Mølmer–Sørensen Gate

The Mølmer–Sørensen gate is the two-qubit entangling operation that made trapped-ion quantum computing practical. Its ancestor, the Cirac–Zoller gate, demanded that the shared vibrational mode of the ion chain sit in a definite Fock state — usually the ground state $|n=0\rangle$ — and stepped quantum information through that single phonon in a fragile sequence of pulses. In 1999 Anders Sørensen and Klaus Mølmer asked whether the motion could be used without ever being genuinely occupied. Their answer: illuminate two ions with two laser tones at once, symmetrically detuned above and below the qubit frequency so one sits near the red motional sideband and the other near the blue. Each tone alone would create or destroy real phonons; together their amplitudes interfere so the phonon appears only virtually. The motion is displaced around a closed loop and returns exactly home, leaving the two ions an entangling phase equal to the geometric area enclosed — a quantity blind to how many phonons were present to begin with. In the Lamb–Dicke regime the bichromatic field is a spin-dependent force: ions in different collective spin states are pushed in opposite directions, tracing trajectories that close after $\tau=2\pi/\delta$ , set by the detuning $\delta$ from the sideband. At that instant spin and motion disentangle and a pure two-qubit phase remains. Tuning the laser power so the enclosed area gives the two-body angle $\pi/4$ turns separable states into Bell states. Because the phase is geometric and $\bar n$ -independent, the gate tolerates thermal motion, works from a Doppler-cooled start, and has been pushed past $99.9\%$ fidelity — the highest of any platform, and the native entangler in today's laboratory and commercial ion processors.

Key takeaways

The Mølmer–Sørensen (MS) gate entangles two trapped ions with a symmetric pair of laser tones (a bichromatic field) detuned by $\pm\delta$ about the qubit transition, exciting a shared motional mode off-resonantly.
Red and blue sidebands act together, so the motion is excited only virtually: the mode returns to its starting state and the entangling phase is independent of the initial phonon number $n$ .
In the Lamb–Dicke regime the drive is a spin-dependent force whose closed loop imprints $U=\exp[-i\tfrac{\pi}{4}\sigma^x_1\sigma^x_2]$ , locally equivalent to CNOT.
Unlike the Cirac–Zoller gate it needs no single-Fock-state preparation and is first-order insensitive to thermal motion, making it the workhorse entangler of modern ion traps, with speed and fidelity set by $\eta$ , $\eta\Omega$ , $\delta$ , and decoherence from scattering and heating.
Key idea: two symmetric tones detuned by $\pm\delta$ from a motional sideband generate a spin-dependent force whose phase-space trajectory closes at $\tau=2\pi/\delta$ , imprinting an $n$ -independent two-qubit phase that — tuned to a two-body angle $\pi/4$ — yields a maximally entangling gate.

The core insight

Drive two ions with a symmetric bichromatic field detuned by $\pm\delta$ from a motional sideband; the red and blue sidebands together generate a spin-dependent force whose phase-space trajectory closes after $\tau=2\pi/\delta$ , imprinting an $n$ -independent two-qubit phase that, tuned to a two-body angle $\pi/4$ , yields a maximally entangling gate.

Where this lives today

The Mølmer–Sørensen interaction is the beating heart of modern ion-trap computing. In machines from Quantinuum and IonQ it is the native two-qubit gate: pairs are addressed with focused Raman beams, entangled by an MS pulse, and shuttled between memory and interaction zones in a quantum charge-coupled-device (QCCD) architecture. Its record fidelities — above 99.9% for hyperfine qubits — are the highest of any platform. Applied globally instead of pairwise, the same Hamiltonian generates the long-range Ising couplings $\sum_{i<j}J_{ij}\sigma^x_i\sigma^x_j$ that Christopher Monroe's group and others use to simulate frustrated magnetism and information scrambling in chains of tens of ions, and to prepare large GHZ states in one shot. The open frontiers are practical: crosstalk-free individual addressing in long chains, robustness to trap-frequency drift (met with shaped pulses that close several phase-space loops at once), and integration with ion–photon entanglement to stitch local registers into a networked, modular machine that scales beyond a single trap.

The physical picture

Treat the two ions' shared vibration as one harmonic oscillator. A spin-dependent force pushes it in a direction fixed by the joint spin state, so the coherent-state amplitude $\alpha$ drifts through phase space. Because the drive rotates at the slow detuning $\delta$ , the amplitude sweeps out a circle rather than running away, and after one full turn — at $\tau=2\pi/\delta$ — it lands back exactly where it started. Like the Aharonov–Bohm phase, the return is not trivial: the spin acquires a geometric phase equal to twice the enclosed area, and that phase, and only that phase, entangles the two ions.

Figure

State-dependent phase-space loops in the plane of Re α and Im α: the collective-spin branches with S_x = +2 and S_x = −2 trace circles of radius ηΩ divided by δ that both return to the origin at τ = 2π divided by δ, while the S_x = 0 branch stays put; twice the enclosed area is the geometric phase Φ = π divided by 8 per unit of S_x squared that entangles the ions

The two laser tones do the real work of hiding the phonon. Detuned just inside the red and blue sidebands, each tone alone would drive a genuine sideband transition and pump quanta into the mode. Driven together and symmetrically, their amplitudes for creating and annihilating a phonon interfere: the resonant part cancels, and what survives is a second-order coupling that carries the internal state from $|gg\rangle$ to $|ee\rangle$ through a virtual phonon, borrowed and immediately returned. Because the phonon never really exists, the process does not count how many were already there — the origin of the gate's thermal robustness. The second figure traces the two interfering two-photon paths that connect $|gg\rangle$ and $|ee\rangle$ while leaving $n$ untouched.

Figure

The bichromatic two-photon mechanism: two tones detuned by δ = 2π×25 kHz inside the red and blue sidebands of a ν = 2π×2 MHz mode carry the pair from ground-ground to excited-excited along two interfering paths — one through the virtual n+1 rung, one through n−1 — so the phonon is only virtual and the net process preserves the phonon number n

The robustness is fundamentally a closed-loop effect. The trajectory closes when $\delta\tau=2\pi$ , a condition on timing alone, independent of the loop's radius: laser-power errors change the radius and phase but cannot stop the loop from closing, so spin and motion always disentangle on schedule — a built-in geometric error-suppression that single-tone schemes lack.

The core equations

The starting point is the interaction of the two ions with the bichromatic field, in the frame rotating at the carrier. Two tones symmetric about the carrier act on the qubit raising operators and on the mode through the recoil phase factor $e^{i\eta(ae^{-i\nu t}+a^\dagger e^{i\nu t})}$ .

H = \hbar\,\Omega\sum_{j=1,2}\sigma^+_j\,e^{\,i\eta(a e^{-i\nu t}+a^\dagger e^{i\nu t})}\Big(e^{-i\delta_b t}+e^{-i\delta_r t}\Big)+\mathrm{h.c.} \qquad (1)

where: $\Omega$ — carrier Rabi frequency of each tone on each ion; $\sigma^+_j=|e\rangle\langle g|_j$ — raising operator of ion $j$ ; $\eta=k\,x_0$ — Lamb–Dicke parameter with zero-point extent $x_0=\sqrt{\hbar/2M\nu}$ ; $a,a^\dagger$ — annihilation and creation operators of the chosen collective mode; $\nu$ — mode frequency; $\delta_b=\nu-\delta$ , $\delta_r=-\nu+\delta$ — blue and red tone detunings. Reading: each tone couples internal flips to phonon creation and annihilation, scaled by $\eta$ . Expanding to first order in $\eta$ and keeping the slowly-rotating sideband terms collapses the two tones into a single spin-dependent force — the operational heart of the gate.

H_{\mathrm{LD}} = i\hbar\,\eta\Omega\,S_\phi\big(a\,e^{-i\delta t}-a^\dagger\,e^{i\delta t}\big),\qquad S_\phi=\sum_{j}\big(\sigma^+_j e^{i\phi}+\sigma^-_j e^{-i\phi}\big) \qquad (2)

where: $S_\phi$ — collective spin operator; at $\phi=0$ , $S_\phi=S_x=\sigma^x_1+\sigma^x_2$ ; $\delta$ — detuning of each tone from its sideband; $\eta\Omega$ — sideband Rabi frequency setting the force amplitude. Reading: the force $\propto S_\phi$ points oppositely for the two spin eigenstates — that is "spin-dependent". Integrating the force gives the mode's coherent displacement — a phase-space circle that returns to the origin once each drive period.

\alpha(t)=\eta\Omega\,\frac{1-e^{i\delta t}}{\delta}\,S_\phi,\qquad \alpha(\tau)=0\ \Longleftrightarrow\ \delta\tau=2\pi\ \Longleftrightarrow\ \tau=\frac{2\pi}{\delta} \qquad (3)

where: $\alpha(t)$ — spin-dependent coherent displacement of the mode; $\tau$ — gate time. Reading: the displacement vanishes at $\delta\tau=2\pi$ for every branch and every initial phonon number — why spin and motion disentangle regardless of temperature. Because the force is linear in the mode operators, the evolution truncates exactly at second order: the motion factors out and a pure two-qubit phase — the enclosed area — remains.

U(\tau)=\exp\!\big[-i\,\Phi\,S_\phi^2\big],\qquad \Phi=\Big(\frac{\eta\Omega}{\delta}\Big)^2\big(\delta\tau-\sin\delta\tau\big)\Big|_{\delta\tau=2\pi}=2\pi\Big(\frac{\eta\Omega}{\delta}\Big)^2 \qquad (4)

where: $\Phi$ — geometric phase, twice the enclosed loop area per unit $S_\phi^2$ ; $S_\phi^2$ — collective two-body coupling. Reading: at closure the $\sin\delta\tau$ term dies and only the area, growing linearly in $\tau$ , survives. For two ions, $S_x^2=(\sigma^x_1+\sigma^x_2)^2=2\,\mathbb{1}+2\,\sigma^x_1\sigma^x_2$ , so the physical entangling angle is $2\Phi$ ; fixing it to $\pi/4$ gives the maximal entangler.

2\Phi=\frac{\pi}{4}\ \Longrightarrow\ \Phi=\frac{\pi}{8}\ \Longrightarrow\ \eta\Omega=\frac{\delta}{4},\qquad U_{\mathrm{MS}}=\exp\!\Big[-i\tfrac{\pi}{4}\,\sigma^x_1\sigma^x_2\Big] \qquad (5)

where: $2\Phi=\pi/4$ — two-body angle for maximal entanglement; $\eta\Omega=\delta/4$ — the laser-power condition. Reading: in collective form $U_{\mathrm{MS}}=\exp[-i\tfrac{\pi}{8}S_x^2]$ , identical to the two-body form up to the global phase $e^{-i\pi/4}$ . On the computational basis it turns product states into Bell states — the signature of a maximal entangler.

U_{\mathrm{MS}}\,|gg\rangle=\tfrac{1}{\sqrt2}\big(|gg\rangle-i\,|ee\rangle\big),\qquad U_{\mathrm{MS}}\,|ge\rangle=\tfrac{1}{\sqrt2}\big(|ge\rangle-i\,|eg\rangle\big) \qquad (6)

where: $|gg\rangle,|ee\rangle$ — both ions in the ground or excited state. Reading: $\exp[-i\tfrac{\pi}{4}\sigma^x_1\sigma^x_2]=\tfrac{1}{\sqrt2}(\mathbb{1}-i\,\sigma^x_1\sigma^x_2)$ , and $\sigma^x_1\sigma^x_2$ flips both spins to build the equal-weight superposition. Applied globally to $N$ ions, the same physics generates the long-range Ising couplings $H_{\mathrm{eff}}=\hbar\sum_{i<j}J_{ij}\,\sigma^x_i\sigma^x_j$ with $J_{ij}=\sum_m\eta_{i,m}\eta_{j,m}\Omega^2/\delta_m$ — a tunable, power-law resource for analog quantum simulation and GHZ-state creation. Finally, the leading imperfections set the fidelity floor — and crucially, not all are suppressed by the gate's thermal robustness.

1-F\ \approx\ \Gamma\,\tau\ +\ \eta^2(2\bar n+1)\,\epsilon_{\mathrm{LD}}\ +\ \dot{\bar n}\,\frac{\eta^2\tau}{2} \qquad (7)

where: $F$ — gate fidelity; $\Gamma$ — spontaneous photon scattering rate; $\bar n$ — mean phonon number; $\dot{\bar n}$ — motional heating rate; $\epsilon_{\mathrm{LD}}$ — dimensionless higher-order Lamb–Dicke coefficient. Reading: photon scattering and heating during the gate are not protected by first-order $\bar n$ -independence.

Derivation AFrom the bichromatic drive to a spin-dependent force

The goal is to show that the full two-tone Hamiltonian of equation (1) reduces, in the Lamb–Dicke regime, to the clean object of equation (2): a force on the motional mode whose sign is fixed by the collective spin operator $S_\phi$ . We expand the recoil exponential, discard the fast carrier and counter-rotating pieces, and use the symmetric tone placement to combine the survivors into one slowly-rotating force.

Step 1. Expand the recoil factor to first order, $e^{i\eta(ae^{-i\nu t}+a^\dagger e^{i\nu t})}\approx 1+i\eta(ae^{-i\nu t}+a^\dagger e^{i\nu t})$ . Why: with $\eta\sqrt{2\bar n+1}\ll1$ the ion barely samples the wavelength's curvature, so only the term linear in the mode operators survives — the whole content of the Lamb–Dicke approximation.

Step 2. The zeroth-order piece drives the carrier at $\pm\delta$ ; with symmetric placement it averages away or is removed by frame choice, leaving the $\eta$ -linear sideband terms. Why: the carrier merely rotates each qubit and does not contribute to the entangling dynamics we are isolating.

Step 3. Insert $\delta_b=\nu-\delta$ , $\delta_r=-\nu+\delta$ : the blue tone's $e^{-i\delta_b t}$ times $a^\dagger e^{i\nu t}$ gives a slow $e^{i\delta t}$ , the red tone's times $a\,e^{-i\nu t}$ gives $e^{-i\delta t}$ , and the fast $e^{\pm 2i\nu t}$ pairings drop by the rotating-wave approximation. Why: the symmetric detuning makes each tone select one mode operator with an envelope rotating at $\delta$ , not $\nu$ .

Step 4. Collect the survivors into $S_\phi=\sum_j(\sigma^+_je^{i\phi}+\sigma^-_je^{-i\phi})$ times $(a\,e^{-i\delta t}-a^\dagger e^{i\delta t})$ , giving $H_{\mathrm{LD}}=i\hbar\eta\Omega\,S_\phi(a\,e^{-i\delta t}-a^\dagger e^{i\delta t})$ . Why: this is a displacement whose phase-space direction is set by the eigenvalue of $S_\phi$ — a genuine spin-dependent force, the object every later step manipulates.

Go deeper · Derivation B · 1 minFrom the spin-dependent force to the entangling phase

Derivation BFrom the spin-dependent force to the entangling phase

The goal is to prove that evolving under the linear force $H_{\mathrm{LD}}$ gives exactly a displacement times a pure two-qubit phase equal to the enclosed area, and that the Magnus expansion truncates after two terms so the result is exact, not perturbative. We write $H_{\mathrm{LD}}/\hbar=f(t)a^\dagger+f^*(t)a$ with $f(t)=-i\eta\Omega\,S_\phi\,e^{i\delta t}$ , compute the first two Magnus terms, and show the rest vanish.

Step 1. The first Magnus term integrates the force: $\Omega_1=-\tfrac{i}{\hbar}\int_0^\tau H_{\mathrm{LD}}\,dt=\alpha(\tau)a^\dagger-\alpha^*(\tau)a$ , a displacement by $\alpha(\tau)=\eta\Omega\,S_\phi(1-e^{i\delta\tau})/\delta$ . Why: a linear force pushes the coherent amplitude by an amount $\propto S_\phi$ , so different spin branches displace differently.

Step 2. At loop closure $\delta\tau=2\pi$ the factor $1-e^{i\delta\tau}=0$ , so $\alpha(\tau)=0$ for every eigenvalue of $S_\phi$ and every initial phonon state. Why: the displacement returning to zero disentangles motion from spin on schedule — the timing, not the power, guarantees it.

Step 3. The second Magnus term is $\Omega_2=-\tfrac{1}{2\hbar^2}\int_0^\tau dt\int_0^t dt'\,[H(t),H(t')]$ ; the commutator $f^*(t)f(t')-f(t)f^*(t')$ is a c-number times $S_\phi^2$ , since $[a,a^\dagger]=1$ . Why: the mode operators contract to a number, so $\Omega_2$ is a spin-only phase — the seed of entanglement.

Step 4. The double integral of $-2i(\eta\Omega S_\phi)^2\sin\delta(t-t')$ gives $\Omega_2=i(\eta\Omega/\delta)^2(\delta\tau-\sin\delta\tau)\,S_\phi^2$ , i.e. $U(\tau)=\exp[-i\Phi S_\phi^2]$ . Why: $\Phi$ is the geometric phase — precisely twice the loop area — the quantity tuned to entangle the ions.

Step 5. At $\delta\tau=2\pi$ , $\sin\delta\tau=0$ , so $\Phi=2\pi(\eta\Omega/\delta)^2$ ; higher Magnus terms vanish because $[H(t),[H(t'),H(t'')]]=0$ for a linear force. Why: the exact truncation makes equation (4) the full evolution, not an approximation.

Go deeper · Derivation C · 1 minWhy the phase is independent of the motional Fock state

Derivation CWhy the phase is independent of the motional Fock state

The goal is to make explicit the property that defines the MS gate — that once the loop closes, the two-qubit phase contains no operator on the vibrational Hilbert space, so it is identical for the ground state, any Fock state $|n\rangle$ , and any thermal mixture. We decompose the spin state into eigenstates of $S_\phi$ , follow each branch's motional trajectory, and reassemble.

Step 1. Write $|\chi\rangle_{\mathrm{spin}}\otimes|n\rangle_{\mathrm{mot}}$ and expand $|\chi\rangle=\sum_s c_s|s\rangle$ in eigenstates of $S_\phi$ with $S_\phi|s\rangle=\lambda_s|s\rangle$ ; each branch feels a displacement $\alpha_s(t)\propto\lambda_s$ . Why: since the force commutes with $S_\phi$ , the spin branches evolve independently, each dragging its own copy of the motion around a loop scaled by $\lambda_s$ .

Step 2. At $\delta\tau=2\pi$ every branch has $\alpha_s(\tau)=0$ , so each component $|n\rangle$ returns to itself exactly — no $\lambda_s$ -dependent displacement survives. Why: loop closure depends only on $\delta\tau$ , not the radius $\propto\lambda_s$ , so the disentangling is exact regardless of $n$ .

Step 3. Each branch collects a geometric phase $\theta_s=-\Phi\,\lambda_s^2$ , set by the loop area and carrying no reference to the Fock index $n$ . Why: the area lives in the $(\mathrm{Re}\,\alpha,\mathrm{Im}\,\alpha)$ plane, fixed by the drive, not by how many quanta the mode held.

Step 4. Reassemble: $U(\tau)|\chi\rangle|n\rangle=(e^{-i\Phi S_\phi^2}|\chi\rangle)\otimes|n\rangle$ , which holds term-by-term for any thermal $\sum_n p_n|n\rangle\langle n|$ . Why: the motion factors out untouched for every $n$ , so a Doppler-cooled chain gives the same gate as a ground-state-cooled one — to first order in $\eta$ .

Limiting cases

The MS Hamiltonian simplifies, or breaks down, in a handful of regimes that map directly onto experimental design choices.

Case	Result	Meaning
Lamb–Dicke regime	$\eta\sqrt{2\bar n+1}\ll 1$	The recoil exponential linearizes; the force is linear in the quadratures and the gate is first-order $\bar n$ -independent — the regime the whole derivation assumes.
Weak-coupling (slow) gate	$\eta\Omega\ll\delta$	Small phase-space loops; population leaves the spin subspace only virtually and the effective Ising coupling is $J\simeq(\eta\Omega)^2/\delta$ , with tiny residual spin–motion error.
Resonant (fast) gate	$\eta\Omega\sim\delta$	Large loops completing in one period $\tau=2\pi/\delta$ ; faster but more sensitive to trap-frequency and detuning noise, so multi-loop pulse shaping is used to restore robustness.
Many-mode / hot limit	$\eta^2(2\bar n+1)\gtrsim 1$	Higher-order Lamb–Dicke and Debye–Waller corrections rescale and dephase the gate; first-order thermal insensitivity fails and ground-state cooling becomes necessary.

Common trapThe MS gate is fully immune to temperature

It is not. The gate is insensitive to the mean occupation $\bar n$ only to first order in the Lamb–Dicke parameter, and only for the loop that closes exactly. Three things it does not protect against: motional heating during the gate, which adds error $\propto\dot{\bar n}\,\eta^2\tau/2$ no matter how cold the start; second-order Debye–Waller corrections $\propto\eta^2(2\bar n+1)$ that rescale the effective Rabi frequency with temperature; and trap-frequency drift or timing error that leaves the loop slightly open ( $\alpha(\tau)\neq0$ ), so residual spin–motion entanglement decoheres the qubits. Treating "hot gate" as "temperature-proof" is the most common misreading of Sørensen and Mølmer.

Worked examples

1 · Designing a maximally entangling gate in ⁴⁰Ca⁺ (fully worked)

A single axial mode of a two-ion ⁴⁰Ca⁺ crystal has $\nu=2\pi\times2\ \mathrm{MHz}$ , addressed on the 729 nm quadrupole transition; choose $\delta=2\pi\times25\ \mathrm{kHz}$ . Find the Lamb–Dicke parameter, gate time, and drive strength for a maximally entangling gate, then estimate the heating-limited infidelity for $\dot{\bar n}=100$ quanta/s.

Step 1. Zero-point extent: $x_0=\sqrt{\hbar/2M\nu}$ with $M=39.96\,u=6.636\times10^{-26}\ \mathrm{kg}$ and $\nu=2\pi\times2\times10^6\ \mathrm{s^{-1}}$ gives $x_0=\mathbf{7.95\ nm}$ .

Step 2. Lamb–Dicke parameter: with $k=2\pi/729\ \mathrm{nm}=8.62\times10^6\ \mathrm{m^{-1}}$ , $\eta=k\,x_0=\mathbf{0.069}$ , comfortably in the Lamb–Dicke regime for a Doppler-cooled $\bar n$ of a few quanta.

Step 3. Gate time from loop closure: $\tau=2\pi/\delta=1/(25\times10^3\ \mathrm{Hz})=\mathbf{40\ \mu s}$ .

Step 4. Drive strength from $\eta\Omega=\delta/4=2\pi\times6.25\ \mathrm{kHz}$ , so the carrier Rabi frequency is $\Omega=(2\pi\times6.25\ \mathrm{kHz})/0.069=\mathbf{2\pi\times91\ kHz}$ . Check: $\Phi=2\pi(\eta\Omega/\delta)^2=2\pi(1/4)^2=\pi/8$ , so the two-body angle $2\Phi=\pi/4$ — exactly the maximally entangling value.

Step 5. Heating-limited infidelity: $1-F\approx\dot{\bar n}\,\eta^2\tau/2=100\times(0.069)^2\times(40\times10^{-6})/2=\mathbf{9.4\times10^{-6}}$ . A 40 μs gate at $\Omega=2\pi\times91$ kHz drives $|gg\rangle\to(|gg\rangle-i|ee\rangle)/\sqrt2$ with a heating floor near $10^{-5}$ — consistent with laboratory MS fidelities once scattering is added.

2 · The Bell state and its entanglement

Verify that the maximally entangling MS gate turns $|gg\rangle$ into a genuine Bell state, and confirm the gate is locally equivalent to CNOT.

Step 1. Since $(\sigma^x_1\sigma^x_2)^2=\mathbb{1}$ , $U_{\mathrm{MS}}=\exp[-i\tfrac{\pi}{4}\sigma^x_1\sigma^x_2]=\tfrac{1}{\sqrt2}(\mathbb{1}-i\,\sigma^x_1\sigma^x_2)$ .

Step 2. Act on $|gg\rangle$ : since $\sigma^x_1\sigma^x_2|gg\rangle=|ee\rangle$ , $U_{\mathrm{MS}}|gg\rangle=\tfrac{1}{\sqrt2}(|gg\rangle-i|ee\rangle)$ . Its reduced single-ion state is maximally mixed ( $\rho_1=\tfrac12\mathbb{1}$ ), so the state is maximally entangled — a Bell state.

Step 3. Local equivalence to CNOT: single-qubit rotations map the $\sigma^x\sigma^x$ interaction onto controlled- $Z$ , and a Hadamard converts controlled- $Z$ to CNOT. The single continuous MS pulse plus local rotations is a complete two-qubit gate, satisfying the DiVincenzo requirement for a universal set.

Physics of the portrait

The MS gate is a theorist's gift to experiment — a scheme, conceived on paper in Aarhus, that dissolved the tyranny of the ground state.

Origins · 1 minKlaus Mølmer and how this was discovered

Klaus Mølmer

b. 1963, Danish physicist · theory of trapped-ion entangling gates

Klaus Mølmer, a theoretical physicist at Aarhus University in Denmark, together with his then-doctoral student Anders Sørensen, published in 1999 a deceptively simple idea that reshaped trapped-ion quantum computing. Ignacio Cirac and Peter Zoller had shown in 1995 that two ions could be entangled by routing quantum information through a single shared phonon, but their scheme demanded the vibrational mode be cooled to its ground state and kept there. Sørensen and Mølmer asked what would happen if two laser tones drove the ions symmetrically about the qubit frequency, and found the motion could serve as a bus without ever being genuinely occupied: the phonon appears only as a virtual intermediary, and the entangling phase becomes independent of the vibrational quantum number. Their two papers — Physical Review Letters 82, 1971 (1999) and the full multi-mode theory in Physical Review A 62, 022311 (2000) — recast the gate as a geometric, area-enclosing operation, robust by construction. The same year, David Wineland's NIST group entangled four beryllium ions with the collective drive, and within two decades several laboratories pushed MS-gate fidelities past 99.9%. It remains Mølmer's most-used contribution — the entangler inside nearly every ion-trap quantum computer built today.

History

1995

— Cirac and Zoller propose the first trapped-ion controlled gate through a single phonon

1999

— Sørensen and Mølmer propose the bichromatic hot gate insensitive to the motional state

2000

— Sackett and colleagues at NIST entangle four ions with the collective bichromatic drive

2008

— Benhelm and colleagues reach 99.3% MS-gate fidelity in ⁴⁰Ca⁺

2016

— Oxford and NIST groups surpass 99.9% two-qubit fidelity with Raman-driven MS gates

Going further

Texts. Nielsen and Chuang, Quantum Computation and Quantum Information, §7.6 (ion traps) · Häffner, Roos and Blatt, "Quantum computing with trapped ions," Physics Reports 469, 155 (2008) · Wineland et al., J. Res. NIST 103, 259 (1998) · Roos, "Ion-trap quantum gates," New Journal of Physics 10, 013002 (2008).

Provenance. The bichromatic Hamiltonian, the Magnus derivation of the closed loop, and the phase $\Phi=2\pi(\eta\Omega/\delta)^2$ follow Sørensen and Mølmer, Phys. Rev. Lett. 82, 1971 (1999) and Phys. Rev. A 62, 022311 (2000); Lamb–Dicke and Debye–Waller corrections follow Wineland and colleagues (1998) and Roos (2008). The maximally entangling condition is stated here as $\eta\Omega=\delta/4$ with two-body angle $2\Phi=\pi/4$ : for two ions $S_x^2=2\,\mathbb{1}+2\,\sigma^x_1\sigma^x_2$ , so the physical angle is $2\Phi$ and $\exp[-i\tfrac{\pi}{4}\sigma^x_1\sigma^x_2]$ maps $|gg\rangle\to(|gg\rangle-i|ee\rangle)/\sqrt2$ . A verifier should recompute $\alpha(\tau)$ and the second Magnus term, confirm loop closure at $\delta\tau=2\pi$ , verify $S_x^2=2\,\mathbb{1}+2\,\sigma^x_1\sigma^x_2$ so $2\Phi=\pi/4$ at maximal entanglement, and check the chain $x_0=7.95$ nm, $\eta=0.069$ , $\tau=40\ \mu$ s, $\Omega=2\pi\times91$ kHz. Quoted 99.9% fidelities are standard reported values; no fabricated numbers.

Prerequisites. Trapped-Ion Qubits · Sideband Cooling · Cirac–Zoller Gate · DiVincenzo Criteria. Leads to. Linear-Optical Quantum Computing · Photonic Qubits.

Ask the tutor: Ask me to walk through why the two bichromatic tones cancel the real phonon while leaving the virtual one, or to design an MS gate for your own ion species, mode frequency, and detuning.

Why does the gate use two laser tones instead of one?

A single tone on a motional sideband drives a genuine sideband transition: it really creates or destroys phonons, so the outcome depends on how many were already present — and you are back to needing the ground state. Two tones placed symmetrically above and below the carrier interfere: the amplitude to add a phonon and the amplitude to remove one combine so the resonant, phonon-changing part cancels, leaving only an off-resonant, second-order coupling that carries the internal state through a virtual phonon, borrowed and instantly returned. Because the phonon never really exists, the number that was there does not matter — and the symmetry of the two tones is what buys the thermal robustness.

What is the difference between the Mølmer–Sørensen and Cirac–Zoller gates?

Both entangle ions through their shared motion, but in opposite spirits. Cirac–Zoller (1995) is sequential and literal: it maps the control qubit onto a single phonon, acts conditionally on the target, then maps the phonon back — every step requiring the mode to start in a definite Fock state, almost always $|n=0\rangle$ . Mølmer–Sørensen (1999) is continuous and geometric: one symmetric bichromatic pulse acts on both ions at once, the motion is only virtually excited, and the entangling phase is the area of a closed phase-space loop, independent of $n$ . That tolerance of thermal motion is why the MS gate, not Cirac–Zoller, is the entangler in essentially every modern trap.

Does the MS gate really work with a hot ion?

To first order in the Lamb–Dicke parameter, yes — the entangling phase does not depend on the mean phonon number $\bar n$ , so a Doppler-cooled crystal gives the same gate as a ground-state-cooled one. But "hot" has limits. Second-order Debye–Waller corrections, scaling as $\eta^2(2\bar n+1)$ , rescale the effective Rabi frequency as the ion warms, so at large $\bar n$ the calibrated power drifts off; and heating during the gate is never suppressed, adding error $\propto\dot{\bar n}\eta^2\tau$ regardless of the start. So the gate is temperature-insensitive, not temperature-proof; the best fidelities still use sideband cooling to a fraction of a quantum.

What exactly is the geometric phase that entangles the ions?

It is the phase a state acquires when a parameter — here the mode's coherent amplitude $\alpha$ — is carried around a closed loop back to its start. A coherent state taken around a loop of area $A$ in the $(\mathrm{Re}\,\alpha,\mathrm{Im}\,\alpha)$ plane picks up phase $2A$ . In the MS gate the spin-dependent force sets the loop size, so different joint-spin configurations enclose different areas and collect different phases — and a phase depending non-separably on the joint spin state is exactly what entanglement is. Being geometric, it depends only on the enclosed region, not on how the loop is traversed.

How is a two-qubit gate built from the collective operator S_x squared?

The gate is $U=\exp[-i\Phi S_x^2]$ with $S_x=\sigma^x_1+\sigma^x_2$ . Squaring gives $S_x^2=2\,\mathbb{1}+2\,\sigma^x_1\sigma^x_2$ : a constant global phase plus the genuine two-body term. So the physical entangling angle is $2\Phi$ , and $2\Phi=\pi/4$ makes $U=\exp[-i\tfrac{\pi}{4}\sigma^x_1\sigma^x_2]$ , locally equivalent to CNOT. Watch the factor of two: writing $\exp[-i\tfrac{\pi}{4}S_x^2]$ would double the angle to $\pi/2$ , giving the local gate $-\sigma^x_1\sigma^x_2$ that does not entangle at all. The maximally entangling collective form is $\exp[-i\tfrac{\pi}{8}S_x^2]$ .

Why can the same interaction make both Bell states and quantum simulators?

Because $S_x^2$ is a sum of pairwise $\sigma^x_i\sigma^x_j$ couplings. On two ions with one mode it produces a single entangling term, hence a Bell state. On $N$ ions across many modes the same drive couples every pair, and summing gives an Ising Hamiltonian $\sum_{i<j}J_{ij}\sigma^x_i\sigma^x_j$ whose couplings — tuned by the detuning relative to the mode spectrum — fall off as a power law of the ion separation. Run briefly and calibrated, it is a clean two-qubit gate; run continuously, it is a programmable quantum magnet. One piece of physics, two uses.