Quantum phase transitions in a chain of fluxonium...
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antum phase transitions in a chain of fluxonium qubits Departments of Physics and Applied Physics, Yale University R. T. Brierley , Hendrik Meier, A. Kou, S. M. Girvin, L. I. Glazman
Transcript of Quantum phase transitions in a chain of fluxonium...
Quantum phase transitions in a chain of fluxonium qubits
Departments of Physics and Applied Physics, Yale University
R. T. Brierley, Hendrik Meier, A. Kou, S. M. Girvin, L. I. Glazman
● Introduction to superconducting qubits Transmon Fluxonium
● Coupled fluxonium chain
● Conclusions
overview
superconducting qubits
● Qubits made from superconducting circuits
● Rapid progress in recent years
● Testing fundamentals of quantum mechanics and measurement
Devoret & Schoelkopf, Science (2013)
superconducting circuits
Quantum LC oscillator
● Useful to describe in terms of phase
● Conjugate to Cooper pair number
● Related to flux in inductor
Girvin, Les Houches (2011)
superconducting circuits
Quantum LC oscillator
● Ladder operators
● Need something nonlinear to make a qubit
● Quantum fluctuations in suppressed by capacitance
Girvin, Les Houches (2011)
superconducting qubits
Transmon qubit
Use a Josephson junction:
Small fluctuations →weakly nonlinear oscillator
Koch et. al., PRA (2007)
superconducting qubits
Add an inductor
To have multiple minima, need
But large inductors introduce extra parasitic → suppresses quantum fluctuations
Girvin, Les Houches (2011)
superconducting qubitsFluxonium qubit
Manucharyan et al., Science (2009)Pop et al., Nature (2014)
5 μm
“Superinductor” made from JJs in series
superconducting qubits
Can couple qubits to cavities or superconducting resonators
Effective Jaynes-Cummings interaction
Can reach strong coupling → “circuit QED”Sensitive probe of qubit states
350 μm
5 μm
Paik et. al., PRL (2011)
Blais et. al., PRA (2004)
Girvin, Les Houches (2011)
quantum simulation
● Large-scale systems being made
● Opportunity to study many-body quantum models in a new environment
● Drive and dissipation → new physics
● What kinds of models can be easily realized?
Houck et. al., Nature Physics (2012)
chain of fluxonium qubits
external flux
chain of fluxonium qubits
● Hard to find conjugate momenta for
● Introduces long range interactions for
classical ground state
Marchand, Hood, Caillé, PRL (1987)Yokoi, Tang, Chou, PRB (1988)
Studies on similar models
Two classical parameters
external flux
classical ground state
higher order fractions
classical ground state
Two regimes:
classical ground state
Two regimes:
kinks
● everywhere
● kept small
may jump by , elsewhere satisfies (continuum limit):
Stable states for small :
kinks● Solutions take the form of kinks
● Single kink cost:
Kardar, PRB (1986)
kinks● Solutions take the form of kinks
● Single kink cost:
Kardar, PRB (1986)
ground state at some
ground state close above
many kinks
Repulsive kink-kink interaction
Overlapping exponential tails
Ground states for larger are superpositions of many kinks
ground state at some
ground state close above
many kinks
kink at link
no kink at link
Introduce pseudo-spin
long-range Ising chain
Bak, Bruinsma, PRL (1982)Aubry, J. Phys. Lett. (1983)
Effective classical Hamiltonian
long-rangeinteraction
eff. magnetic field
Frenkel-Kontorova-type model
soliton ground states
● When kinks are sparse, continuum calculation gives
● Kinks live on discrete lattice, can't have ideal separation● e.g. for
devil's staircase
step width:
Kink density vs. magnetic fluxBak & Bruinsma, PRL (1982)
excitation spectrum
Adding an antenna
● Couples to phase across single link
● Absorption of photon cavity
Creation / annihilationof local kinks
excitation spectrum
single-kinkexcitations
kink creation
kink annihilation
quantum effects
Quantum fluctuations
Quantum long-range Ising model
overlap (c.f. fluxonium)
quantum correctionsto single-kink excitations
Quantum bandLong-range quantum hopping
Two processes:
Destructively interfere if
quantum correctionsto single-kink excitations
Quantum band
Quantum effects perturbative if
Long-range quantum hopping
quantum critical
quantum phase diagram
quantum phase diagram
Devil's staircase has infinite number of pinned phases, period
...
Bak & Bruinsma, PRL 49, 249 (1982)
Classical phase transition to state with one extra/fewer kink
e.g.:
quantum phase diagram
Adding/removing kink in pinned phase → defects
cf. domain walls in AFM
Repel each other to give new classical ground state
...
See also: Odintsov, PRB 54, 1228 (1996); Sela et. al., PRB 84, 085434 (2014)
quantum phase diagram
Adding/removing kink in pinned phase → defects
cf. domain walls in AFM
Repel each other to give new classical ground state
...
See also: Odintsov, PRB 54, 1228 (1996); Sela et. al., PRB 84, 085434 (2014)
quantum phase diagram
Adding/removing kink in pinned phase → defects
cf. domain walls in AFM
Repel each other to give new classical ground state
...
See also: Odintsov, PRB 54, 1228 (1996); Sela et. al., PRB 84, 085434 (2014)
quantum phase diagram
Adding/removing kink in pinned phase → defects
cf. domain walls in AFM
Repel each other to give new classical ground state
...
See also: Odintsov, PRB 54, 1228 (1996); Sela et. al., PRB 84, 085434 (2014)
quantum phase diagram
Adding/removing kink in pinned phase → defects
cf. domain walls in AFM
Repel each other to give new classical ground state
...
See also: Odintsov, PRB 54, 1228 (1996); Sela et. al., PRB 84, 085434 (2014)
quantum phase diagram
defects each gain kinetic energy
Pinned order destroyed if
...
See also: Odintsov, PRB 54, 1228 (1996); Sela et. al., PRB 84, 085434 (2014)
quantum phase diagram
Described by Luttinger parameter
Gapless excitations
KT KT IsingC-IC
...
See also: Odintsov, PRB 54, 1228 (1996); Sela et. al., PRB 84, 085434 (2014)
Mobile defects → incommensurate Luttinger liquid
quantum phase diagram
At boundaries
KT transition to gapped homogeneous phase if(effect of )
Ising transition for
KT KT IsingC-IC
...
See also: Odintsov, PRB 54, 1228 (1996); Sela et. al., PRB 84, 085434 (2014)
quantum phase diagram
Power law interactions: Sela et. al., PRB 84, 085434 (2014)
other regime
other regime
Kink picture not valid: small everywhere
Second order transition to state
Ising transition
expand potential and take continuum limit
Ising transition
Quantum (1+1)-dimensional Ising model
Ginzburg region:
Quantum critical behaviour accessible for
Ising transition
Excitation spectrumSmall oscillations in
Ising transition
Gap closes at transition
Small oscillations in Excitation spectrum
Ising transition
Unit cell doubled in ordered phase
Excitation spectrum
Ising transition
: gap closes at some
Unit cell doubled in ordered phase
Excitation spectrum
Ising transition
Ising transition to phase
● Devil's staircase not present above
● First order transitions
● How do homogeneous phases join?
intermediate regime
Marchand et. al, PRB 37, 1898 (1988)Yokoi et. al., PRB 37, 2173 (1988)
conclusions● Chain of fluxonium qubits with realistic parameters● Several classes of phase transition● Observable in excitation spectrum
● C-IC● Kosterlitz-Thouless
● Ising● First order
● Luttinger Liquid● Ising
