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Quantum �network technology

The second life of rare-earth crystals

Wolfgang Tittel

University of Geneva

Constructor Institute of Technology & Constructor University

As you certainly all know, the development of the Internet over the past 50 years has revolutionized our lifestyle. There is no day during which we don’t exchange emails, fill out online forms, look up information, or or post information using the latest social media platform.

Interestingly, as we change the unit of information from bits (0 or 1) to qubits, we discover fundamentally more powerful functionalities, including the distribution of secret encryption keys …, the possibility for networked quantum computing, and the possibility for blind quantum computing.

Without surprise, new capabilities require new technology and materials, or, if you are lucky, the use of well-known materials but under new conditions.

In this presentation I will elaborate on how rare-earth doped solid-state materials, here shown are Nd:YAG laser rods, which are at the heart of the well-known 1064 nm laser allow building the required quantum technology when cooled to temperatures of a few Kelvin and with cavity-enhanced coupling to light—often nano-photonic cavitiess.

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Elements of a quantum network

Interacting centers for matter-based qubits
Individual optical centers for single and entangled photons
Large ensembles for high-capacity, feed-forward-controlled optical quantum memories
For compatibility and scalability, these components should be based on the same type of defect and be on-chip integrable using standard photonics technology.

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Outline

Rare-earth-ion-doped crystals
Single photons based on individual rare-earth ions
Quantum repeater – a brief introduction
Optical quantum memory using ensembles of rare-earth ions
Outlook and conclusion

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Rare-earth-ion-doped crystals

G.H. Dieke, Spectra and Energy Levels of Rare Earth Ions in Crystals, Wiley Interscience, New York, 1968.

1064 nm

795 nm

1532 nm

1550 nm

883 nm

606 nm

1030 nm

580 nm

2100 nm

1900 nm

Energy of electronic levels [x10³ cm^-1]

Commercial solid-state lasers

Quantum technology (memories, single emitters)

Saglamyurek, WT et al., Nature 469, 512-515 (2011).

Saglamyurek, WT et al., Nature Phot. 9, 83 (2015).

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Rare-earth crystals: a brief introduction�

Simplified level structure of Tm³⁺-doped crystals

1) Transitions (zero-phonon lines) in the visible and near infrared -> quantum communication

2) Γ_inhom ≈ 100 MHz – 500 GHz -> broadband quantum memory

3) Excited states with very long lifetimes (ms)

-> difficult to observe single-photon emissions

4) At T< 2 K: Γ_hom^opt ≈ 50 Hz – 100 kHz -> T₂ = 4 ms -> high-capacity and long-lived quantum memory

5) At T< 2 K: ground states with long T₁ (d) and long T₂ (h) -> long-lived quantum memory and qubits

6) Electric dipole-dipole interaction between neighboring ions -> quantum gates

Promising for optical quantum memory and QIP. But not for single-photon emitters.

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Tm:YAlO₃

How to create a single photon? Spontaneous emission from a single emitter�

The long optical lifetimes in rare-earth ions result in very small decay rates
Photons will be emitted into random directions

γ

Single rare-earth ion

Yb:YVO₄

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Tm:YAlO₃

For κ >> g >> γ₀ (weak coupling regime):

Mode volume

Quality factor

# for atom in max field region

γ₀: vacuum emission rate

∝ (energy loss)^-1

κ ∝ L^-1

₀

Creating (and observing) single photons

Purcell factor

Needed: cavity with small V and large Q

Effective Purcell factor (which describes the actual increase of emission rate) is smaller by a factor given by the branching ratio. We have observed F_P(effective) of 60.

Behind this development is the so-called Purcell effect, which I will briefly explain on this slide.

Instead of interrogating the emitter in free-space (or, in our case, a crystal), let’s assume we put it inside an optical cavity as sketched here. Now, in addition to the spontaneous decay, we have to consider two more processes: The possibility that the emitter echanges its excitation with the cavity, which is characterized by the coupling constant g. And the possibility that an excitation in the cavity – a photon – escapes in the mode defined by the cavity, gamma. What we want is that, after excitation of the emitter by a laser pulse, the ewxcitation is rapidly transferred to the cavity mode, i.e. g >>gamma, and that it leaks out of the cavity before it can hop back onto the emitter, i.e. gamma >> g. For this scenario one could conclude that this may lead to a rapid and efficient creation of a freely propagating photon, and this intuition is correct.

More precisely, the enhancement of the emission rate—the Purcell factor—is given by the ratio of the cavity quality factor Q, which is related to the time a cavity can store energy, and the mode volume- roughly the integration of the intensity. `this was difficult as it requires cavities with a highly confined mode – note here that it is measured with respect to lambda ^3!

Such cavities can be created using nano-photonic crystals. What you see here is a so-called nano-beams …The holes serve as mirrors—or Bragg reflectors– but the hole pattern is slightly changed in the middle part, allowing a highly confined optical mode to exist. Such a cavity can be placed on top of a rare-earth crystal, and the evanescent field then creates the desired conditions of a neighboring rare earth ion to become a good photon source with 1000-fold increased emission rate.

-------

Q factor describes the storage time of a photon (or classically, of electromagnetic field energy) inside a cavity; the energy stored inside the mode with frequency ω and quality factor Q : W(t) =W0 e^- Qt =W0e−2κt , where κ = 2Q is the cavity field

decay rate.

F_P can also be expressed using g,kappa and Gamma: F_P=4g(r)^2/kappa Gamma (cooperativity), with g= dE/hbar

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The Purcell effect: atom-light interaction in the weak coupling regime

Nano-photonic crystal cavities

Deotare et al. Appl. Phys. Lett. 94, 121106 (2009)

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Single photons from individual rare-earth ions

Purcell-entangled light-matter interaction has recently enabled the observation of true single photons from individual rare-earth ions coupled to nanocavities.

Thompson group: Silicon photonic crystal (nano) cavity stamped on top of Er:YSO (PRL 2018)

Faraon group: photonic crystal nano-cavity milled out of Nd:YVO₄ (PRL 2018)

Homogeneous approach

Heterogeneous approach

F_P=650

F_P=110

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A single-photon source based on Er:LiNbO₃

Design and Simulations

Patterning:

E-beam lithography

Patter transfer:

Reactive ion etching

Under-cutting:

Hydrofluoric acid

Inspection:

Optical microscope & SEM

Optical characterization in air, and after transfer on Er:LiNbO₃, at T=293K and T=4K

Spectra and quality factor

Si (250 nm)

Si (0.5 mm)

SiO₂ (3 μm)

Er:LiNbO₃

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Design

Inverse taper coupler: allows coupling light in and out

A “bus” waveguide (with Bragg reflector): couples to 2 nanobeam cavities

Tether: A weak point that can be broken to detach the device

Joint work with Gröblacher group

Photonic crystal cavity

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A single-photon sources based on Er:LiNbO₃

Design and Simulations

Patterning:

E-beam lithography

Patter transfer:

Reactive ion etching

Under-cutting:

Hydrofluoric acid

Inspection:

Optical microscope & SEM

Optical characterization in air, and after transfer on Er:LiNbO₃, at T=293K and T=4K

Spectra and quality factor

Si (250 nm)

Si (0.5 mm)

SiO₂ (3 μm)

Er:LiNbO₃

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Cavity characterization

Mode volume	calculated	0.1 μm³
Quality factor (on Er: LiNbO₃)	measured	50 000
Purcell factor (E=E_max, β=1)	predicted	1 000

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Purcell-enhanced emission

Si cavity on 0.005% Er:LiNbO₃
T approx. 50 mK
Observation of isolated photoluminescence lines off line-center
13-fold(144-fold) reduction of decay constant from 1.8 ms to 134 μs (12.5 μs)
T₁ <10 μsec and radiatively limited emission (T₁=T₂/2) seem possible

-> Fourier-limited photons

Joint work with Gröblacher group, TU Delft

T₁=

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Purcell-enhanced emission�

Measurement of auto-correlation coefficients shows non-classical (single-photon) nature of emissions and confirms interaction with individual erbium ions

-> single-photon source and possibility for qubit read-out

g²(0) = 0.190+-0.02

in₁

out₁

out₂

&

Detection time difference [bins]

Experimental setup

Joint work with Gröblacher group, TU Delft

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Purcell-enhanced emission�

Demonstration of Stark tunability of single ion
Single-photon character not affected
Feedback mechanism to counter spectral diffusion

-> indistinguishable single photons

-> distant spin-spin entanglement

-> heralded entangled photon pairs

Energy

Δν=(d_e-d_g)E / h

Joint work with Gröblacher group, TU Delft

Y. Yu et al., Phys. Rev. Lett. 131, 170801 (2023)

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Quantum repeater - how to mitigate loss

Goal: Overcome the exponential scaling of photon transmission over a long (lossy) quantum channel

Note: multiplexing does not lead to better scaling

Solution

1) Break long link into shorter elementary links.

N Sinclair, WT et al., Phys. Rev. Lett. 113, 053603 (2014)

2) Distribute heralded and long-lived entanglement across each elementary link.

3) Multiplex distribution (any degree of freedom) to make it efficient.

4) Mode mapping based on feed-forward info allows connecting “good” links using Bell-state measurements.

No need for photons to travel in one go over the entire link.

Exponential scaling

Same scaling

Better scaling

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Quantum memory requirements

Large storage efficiency
Sufficient storage time
Fidelity ->1
Feed-forward mode mapping
High multiplexing capacity
Wavelength of operation
Bandwidth per qubit
Integrability

BSM

QM

Add info about necessary storage efficiency

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How to store photonic quantum states in a multiplexed manner? Use large ensembles of atoms�

In

out

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Photon-echo quantum memory (CRIB)

1. Preparation of an optically thick, single absorption line

2. Controlled reversible inhomogeneous broadening (CRIB)

3. Absorption of light in arbitrary quantum state

-> fast dephasing

4. Reduction of broadening to zero

5. Phase matching: φ(z) = -2kz ; E_in ∝ e^ikz→ E_out ∝ e^-ikz

6. Re-establishment of broadening, but with reversed sign

frequency

opt. depth

(interaction with external electric field)

-> Time reversed evolution of atomic system and re-emission of with unity efficiency and fidelity

Ω

frequency

opt. depth

Γ_hom

frequency

opt. depth

Moiseev et al., PRL (2001); Nilsson et al., Opt. Comm. (2005); Kraus, WT et al., PRA(2006); Alexander et al., PRL (2006)

Δ_i -> -Δ_i ∀ i

Experiments: Canberra, Geneva

|g>

|e>

|a>

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Photon echo quantum memory (AFC)

1. Preparation of an atomic frequency comb

2. Absorption of a photon -> fast dephasing

3. Rephasing at t=1/ν_comb: 2πΔ_jt=2π(nν_comb)/ν_comb = n 2π

-> Re-emission of photonic qubits with unity efficiency* and fidelity.

frequency

absorption

Γ_hom

Afzelius et al., PRA (2009); De Riedmatten et al., Nature (2008)

Experiments: Geneva, Lund, Paris, Calgary, Delft, Barcelona, Hefei, Caltech

Needed: inhomogeneously broadened transition, long-lived auxiliary state, narrow homogeneous linewidth

-> long storage times require small ν_comb, i.e. narrow homogeneous lines

* requires phase matching or cavity

frequency Δ

absorption

ν_comb

Ω

|g>

|e>

|a>

|s>

M. Afzelius et al. PRA 79, 052329 (2009)

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The atomic frequency comb (AFC) protocol in rare-earth crystals

2-level AFC protocol with fixed storage time: τ=1/ν_comb
Feed-forward mode-mapping possible using external devices
Well suited for multiplexing
Efficiency (1/e) limited by T₂^opt : τ_M= T₂^opt/4

Extension to spin-wave storage allows memory-internal read-out on demand
Storage time (efficiency) then limited to spin level broadening (<- refocusing pulses)
Memory is locked after first control pulse

Impedance-matched cavities allow high-efficiency storage despite small optical depth αL

M. Afzelius et al. Phys. Rev. A 79, 052329 (2009); N. Sinclair, WT et al., PRL113, 053603 (2014); M. Afzelius and C. Simon, PRA 82, 022301 (2010).

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Towards efficient quantum memory

M. Afzelius and C. Simon, Phys. Rev. A 82, 022310 (2010)

The efficiency of the AFC quantum memory is limited by its optical depth

Using an impedance-matched cavity allows in principle to increase the efficiency to 1 despite small single-pass absorption e^-^α^l

Condition: R₁=e^-2^α^l R₂with R₂= 1

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Towards efficient quantum memory

J. Davidson, WT et al., PRA 101, 042333 (2020)

Reflection-coated Tm:Y₃Al₅O₁₂ (YAG) crystal

R₂=0.99

R₁=0.40

Cavity transmission spectrum (outside Tm resonance)

Cavity reflection spectrum (within Tm resonance)

uncoated

Approx. impedance matching

4mm

T₁=1.1 ms

FSR~20 GHz, F~7

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Towards efficient quantum memory

J. Davidson, WT et al., PRA 101, 042333 (2020)

Setup allows creating and measuring

the AFC,
attenuated laser pulses in time-bin qubit states
time-bin qubits encoded into heralded single photons

Preparation of light

Quantum memory

detection

|ψ>=α|e>+βeⁱ^φ|l>

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Results

J. Davidson, WT et al., PRA 101, 042333 (2020)

AFC created using the non-coated part of the crystal. η~1%

AFC-based storage of attenuated laser pulses (μ=0.7) using coated part of crystal

back-reflected

recalled

η_max =27%

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More Results

J. Davidson, WT et al., PRA 101, 042333 (2020)

Quantum process tomography of time-bin qubits encoded into attenuate laser pulses (μ=0.7)

-> F=(85 ± 1)% >> 0.66

Quantum state tomography of time-bin qubits encoded into heralded single photons

Measurement of non-classical cross-correlations with photon pairs before and after storage:

g⁽²⁾_before=61.8 ±3.8, g⁽²⁾_after = 9.1 ±1.2

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State-of-the-art

Comparing results taken under different conditions… But while not all experiments demonstrate quantum nature, all used a quantum protocol

2-level AFC: not yet better than fiber, but close. Materials with sufficient T₂^opt for τ = 1ms exist. Need to reduce technical noise and add cavities.

AFC spin-storage: scaling already better than fiber, but efficiencies still small. Materials with sufficient T₂^spin exist. Need to improve efficiency of π-pulses and noise, and add cavities.

Three use cases (assuming sufficient multiplexing)

Quantum repeaters for fiber networks
Quantum repeaters for satellite networks
Physical qubit transport

Fiber-based networks

Satellite-based networks

Physical qubit transport-based networks

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Towards fully quantum-enabled networks based on an integrated platform

Single (and entangled) photons based on Purcell-enhanced emission from single rare-earth ions
Compatible quantum memories based on large ensembles of rare-earth ions
(Quantum computing nodes using interacting rare-earth ions coupled to nano-cavities for readout)
Exploit maturity of Si/SiN/LiNbO₃ photonics (foundries) to create scalable and integrated devices
More fundamental research into materials, protocols, applications,…

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Thank you!

Gröblacher lab (TU Delft)

Simon Gröblacher

Rob Stockill

Yong Yu

Gaia Da Prato

Emanuele Urbinati

Tittel lab (TUD & UNIGE)

Dorian Oser

Sara Marzban

Patrick Remy

Erell Laot

Javier Carrasco

Deeksha Gupta

Archi Gupta

Akshay Karyath

Luozhen Li

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Qubits and single-qubit gates

Exploit ultra-long coherence times of nuclear spin transitions (e.g. 6h in Eu:Y₂SiO₅)
To allow qubit-selective spin manipulation in densely-doped crystals, take advantage of inhomogeneously broadened optical transition
Demonstrated with ensembles (Kröll group) and a single optical transition

Sellars group, 2015; Kröll & Molmer groups, since 2002; Thompson & Faraon groups, since 2018

Purcell-enhance readout transition of a single ion using evanescently-coupled nanocavity (Thompson and Faraon groups)

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�Remote spin-spin entanglement

Bell-state measurement

The Barrett–Kok scheme allows creating entanglement between spin state and flying time-bin qubit
Entanglement swapping heralds “event-ready” entanglement between distant spins
Demonstrated with NV centers (Hanson group)
Purcell-enhanced photon emission will allow demonstration with single rare-earth ions

|ψ> = (|↓e> + |↑l>)2^-12

And finally, we will investigate the interaction between stationary qubits encoded into nuclear spin states of individual Tm ions.

First, we will demonstrate single-qubit gates, for which we will use two-color optical pulses as shown here. For optical read-out, we will use the second ground state transition–which is a unique feature of Tm:YALO--, again together with a nano-cavity.

Next, we will use two such spin qubits, but instead of detecting the photons emitted from each qubit individually, we will make a joint measurement with the two photons. This allows entangling the two distant thulium qubits.

This is known from NV centers, but has not yet been demonstrated with rare-earth ions. Again, I anticipate demonstrating this powerful primitive over deployed fiber.

And finally, we will exploit the electric dipole-dipole interaction between neighboring Tm spins, which allows changing the state of the target qubit conditioned on the state of the control qubit.

Such a CNOT gate has been demonstrated before with rare-earth ions, but only in a proof-of-principle demonstration using large ensembles – not individual rare-earth ions as we propose here.

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Conditional spin-spin quantum gates

Create CNOT gate between two neighboring solid-state qubits using electric dipole-dipole interaction
Proof-of-principle demonstration using ensembles by Kröll group
Purcell-enhanced photon emission should allow demonstrations using single rare-earth ions

Stark shifts of XXXXXX in Eu:YSO observed

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Thank you!

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A reminder: quantum state, density matrix and Bloch vector

T

Γ_hom

Γ_hom ∝ T⁷

4f

5s, 5p

Bloch sphere

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Storage of light in an atomic ensemble using two-pulse photon-echoes

Kurnit, Abella & Hartmann, Phys. Rev. Lett. (1964), Mossberg, Opt. Lett. (1982)

Γ_hom

frequency

opt. depth

t

electric field amplitude

|e>

|g>

E

ω

|g⟩

|e⟩

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Storage of light in an atomic ensemble using two-pulse photon-echoes

Kurnit, Abella & Hartmann, Phys. Rev. Lett. (1964), Mossberg, Opt. Lett. (1982)

Γ_hom

frequency

opt. depth

π/2-pulse

t

electric field amplitude

|e>

|g>

E

ω

|g⟩

|e⟩

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Storage of light in an atomic ensemble using two-pulse photon-echoes

Kurnit, Abella & Hartmann, Phys. Rev. Lett. (1964), Mossberg, Opt. Lett. (1982)

Γ_hom

frequency

opt. depth

π/2-pulse

t

electric field amplitude

|e>

|g>

E

ω

|g⟩

|e⟩

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Storage of light in an atomic ensemble using two-pulse photon-echoes

Kurnit, Abella & Hartmann, Phys. Rev. Lett. (1964), Mossberg, Opt. Lett. (1982)

Γ_hom

frequency

opt. depth

π/2-pulse

t

electric field amplitude

dephasing

ω=ω₀

ω>ω₀

ω<ω₀

|e>

|g>

E

ω

|g⟩

|e⟩

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Storage of light in an atomic ensemble using two-pulse photon-echoes

Kurnit, Abella & Hartmann, Phys. Rev. Lett. (1964), Mossberg, Opt. Lett. (1982)

Γ_hom

frequency

opt. depth

π/2-pulse

t

electric field amplitude

π−pulse

τ₁

dephasing

ω=ω₀

ω>ω₀

ω<ω₀

|e>

|g>

E

ω

|g⟩

|e⟩

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Storage of light in an atomic ensemble using two-pulse photon-echoes

Kurnit, Abella & Hartmann, Phys. Rev. Lett. (1964), Mossberg, Opt. Lett. (1982)

Γ_hom

frequency

opt. depth

π/2-pulse

t

electric field amplitude

π−pulse

τ₁

dephasing

ω=ω₀

ω>ω₀

ω<ω₀

π-pulse

ω>ω₀

ω<ω₀

|e>

|g>

E

ω

|g⟩

|e⟩

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Storage of light in an atomic ensemble using two-pulse photon-echoes

Kurnit, Abella & Hartmann, Phys. Rev. Lett. (1964), Mossberg, Opt. Lett. (1982)

Γ_hom

frequency

opt. depth

π/2-pulse

t

electric field amplitude

π−pulse

τ₁

dephasing

ω=ω₀

ω>ω₀

ω<ω₀

π-pulse

ω>ω₀

ω<ω₀

rephasing

|e>

|g>

E

ω

|g⟩

|e⟩

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Storage of light in an atomic ensemble using two-pulse photon-echoes

Kurnit, Abella & Hartmann, Phys. Rev. Lett. (1964), Mossberg, Opt. Lett. (1982)

Γ_hom

frequency

opt. depth

π/2-pulse

t

electric field amplitude

π−pulse

τ₁

dephasing

ω=ω₀

ω>ω₀

ω<ω₀

π-pulse

ω>ω₀

ω<ω₀

rephasing

|e>

|g>

E

ω

|g⟩

|e⟩

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Storage of light in an atomic ensemble using two-pulse photon-echoes

Kurnit, Abella & Hartmann, Phys. Rev. Lett. (1964), Mossberg, Opt. Lett. (1982)

Γ_hom

frequency

opt. depth

π/2-pulse

t

electric field amplitude

π−pulse

τ₁

dephasing

ω=ω₀

ω>ω₀

ω<ω₀

π-pulse

ω>ω₀

ω<ω₀

rephasing

|e>

|g>

E

ω

|g⟩

|e⟩

echo-pulse

2τ₁

echo at t=2τ

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Storage of light in an atomic ensemble using two-pulse photon-echoes

Kurnit, Abella & Hartmann, Phys. Rev. Lett. (1964), Mossberg, Opt. Lett. (1982)

Γ_hom

frequency

opt. depth

t

electric field amplitude

π−pulse

τ₁

dephasing

ω=ω₀

ω>ω₀

ω<ω₀

π-pulse

ω>ω₀

ω<ω₀

rephasing

2τ₁

echo at t=2τ

allows data storage

|e>

|g>

E

ω

|g⟩

|e⟩

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Storage of light using two-pulse photon-echoes

Ruggiero et al, PRA (2009); Sanguard, WT et al., PRA (2010), Massar &Popescu, PRL (1995)

Time-bin qubit (single photon) input

P_echo= P_noise
ρ_out= Fρ_in+(1-F) ρ_in

- F= tr(ρ_inρ_out)

= (P_echo + P_noise)/(P_echo + 2P_noise) = 2/3

= F_classical^(max)

p(t)

t

|e>

|l>

|e>

|l>

Where does this come from? Einstein coefficients?

Revise the ROSE protocol

Not a quantum memory!

|g⟩

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Hidden memory deadtime

Readout-on demand only for one qubit per train (sufficient for scheme on slide 3)

Memory cannot store additional qubits after 1st control pulse -> deadtime and duty cycle τ_M^opt/τ_M^total<<1

Potential solution : Shift subsequent qubit trains in spectrum so that control pulses only interact with specific trains

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Quantum memory requirements

Large storage efficiency
Sufficient storage time
Fidelity ->1
Feed-forward mode mapping
High multiplexing capacity
Wavelength of operation
Bandwidth per qubit
Integrability

BSM

QM

Necessary criterion for QM: storage efficiency better than using a fiber creating same delay

t_fiber = 10^-0.02・L L in km; t=L*5 μs/km

η_QM = e^-t/^τ

-> τ_M (min) = 108 μs

M

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