Simulating relativistic physics in superconducting · PDF fileSimulating relativistic physics in superconducting circuits! ... G. Johansson, P. Delsing, E ... Joel Lindkvist,

Simulating relativistic physics in

superconducting circuits !

Advanced many-body and statistical methods

in mesoscopic systems Brasov, 1-5th of September 2014

Göran Johansson Chalmers University of Technology

Gothenburg, Sweden

Applied Quantum Physics Laboratory:!Theoretical Physics - Solid State Physics, Graphene,

Mesoscopic Physics, Quantum Information, Superconductivity

Close collaboration with experimentalists in e.g.!Quantum Device Physics Laboratory:

Per Delsing

Quantum Mechanics and Electrical Circuits

An LC-oscillator in the microwave regime

A QM harmonic oscillator:!- Quantized Amplitudes!- Vacuum Fluctuations

f=5 GHz --> hf / kB = 240 mKLow temperatures needed!

(300 K --> 6.3 THz)


Low temperatures – also with microwave equipment installed

Resistance/dissipation gives level broadening ->!

Minimize dissipation!An LC-oscillator in the microwave regime



Nonlinearity needed for quantum effects in average quantities.

Low temperatures – also with microwave equipment installed

Resistance/dissipation gives level broadening ->!

Minimize dissipation!An LC-oscillator in the microwave regime


The Josephson Junction

S I S

'1 '2

- Tunnel junction between superconductors!

- Current determined by phase difference of wave function on each side

Josephson Junction:I = I0 sin ('2 � '1) = I0 sin

✓2⇡

�

�0

◆


- A nonlinear (almost) dissipationless inductor

V = L I

�0 =h

2e

Inductor:

S I S

'1 '2

I =1L

ZV dt0 =

�L




✓2⇡

�

�0

◆

I0


- A nonlinear (almost) dissipationless inductor

V = L I

�0 =h

2e

Inductor:

S I S

'1 '2

I =1L

ZV dt0 =

�L




✓2⇡

�

�0

◆

Josephson Inductance:

I ⇡ I02⇡�

�0) LJ =

�0

I02⇡

� ⌧ �0

I0

The SQUID - a Tunable InductanceThe Superconducting Quantum Interference Device: a tunable Josephson junction

The external flux forces a circulating current. Effectively reducing the critical current through the SQUID.

'ext

=2⇡�

ext

�0

SQUID inductance:

A tunable dissipationless inductance

LJ =

�0

2⇡

1

2I0��cos

'ext

2

��

Microwave Transmission line1D open space: Coplanar waveguide (a squashed coaxial cable)

Transmission line - 1 D massless Klein-gordon

equation

H =X

n

q2n2�xC0

+(�n � �n�1)

2

2�xL0

@

2�(x, t)

@t

2� 1

L0C0

@

2�(x, t)

@x

2= 0

�x ! 0

Quantum Network Analysis:!Wallquist et al, PRB 2006 Yurke and Denker PRA 1984 Devoret, Les Houches 1997

Specify in-field and calculate the out-field.

A Transmission Line with tunable boundary (SQUID)

SQUID Boundary ConditionC

@

2�(0, t)

@t

2+

�(0, t)

LJ(t)+

1

L0

@�(0, t)

@x

= 0

Fixed EJ - The effective length of the SQUID

Effective velocity:

k!Le↵ < 1

Mapping to length works for:

ve↵ = �Le↵!d

�Le↵ < Le↵

k!dLe↵ < 1 , ve↵ < c

Fixed EJ - The effective length of the SQUID

Effective velocity:

k!Le↵ < 1

Mapping to length works for:

ve↵ = �Le↵!d

�Le↵ < Le↵

k!dLe↵ < 1 , ve↵ < c

Harmonic Drive

Small Amplitude Drive:

Harmonic DriveHarmonic Drive

Large Amplitude Drive: solve for cn numerically

1.0

0.5

0.0

-0.5

-1.0

Volta

ge

Length

Changing Position

-1.0

-0.5

0.0

0.5

1.0

Volta

ge

Length

Changing Inductance

Changing position and changing impedance are equivalent for the EM mode. Effective velocity: 1mm x 10 GHz = 3% c0 -> Relativistic Effects

Tunable electrical length

1.0

0.5

0.0

-0.5

-1.0

Volta

ge

Length

Changing Position

-1.0

-0.5

0.0

0.5

1.0

Volta

ge

Length

Changing Inductance

Changing position and changing impedance are equivalent for the EM mode. Effective velocity: 1mm x 10 GHz = 3% c0 -> Relativistic Effects

Tunable electrical length

The first report on experimental observation of the dynamical Casimir effect

C.M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J.R. Johansson, T. Duty, F. Nori & P. Delsing, Nature 479, 376-379 (2011)

http://www.nature.com/nature/journal/v479/n7373/full/nature10561.html

The (static) Casimir effect


Quantum field theory (QED): Different density of states outside/between the mirrors




Theory by Casimir (1948), later experimentally verified


Single Oscillating MirrorMoore (1970), Fulling-Davies (1975)

Recent review: Dodonov (2010)

The dynamical Casimir effect

Single Oscillating MirrorMoore (1970), Fulling-Davies (1975)

Recent review: Dodonov (2010)

The dynamical Casimir effect

Related to Hawking radiation and the Unruh effect.

No experimental confirmation - for 40 years

Lambrecht, Jaekel, Reynaud, PRL (1996)

�/⌦

Out

put p

ower

Broadband photon spectrum

21

Overview of the dynamical Casimir effect

Examples of DCE photon production rates for some naïve systems

Lambrecht et al., PRL 1996.

Photon production rate:

Case Frequency (Hz)

Amplitude (m)

Maximum velocity (m/s)

Photon production rate (# photons / s)

moving a mirror by hand

“handwaving”

1 1 2 ~2e-18

nanomechanical oscillator

1e+9 1e-9 2 ~2e-9

21

Overview of the dynamical Casimir effect

Examples of DCE photon production rates for some naïve systems



Case Frequency (Hz)

Amplitude (m)




“handwaving”

1 1 2 ~2e-18


1e+9 1e-9 2 ~2e-9

The very low photon-production rate makes the DCE very difficult to detect experimentally in systems with mechanical modulation of the boundary condition.

What do we mean by a mirror?

It is almost impossible to move a massive mirror close to speed of light.

C. Braggio et al., Europhys Lett. 70, 754 (2005).

By "ideal mirrors" we mean mirrors which are perfectly conducting and whose effects may, therefore, be described by means of appropriate

boundary conditions on the electromagnetic field at the ends of the cavity.

The dynamical Casimir effect in a coplanar waveguide



Case Frequency

(Hz)

Amplitude

(m)




1 1 2 ~2e-18


1e+9 1e-9 2 ~2e-9

SQUID in coplanar waveguide

18e+9 ~1e-4 ~4e6 ~2e5

THE DILUTION FRIDGE

The variable length line



• Drive boundary condition at ~10 GHz!• Starting from vacuum, see broad band photon flux

increasing with effective velocity

25

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10080604020Pump Power [pW]

Data Theory

THE DCE

MIRROR

E-E

Vacuum fluctuations = Virtual photon bubbles

More properties of DCE photons

DCE = Broken bubbles

MIRROR

More properties of DCE photons

photon Symmetry

photon Symmetry

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Photon Flux D

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MIRROR

photon Symmetry

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Digitizer Detuning [MHz]

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quantitative Symmetry

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350-500 MHz Negative Positive

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650-850 MHz Negative Positive

E-mail from Moore 2011:

Dear Dr. Wilson, I would like to congratulate you and your co-‐workers for your experimental observa>on of the dynamical Casimir effect. This was a great surprise so many years aCer my 1969 Ph.D. disserta>on. Regards, Gerald T. Moore Air Force Research Laboratory

Observation of the Dynamical Casimir Effect in a

Superconducting CircuitC.M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J.R. Johansson, T. Duty, F. Nori & P. Delsing, Nature 479, 376-379 (2011)

Chris Wilson Tim DutyMichaël SimoenArsalan

Pourkabirian Robert Johansson

Franco Nori

Per DelsingGöran Johansson


Observation of the Dynamical Casimir Effect in a

Superconducting CircuitC.M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J.R. Johansson, T. Duty, F. Nori & P. Delsing, Nature 479, 376-379 (2011)

Chris Wilson Tim DutyMichaël SimoenArsalan

Pourkabirian Robert Johansson

Franco Nori

Per DelsingGöran Johansson


More relativistic physics in superconducting circuits (by workshop participants)

• ”Analogue Hawking Radiation in a dc-SQUID Array Transmission Line”, P. D. Nation, M. P. Blencowe, A. J. Rimberg, E. Buks, Phys. Rev. Lett. 103, 087004 (2009) !

• ”Dynamics of entanglement via propagating microwave photons”, C. Sabin, J. J. Garcia-Ripoll, E. Solano, J. Leon, Phys. Rev. B 81, 184501 (2010)!

• ”Photon production from the vacuum close to the super-radiant transition: When Casimir meets Kibble-Zurek”, G. Vacanti, S. Pugnetti, N. Didier, M. Paternostro, G. Massimo Palma, R. Fazio, V. Vedral, Phys. Rev. Lett. 108, 093603 (2012)!

• ”Colloquium: Stimulating uncertainty: Amplifying the quantum vacuum with superconducting circuits”, P. D. Nation, J. R. Johansson, M. P. Blencowe, F. Nori, Rev. Mod. Phys. 84 (2012)!

• ”Dynamical Casimir effect in a Josephson metamaterial”, P. Lähteenmäkia, G. S. Paraoanu, J. Hassel, P. J. Hakonen, PNAS 110, 4234 (2013) varying refractive index in a cavity!

• "Dynamical Casimir effect entangles artificial atoms", S. Felicetti, M. Sanz, L. Lamata, G. Romero, G. Johansson, P. Delsing, E. Solano, Phys. Rev. Lett. 113, 093602 (2014)

Incomplete!

The twin paradox

x

t

P. Langevin, Scientia 10 31 (1911)

The traveling clock will measure a smaller elapsed time between events A and B

A. Einstein, Annalen der Physik, 17 (1905)

A

B

The twin paradox

P. Langevin, Scientia 10 31 (1911)

The accelerated clock will measure a smaller elapsed time between events A and B

A. Einstein, Annalen der Physik, 17 (1905)

x

t

some experimental verifications of time dilationMeasuring cosmic radiation at different altitudes. Slow particles decay faster. B. Rossi, D. B. Hall, Phys. Rev. 59, 223 (1941)

Measuring decay times of relativistic muons in a CERN storage ring. Increase from 2.2 to 64 microseconds. J. Bailey et al., Nature 268, 301 (1977)

Flying atomic clocks eastward and westward around the world. Eastward lost 59 ns, while westward gained 273 ns compared to a static clock. J.C. Hafele and R.E. Keating, Science 177, 166-170 (1972)

Measuring gravitational time dilation with 1m height difference and time dilation due to 10 m/s velocity difference. C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland, Science 329, 1630 (2010)

Simulating a relativistically moving cavity

See also: ”Teleportation in motion with superconducting microwave circuits”, N. Friis, A. Lee, K. Truong, C. Sabín, E. Solano, G. Johansson, I. Fuentes, Phys. Rev. Lett. 110, 113602 (2013).

Rob’s cavity in the lab frame

x

t

Alice’s cavity

Rob’s cavity

Joel Lindkvist, C. Sabín, Ida-Maria Svensson, A. Dragan, P. Delsing, I. Fuentes, G. JohanssonarXiv:1401.0129

Rob’s cavity in the lab frame

x

t

Alice’s cavity

Rob’s cavity


The twin paradox on chip

L

⌘ = const.

x

tAlice and Rob are given identical rigid cavities of rest length L Moving cavities

�✓ = !�⌧

A coherent state as a clock

✓

Experimentally realizable!

Joel Lindkvist, C. Sabín, Ida-Maria Svensson, A. Dragan, P. Delsing, I. Fuentes, G. Johansson

arXiv:1401.0129

Alice’s cavity

Rob’s cavity

Theoretical framework

bm =X

n

�↵⇤mnan � �⇤

mna†n

�

Determining the phase shift in Rob’s cavity

The modes in the cavity before and after the trip are related by the Bogoliubov transformation

h ⌘ aL/c2↵mn �mnand are computed to second order in

✓ tan ✓ =�Im(↵11 � �11)

Re(↵11 � �11)The phase shift is given by

⌧ = ✓/!Elapsed proper time:

D. E. Bruschi, I. Fuentes, J. Louko, Phys. Rev. D (2012)

Includes effects of mode-mixing and particle creation.

Large phase shift due to time dilation

⌘ = const.


1.1 cm cavity, 4 ns roundtrip, 500 repetitions, maximal velocity 1.4 % of c

Tidal effects - Large clocks tick slower


Measurable difference between point like and extended clock. a=1.7 x 1015 m/s2

Collaborators on Twin Paradox

Chalmers Theory J. Lindkvist G. Johansson !Chalmers Experiment I.-M. Svensson, P. Delsing !Nottingham University C. Sabín, I. Fuentes, !Warsaw University A. Dragan

Summary

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Thank you for your attention!

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Documents

Simulating relativistic physics in superconducting · PDF fileSimulating relativistic physics in superconducting circuits! ... G. Johansson, P. Delsing, E ... Joel Lindkvist,