuum modes) c (real photons) Interes:ng new aspect: SQUID directly coupled to TL: no oscillator, no modula6on of resonance Temporal modulation of boundary condition (reflection phase) Ν Thy: Johansson, PRL 103, (2009) Exp: Wilson, Nature 479, 376 (2011) 0 Ω/2 ω

25 Dynamical Casimir Effect (original) b (vacuum modes) c (real photons) Ω DCE: parametric pump - moving mirror Moore, Math. Phys. 11, 2679 (1970); Lambrecht, PRL 77, 615 (1996)

26 Related rela:vis:c effects Unruh radia6on: effec6ve pump accelerated reference frame (horizon) Hawking radia6on: effec6ve pump gravita6onal field of black hole Nation, RMP 84, 1 (2012) Hawking, Nature 248, 30 (1975) 26

27 NONLINEAR PARAMETRIC EFFECTS 27

28 a Φ(t) JJ = Nonlinear element b (input = signal + noise) c (output) Resonance approximation nonlinear frequency shift linear resonance a 2 non-linear resonance a 2 bistability -δ/α 0 δ 0 δ 28

29 Tunable cavity Φ(t) a cavity TL b c Sandberg, APL 92, (2008) Langevin equation Effective Hamiltonian Deriva6on procedure preserves commuta6on rela6on Dykman, Sow. Phys. JETP 50, ; Wustmann, PRB 87, (2013) 29

30 Parametric oscilla:on Classical solu6on ε Α 0 Α = 0 0 δ -δ th Duffing effect δ th δ Basins of attraction A=0 H p q p q q δ/γ = -30 δ/γ = -4 p A 0 Wustmann, PRB 87, (2013) 30

31 Parametric radia:on Parametric radia6on from tunable cavity vs δ and ε Quadrature histograms for one- bi- and tri- stability regions Wilson, PRL 105, (2010) 31

32 Oscillator quantum state Closed cavity: CS are eigen states of parametric oscillator δ = 0 coherent state eigen state Average amplitude = classical amplitude 0 Puri, arxiv:

33 Photonic condensa:on What is the parametric oscillator regime from the condensed mamer physics view point? a 2 noise coherent Below threshold no coherent field, only noise, a = 0 Above threshold coherent, classical field emerges a = A This classical field obeys equa6on that resembles Gross- Pitaevsky equa6on for BEC, and (nonlinear) BdG equa6on for superconductor ε Cri6cal fluctua6ons at threshold Transi:on to oscillator regime = photonic condensa:on, a quantum phase transi6on from disordered phase (noise) to ordered phase (oscilla6on) whose complex amplitude A plays role of order parameter. Physics here is different from BEC: although we are dealing with bosons, two- mode squeezing introduces pair wise correla6on quan6ta6vely similar to those for Cooper pairs. 33

34 Cavity nonlinear response Below threshold Narrowing of resonance and strong nonlinearity close to threshold Above threshold Driving removes degeneracy (tilts metapotential) Wustmann, PRB 87, (2013) 34

35 Nonlinear amplifica:on ε=γ ε=0 B 2 /Γ Max differen6al gain: at threshold for small input ε Γ, B 0 Linear gain diverges, nonlinear gain is bound at threshold Maximum amplifica6on is controlled by parameter Γ/α Same refers to squeezing parameter hence noise squeezing and entanglement Signal and noise are differently amplified and squeezed SNR > 1 35

36 Squeezing ε/γ=0.2 ε/γ=0.7 Nonlinear gain for phase sensi6ve amplifica6on G max G min (α/4γ) 1/2 ( Β 2 /Γ) 1/2 < ε = Γ 36

37 Non- degenerate resonance ω 2 ω 1 Cavity contains many modes Non- equidistant spectrum Two dis:nctly different regimes down-conversion: amplification, oscillation up-conversion: frequency conversion 37

38 Frequency conversion reflection S 11 one cavity Zakka-Bajjani, Nature Phys 7, 599 (2011) transmission S 12 two cavities Abdo, PRL 110, (2013) Wustmann, arxiv:

39 Frequency conversion Linearized Langevin equations Transmission coefficient In absence of internal losses conversion is unitary Full conversion G S,I at Δ = 0, = instability threshold in amplification regime S 12 S 11 Wustmann, arxiv:

40 Frequency conversion: optomechanics Parametric conversion is able to couple vastly different EM frequencies Holy Grail: MW optic quantum convertor for long distance communication SQUID is not suitable for optical frequencies Mechanical oscillator instead of SQUID (DCE) cavity 2 cavity 1 MO Lecocq, PRL 116, (2016) Cavity Optomechanics 2014; Aspelmeyer RMP (2014) pump 1 pump 2 40

41 Summary Parametrically driven EM oscillator is a quantum limited amplifier is capable to squeeze noise creates entangled photon pairs out of vacuum Parametrically squeezed vacuum exhibits two- mode entanglement; single mode detector sees it as black body radia6on The largest parametric effects are at the instability threshold; nonlinearity bounds amplifica6on and squeezing at the threshold Parametric instability indicates dissipa6ve phase transi6on of photonic pairs to parametric oscillator state Non- degenerate parametric resonance allows for frequency conversion Parametric resonance in EM- mechanical hybrids gives solu6on to problem of long distance quantum communica6on between SC qubit clusters 41

42 Reading Quantum circuits B. Yurke and J. S. Denker, Phys. Rev. A 29, 1419 {1984) M. Devoret, in Quantum Fluctuations, Les Houches, 1995 (Elsevier 1997) G. Burkard, R. H. Koch, and D. P. DiVincenzo, Phys. Rev. B 69, (2004) SC qubits and c-qed G. Wendin and V. Shumeiko, Low Temp. Phys (2007) R. J. Schoelkopf and S. M. Girvin, Nature 451, 664 (2008) Parametric oscillator Fluctuating Nonlinear Oscillators, ed. M.I. Dykman (Oxford, 2012). Tunable cavity M. Sandberg, et al., APL 92, (2008) W. Wustmann and V.Shumeiko, PRB 87, (2013) W. Wustmann and V.Shumeiko, arxiv: Quantum noise A.A. Clerk, M.H. Devoret, S.M. Girvin, F. Marquardt, and R.J. Schoelkopf, RMP 82, 1155 (2010) C. M. Caves, PRD 26, 1817 (1982) Quantum optics, squeezing D.F. Walls and G.J. Milburn, Quantum Optics (Springer, 2008) Quantum Squeezing, ed. P.D. Drummond and Z. Fizek (Springer, 2004). 42

43 DCE G. T. Moore, J. Math. Phys. (N.Y.) 11, 2679 (1970). A. Lambrecht, M.-T. Jaekel, and S. Reynaud, Phys. Rev. Lett. 77, 615 (1996) J. R. Johansson, G. Johansson, C. M. Wilson, and F. Nori PRL 103, (2009. C.M. Wilson, et al., Nature 479, 376 (2011). Relativistic effects P.D. Nation, J.R. Johansson, M.P. Blencowe, F. Nori, RMP 84 (2012) S.W. Hawking, Nature 248, 30 (1974) Optomechanics M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, RMP 86, 1391 (2014) 43